Aurélie Bonein

| Stéphane Turolla

Stéphane Turolla
email: stephane.turolla@inrae.fr

Spatial competition with demand uncertainty: A laboratory experiment

Motivated by recent research on product differentiation, we conduct laboratory experiments to study how demand uncertainty influences firms' incentives to differentiate. We ground our experiment on a discrete version of the standard location-then-price game introduced by Hotelling (1929), and we consider different levels of demand uncertainty. We first derive the game equilibrium assuming risk-neutral firms, and obtain the standard prediction that a high level of demand uncertainty generates more differentiation.

Second, we extend the analysis to consider non-risk neutral firms and markets with asymmetric risk profiles. We show that the game equilibrium can differ substantially according to the attitude to risk. Third, we compare our predictions with the experimental data and find that demand uncertainty acts as a differentiation force in the context of both symmetric markets composed of risk-neutral or risk-lover subjects and asymmetric markets. We find support also for the agglomeration effect arising from demand uncertainty for sufficiently risk-averse subjects. Overall, these results might explain the opposite product differentiation strategies frequently observed in markets with fast-evolving tastes (i.e., minimum or maximum differentiation). Finally, the data confirm that subjects differentiate to relax price competition and provide evidence of a strong positive relationship between differentiation and prices.

| INTRODUCTION

Product success hinges principally on the firm's ability to meet demand. Despite this simple statement, firms encounter considerable practical difficulties in relation to envisioning how consumers will appreciate a brand new product. This could be due to financial constraints which prevent the firm from conducting in-depth market studies and to the difficulties involved in predicting rapidly evolving consumer tastes. Opening a restaurant, or launching a new line of apparel, a new version of an electronic device (smartphone and camera), or a new mineral water flavor are a few examples of initiatives where firms are uncertain about consumer tastes for product characteristics. This uncertainty is accentuated by the current rapid evolution of consumer tastes. There are several factors such as the growing integration of economies, the existence of social networks that favor fads and their dissemination, and the occurrence of global shocks (e.g., recessions, climate events, and covid-19 crisis) that give rise to new lifestyle aspirations which seem to have accelerated renewal of consumer tastes, and by extension demand uncertainty. 1 Thus, firms cite demand uncertainty as one of the main challenges on the road to success (see, e.g., Capgemini Consulting, 2012). locations simultaneously and before the consumer location distribution is revealed, and then compete in prices with perfect knowledge about the location of demand. This location-then-price game reflects the firm's long-term decision about positioning. It also implicitly assumes that once settled in a location firms can learn quickly about demand characteristics and can rapidly adjust their prices at no cost. Contrary to the literature, we develop a model for different firm risk profiles. 6 In addition to the standard case of risk neutrality, we suppose that firms can be risk-averse or risklover. This departs from the economics and management literatures which generally assume that faced with demand uncertainty firms are risk-averse. We are not suggesting that this hypothesis is irrelevant since it is supported by a large number of studies. 7 However these findings were generally drawn "on average," and all of those studies that rely on a measure of risk aversion report a fraction of non-risk averse decision-makers in both the lab and in the field (e.g., Caliendo et al., 2009; Cramer et al., 2002; Morgan et al., 2016). Therefore, we make no assumptions about the firm's risk attitude and we investigate the effect of demand uncertainty by considering different risk profiles. We also consider different market configurations depending on whether two firms have the same risk profile (i.e., symmetric market) or not (i.e., asymmetric market). Using this model, we implement three experimental treatments. In the baseline treatment, subjects have perfect information on demand location; in the other two treatments, we introduce different levels of demand uncertainty.

The model delivers clear predictions. First, we obtain the standard result that high demand uncertainty acts as a differentiation force in markets composed of risk-neutral firms (see, e.g., Meagher & Zauner, 2005). This result applies also to risk-lover firms, and we find that the differentiation effect increases with the degree of risk-loving. However, if we assume sufficiently risk-averse firms the predictions change. The differentiation force entailed by demand uncertainty is counterbalanced by the attraction of the city center which ensures a smaller but safer share of demand. Therefore, we find that for sufficiently risk-averse firms, demand uncertainty acts as an agglomeration force regardless of the size of the uncertainty. Finally, we extend the model to the case of asymmetric markets composed of a risk-averse and a non-risk averse firm.

Subjects in the lab play one of three treatments of the location-then-price game, and participate in a second experiment which allows us to elicit their risk preferences. The experimental data confirm that on average demand uncertainty acts as a differentiation force which supports the central prediction in the literature. Although subjects do not differentiate as much as predicted by the theory, we observe a significant increase in the level of differentiation as demand uncertainty rises. This result also appears robust to the way demand uncertainty is modeled. When controlling for subjects' risk attitudes, the prediction that high demand uncertainty yields a differentiation force in symmetric markets composed of risk-neutral or risk-lover subjects is confirmed and applies also to the case of asymmetric markets. In addition, we show that demand uncertainty might generate an agglomeration force for sufficiently riskaverse competing subjects together. This new finding highlights the importance of considering the risk preferences in the presence of demand uncertainty and might explain some of the agglomeration patterns observed in markets featuring fast-evolving tastes (e.g., bars and restaurants) and risk-averse decisions makers. Finally, our experimental data show that when differentiation increases subjects succeed in mitigating price competition; a relationship at the heart of addressing models that is rarely verified empirically.

The remainder of the paper is organized as follows. Section 2 presents the discrete location-then-price game framework and reports the game equilibria for the different treatments. Section 3 describes the experimental procedure, and Section 4 presents the results of the experiments. Section 5 concludes the paper.

| A DISCRETE VERSION OF THE HOTELLING GAME

Section 2.1 provides a discrete version of the standard Hotelling's model. In Section 2.2 we derive the game equilibria under the assumption of perfect knowledge about demand location and risk-neutral firms. In Section 2.3 we introduce demand uncertainty and in Section 2.4, we relax the hypothesis of risk-neutral firms.

| Basic settings

As a natural benchmark, we consider the standard Hotelling's model in which firms play a location-then-price game under the canonical assumptions of risk-neutral firms and perfectly known demand (hereafter the demand certainty [DC] case) before the firms' location and price decisions. 8 To conduct the laboratory experiments, we recast Hotelling's (1929) model in a discrete framework by adopting the game proposed by Barreda-Tarrazona et al. (2011). While a discrete framework makes it difficult to derive closed-form solutions it allows us to limit both the number of locations and the number of price offers. This favors the replication of the game in the lab, and facilitates comparison with real-world situations in which the sets of actions are reduced. In the following paragraphs, we present the key features of the game proposed by Barreda-Tarrazona et al. (2011) while introducing specific notations that will be useful for introducing the case of demand uncertainty.

Let us consider two firms ∈ i {1, 2} playing a location-then-price game. In the first stage, the firms with perfect information about demand choose their locations, x simultaneously. Once the locations are set, the firms compete in prices simultaneously. The solution to this game can be derived from the resolution to the firm's profit maximization problem given by ⋅ x x p p p q Π ( , , , ) = max{ }

i i i i i i i - -
, where p i is the price charged by firm i and q i is its residual demand. Assume there is a unit-length linear segment populated by seven consumers j = {1, …, 7} uniformly distributed over M M [ -3, + 3], with ∈ M ℜ. Figure 1 shows that each consumer is located on one of the seven equidistant slots denoted x j and separated from its closest neighbors by a distance of 1/6. 9 A consumer can either buy a maximum of 1 unit of the good sold by one of the two firms labeled i and located at ∈

x M M M { -3, -2, …, + 3} i or can abstain from buying. Without loss of generality, we assume that firm 1 is located to the left of firm 2, that is, ≤

x x 1 2 . The firms sell a homogeneous good produced with constant marginal costs normalized to 0. Since products are not vertically differentiated, the difference in consumer utility between two products offered at the same price will depend on the taste difference between the consumer's ideal product and the product offered. In the case of geographical differentiation, this corresponds to the difference in transportation costs. Hence, for a consumer located at x j who visits firm i, his indirect utility function is given by

V R p tx = -- , ji i ji (1)
where R is the basic reservation utility which is assumed to be positive, p i is the price charged by firm i, x ji is the distance between consumer j and firm i

  x x x (i. e., = -) ji j
i , and t is the transportation cost per unit distance. According to the highest utility rule, consumer j purchases the product from firm

i if V V > ji jk with ≠ k i.
In other words, as long as his payment does not exceed his reservation utility R consumer j chooses the cheapest firm (i.e., the firm minimizing mill price plus transportation cost). In the event of ties between V ji and V jk , consumer j is assumed to randomly choose one of the two firms with probability 1/2. Bearing in mind that the experiment should be kept as simple as possible, we adopt the same parameter values as in Barreda-Tarrazona et al. (2011), which entails clear-cut solutions for experimental subjects. Therefore, we set R = 10, p = {0, …, 10} i and t = 6. 10 At this point, we need to consider two hypotheses of the model. First, rather than the standard hypothesis of quadratic transportation costs we chose a linear form to model transportation costs which reduces the cognitive burden of the experimental subjects in the lab. Also, there is no economic reason for choosing a quadratic rather than a linear formulation. The use of linear transportation costs in Hotelling's location-then-price game tends to be dismissed because it leads to the non-existence of a price equilibrium in pure strategies with firms located inside the market quartiles (see D 'Aspremont et al., 1979). In fact, for location configurations close to but not at the center, there is an incentive for the firm to undercut its rival to capture the entire demand. This implies that the profit functions are not quasi-concave and that the best response functions do not intersect due to discontinuity. Obviously, in a discrete game context, this issue does not arise since by nature the profit functions are not quasi-concave.

Second and in contrast to experiments based on spatial competition models (see, e.g., Brown-Kruse & Schenk, 2000; Collins & Sherstyuk, 2000; Huck et al., 2002; Orzen & Sefton, 2008), the resolution of the game yields possible outcomes in which the market is not fully covered. This particularity originates from the model parametrization which allows the firms to set delivered prices that exceed the consumer's reservation utility. Therefore, for those particular prices, consumers are better off not buying the good which results in an uncovered market. Thus, in the particular case where non-buying consumers are located between the two firms, the firms do not compete directly with one another but instead behave like local monopolists.

F I G U R E 1 Demand certainty treatment L ( = 0): Linear city
with seven consumers and seven potential locations.

| Location-then-price game equilibria

Throughout the paper, we study the non-cooperative game equilibrium and consider the subgame perfect Nash equilibrium (SPE) as the equilibrium concept. 11 We first consider risk-neutral firms, and determine the price equilibrium for each pair of locations x x ( , ) 1 2 . Whenever a consumer is indifferent between firms 1 and 2, we adopt an arbitrary sharing rule such that the marginal consumer chooses one of the two firms with the probability 1/2. 12 In the case of a multiplicity of price equilibria, we assume that firms choose the riskless option, meaning the price equilibrium which ensures an equivalent level of profit with a larger number of safe consumers. If the multiplicity of equilibria persists, we consider the strict Nash equilibrium. However, if there is no strict Nash equilibrium, we assume that firms choose the Pareto superior equilibrium and otherwise the joint profit-maximizing solution. Assuming that firms are risk-neutral and perfectly informed about demand location, the payoffs in Table 1 are derived from the price equilibrium for each pair of locations. 13 Given this matrix of payoffs, it is now easy to determine the unique SPE of the location-then-price game under demand certainty.

Proposition 1. The unique SPE for the location-then-price game under demand certainty and risk neutrality is

⋆ ⋆ ( ) x x , = (2,6) 1 2
with equilibrium prices ⋆ ⋆ ( )

p p , = (7,7) 1 2
. This result first established by Barreda-Tarrazona et al. (2011), implies a degree of differentiation equal to

≡ ⋆ ⋆ x x Δ - = 4 DC 2 1
.

| Demand uncertainty

We now introduce demand uncertainty into the location-then-price game under the assumption that firms are risk-neutral. This extension corresponds to the central issue which we examine experimentally in this paper. The introduction of demand uncertainty resonates with a stream of research that investigates to what extent uncertainty over (aggregate) demand location mitigates the firm's positioning in geographical space. In brief, this literature considers a modified version of the (continuous) Hotelling game in which uncertainty applies to consumer locations through a random shift of the linear city. 14 For instance, Casado-Izaga (2000) uses the standard location-then-price game with quadratic transportation costs, and models demand uncertainty as a common draw for the mean of the consumers' distribution which is assumed to be uniformly distributed. 15 For simplicity, the center of the linear city is assumed to be drawn from a uniform distribution on the closed unit interval [0,1] and revealed to firms once the simultaneous location game has ended. Meagher and Zauner (2005) generalize this setting by extending the support of the distribution of the city center to

                -, L L 2 2
, where L represents the size of the uncertainty. They show that a unique pure strategy SPE exists which leads to higher differentiation, higher expected prices, and higher expected profits as L increases. Notes: ⋆ ( ) and e ( ) denote, respectively, the mixed-strategy equilibrium and the expected payoffs due to indifferent consumers.

The intuition for this result is simple. Under demand certainty, the location equilibrium in the Hotelling game is shaped by the basic tradeoff firms face between securing a larger fraction of the demand by locating closer to the center (demand effect) or relaxing the price competition by moving farther away from competitors (strategic effect). Because demand uncertainty reduces the magnitude of the demand effect but does not affect the strategic effect, firms choose to differentiate more compared with the certainty case. More precisely, with a uniform distribution of the demand center, extreme realizations of demand are equally likely as central realizations to occur. This implies that firms are no longer certain to lose consumers if located far from the city center. Consequently, the demand effect is weakened and firms choose to differentiate to reduce price competition. This result becomes more pronounced with greater levels of uncertainty.

Although established for a given set of assumptions, the differentiation force entailed by demand uncertainty remains valid under alternative assumptions: a non-uniform distribution of the random shock (Meagher & Zauner, 2004, 2008, 2011), sequential location choices (Bonein & Turolla, 2009; Casado-Izaga, 2000; Harter, 1996), and pricing decisions under demand uncertainty (Meagher & Zauner, 2004, 2005). 16 To test the effect of demand uncertainty on firm differentiation experimentally, we extend the baseline model by assuming that firms are uncertain about consumer locations. The timing of the game is as follows. First, firms simultaneously choose their location in the absence of knowledge about the location of demand. Then the location of demand is revealed, and firms' locations are announced publicly. Finally, firms simultaneously set their prices. This timing corresponds to the ex-post pricing timing in Meagher and Zauner (2004, 2005). It suggests that firms can learn rapidly about demand characteristics and that it is easier for firms to change their prices than their locations (or any other product characteristic). Implicitly the chosen price is not the introductory price but rather a short-term price. We also follow Meagher and Zauner (2005) by assuming that firms are uncertain about the realization of the first moment of the consumer distribution while they are perfectly informed about the higher-order moments of the distribution and the definition of its support. Formally, let us assume that the demand center M is distributed according to a discrete uniform distribution on { }

-+ 4, …, 4, …, + 4 L L 2 2
, with L > 0. Implicitly, this means that firms' beliefs can be represented by a common prior. 17 To facilitate comparison among treatments, we impose that the city center is always the integer 4 regardless of the level of uncertainty. The expected profits of firm i for a given pair of locations

                           x x p p M p M q x x p p M [Π ( , , , )] = × 1 × ( , , , ) , i i i i i i C c ic i i i i - - - - (2)
where c is a realization of the random variable M, M C is the number of possible realizations of M, and ⋅ q ( ) ic is the number of consumers i attracts for realization c of the demand center. Note that since the demand center M is distributed uniformly, the formulation of the expected profits gives equal weight to each realization of M in the computation of the firm's demand.

In the experiment, we consider two treatments that embody demand uncertainty and differ in the level of uncertainty. In the first treatment, we study firm decisions under a high level of uncertainty about consumer locations by setting L = 8. This treatment, labeled High Demand Uncertainty (HDU), increases the support of the distribution of consumers, thus offering nine possible realizations for the demand center M. 18 Figure 2 shows that this treatment extends the set of potential firm locations to 15 slots while holding the number of consumers populating the linear city at 7. In other words, demand is unchanged (i.e., seven consumers), but before locating, firms now are uncertain about the location of demand, which could be between sites -3 and 11.

In the second treatment, we choose a lower level of uncertainty such that we obtain an identical location equilibrium with respect to the DC treatment. The rationale for this treatment is that it allows us to analyze whether the introduction of a low level of demand uncertainty leads risk-neutral subjects to adopt different decisions compared with the DC treatment while the theoretical predictions are unchanged. This treatment, which is labeled Low Demand Uncertainty (LDU) involves the assumption that L = 4, which offers 11 potential firm locations (see Figure 3; the locations are labeled from -1 to 9). Similar to the baseline case, a unique SPE in a pure strategy can be characterized by solving these two variants of the game by backward induction. As before, we first determine the price equilibrium for each pair of locations. Then, given the matrix of payoffs for each pair of locations, we derive the unique location equilibrium in a pure strategy. 19 As noted above, the location and price equilibria obtained in the LDU treatment are similar to those derived under perfect knowledge about the location of demand.

F I G U R E 2
Proposition 2. The unique SPE for the location-then-price game under low demand uncertainty and risk neutrality is ⋆ ⋆ ( )

x x , =(2,6) L L 1 2
with equilibrium prices ⋆ ⋆ ( )

p p , =(7,7) L L 1 2
.

In contrast, the resolution of the game in the HDU treatment leads to a different location equilibrium.

Proposition 3. The unique SPE for the location-then-price game under high demand uncertainty and risk neutrality is ⋆ ⋆ ( )

x x , =(1,7) H H 1 2
with equilibrium prices ⋆ ⋆ ( )

p p , =(7,7) H H 1 2
.

When firms face a high level of demand uncertainty, they choose to locate at a greater distance from one another relative to the DC and LDU treatments. We obtain the central result in Meagher and Zauner (2005), which stresses that demand uncertainty acts as a differentiation force. Above a certain threshold of demand uncertainty, firms are no longer sure to capturing a larger share of demand than when located close to the center. Consequently, the firm's best response consists of reducing the intensity of the price competition by locating farther away from its rival and thereby maximizing its expected profit. However, unlike Meagher and Zauner (2005), we do not predict that firms raise their prices if demand uncertainty increases. This is due to a combination of three factors which make a deviation from the initial price equilibrium unprofitable: (i) adoption of a linear transportation cost which strengthens price competition compared with the case with quadratic transportation costs, (ii) a discrete framework which introduces threshold effects in firms' actions, and (iii) a level of demand uncertainty sufficient to increase differentiation but not enough to induce softer price competition.

Following the resolution of the games, the following theoretical prediction can be brought to the data:

Prediction 1. A high level of demand uncertainty acts as a differentiation force.

| Behavioral hypotheses

So far our theoretical predictions have been derived under the assumption of risk-neutral firms. However, in our model two sources of uncertainty combine, and decision-makers in both the lab and the firms are not necessarily risk-neutral.

The first source of uncertainty stems naturally from the existence of indifferent consumers. With a sharing rule of 1/2, a firm is not certain to be chosen by an indifferent consumer, and thus has to form expectations about the total number of consumers it will attract. We can illustrate the importance of risk preferences when some consumers are indifferent between two firms and demand is certain. We employ an expected utility theory framework and a standard isoelastic utility function characterized by constant relative risk aversion (CRRA). 20 The expected utility of the firm i's profit can be written as

U p q r (Π ) = ( ) 1 - , i i i r 1- (3)
where r is the coefficient of constant relative risk aversion. If r = 0, we return to the situation with risk-neutral firms; r > 0 and r < 0 denote risk-averse and risk-loving behaviors, respectively. 21 In the presence of indifferent consumers, the expected utility of the profit is

U p q p q q r (Π ) = 1 2 [ ¯] + [ (¯+ )] 1 - , ͠ i i i r i i i r 1- 1- (4)
where q ¯is the sure demand and q͠ is the number of indifferent consumers. Assume that firms are located in x x ( , ) = (3, 5)

1 2

and that they choose to set identical prices equal to 8. These symmetric outcomes lead to each firm being visited by three consumers, with one consumer remaining indifferent. It follows that firm 1 can either obtain a profit of 24 or 32 with a probability of 1/2 for each outcome. Conversely, if firm 1 chooses to deviate and sets a price equal to 7, it will attract four consumers with certainty and will obtain a profit of 28. Therefore, a risk-averse firm will always prefer to secure a profit of 28 and will set a price of 7, while a risk-lover firm will prefer to take the risk of an indifferent consumer and will set a price of 8.

The second source of uncertainty emerges from the random allocation of the demand center in the LDU and HDU treatments. Both sources of uncertainty could lead the adoption by the firms of different location choices and price decisions depending on their risk preferences.

If the economic environment is uncertain, a standard approach might consist of considering risk-averse rather than risk-neutral firms (see, e.g., Asplund, 2002; Gervais, 2018; Leland, 1972; Sandmo, 1971). Although there is a large body of anecdotal evidence which tends to confirm that in general firms are risk-averse, the empirical evidence based on a measure of risk aversion also shows the presence of a non-negligible fraction of non-risk averse decision-makers. We account for this diversity of risk profiles and assume that the firm might be a risklover, be risk-neutral, or risk-averse. Since this generates a high number of market configurations, in the first step we restrict the analysis to markets where both firms have the same risk profile (hereafter symmetric markets). 22 We assume also that the firms' risk preferences are common knowledge. This can be justified by the existence of exogenous market features which force firms to behave in certain ways (e.g., corporate tax systems, failure rate, other market-specific regulations, or economic shocks).

Again, we solve the game by backward induction to derive the game's theoretical predictions. Given the size of the game's monetary stakes, the behaviors obtained may be close to those obtained for the risk-neutral case due to the concavity of the utility function. To examine how risk-aversion can affect the game equilibrium, we use an extremely concave expected utility function to model risk aversion (assuming r = 0.9; see Rabin, 2000 for a discussion of this issue). For risk-loving behavior, we use a CRRA parameter of -0.2, a value that is sufficient to observe an effect that differs from the risk-neutral case. In the main text we report the location and price equilibria by treatment for these particular values. We also examine the relationship between the degree of risk aversion and the firm's decisions. For space reasons, the location and price equilibria derived under other values of the CRRA parameter are reported in Appendix B.

| DC treatment

As a natural benchmark, we examine the SPE of the game where firms have perfect information about consumer locations. We find that firms' risk preferences do not affect the location equilibrium (i.e., ⋆ ⋆ ( )

x x , = (2,6) 1 2
), and therefore do not affect the level of differentiation regardless of the degree of risk aversion. Only the price equilibrium differs according to the firm's risk preferences. For risk-neutral and risk-averse firms, and even if the price strategy p p ( , ) = (8, 8) 1 2

Pareto dominates the outcome p p ( , ) = (7, 7)

1 2

, they will always be better off by deviating unilaterally and choosing a price equal to 7 to secure safe demand. The unique price equilibrium then is

⋆ ⋆ ( ) p p , = (7,7) 1 2
for all risk-neutral and risk-averse subjects. Conversely, for risk-lover firms, a unilateral deviation from a price equal to 8 is never profitable, resulting in a price equilibrium equal to ⋆ ⋆ ( )

p p , = (8,8) 1 2
. At equilibrium, firms that enjoy risk charge the highest possible price that ensures market coverage and the price competition then softens for these risk-lover firms. 23

| LDU and HDU treatments

Let us first consider the case of risk-lover firms. Intuitively, we might expect risk-lover firms faced with uncertainty to be more inclined to differentiate compared with the DC treatment for two reasons. First, we have seen that demand uncertainty weakens the demand effect and pushes (risk-neutral) firms to differentiate more compared with the DC treatment. Second, we have demonstrated that risk-loving preferences favor the adoption of riskier but potentially more profitable choices. Even though expected demand is lowest at the edges of the city, the profits obtained in these locations might be the highest depending on realization of the demand center and the location of the rival. In other words, risk-lover preferences may induce firms to differentiate more. This intuition is confirmed by computation of the SPE under different values of the CRRA parameter. It appears that high demand uncertainty acts as a differentiation force for risk-lover firms. For example, with a CRRA parameter of r = -0.2, we obtain a location equilibrium of

⋆ ⋆ ( ) x x , =(1,6) H H 1 2
in the HDU treatment (see Table 2). 24 This location equilibrium leads to a higher level of differentiation than in the DC treatment. More generally, varying the value of the CRRA parameter reveals that the differentiation force yielded by a high demand uncertainty is even stronger for highly risk-lover firms (see Appendix B).

We observe also that similar to risk-neutral firms, a low level of demand uncertainty is not enough to induce a location equilibrium different from that in the DC treatment (i.e., ⋆ ⋆ ( )

x x , =(2,6) L L 1 2
).

Turning now to the price equilibria, fiercer price competition than in the DC treatment is predicted under low and high demand uncertainty. While in the DC treatment it is optimal for risk-lover firms to set high prices, the introduction of demand uncertainty mitigates this riskier strategy. Indeed, in the LDU and HDU treatments, the support of demand is increases which in turn increases the average distance between firm and consumers. For some price strategies this could result in an uncovered market. Then the price raising effect of risk-loving preferences is attenuated in the presence of demand uncertainty. This results in lower prices at the equilibrium compared with the DC treatment, regardless of the value of the CRRA parameter of risk-lover firms.

The theoretical predictions derived under the assumption of risk-lover firms suggest the following location decisions.

Prediction 2. A high level of demand uncertainty acts as a differentiation force for risk-lover firms and this effect increases with the degree of risk-taking. Notes: This table reports location equilibrium, price equilibrium, demand and equilibrium profits by treatment and by risk preferences. To allow comparison, the outcomes for risk-neutral firms are also reported.

Abbreviations: DC, demand certainty; LDU, Low demand uncertainty; HDU, high demand uncertainty.

We now turn to the case of risk-averse firms. As previously shown, risk-averse firms prefer safer outcomes. Demand support increases with the introduction of demand uncertainty, which should induce firms to locate closer to the city center to maximize their expected number of consumers. Such an agglomeration force entailed by risk aversion could offset (or even reverse) the differentiation force of demand uncertainty. Consequently, the SPE of the game will depend on the magnitude of each force. To examine this issue, we compute the game equilibrium for several values of the CRRA parameters (see Appendix B).

Following the terminology in Holt and Laury (2002), we find that the agglomeration force predominates for highly risk-averse firms ( r 0.97 < < 1.37), very risk-averse firms ( r 0.68 < < 0.97), and part of risk-averse firms ( r 0.41 < < 0.68). In the context of demand uncertainty, these firms are better off from locating closer to the city center, which reduces the level of differentiation. For these firm categories the location equilibrium is unchanged in the case of LDU (e.g., ⋆ ⋆ ( )

x x , =(2,5) L L 1 2
for r = 0.9). 25 However, in the HDU treatment the agglomeration effect is more pronounced for the most risk-averse categories ( ⋆ ⋆ ( )

x x , =(3,5) H H 1 2
for r = 0.9). In both treatments, the decrease in differentiation leads logically to fiercer price competition relative to the DC treatment. This finding is consistent with the prediction in Asplund (2002), who shows that risk aversion increases the price competition in the presence of demand uncertainty.

For lower degrees of risk aversion (i.e., ≤ r 0.3 < 0.45), the marginal utility of small stakes decreases so that the forces cancel out, regardless of the size of uncertainty. We return then to the location equilibrium obtained in the DC treatment (i.e., ⋆ ⋆ ( )

x x , = (2,6) 1 2
). Also, for slightly risk-averse firms (i.e., ≤ r 0.18 < 0.3), the incentive to secure safer demand is so low that a high demand uncertainty is sufficient to generate a differentiation effect (see Appendix B).

The theoretical predictions derived assuming risk-averse firms lead to the following location decisions.

Prediction 3. Demand uncertainty acts as an agglomeration force for sufficiently risk-averse firms, regardless of the size of the uncertainty. For lower degrees of risk aversion, the incentive to agglomerate diminishes such that firms do not react to demand uncertainty. In case of high demand uncertainty and slightly risk-averse firms the agglomeration force is even greater than the differentiation force.

| EXPERIMENTAL PROCEDURE

Our experimental design is intended to test the differentiation force of demand uncertainty while controlling for subjects' risk attitudes. Each experimental session consists of two experiments, in which all subjects are required to participate. To avoid any order effects the order of the two experiments was reversed in half of the sessions. The main experiment refers to the discrete location-then-price game presented above, while the second experiment measures subjects' risk preferences using the design proposed by Drichoutis and Lusk (2016). 26 In the location-then-price game, each subject interacts with rival subjects over 30 periods, t = 1, 2, …, 30. Specifically, the experiment simulates six markets of five periods each, with a subject matched with the same rival during the five periods of the same market but is paired with a new rival at the beginning of each new market. In our design, we introduce a series of periods where the location decisions are given and during which subjects can modify only their prices. This means that in the first period of each market, the subject chooses both his location and price but, in the remaining four periods of the market, the subject competes only in prices. This setup has the advantage of reflecting the long-term real life location decision whereas price changes are deemed less costly and can be realized more frequently. Thus in the experimental design we assume that location decisions are more rigid than pricing decisions. Note that repetition of the price subgame is not included in the theoretical model but that does not change the model predictions. 27 Specifically, the timing of the decisions in the first period of each market (i.e., t = 1, 6, 11, 16, 21, 26) is as follows. First, the two subjects simultaneously choose their locations, then the rival's location is revealed, and, the subjects compete simultaneously in prices; finally, each simulated consumer decides whether to buy from the cheapest firm. At the end of the period, each subject learns (i) his demand level, (ii) his profit, and (iii) the price of his rival. Then four periods of price competition follow. Once the five periods have elapsed, a new market begins and subjects are randomly reshuffled, under the constraint that each subject is matched exactly once with the same rival. This stranger matching protocol is common information. To achieve this, we form random groups of subjects, and the rematching is conducted within each group. This stranger design limits end-game effects on both location and price strategies, which enables us to end the experiment with the last period of price competition in the market. Of course, consumer locations remain unchanged during the five periods of a market but in the LDU and HDU treatments, can vary from one market to another according to the random draw of the demand center. The draws of the demand center were made for the first session and replicated in all other experimental sessions.

Experimental sessions were conducted at the University Rennes 1 LABEX-EM (CREM-CNRS) Institute. The subjects were students with different backgrounds. The experiment was programmed and conducted using the Z-tree software (Fischbacher, 2007). Participants were invited using Orsee (Greiner, 2015). Before the start of the game, subjects were informed that (i) there would be two independent experiments, (ii) money earned in the experiments would depend on their decisions and the decisions of their rivals, and (iii) they would be paid the earnings they accrue in the two experiments. It was made clear that information about the earnings obtained in each experiment would be given only at the end of the experimental session. We set this condition to reduce potential earnings spillover effects from one experiment to the next. A total of six sessions were conducted for each treatment, with 18 subjects per session. No subject had previously participated in a similar experiment, and none of them participated in more than one session, which results in 324 subjects.

To ensure comparability across sessions and treatments, before the first session we randomly formed pairings within each matching group and used the same pairings for all the sessions. Upon arrival, the subjects were randomly allocated to visually separated cubicles so that neither during nor after the experiment were subjects informed about the identities of the other subjects with whom they were matched. The subjects were provided with written instructions which were read aloud by the experimenter (see the Supporting Information Appendix). 28 To ensure that all subjects completely understood the instructions, they were required to calculate firm profits in hypothetical exercises. Although this may have introduced some bias, this risk was limited by the fact that the exercises reflected representative contingencies. The responses to these exercises were checked privately before the experiment began to ensure that all the subjects understood the experiment. As a further check, we ran a practice round representing one market (i.e., one period of location-then-price decisions followed by four periods of price decisions) before the experiment began.

At the end of the two experiments (i.e., the location-then-price game and the risk experiment), the subjects were informed about their earnings and were asked to complete a brief post-experimental questionnaire which asked about personal characteristics (e.g., age, gender, and field of study; see Appendix D). From the outset, it was made clear that the subjects' payoffs could be equal to their earnings in the two experiments. For the location-then-price game, subjects' earnings were proportional to the sum of their profits over the 30 periods. Subjects were paid based on the following conversion rate: 25 Experimental Currency Units = 1 euro. The conversion rate was chosen such that at equilibrium, a risk-neutral subject earned 29.40 euros for the 30 periods in the DC and LDU treatments and 28.47 euros in the HDU treatment and could obtain only 4.20 euros for the 30 periods in the DC treatment in the case of no differentiation and non-cooperative prices.

In the risk experiment, the subjects completed 10 decision tasks. At the end of the experimental session, a decision task was randomly selected for payment, and subjects were paid according to their choice (lottery A or lottery B). The selected decision task was the same for all subjects. Subjects knew that at the end of the experimental session, a random device would determine whether they were actually paid. Each subject was given a 1/9 chance of receiving the payment associated with his decision. This procedure provided the subjects with an incentive in each decision task to choose according to their true preferences and is thus incentive compatible. Moreover, as Harrison et al. (2009) argue, stochastic fees allow for the generation of samples that are less risk-averse than would otherwise be observed. The exchange rate used was 0.4 Experimental Currency Units = 1 euro, with the highest earnings equal to 11.75 euros. Further details on the risk experiment are provided in Appendix C. The average length of an experimental session was 80 min, and the average participant earning was 17 euros.

| EXPERIMENTAL RESULTS

In Section 4.1 we examine whether subjects differentiate more in the presence of demand uncertainty. In Section 4.2 we analyze how demand uncertainty impacts location decisions depending on the subject's attitude to risk. In Section 4.3 we examine the relationship between the level of differentiation and the price.

| Location choices and demand uncertainty

We begin our analysis of location decisions by pooling the data. During the 18 sessions a total of 1944 location decisions were recorded. The first three panels in Figure 4 plot the location choices for each treatment. It is striking that the center of the support of the linear city is the most frequent location choice regardless of the level of uncertainty. In over 55% of all cases in the DC treatment subjects locate at the center of the demand support and the prevalence of this central position is confirmed in the LDU and HDU treatments, with, respectively, 36% and 24% of the location choices. Overall, few subjects decide to locate at the edges of the support of the linear city. The tendency to cluster at the center is in line with the results of past experiments on spatial duopoly markets. 29 Table 3 reports the most frequent location pairs observed for each treatment and their average level of differentiation. Differentiation is measured as the distance between the two subjects competing in a given market. We observe that the subjects' location decisions yield less differentiation than the location equilibria. This is especially clear in the DC treatment where in a given market the subjects are both located at the city center (4, 4) in more than 32% of the location games, while the second-most preferred location configuration yields only a low level of differentiation.

Thus the preferred pairs of locations are far away from the non-cooperative location equilibrium ⋆ ⋆ ( )

x x , = (2,6) 1 2
.

Nevertheless, we find that the attractiveness of the center of the demand support decreases if demand location is uncertain (LDU and HDU). While the pair of locations (4, 4) is always the most frequent choice in the LDU (12.65%) and HDU (5.56%) treatments, Figure 4 shows that subjects are more prone to locate far from the center of the demand support in these two treatments. The prevalence of central location and the resulting low level of differentiation explain why the average earnings are well below the amount the subject would obtain were they to play the equilibrium strategies.

Turning now to the main proposition of the paper, the left panel in Figure 5 shows the distribution of the levels of differentiation obtained in each treatment to verify whether a high level of demand uncertainty yields a higher differentiation. For each treatment, we construct a Box and Whisker plot that summarizes the distribution of the levels of differentiation computed on all location pairs. It is clear that subjects differentiate more in contexts of a high level of demand uncertainty. Computation of the average level of differentiation by treatment highlights that differentiation more than doubles between the DC and HDU treatments (see Table 3, Column 4). The statistical significance of this difference is confirmed by comparing the average levels of differentiation among independent matching groups. Using Mann-Whitney (M-W) U tests and the alternative hypothesis delivered by the model predictions, we find that the differentiation in the HDU treatment is statistically significantly higher than in the DC (one-tailed M-W test: p<0.0001) and the LDU treatment (one-tailed M-W test: p = 0.0002). 30 In other words, subjects faced with high demand uncertainty choose to differentiate more. These findings corroborate Prediction 1 and testify to the differentiation force exerted by demand uncertainty. 31 This provides a convincing empirical test of the attractiveness of more remote locations under demand uncertainty.

Result 1. On average, high demand uncertainty yields higher differentiation than in the case with perfect information about demand location.

However, the results obtained under a low level of demand uncertainty are less in line with the model predictions. As suggested by Figure 5, we find a significantly higher level of differentiation in the LDU treatment than in the DC treatment. We reject the null hypothesis of an equal level of differentiation between the LDU and DC treatments (two-tailed M-W test: p = 0.0001). Although, the model predicts that a low level of demand uncertainty does not change the level of differentiation, we observe empirically that subjects differentiate more than in the DC treatment.

| Robustness analysis

Because the firms are constrained to locations in the demand support, introducing demand uncertainty necessarily increases the number of location slots. It could be argued that even with perfect information about demand location, some firms might be tempted to exploit this larger differentiation space to relax the price competition. 32 Then the higher level of differentiation observed in the presence of demand uncertainty could be explained by this longer support effect. To test whether Result 1 is driven by this longer support effect, we implement an additional treatment in which the demand location is known by the firms (as in the DC treatment) but the range of the firms' possible locations is similar to the HDU treatment (i.e., 15 slots). This robustness treatment is labeled demand certainty-long support (DC-LS). It should be emphasized that the theoretical predictions of the DC-LS treatment are identical to those of the DC treatment, regardless of the risk preferences considered. 33 Similar to the previous treatments, we find that most subjects are located at the center of the demand support (53% of location choices) with a few subjects choosing a location at the edges of the linear city support. The distribution of location choices is plotted in the last panel of Figure 4 and is quite close to the distribution of locations obtained in the DC treatment (Kruskal-Wallis test, p = 0.5144). If we compute the average level of differentiation, we find that subjects do not differentiate significantly more compared with the DC treatment: 1.01 (DC) versus 1.29 (DC-LS). Using a Mann-Whitney U test, we are not able to reject the null hypothesis of identical average levels of differentiation by matching groups between treatments at the 5% significance level (two-tailed M-W test: p = 0.0667). In contrast, we observe notable differences in the levels of differentiation between the DC-LS and HDU treatments, see Figure 5, confirmed statistically by a Mann-Whitney U test (one-tailed M-W test: p< 0.0001). This robustness analysis confirms that the higher level of differentiation observed in the HDU treatment is the result mainly of demand uncertainty.

| Effect of risk attitudes on location choices

Because uncertainty about demand location can have different effects on subjects' location choices depending on their attitude to risk, we next analyze how demand uncertainty affects location decisions by risk profile. In a preliminary step we characterize subjects' risk attitude. Using data from the risk experiment, we are able to provide an estimate of the CRRA parameter for each subject by estimating a structural model of decision-making following the modeling strategy in Harrison and Rutström (2008) and Andersen et al. (2010). We use the CRRA estimates to determine the subjects' risk profiles. Subjects are risk-lovers, risk-neutral, or risk-averse. The classification is derived by comparing the CRRA estimates with the open CRRA intervals from the risk experiment (see Appendix C for more details on the estimation method and the classification). 34 Using this classification, we first examine whether high demand uncertainty acts as a differentiation force for both risk-lovers and risk-neutral subjects as per the model predictions for the case of symmetric markets. In the next step, we compare the model predictions for symmetric markets composed of risk-averse subjects with the data. This allows us to examine how demand uncertainty affects the level of differentiation depending on the degree of subjects' risk aversion. In the last step, we study the effect of demand uncertainty on differentiation for markets in which a risk-averse subject competes with a non-risk averse subject (i.e., asymmetric markets).

In line with the model setting, these analyses use observations corresponding to either symmetric markets or asymmetric markets. This significantly reduces the size of the sample: 38.58% of the pairs of subjects (375 in a total of 972 pairs) are retained for symmetric markets and 50.21% (488 in a total of 972 pairs) are retained for asymmetric markets. It also means that we cannot compare the average level of differentiation among matching groups. We therefore conducted nonparametric tests using one observation of differentiation per pair of subjects (see Table C3 in Appendix C for detailed information on the number of pairs of subjects by risk profile). 35 This reduces the power of the statistical tests, especially when we compare the level of differentiation within a risk profile.

| Demand uncertainty and non-risk averse subjects

Figure 6 depicts the distributions for the levels of differentiation by treatment and by non-risk averse profiles. A first insight is that a high level of demand uncertainty significantly increases the level of differentiation for risk-lover and risk-neutral subjects. On average, the level of differentiation is always higher in the HDU treatment than in the DC treatment (one-tailed M-W tests: p = 0.0260 for risk-lover subjects and p = 0.0276 for risk-neutral subjects). Also, we observe that, on average, the introduction of a low level of demand uncertainty does not allow risk-lover and risk-neutral subjects to differentiate more than in the DC treatment (two-tailed M-W tests: p = 0.8101 for risk-lover subjects and p = 0.4315 for risk-neutral subjects). These results are consistent with Predictions 1 and 2.

Result 2. In line with the model predictions, only a high level of demand uncertainty acts as a differentiation force for both risk-lovers and risk-neutral subjects.

| Demand uncertainty and risk-averse subjects

The introduction of demand uncertainty for risk-averse firms generates opposite location forces: an agglomeration force and a differentiation force. As before, demand uncertainty lessens the demand effect, which pushes firms to differentiate more. However, the introduction of demand uncertainty also leads risk-averse firms to secure a smaller share of demand by locating closer to the center of the demand support. Then the global effect of demand uncertainty on differentiation strategies will depend on the size of the uncertainty and the degree of the firm's risk aversion. For firms that are at least risk-averse according to the terminology in Holt and Laury (2002), the model predicts that the agglomeration force predominates regardless of the size of uncertainty. However, for lower degrees of risk aversion, the marginal utility of small stakes decreases so that the forces cancel out. Further, under a high level of demand uncertainty the differentiation force may even overcome the effect of risk aversion.

We compare these predictions with subjects' decisions. Table 4 reports the average level of differentiation observed by treatment for different ranges of the CRRA parameter. Panel A compares the levels of differentiation observed in the LDU treatment relative to the DC treatment. We find that risk-averse subjects differentiate less, on average, compared with the DC treatment (all the one-sided tests p-values are statistically significant for ≥ r 0.7). Conversely, for slightly risk-averse subjects, there is no significant difference in the average differentiation between the DC and LDU treatments (we are not able to reject the null hypothesis of an equal level of differentiation for ≤ r 0.18 < 0.53). However, we observe that the test of an agglomeration effect (or the absence of effect) is barely significant if we include subjects with a degree of risk aversion of around 0.55. This applies to a small number of subjects who inexplicably differentiate more when faced with demand uncertainty. Overall, our results support our predictions and are confirmed by the parametric analyses which use a larger number of observations defined at the individual level (see Appendix E).

Result 3. A large majority of subjects behave in line with our theoretical prediction: compared with the certainty case, low demand uncertainty induces risk-averse subjects to agglomerate but does not impact the level of differentiation of slightly risk-averse subjects.

Panel B displays the average levels of differentiation between the high demand uncertainty and DC treatments. We find that only the most risk-averse subjects ( ≥ r 0.9) behave as predicted and differentiate less compared with the DC treatment (one-tailed M-W test: p = 0.0921). However, the result is barely statistically significant, certainly due to the small number of observations. For lower degrees of risk aversion, we show that the level of differentiation in the HDU treatment decreases as subjects' risk aversion increases. For instance, the average level of differentiation goes from 1.95 to 1.50 moving from the less to the more risk-averse pairs of subjects. However, comparison with the levels of F I G U R E 6 Levels of differentiation by treatment and non-risk averse profiles. Taking account only of those markets where the subject and its rival have the same risk profile, we obtain 19, 31, and 14 pairs of observations for risk-lovers and 20, 18, and 16 pairs of observations for risk-neutral subjects for the DC, LDU, and HDU treatments, respectively. DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty. differentiation obtained in the DC treatment does not support the prediction of an agglomeration effect (or the absence of an effect) entailed by HDU. 36 Result 4. In line with the model predictions, a high level of demand uncertainty pushes the highest risk-averse subjects to agglomerate. However, for lower degrees of risk aversion, subjects differentiate too much compared with the certainty case which rejects the model predictions.

In what follows we propose several explanations for the discrepancy between the behavior of risk-averse subjects facing a high level of demand uncertainty and the model predictions.

Biased estimate of the CRRA parameter

The results of Panel B show that in a presence of a high demand uncertainty risk-averse subjects do not behave as predicted, except for the most risk-averse subjects (i.e., ≥ r 0.9). A quite high level of differentiation is observed in the HDU treatment compared with the DC treatment, especially for low levels of risk-aversion. However, the reported average levels of differentiation are below the average level computed for risk-lovers (3.14) and riskneutral subjects (1.94) in the HDU treatment. There are two elements that might explain this tendency of riskaverse subjects to differentiate. First, it is possible that risk-averse subjects do not evaluate the opposite forces induced by high demand uncertainty correctly, and thus either underestimate the risk of loosing demand or overestimate the differentiation force of demand uncertainty. However, despite this tendency to differentiate, we can confirm that the average level of differentiation is decreasing with the degree of risk-aversion. Second, it might be that this simply indicates that the estimated CRRA parameters are only a proxy for the true individual risk preferences and therefore may be biased upward. The latter explanation seems credible since we observe a similar level of differentiation between the lowest risk-averse (0.30≤r < 0.45 ) and the entire sample of riskneutral subjects (1.95 vs. 1.94 on average).

Sensitivity to location choice set

It is possible also that risk-averse subjects are sensitive to the size of the location choice set. Because the introduction of demand uncertainty enlarges the support of the demand, it is possible that some subjects are attracted by more remote locations. We showed previously that in the case of perfect information about demand location, subjects do not differentiate more in the presence of longer demand support, irrespective of the degree of risk aversion. However, it is possible that depending on their risk profile subjects react differently to the enlarged location choice set. If risk-averse subjects are more attracted by these remote locations, this might explain why they differentiate more when faced with demand uncertainty. To test this conjecture, we first compare the levels of differentiation between the DC and DC-LS treatments for subjects whose behavior differs, on average, from our theoretical predictions (i.e., ≤ r 0.30 < 0.9). The result of the Mann-Whitney U test rejects the hypothesis of an equal level of differentiation between these two treatments (two-tailed M-W test: DC vs. DC-LS p = 0.0164), meaning that a support effect is at play for this group of risk-averse subjects. Using the DC-LS treatment as the comparison group, we next test whether these subjects differentiate more in the HDU treatment. The result of the Mann-Whitney U test indicates that there is no significant difference in the levels of differentiation between the two treatments (two-tailed M-W test: DC-LS vs. HDU p = 0.1578). We can conclude that the differentiation observed for both slightly risk-averse and risk-averse subjects (i.e., ≤ r 0.30 < 0.9) is explained in part by the enlarged location choice set. This finding is reinforced if we restrict the range of the CRRA parameter to ≤ r 0.30 < 0.45 (two-tailed M-W test: DC-LS vs. HDU p = 0.5415). 37

Demand uncertainty and cognitive burden

As mentioned above, demand uncertainty yields opposite differentiation forces for risk-averse subjects. Therefore, it is likely that subjects need time to correctly assess the intensity of each force. If so, subjects should behave more in accordance with the model predictions in the final market. To test this conjecture, we compare levels of differentiation between the DC and HDU treatments, for the first and the last markets separately. Again, we focus on subjects whose behavior on average differs from the model predictions (i.e., ≤ r 0.30 < 0.9). We find that the average level of differentiation is substantially reduced in the last compared with the first market in the HDU (1.55 vs. 2.45, respectively) but the Mann-Whitney U tests reject the null hypothesis of an equal level of differentiation between the DC and HDU treatments in both markets (two-tailed M-W tests: DC vs. HDU p = 0.0078 for the first market and DC vs. HDU p = 0.0405 for the last market). However, as long as we include the least risk-averse subjects in the sample (i.e., ≤ r 0.18 < 0.9), the difference appears no more significant in the final market (two-tailed M-W tests: DC vs. HDU p =0.0009 for the first market and DC vs. HDU p = 0.1599 for the last market). These results highlight the difficulty encountered by slightly risk-averse subjects in making a correct assessment of the opposite forces of demand uncertainty.

| Demand uncertainty in asymmetric markets

The previous findings are derived from markets where both subjects have the same attitude to risk. However the location-then-price game experiment generates a larger amount of data due to the pairing of subjects with heterogeneous risk profiles (asymmetric markets). In this section, we leverage the location choices observed in these asymmetric markets to derive some more general conclusions on the effect of demand uncertainty on differentiation strategies. First, we formulate new predictions by relaxing the assumption that a subject competes with a rival with the same risk profile. We concentrate on markets composed of a pair of subjects with heterogeneous risk profiles. As before, we assume that exogenous and perfectly known risk attitudes of the subject and its rival. This could be rational for exogenous firm characteristics (e.g., owner vs. manager or entrepreneurship vs. franchisee) or observation of firm decisions in other (independent) markets, for instance. As there are numerous risk attitude combinations, we simplify the analysis by aggregating risk-lover and risk-neutral firms in a non-risk averse category. 38 The modeling framework boils down to a location-then-price game between a non-risk averse firm and a risk-averse firm which is representative of 50.21% of the experimental markets.

For low degrees of risk aversion, the model predicts that firms differentiate more (equally) when confronting high (low) demand uncertainty. We revert to the situation of both firms assumed to be risk-lovers or risk-neutral. The presence of a slightly risk-averse firm is not sufficient to deviate from the equilibria observed with symmetric non-risk averse firms. This is intuitive since we have shown that in the case of symmetric markets, in a context of demand uncertainty slightly risk-averse subjects in many cases behave similarly to non-risk averse subjects (see Table 4).

If we compare these predictions with the data, we find that high demand uncertainty yields a differentiation effect that is statistically significant at the 1% level for ≤ r 0.18 < 0.41. We confirm also that a low level of demand uncertainty has no effect on the level of differentiation compared with the DC treatment, for a definition of slightly risk-averse subjects up to ≤ r 0.18 < 0.41 (two-tailed M-W tests: DC vs. LDU p = 0.1129).

Result 5. In line with the model predictions derived for asymmetric markets, a high (low) demand uncertainty drives (does not drive) differentiation for slightly risk-averse subjects competing with a non-risk averse subject.

For a high level of risk aversion (i.e., ≥ r 0.7), the model predicts that compared with the certainty case demand uncertainty yields an agglomeration force regardless of the size of the uncertainty. This result is driven by the location of the risk-averse firm close to the center of the demand support and the move away from the center of the demand support by the non-risk averse firm in the case only of high demand uncertainty. This results in the respective location equilibria ⋆ ⋆ ( )

x x , =(2,5) L L 1 2
and ⋆ ⋆ ( )

x x , =(1,4) H H 1 2
under a low and high demand uncertainty, respectively, whereas in the case of demand certainty the equilibrium remains ⋆ ⋆ ( )

x x , = (2,6) 1 2
. These strategies are in line with the predictions derived from symmetric markets. However, in the asymmetric markets case, the risk-averse firm has a higher incentive to locate closer to the center of the demand support because it internalizes that the non-risk averse firm will locate away from the center under demand uncertainty.

The data clearly reject the prediction that compared with the certainty case demand uncertainty reduces the level of differentiation in asymmetric markets. For a high level of demand uncertainty, we reject the null hypothesis of a lower average level of differentiation in the HDU treatment compared with the DC treatment for all possible CRRA parameter ranges with a lower bound at 0.7 (e.g., with ≥ r 0.7, one-tailed M-W test: p< 0.0001; the average level of differentiation is 1.10 and 2.42 in the DC and HDU treatments, respectively). This finding is not a surprise since we showed previously that risk-averse subjects that compete with a rival with the same risk profile differentiate more than expected (see Section 4.2.2). It would seem that the results derived so far for asymmetric markets could be rationalized by the fact that in the absence of any information on rivals' risk preferences, subjects' location is not altered by them. This hypothesis is confirmed by running a Tobit regression by treatment where we regress the distance (in absolute value) between the center of the demand support and the subject's location on dummies for the rival's risk profile while controlling for subjects characteristics (see Appendix F). None of the dummy variables are statistically significant meaning that on average the distance between the center of the demand support and the subject's location is the same, regardless of its rival's risk preferences. Thus, it is highly likely that without additional information on rivals' characteristics subjects will assume that their rivals behave as they do.

Last, for a low level of demand uncertainty, we find that risk-averse subjects also differentiate more in the LDU treatment compared with the DC treatment (one-tailed M-W test: p< 0.0235 with ≥ r 0.7) and this result is robust to other CRRA parameter ranges. This finding is at odds with the predictions based on both asymmetric and symmetric markets. The empirical results obtained using symmetric markets and non-risk averse subjects suggest that at worst there would be the same absence of an effect of low demand uncertainty. It is therefore important to not generalize the findings obtained in a symmetric market and low demand uncertainty context to other market configurations.

Result 6. Contrary to the model predictions derived with asymmetric markets, both low and high demand uncertainty act as a differentiation force for risk-averse subjects competing with non-risk averse subjects.

| Price competition

Because demand uncertainty is revealed before the price subgame begins, the only source of uncertainty when fixing prices arises from the random allocation of the indifferent consumer(s). It follows that addressing the issue of risk attitudes in the price competition stage is now less crucial. We therefore examine how subjects set prices conditional on their location choices by pooling all the price data, regardless of subjects' risk attitudes. The analysis is conducted separately for each treatment, since the distance between subjects and the demand center might vary significantly among treatments. Finally, recall that in the experiment, subjects compete in prices with the same rival during five periods. These repeated interactions might lead prices in a given market to evolve over time. The results presented below account for all the periods of the price game but, are robust to considering only the first or the last period. 39 One of the advantages of our study is that it enabled us to investigate the relationship between the level of differentiation and price competition which has been widely debated in the theoretical literature. Table 5 reports the average price by the level of differentiation which allows us to examine whether high levels of differentiation enable subjects to charge higher prices. In the case of the DC treatment (Panel A), we can confirm that on average, subjects are able to reduce the price competition when differentiation increases (Spearman rank correlation coefficient, ρ = 0.3588; p < 0.0001). This result supports the positive relationship between differentiation and price and corroborates the finding in Barreda-Tarrazona et al. (2011). However, this relationship does not hold if we consider the two highest levels of differentiation observed in the LDU and HDU treatments (see Panels B and C).

One explanation might be that some subjects are too far away from the demand center for certain realizations of the demand center. Although highly differentiated from their rivals, they prefer to set low prices and to engage in a greater price competition to attract distant consumers. To control for distance from the demand center in the price-setting Notes: The first columns report some summary statistics on prices by the level of differentiation and by treatment. The seventh column reports the average distance between a subject and the demand center.

Abbreviations: DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty.

decision, we conduct a more formal empirical analysis where the price chosen by a subject is modeled as a function of the level of differentiation, the absolute value of the distance between the subject's location and the demand center, and the subjects' characteristics while controlling for the price competition period within a market and the time-invariant effect of the matching group. We estimate this specification for each treatment, separately. Table 6 reports the estimates with robust standard errors clustered at the matching group level.

The results of the estimations show that by controlling for the distance to the demand center, we obtain the standard positive relationship between differentiation and prices, regardless of the treatment. Furthermore, it is interesting to note that the delicate tradeoff between relaxing the price competition and securing a larger share of the demand is even more altered as the subject is located farther away from the demand center. This is particularly evident in the HDU treatment, where the positive effect of differentiation on price is counterbalanced by the negative impact of distance to the demand center.

Result 7. Conditional on the distance to the demand center, we observe a positive relationship between differentiation and price.

| CONCLUSION

Since the end of the 20th century, how to position a product in the characteristics space has become a central issue for firms offering brand new products. Due to ever-evolving consumer demand, firms face considerable uncertainty about consumer tastes. In response to this demand uncertainty, multinational firms spend billions of dollars on market research to gather information on consumer preferences.

Previous research on product differentiation predicts that firms will differentiate more as uncertainty increases. The reason for this is that it is more profitable for firms to differentiate in the context of demand uncertainty because price competition is relaxed and there is no guarantee that demand will decrease.

In this paper, we empirically tested the differentiation force of demand uncertainty based on a laboratory experiment. Our laboratory setting corresponds to a discrete version of the location-then-price game in Hotelling (1929). We follow previous works and model demand uncertainty as a random shift in the support of the demand (Casado-Izaga, 2000; Meagher & Zauner, 2004, 2005). Subjects simultaneously choose their location with imperfect information on the location of demand. Then demand is revealed and the subjects set prices. The experimental data confirm the central prediction in the literature that demand uncertainty acts as a differentiate force. However, this might be hiding some heterogeneous behaviors since it is expected that a risk-averse subject is likely to react differently from a risk-lover when confronted by a random outcome. We extended the model by considering different market configurations according to subjects' risk preferences: symmetric markets in which the subject competes with a rival with the same risk profile (risk-lover, riskneutral, or risk-averse), and asymmetric markets in which a non-risk averse subject competes with a risk-averse subject. We tested the model predictions with the data, and confirmed that high demand uncertainty yields a differentiation force for both symmetric and asymmetric markets composed of risk-neutral or risk-lover subjects. We highlighted that demand uncertainty may not necessarily lead subjects to differentiate more. In the case of sufficiently risk-averse subjects competing together, we show that demand uncertainty pushes them to agglomerate, as predicted by the model. These findings point to the importance of the attitude to risk in analyses of differentiation strategies under demand uncertainty. Depending on their risk preferences, subjects can react in opposite directions when faced with demand uncertainty: they may locate further away from the center of the demand support thus giving more importance to the strategic effect or might locate closer to the center of demand thus giving more importance to the demand effect. These maximal and minimal differentiation strategies are observed frequently in markets with important demand uncertainty, such as new growth markets (e.g., mobile apps or social media) or industries with fast-evolving tastes (e.g., food and beverages). Therefore, our findings suggest that risk preferences might provide a valid explanation for the product positioning strategies observed in these markets. We would encourage future empirical studies to control for decision-makers' risk attitude. However, even in a laboratory setting measuring risk preferences is difficult due to the existence of several factors that can affect risk preferences (e.g., monetary stakes play) and that might be unobservable (e.g., wealth).

Finally, we exploited our experimental setting to revisit the standard result of industrial organization models that price competition softens as differentiation increases. Our empirical analysis of subjects' pricing decisions confirms a positive relationship between differentiation and price. Conditional on the distance to the demand center, we find that subjects set higher prices if differentiation increases. demand uncertainty. However, in their model prices are assumed to be exogenous which entails the standard result for firms located at the city center in the absence of uncertainty. When uncertainty arises, an increase in risk aversion drives firms to move closer to the edges of the city to secure a smaller share of demand and to avoid risky realizations of a random shock which decreases the expected payoff at the city center.

7 Numerous empirical studies find risk-averseness among firms in the presence of uncertainty. This has been shown for large companies with hierarchical organizations operating in the financial, media, and technology sectors (see, e.g., Lovallo & Kahneman, 2020) and also for entrepreneurial ventures where demand uncertainty arises mostly from heterogeneous and evolving consumer tastes (e.g., retail, motel, and manufacturing industries; see Mazzeo, 2004; Sharma & Tarp, 2018; Wu & Knott, 2006). Several reasons have been proposed to explain firm risk aversion: delegation of firm control to a financially constrained manager, fear of losing job or reputation, financial distress, composition of corporate governance, non-linear corporate tax system, or particular personality traits (see, e.g., Asplund, 2002; Bernhardt & Rastad, 2016; Spagnolo, 1999, and the references cited therein).

which results in ambiguous consumer demand. Because firms are no longer able to compute the expected profits in the location subgame, Król (2012) analyzes product differentiation decisions when firms adopt pessimistic decision criteria. The resolution of the game reveals that the optimal strategy consists of adopting an extreme form of pessimism which when demand location is uncertain leads firms to agglomerate at the city center, a result similar to our theoretical prediction for the case of high risk aversion. 18 Although Meagher and Zauner (2005) examine higher levels of uncertainty, the choice of L = 8 is the result of a delicate tradeoff between adopting a sufficiently high level of uncertainty to observe a change in the equilibrium outcome and limiting the support of the consumer distribution (i.e., reducing L) to avoid an excessive number of equilibria with a large share of non-buying consumers. A disadvantage when increasing the size of the uncertainty is that we significantly increase the choice set of locations, thereby complicating the subjects' task and encouraging them to behave as local monopolists. 19 The price equilibrium payoffs and the equilibrium prices for each pair of locations are provided in the Supporting Information Appendix. 20 The power functions family has been used frequently to model risk aversion in the fields of economics (Holt & Laury, 2002; Palacios-Huerta & Serrano, 2006), psychology (Luce & Krumhansl, 1988), and health (Bleichrodt et al., 1999). In our experiment, the domain of interest involves only positive outcomes. In this case, this specification of the utility curvature has been shown typically to better fit the experimental data than alternative families (see, e.g., Camerer & Ho, 1994; Holzmeister & Stefan, 2021; Wakker, 2008).

21 When r = 1, ⋅ ⋅ ln Π ( ) = ( ) i .
22 In Section 4.2 we relax the assumption of symmetric markets which allows us to extend our conclusions to a larger set of real-world situations. 23 The price equilibrium payoffs and the equilibrium prices are reported in the Supporting Information Appendix for each pair of locations and different values of the CRRA parameter. 24 And its symmetric ⋆ ⋆ ( )

x x , =(2,7 ) H H 1 2
. 25 And its symmetric ⋆ ⋆ ( )

x x , =(3,6) L L 1 2
. driven by the longer support effect. Comparison between the DC-LS and HDU treatments shows that subjects differentiate more when faced with a high level of demand uncertainty (one-tailed M-W tests: DC-LS vs. HDU p =0.0030 for risk-lover subjects and p =0.0003 for risk-neutral subjects). Finally, note that because we have only one pair of highly risk-averse subjects ( ≥ r 0.9) in the DC-LS treatment, we are not able to conduct this robustness test for this risk profile. resulting expected profit. Otherwise, we proceed in exactly the same way as in the computation of the price Nash equilibria in the three experimental treatments. Table A1 gives the equilibrium prices, the firm's demand, and the firm's profit for each pair of locations for the DC treatment under the assumption that firms are risk-neutral. The firms' profits correspond to those reported in Table 1in the manuscript. To save space, predictions for risk-lover and high-risk averse firms in the DC treatment as well as those derived for each risk profile in the LDU and HDU treatments are reported in the Supporting Information Appendix.

) = -1) +( (1 2Φ( ) + 1 2Φ(1 - )) = 0)]. i i i i (C5)
The distribution of the CRRA estimates obtained in each treatment is plotted in Figure C2. We observe that the densities of the CRRA estimates are similar across treatments (Kruskal-Wallis test, p = 0.3324). Furthermore, for each treatment, the main part of the density function lies to the right of the risk-neutral prediction, revealing a tendency toward risk-averse behavior among our participants. Using the CRRA estimates and the CRRA

APPENDIX E: TOBIT ESTIMATES AND SYMMETRIC MARKETS-EFFECT OF DEMAND UNCERTAINTY ON LOCATION CHOICES

We provide in this section empirical evidence that gives support to the results of the nonparametric tests regarding the effects of risk attitudes on differentiation with symmetric markets (see Section 4.2). Indeed, for some risk profiles, the nonparametric tests are performed with a low number of observations which makes questionable the statistical power of the tests. We therefore conduct additional analyzes. Notes: The Tobit regressions also include round dummies to control for potential learning or period effects. The estimated coefficients of the dummy variables are not shown to save space. Only the point estimate of the fourth round is significant (at the 10% level) in Columns ( 4) and ( 5). Robust standard errors are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

Abbreviations: LDU, low demand uncertainty; HDU, high demand uncertainty.

To overcome the problem of the low number of observations by market, we propose/construct a new measure of differentiation defined at the individual level which allows us to increase the size of the sample. This measure corresponds to the distance (in absolute value) between the center of the demand support and subject's location. It informs about the choice of the subject to locate further from the center in each market. We use this individual variable and conduct parametric analyzes by running Tobit models for different risk profiles. Precisely, we regress this distance measure against the level of demand uncertainty (a binary variable for each experimental treatment and the DC treatment taken as the reference) while controlling for subjects characteristics (CRRA parameter, age, gender, work, undergraduate, and economics student) and round periods. As before, the regressions are performed using samples of subjects that compete with a rival of the same risk profile.

The estimates of the Tobit models are reported in Table E1 with robust standard errors in parentheses. The first two columns present the results of the estimation conducted with risk-lover and risk-neutral pairs of subjects. They both confirm that a high demand uncertainty significantly increases the distance variable (i.e., a proxy of the level of differentiation) for these risk profiles. Furthermore, we find no significant effect of a low demand uncertainty on differentiation. These results corroborate the findings of the nonparametric tests conducted in Section 4.2.

Columns (3)-( 5) contain the estimation results for risk-averse subjects. We first consider the group of risk-averse subjects for which demand uncertainty does not lead to an agglomeration effect. Since there is no market in the data where both subjects have an estimated CRRA parameter below 0.3, this boils down to the case where the differentiation force of demand uncertainty is predicted to be counterbalanced by the agglomeration force of risk aversion. The sample is thus composed of subjects with a CRRA parameter between 0.30 and 0.45. The estimation results are reported in Column (3) and show that subjects confronted with a high demand uncertainty differentiate more than that in the DC treatment. This is consistent with the results of the nonparametric tests. 40 In Column (4), we consider the sample of subjects that are predicted to agglomerate when facing demand uncertainty (i.e., ≥ r 0.55). In contrast to the model predictions, we find that, on average, subjects locate further away from the center of the demand support when facing demand uncertainty compared with the DC treatment. However the negative sign of the estimated coefficient(s) of the interaction terms between the LDU (HDU) treatments and the CRRA parameter indicates that the more averse subjects are, the more they locate closer to the center when facing demand uncertainty. Again, these results are in line with the results of the nonparametric tests computed with market-level data. Finally, we estimate in Column (5) a Tobit model for which we only consider high-risk averse subjects with an estimated CRRA parameter equal or above 0.90. In line with the model prediction and the nonparametric analysis, we find that on average high-risk averse subjects differentiate significantly less when facing demand uncertainty (LDU and HDU treatments) compared with the DC treatment.

APPENDIX F: TOBIT ESTIMATES AND ASYMMETRIC MARKETS

We present in Table F1 the Tobit estimates where we regress the distance (in absolute value) between the center of the demand support and subject's location on its rival's risk profile. The following regressions are performed using the sample of asymmetric markets. The aim of these parametric analyzes is to test whether a subject modifies its differentiation strategy (captured here by its distance to the center of the demand support) depending on its rival's risk profile. As a subject participates in only one treatment, we perform the regressions by treatment which allows us to fully control for the effect of demand uncertainty. The specification also includes subjects characteristics (CRRA parameter, age, gender, work, undergraduate, economics student) and round periods. The size of the samples allows us to cluster standard errors at the independent matching groups level. As expected, the estimates reveal that subjects adopt the same differentiation strategy regardless of their rival's risk profile. Note: The Tobit regressions also include round dummies to control for potential learning or period effects. The estimated coefficients of the dummy variables are not shown to save space. The point estimate of the fourth, fifth and sixth round is significant in Column (1) as well as the fifth round in Column (3).

Clustered standard errors at the independent matching groups level are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

Abbreviations: DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty.

  High demand uncertainty treatment L ( = 8): Linear city with seven consumers and 15 potential locations.

F

  I G U R E 3 Low demand uncertainty treatment L ( = 4): Linear city with seven consumers and 11 potential locations.

  T A B L E 2 Location and price equilibria by treatment and risk preferences.

F

  Abbreviations: DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty.

F

  I G U R E 5 Levels of differentiation by treatment. The distribution of the levels of differentiation for each treatment is represented by means of a Box and Whisker plot. The closed boxes are constructed from the first and third quartiles and the middle bold line indicates the median. Thus, the length of the box represents the interquartile range (IQR). The whisker lines delimit all observations within the IQR 1.5 of the nearest quartile. The circles denote outliers. DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty; DC-LS, demand certainty-long support.

Notes:

  Payoffs are in Experimental Currency Units. The last three columns report the expected payoff differences and intervals for constant relative risk aversion (CRRA) estimates and were not shown to subjects. F I G U R E C1 Computer screen for a decision task in the risk experiment. [Color figure can be viewed at wileyonlinelibrary.com]

F

  I G U R E C2 Estimated CRRA with a power function and Fechner noise. The dashed lines represent the interval of the values of the CRRA parameter for risk-neutral subjects. Density at the left (resp., right) represents values for risk-lover (resp., risk-averse) subjects. CRRA, constant relative risk aversion; DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty. [Color figure can be viewed at wileyonlinelibrary.com] T A B L E C2 Share of risk preferences by treatment (in %).

  T A B L E F1 Tobit models and asymmetric markets.

  Equilibrium payoffs in the price subgame (demand certainty).

		1	2	3	4	5	6	7
	1	(3.5, 3.5) e	(1, 6)	⋆ (4.71, 15.88)	⋆ (10, 22.42)	(18, 28)	(21, 28)	(24.5, 24.5) e
	2	(6, 1)	(3.5, 3.5) e	(2, 5)	⋆ (10, 17)	⋆ (20, 23.65)	(24.5, 24.5) e	(28, 21)
	3	⋆ (15.88, 4.71)	(5, 2)	(3.5, 3.5) e	(6, 8)	⋆ (18.2, 18.2)	⋆ (23.65, 20)	(28, 18)
	4	⋆ (22.42, 10)	⋆ (17, 10)	(8, 6)	(3.5, 3.5) e	(8, 6)	⋆ (17, 10)	⋆ (22.42, 10)
	5	(28, 18)
	6	(28, 21)	(24.5, 24.5) e	⋆ (20, 23.65)	⋆ (10, 17)	(2, 5)	(3.5, 3.5) e	(6, 1)
	7	(24.5, 24.5) e	(21, 28)	(18, 28)

T A B L E 1 ⋆ (23.65, 20) ⋆ (18.2, 18.2) (6, 8) (3.5, 3.5) e (5, 2) ⋆ (15.88, 4.71) ⋆ (10, 22.42) ⋆ (4.71, 15.88) (1, 6) (3.5, 3.5) e

  Risk-averse subjects and levels of differentiation.

				Predicted effect	Empirical data
	CRRA		of demand	Mean differentiation		M-W test
	parameter	uncertainty	DC treatment	Uncertainty treatment	(p-value)
	Panel A: LDU treatment
	≥ r 0.90		Agglomeration	2.43 (7)	0.67 (3)	0.0500
	≥ r 0.70		Agglomeration	1.62 (13)	0.89 (9)	0.0795
	≥ r 0.55		Agglomeration	1.62 (13)	1.00 (14)	0.1126
	0.18	≤ r	< 0.55	No effect	0.85 (40)	1.40 (20)	0.0693
	0.18	≤ r	< 0.53	No effect	0.86 (29)	1.27 (15)	0.1675
	0.18	≤ r	< 0.40	No effect	0.91 (11)	1.29 (7)	0.4847
	Panel B: HDU treatment
	≥ r 0.90		Agglomeration	2.43 (7)	1.50 (6)	0.0921
	≥ r 0.70		Agglomeration	1.62 (13)	1.50 (8)	0.5000
	≥ r 0.45		Agglomeration	1.22 (18)	1.59 (29)	0.1976
	0.30	≤ r	< 0.45	No effect	0.73 (11)	1.95 (21)	0.0225
	0.18	≤ r	< 0.30	Differentiation	No observation		-

T A B L E 4

Notes: This table reports the predicted effect of demand uncertainty relative to the DC treatment, and the level of differentiation observed in the data for different values of the CRRA parameter. Panels A and B presents the respective results for the LDU and HDU treatments. The numbers in parentheses correspond to the number of observations for the considered case. Column (5) presents the p -values for the Mann-Whitney (M-W) U test. We conducted one-tailed tests except in the case of the null hypothesis of "No effect" which required a two-tailed test.

Abbreviations: CRRA, constant relative risk aversion; DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty.

  T A B L E 5 Differentiation and price competition.

		All periods of the price games
			Average					Distance to
	Differentiation	%	price	S.D.	P25	P50	P75	demand center
	Panel A: DC treatment
	0	36.73	2.62	1.47	1	2	3	0.11
	1	38.27	3.44	1.50	2	3	5	0.60
	2	16.05	3.90	1.70	3	4	5	1.10
	3	5.86	4.39	1.70	3	4	5	1.50
	4	3.09	4.67	1.84	3	5	6	2.00
	Panel B: LDU treatment
	0	23.46	2.38	1.42	1	2	3	1.16
	1	33.02	2.80	1.34	2	3	4	1.14
	2	24.69	3.25	1.46	2	3	4	1.48
	3	8.95	3.42	1.66	2	3	5	2.12
	4	5.86	3.48	1.61	2	3	5	2.42
	5	3.09	3.91	1.65	3	4	5	2.50
	6	0.62	3.50	1.85	2	3	5	3.00
	7	0.31	3.40	1.71	2	3	4	3.50
	Panel C: HDU treatment
	0	14.81	2.09	1.31	1	2	3	2.48
	1	26.85	2.59	1.46	2	2	3	2.56
	2	24.38	2.49	1.40	1	2	3	2.78
	3	15.12	2.91	1.52	2	3	4	2.50
	4	11.73	3.32	1.74	2	3	5	3.24
	5	2.78	3.83	2.06	2	4	5	4.06
	6	2.47	3.88	1.96	2	4	5	4.13
	7	1.23	4.13	1.57	3	4.5	5	4.50
	8	0.31	3.50	2.68	1	3	6	5.00
	9	0.31	2.40	0.84	2	3	3	4.50

  T A B L E 6 Price regressions. All the regressions include subjects sociodemographic characteristics (age, gender, work, undergraduate degree, and economics student). Clustered standard errors (at the group matching level) are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively. Abbreviations: DC, demand certainty; LDU, low demand uncertainty; HDU, high demand uncertainty; FE, fixed-effects.

	Dependent variable: Price	DC treatment	LDU treatment	HDU treatment
	Differentiation	0.6231***	0.3484***	0.3229***
		(0.0448)	(0.0415)	(0.0228)
	Distance to demand center	-0.1305	-0.2833***	-0.3300***
		(0.0915)	(0.0205)	(0.0251)
	Coefficient risk aversion	-0.0119	0.0669	-0.0026
		(0.1122)	(0.1001)	(0.0778)
	Constant	5.7286***	3.3390***	3.3038***
		(0.8024)	(0.3000)	(0.2952)
	Price period FE	Yes	Yes	Yes
	Matching group FE	Yes	Yes	Yes
	R 2	0.3428	0.2816	0.3482
	Observations	3240	3240	3240

Notes:

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The result is similar to[START_REF] Cheng | Vertical product differentiation under demand uncertainty[END_REF] for the case of firms that are vertically differentiated and uncertain about how consumers perceive product quality.

Nonetheless, it has been shown that in the one-dimensional case it is possible to obtain a range of equilibrium outcomes, from minimal to maximal differentiation, depending on the model assumptions (see, e.g.,[START_REF] Brenner | Determinants of product differentiation: A survey[END_REF] for a survey of Hotelling-style models of product differentiation).

These studies typically focus on markets where the geographical dimension is predominant. For instance,[START_REF] Schuetz | Why are walmart and target next-door neighbors?[END_REF] shows that Big Box retailers prefer to agglomerate in desirable locations rather than cede market share to rivals (see also[START_REF] Eckert | An empirical examination of clustering and dispersion within Canadian shopping centers[END_REF] for a study of shopping center locations), whereas[START_REF] Netz | Maximum or minimum differentiation? Location patterns of retail outlets[END_REF] find that gasoline stations increase their spatial differentiation as the intensity of competition increases (see also[START_REF] Seim | An empirical model of firm entry with endogenous product-type choices[END_REF] for a study of the video retail industry). The results in[START_REF] Elizalde | Competition in multiple characteristics: An empirical test of location equilibrium[END_REF] are less extreme and suggest both location equilibria in the context of the movie theater industry.

It is especially difficult to define a relevant proxy for product differentiation while controlling for exogenous market constraints (e.g., geographical boundaries or demographic factors in the retail industry; see[START_REF] Picone | Distance decreases with differentiation: Strategic agglomeration by retailers[END_REF] which could bias the measure.

To remain as close as possible to the pioneering work of[START_REF] Hotelling | Stability in competition[END_REF], we assume also that consumers are uniformly distributed along the linear city, that firm location is restricted to the inner city, that travel costs are proportional to the distance traveled, and that demand is inelastic up to a reservation price.

In our study, we consider only the Simulated Consumers Treatment of[START_REF] Barreda-Tarrazona | An experiment on spatial competition with endogenous pricing[END_REF], which implies odd numbers of locations and exogenous consumers.

To facilitate the exposition of the game in the lab, we fix to one the distance between two neighboring locations and set to one the transport cost per unit distance so that transportation costs are unchanged compared with those of the model.

In the Supporting Information Appendix, we discuss another strategy which allows firms to reach the collusive outcome.

In the case of multiple indifferent consumers, we consider the polar cases of none or all indifferent consumers visiting a firm. We would stress that location and price equilibria do not depend on this sharing rule. For instance, assuming that half of the indifferent consumers visit a firm leads to the same equilibria.

The multiplicity of equilibria exists in only 15.69% of the price combinations and the equilibrium of the location-then-price game is insensitive to this selection rule. In Appendix A we explain how we determined the unique price equilibrium for each pair of locations, and report the corresponding equilibrium prices.

Throughout the paper, we use the terms linear city and demand interchangeably.

Readers interested in a more general setting where uncertainty applies at the distributional level and does not only rely on the first moment of the distribution should consultMeagher and Zauner (2004, 2008).

Recently, Meagher et al. (2020) use a Salop-style model to study firms' entry/exit decisions over an industry life cycle and confirm the differentiation force of demand uncertainty.

[START_REF] Król | Product differentiation decisions under ambiguous consumer demand and pessimistic expectations[END_REF] examines the case where firms do not know the exact probability distribution of the mean of the consumer distribution

The procedure for the DC-LS treatment is similar to that implemented in the previous treatments. Overall, six sessions were conducted with 18 subjects per session, resulting in 108 subjects. No subject had previously participated in a similar experiment. Subjects' characteristics and risk preferences were similar to those in the previous treatments. Additional information about this treatment is available upon request from the authors.

Note that the classification holds if we use the first switching point to determine subjects' risk profiles but this method removes the heterogeneity of individual risk parameters which is needed for the subsequent analysis. Nonetheless, it should be remembered that the measure of risk preferences we obtained is only a proxy for the subjects' true risk preferences which are not observable.

To further check the robustness of the findings from our non-parametric tests, we also ran Tobit regression analyses at the individual level controlling for subjects' characteristics (see Appendix E).

Note that we cannot test the prediction that high demand uncertainty generates a differentiation effect for the least risk-averse subjects since we do not observe this market configuration in our data.

It is important to note that the differentiation force entailed by high demand uncertainty for risk-lovers and risk-neutral subjects is not

This makes sense since we show that both risk profiles differentiate more (equally) when faced with high (low) demand uncertainty compared with the certainty case.

The robustness tests and an analysis of the dynamics of price competition are provided in the Supporting Information Appendix.

This finding is robust if we include in the sample all the subjects that are supposed to not agglomerate when facing a high level of demand uncertainty (i.e., with a CRRA parameter between 0.30 and 0.55).

ACKNOWLEDGMENTS

We are grateful to two anonymous referees for suggestions and comments that have helped to improve the paper. We also thank Iván Barreda-Tarrazona, Léa Bousquet, Carl Gaigné, Stéphane Robin, and participants at the AFSE, ASFEE, JMA, EEA-ESEM, and EARIE conferences for valuable comments. Special thanks go to Elven Priour for valuable assistance in the programming of the experiment. We are grateful for support from the LOCEX program. All errors are our own.

and Lusk ( 2016) is that it removes non-linear probability weighting as the explanation for switching. 27 The repetition of the price subgame over a finite number of periods does not change the theoretical predictions of the game in gametheoretic terms. However, in a given market, the repetition of the price subgame may lead to lower (or higher) prices in the last period compared with the first period, perhaps reflecting the existence of some learning effects or collusive behaviors (see Camacho-Cuena et al., 2005 for a discussion of this issue). However, it is important to stress that thanks to our stranger matching protocol it cannot affect firms' decisions in the subsequent location-then-price game. Readers interested in the evolution of the price decisions over time should refer to the Supporting Information Appendix. 28 Note that for simplicity and to facilitate subjects' understanding, the demand support is only expressed in positive integers in the instructions. 29 For instance, Brown-Kruse et al. (1993) demonstrate that in their setting central locations are consistent with min-max behavior; risk aversion has been suggested as a possible explanation for central location (see, e.g., Collins & Sherstyuk, 2000; Georgantzis, 2006; Mangani & Patelli, 2002). In Section 4.2, we explore this explanation further. 30 Since we conduct multiple comparisons among the treatments, there is a risk of false positives. A Bonferroni correction to control for the familywise error rate confirms that all of our results are robust. 31 In Appendix C we show that the risk profile shares are similar among treatments which rules out composition effects as an explanation for the observed differences among treatments (Pearson χ 2 test: p = 0.343). 32 The existence of additional location slots may encourage some subjects to choose them even if they do not correspond to the equilibrium locations. This larger choice set could lead to an adverse effect known widely as "choice overload" (see Chernev et al., 2015 for a recent meta-analysis).

SUPPORTING INFORMATION

Additional supporting information can be found online in the Supporting Information section at the end of this article.

How to cite this article: Bonein, A., & Turolla, S. (2023). Spatial competition with demand uncertainty: A laboratory experiment. Journal of Economics & Management Strategy, 1-34. https://doi.org/10.1111/jems.12517

APPENDIX A: PRICE EQUILIBRIUM

To determine the SPEs of the location-then-price game, we have to derive the price equilibrium for each pair of locations x x ( , ) 1 2 . To this end, for each pair of locations, we first compute the firm's demand and the resulting profit for every possible price combination; then, we determine the Nash equilibrium in prices. With all the Nash equilibrium in prices in hand, we finally determine the price equilibrium of the location-then-price game.

It is, however, possible to obtain multiple price equilibria for some pairs of locations. To illustrate how we select the unique price equilibrium, we provide below a numerical example. We assume that non-risk lover firms choose the riskless option, meaning the price equilibrium which ensures an equivalent level of profit with a safe number of consumers. If the multiplicity of equilibria persists, we consider the strict Nash equilibrium. However, if there is no strict Nash equilibrium, we assume that, we select the Pareto price equilibrium. For instance, assume risk-neutral firms and the pair of locations (3,4) in the DC treatment. There are two potential Nash price equilibria. In the first equilibrium, each firm sets a price equal to 1, and the corresponding profits are (3,4). In the second one, each firm sets a price equal to 2, and the corresponding profits are (6,8). The result is that the selected price equilibrium is (2,2), as it is Pareto optimal. When no Pareto price equilibrium exists, we select the joint profit-maximizing equilibrium. Finally, it is noteworthy that a price equilibrium in pure strategy does not exist for each pair of locations. For these particular price-setting games, we thus calculate a mixed-strategies Nash equilibrium by considering all the plausible price supports, that is, all the prices that have a positive probability of being played in an equilibrium strategy.

Note that while firms choose their location with perfect knowledge about the location of demand in the DC treatment, it is unknown in the LDU and HDU treatments. The uncertainty about demand location implies several potential demand locations that we assume uniformly distributed in the computation of the expected demand and the T A B L E A1 Equilibrium prices for each pair of locations and risk-neutral firms (DC treatment).

Pair of locations Prices Demands Profits

Level of differentiation equal to 0

(1, 1) (1, 1) (3.5, 3.5) (3.5, 3.5)

(2, 2) (1, 1) (3.5, 3.5) (3.5, 3.5) (3,3) (1,1) (3.5,3.5) (3.5,3.5) (4,4) (1,1) (3.5,3.5) (3.5,3.5) (5,5) (1,1) (3.5,3.5) (3.5,3.5) (6,6) (1,1) (3.5,3.5) (3.5,3.5) (7,7) (1, 1) (3.5, 3.5) (3.5, 3.5)

Level of differentiation equal to 1

Level of differentiation equal to 2 Level of differentiation equal to 6

(1, 7) (7, 7) (3.5, 3.5) (24.5, 24.5)

Notes: For price equilibria in mixed strategies we report into brackets both the prices and the associated probabilities.

Abbreviation: DC, demand certainty.

APPENDIX B: GAME EQUILIBRIUM DEPENDING ON DIFFERENT DEGREES OF RISK AVERSION

In this section, we report in Table B1 the location and price equilibria by treatment and for different degrees of risk aversion. This enables us to examine the effects of demand uncertainty when considering different levels of risk aversion. For comparison purposes, we also report the equilibria obtained with the values of the CRRA parameter considered in the manuscript (see Table 2). A wide variety of methods could be used to elicit individuals' risk preferences. In our design, we follow the experiment of Drichoutis and Lusk (2016). Each subject completes 10 decision tasks where each task represents a choice between two binary lotteries, A and B. As shown in Table C1, the probabilities remain constant and equal to 1/2 across the decision tasks and in the lowest payoff for each lottery.

T A B L E C1 Elicitation task for risk preferences. Compared with the well-known Holt and Laury (2002) task, the probabilities remain constant across the 10 decision tasks, and the euro payoffs change such that the switch point from lottery A to lottery B can only be explained by the shape of the utility function and, in turn, risk attitudes. This method has the clear advantage of removing non-linear probability weighting as an explanation for the switch between lottery A and lottery B.

Previous experimental studies have underlined several switching back behaviors from lottery B to lottery A after the first switch. As noted by Andersen et al. (2006), it is quite possible that such switching behaviors are the result of subjects being indifferent between lotteries. To limit this behavior and obtain a more precise estimation of risk attitudes, we allow an explicit indifferent option for each pairwise lottery. Although it is possible for them to switch back, they also have the option of explicitly selecting indifference. The decision task is thus presented in Figure C1.

Subjects' choices are used to determine their risk preferences. For instance, a risk-neutral subject would choose lottery A for the first four decisions listed in Table C1 because the expected value of lottery A exceeds the expected value of lottery B for the first four choices. As one moves down each row of Table C1, the expected value of lottery B exceeds the expected value of lottery A. Thus, responses to these pairwise lottery choices allow us to estimate the coefficient of constant relative risk aversion for each subject.

C.2 | Structural estimation of risk aversion

We estimate a structural model while assuming the expected utility theory. We follow the modeling strategy of Harrison and Rutström (2008) and Andersen et al. (2010) to identify risk aversion parameters for the subjects in our sample.

From Table C1, we observe that subjects face a series of lottery choices j, where a choice has to be made between two lotteries A and B: { } ( )( )

) and a low outcome y l A (resp., y l B ) with probability

). Note that lottery B has a larger variance than lottery A. We model individual utility as suggested by the expected utility. The value function is written as follows:

where y is the lottery prize and ≠ r 1 is a parameter to be estimated. For r = 1, we assume that u y ln y ( ) = ( ) if needed. Thus, r is the coefficient of CRRA, with r < 0, r = 0, and r > 0 yielding convex, linear or concave value functions, respectively, and thus risk-lover, risk-neutral, or risk-averse subjects, respectively.

We model the decision using a discrete choice model in which we consider a latent variable d* associated with the decision process. We do not observe d* but only the choices that subjects make:

It follows that for subject i and for a given lottery ∈ k A B { , }, the expected utility is written as

Finally, we allow subjects to make some errors; that is, the probability of choosing a lottery is not one when the expected utility of that lottery exceeds the expected utility of the other lottery. We consider the Fechner specification popularized by Hey and Orme (1994) that implies a simple change in the difference in expected utility:

We estimate the CRRA parameter for each subject using a maximum likelihood estimator. The conditional log-likelihood function is written as in Equation (C5), where ⋅ Φ( ) is the standard normal distribution function, y = 1 i when lottery B is chosen, y = -1 i when lottery A is chosen, y = 0 i when the subject is indifferent and ∇EU is the difference in expected utility between the two lotteries. intervals reported in Table C1, we are able to determine the risk profile of each subject. The resulting repartition of risk profiles is reported in Table C2 by treatment. We note that the frequency of risk-lover, risk-neutral and risk-averse subjects does not notably differ across the experimental treatments (Pearson χ 2 test: p = 0.343). Furthermore, in each treatment, onehalf of subjects is risk-averse, and the remaining subjects are almost equally divided between risk-lover and risk-neutral.

Finally, in Section 4.2 of the manuscript, we analyze how demand uncertainty impacts location decisions by risk profile. We first conduct the analysis by considering markets in which a subject competes with a rival of the same risk profile (i.e., symmetric markets). This selection process leads us to consider 38.58% of the pairs of subjects (375 in a total of 972 pairs). In the second step, we consider asymmetric markets composed of non-risk averse and risk-averse subjects. This corresponds to 488 pairs of subjects (50.21% of the pairs of subjects). The number of pairs of subjects by market configuration and by treatment is reported in Table C3. Note that a last market configuration, corresponding to a pairing between a risk-lover and a risk-neutral subject, is not represented in this a For the educational level, once students receive their baccalaureate at the end of secondary school, they can enter the postsecondary education system (e.g., university). At this stage, they could be in the first year of study (Licence 1), the second year of study (Licence 2), and the third year of study (Licence 3; designated as a bachelor's degree). These three levels correspond to the undergraduate level. They could also be in graduate studies (first year as Master 1 or second year as Master 2).