Felipe Maciel Cardoso

Carlos Gracia-Lázaro

Frederic Moisan

Sanjeev Goyal

Ángel Sánchez

Yamir Moreno

Effect of network topology and node centrality on trading

Global supply networks in agriculture, manufacturing, and services are a defining feature of the modern world. The efficiency and the distribution of surpluses across different parts of these networks depend on the choices of intermediaries. This paper conducts price formation experiments with human subjects located in large complex networks to develop a better understanding of the principles governing behavior. Our first experimental finding is that prices are larger and that trade is significantly less efficient in small-world networks as compared to random networks. Our second experimental finding is that location within a network is not an important determinant of pricing. An examination of the price dynamics suggests that traders on cheapest-and hence active-paths raise prices while those off these paths lower them. We construct an agent-based model (ABM) that embodies this rule of thumb. Simulations of this ABM yield macroscopic patterns consistent with the experimental findings. Finally, we extrapolate the ABM on to significantly larger random and smallworld networks and find that network topology remains a key determinant of pricing and efficiency. open

Globalization is a prominent feature of the modern economy [START_REF] Rodrik | The Globalization Paradox: Democracy and the Future of the World Economy[END_REF] . Nowadays, supply, service and trading chains [START_REF] Anderson | Trade costs[END_REF][START_REF] Antràs | Organizing the global value chain[END_REF][START_REF] Antràs | Organizing the global value chain[END_REF][START_REF] Hummels | The nature and growth of vertical specialization in world trade[END_REF][START_REF] Antras | Intermediated trade[END_REF][START_REF] Chaabane | Design of sustainable supply chains under the emission trading scheme[END_REF] play a central role in different contexts such as agriculture [START_REF] Fafchamps | Relationships and traders in madagascar[END_REF][START_REF] Bayley | A Revolution in the Market: The Deregulation of South African Agriculture[END_REF][START_REF] Meerman | Reforming Agriculture: The World Bank Goes to Market[END_REF][START_REF] Traub | The effects of price deregulation on maize marketing margins in south africa[END_REF] , transport and communication networks [START_REF] D'ignazio | Antitrust analysis for the internet upstream market: a border gateway protocol approach[END_REF][START_REF] D'ignazio | Asymmetry and discrimination in internet peering: evidence from the linx[END_REF] , international trade [START_REF] Dicken | Chains and networks, territories and scales: towards a relational framework for analysing the global economy[END_REF] and finance [START_REF] Li | Dealer Networks: Market Quality in Over-the-Counter Markets[END_REF][START_REF] Gale | Trading in networks: a normal form game experiment[END_REF] . One key question on these systems is how pricing dynamics by intermediaries of the economy impacts both efficiency and surpluses. The purpose of this paper is to develop a better understanding of the forces that shape intermediary pricing behavior in such complex networks.

Game theory constitutes a useful framework to study competition among trading agents [START_REF] Osborne | Bargaining and Markets[END_REF] . In this context, the Nash Bargaining Game [START_REF] Nash | The bargaining problem[END_REF] studies how two agents share a surplus that they can jointly generate. In the Nash Bargaining Game, two players demand a portion of some good. If the total amount requested by both players is less than the total value of the good, both players get their request; otherwise, no player gets their request. There are many Nash equilibria in this game: any combination of demands whose sum is equal to the total value of the good constitutes a Nash equilibrium. There is also a Nash equilibrium where each player demands the entire value of the good [START_REF] Nash | Two-person cooperative games[END_REF] .

As a generalization of Nash demand game to n players, Choi et al. [START_REF] Choi | Trading in networks: theory and experiments[END_REF] proposed and tested in the laboratory a model of intermediation pricing. In this model, a good is supposed to go from a source S to a destination D. Intermediaries, which are located in the nodes of a network, may post a price for the passage of the good. Trading occurs if there exists a path between S and D on which the sum of prices is smaller than or equal to the value of the good. The key finding was that the pricing and the surpluses of the intermediaries depends on the presence of "critical" nodes: a node is said to be critical if it lies on all possible paths between S and D. Condorelli and Galeotti [START_REF] Galeotti | Strategic models of intermediation networks[END_REF] provide an overview of the literature on intermediation and argue that the criticality of a node is an important determinant of pricing behavior, intermediation rents, and the efficiency of trading, in a wide class of models of auctions and bargaining. The goal of the present paper is to investigate this claim in large scale networks, so as to develop a better understanding of the role of network topology in commerce.

We conduct experiments with human subjects embedded in complex networks: specifically, we consider a random network and a small-world network each with 26 subjects (and the same level of average connectivity).

In these networks there are no critical nodes: the results of Choi et al. would suggest that intermediary prices must be close to zero and that their surpluses must also be close to zero. As we will show below, our first finding is that, in all the networks studied, when there is not total but partial criticallity, intermediaries set positive prices and they make large profits. Moreover, network topology has powerful effects: in particular, in the random network, intermediaries set lower prices as compared to a small-world network. As a consequence, there is full trading efficiency in the random network, but trade breaks down in almost one third of the cases in the smallworld network.

This striking difference leads us to an examination of how location within a network affects pricing: our second finding is that within a given network, standard measures of network centrality appear to have no significant effect on pricing behavior. As network location does not matter for prices, the presence on the cheapest and active path must be crucial for profits. And indeed, this is what we observe: intermediaries' earnings are positively related to their betweenness weighted by the path length.

Turning to the dynamics of price setting, we observe that traders raise prices if they lie on the successful trade path (i.e., the least-cost path), and that they lower prices when they are off the least-cost path. Based on these observations, we build an agent based (ABM) model that reproduces qualitatively the experimental results. We then use simulations to extrapolate our findings to larger networks: our third finding is that network topology continues to matter and that random networks exhibit lower prices and higher level of efficiency even when there are 100 traders. Finally, our forth finding uncovers the role of node-disjoint paths-two paths are disjoint if and only if they do not share any node-and of the average path length in shaping level of pricing in the simulations: networks that have a larger number of node-disjoint paths exhibit lower prices and higher efficiency. Among networks with the same number of node-disjoint paths, average path length is an important driver of costs.

Experimental setup

We consider a simple game of price setting in networks to study supply, service and trading chains taken from Choi et al. [START_REF] Choi | Trading in networks: theory and experiments[END_REF] . Let N be a set of nodes N = {S, D, 1, 2, . . . , n} , where S is a source and D a destination; and L a set of pairs of elements of N . N and L define a trading network where the elements of L are the links. A path between S and D is a sequence of distinct nodes {i 1 , i 2 , ..., i l } such that {(S, i 1 ), (i 1 , i 2 ), (i 2 , i 3 ), . . . , (i l-1 , i l ), (i l , D)} ⊂ L.

Each experiment consists of 4 series of 15 rounds each, and it involves n human subjects that will play the role of intermediaries. Before starting the first round of a series, each subject is randomly assigned to a node in {1, 2, . . . , n} . The positions of S and D are also assigned at random. These positions (players, S and D) remain constant over the 15 rounds. Subjects are always informed about the network and their position in it, that is, they can see the whole network including S and D. At each round, every subject has to make a decision; namely, she has to post a price from 0 to 100 tokens for the passage of a good by her node. The prices determine a total cost for every path between S and D. A path is feasible if its cost is not greater than a given threshold (100 tokens) that represents the value of the economic good generated by the path. After all players have made their choices, the cheapest path is selected: if it is feasible, each player located in this path receives her proposed price as a payoff. Otherwise, no trade takes place and payoffs are zero. Players who are out of the selected path do not get any payoff in that round. In the case of more than one cheapest path, the tie is resolved through a random choice among cheapest paths. From the second round onward, players are informed about the existence of a trade in the previous round, about the previously selected path, and about the prices and payoffs of all the players in the previous round together with their positions in the network.

We have conducted two experimental sessions in a random network of 26 nodes with �k� = 3 and two more sessions in a small-world-like network of 26 nodes with �k� = 3 . Additionally, we have conducted another experimental session in a random network of 50 nodes with �k� = 4 that will allow us to check the robustness of the results against the size and connectivity of the network. These results can be found in SI sections 1-2 and SI Table S2. Further details can be found in Materials and Methods section.

Results and discussion

The networks used in the experiment allow for coexisting paths with a different number of intermediaries, where theory predicts both efficient and inefficient (Nash) equilibria. Furthermore, these networks present different characteristics, such as degree and centrality distributions, that may affect the bargaining power of the intermediaries. These facts motivate our first question: How does the network topology affect costs and prices? Figure 1A shows the cheapest path cost in each of the networks considered. As shown, the small-world networks exhibit higher costs than random networks (t(232.41) = 15.5, p < 0.001 ). Figure 1B instead displays the costs of the cheapest path normalized by the number of nodes on it, i.e., the mean price of nodes along the cheapest path. The differences between networks persist, indicating that prices and costs strongly depend on the topology of the network. These results, separated by rounds, are shown in Fig. S4 of the SI. Table 1 shows that there is a very large effect of topology on efficiency: in the random network trade is realized in practically all the cases, while in the small-world network trade breaks down in almost one third of the cases (binomial-test, 0.95 CI=(0.76, 0.90), p < 0.001 ). However, small-world networks involve higher costs and profits than random networks, since the higher posted prices compensate for the lower efficiency. Therefore, we conclude that the topology of the network matters for intermediation: the surpluses of the intermediaries vary significantly from one network class to the other in the experiments.

Profit is only obtained when the subjects are on the cheapest path, i.e., when they are on the path through which the trading is realized. Thus, it is of interest to examine what is the role of the location of intermediaries in the network in shaping their behavior, which we do next. First, we observe that the networks in our experiment do not contain any critical nodes and yet they generate large rents. So the results from the small scale experiments by Choi et al. [START_REF] Choi | Trading in networks: theory and experiments[END_REF] do not apply to complex larger scale networks, the current study being an extension of work by Choi et al. [START_REF] Choi | Trading in networks: theory and experiments[END_REF] to more complex networks. It seems likely then that nodes that are present on more paths have greater market power. This motivates a generalization of the notion of criticality as follows:

where sd(v) is the partial criticality of node v, |P SD (v)| stands for the number of paths between the source and destination containing a given node v, and |P SD | for the total number of paths between the source and the desti- nation. Following this line of thought, a higher partial criticality may indicate a potential for greater bargaining power and therefore nodes with a higher partial criticality should show higher prices and profits. Figure 2A shows the accumulated prices of the intermediaries as a function of their partial criticality. There is no significant relation between partial criticality and the prices posted by participants. Even more strikingly, as illustrated in Fig. 2B, there is no relationship between the accumulated payoff obtained and a node's partial criticality. Figure 2C shows the frequency that each player is on the cheapest path versus her partial criticality. Again, there is no relation between these variables. This lack of correlation may be due to the equal weighting of paths with different length. In order to address this point, we refine our generalized notion of partial criticality to take path length into account:

where the summations are over all the paths between S and D containing v (numerator) and over all the paths between S and D (denominator). l(p) represents the length of path p and α stands for an arbitrary weight: as α increases, more importance is given to shorter paths. Specifically, when α → ∞ it will consider only the shortest paths, sd ∞ (v) being a measure of the source-destination betweenness of node v (SD-betweenness(v)). On the opposite side, for α = 0 the partial criticality of Equation 1 is recovered, that is, sd 0 (v) = sd(v).

Figure 2D shows the accumulated prices of the intermediaries as a function of their SD-betweenness. Again, there is no relation observed between pricing behavior and betweenness. However, as shown in Fig. 2E, there is a positive correlation between the accumulated payoff obtained by intermediaries v and their sd ∞ (v) . That is, although pricing is uncorrelated with SD-betweenness centrality, profits are positively correlated with it. The (1) reason behind this difference must therefore lie in how the presence of v on the least-cost path is correlated with sd ∞ (v) . This is displayed in Fig. 2F, which represents the fraction of times that an intermediary is on the cheap- est path versus her SD-betweenness. As shown, there is a positive correlation between these measures, which explains why-in a situation where prices are largely insensitive to network location-profits will be correlated with sd ∞ (v) . The robustness of these results against the size and connectivity of the network is discussed in the SI, Section 2.A. So far, we have seen that node centrality does not influence earnings when we equally consider all the paths from S to D to compute it, but it does when we consider only the shortest paths. This fact indicates that the weight given to paths length is important to study the capacity of the nodes to extract surpluses. In order to verify this hypothesis, Fig. 2G shows the coefficient of determination R 2 of the regression of intermediaries payoffs on sd α as a function of α . The best fit is obtained for α ∼ 12 , which indicates that longer paths should have significantly smaller weight than shorter ones. As the number of paths grows exponentially with network size, SD-betweenness seems to be a feasible and good descriptor of participants' earnings.

sd(v) = |P SD (v)| |P SD | , (2) sd α (v) = [S,v,D] l(p) -α [S,D] l(p) -α , Figure
Behavioral rules. We have noted that participants' behavior is not determined by network position: criticality and classical measures of centrality are not good predictors of the prices posted by intermediaries. Nonetheless, results show differences in the prices posted by traders across different networks. Even if these networks might seem relatively small and similar, they are not. The environment (defined as the set of all the information that the individuals need to factor in their decisions) is very complex: there are many different paths passing through most of the traders, they need to take into account their price as well as those of other players, etc. It is thus reasonable to assume that the traders confronting such a complex and dynamic environment use rules of thumb, which on the other hand, should not depend on the network. In what follows, we develop a model that accounts for individual behavior and for the differences observed experimentally.

Together with the network information, the other information shown to subjects is whether they were on the selected trading path. Figure 3A shows, for each one of the networks considered, the mean change in price for the cases when the participant was or was not along the cheapest path in the previous round. In the same way, Fig. 3B shows the probabilities to increase and to decrease the posted price conditioned to have been (Y) or not (N) in the cheapest path. Players appear to follow a simple rule, namely, to increase their price if they were on the cheapest path in the previous round and to decrease it otherwise. Furthermore, the expected values shown in Fig. 3A point out that successful intermediaries keep increasing their prices and therefore, without sufficient competition, costs and prices would always grow.

We now build a simple agent based model (ABM) [START_REF] Goldstone | Computational models of collective behavior[END_REF] , as described below:

i) If node u belongs to a cheapest path at time t, it will change its posted price on time t + 1 by σ; ii) If node u does not belong to a cheapest path at time t, it will change its posted price on time t + 1 by -ρ;

iii) The minimum price a node can post is 0. To validate this model we executed it by bootstrapping the initial prices, the value of changes if on the cheapest path ( σ ) and the value of changes if not ( ρ ). The results, shown in Fig. 4, indicate that costs from simulations (resp. efficiency) are higher (resp. lower) in small-world networks than in random networks (t(9659.3)=68.33, p < 0.001 ), in agreement with our experimental results. Costs reached relatively high values in some rounds, as the model does not incorporate participants direct response to the maximum cost threshold. Table 2 also confirms that topological differences between the networks are driving the differences in cost.

Once we have shown that the model captures very well the experimental observations, we verify if the same phenomena are observed in larger networks. Results for networks of size 50 and 100, shown in Fig. 5 and Table 3, are also consistent with the experimental data, confirming that the network topology has a significant effect on trading outcomes: small-worlds lead to higher costs and lower efficiency. A similar analysis with random initial prices, thus unlinking numerical results from those obtained from the experiments, can be found in SI section C3 and SI Fig. S7. Results in Fig. S7 are compatible with those shown in Fig. 5, providing more evidence about the effects of the network structure on prices and costs.

Topological properties behind the differences in cost. Finally, we go one step further in order to explain what lies behind the differences found in costs. One possible theoretical hypothesis could be that costs depend on competition between paths. In our setup, this would be equivalent to assume that costs should decrease with the number of possible ways to reach the destination, i.e., the number of independent (sets of) paths from S to D. Specifically, we expect competition to be proportional to the number M of node-disjoint paths [START_REF] Van Steen | Graph Theory and Complex Networks: An Introduction[END_REF] , as it captures the possible number of simultaneous independent trades (see SI Section II.C.1 for a deeper discussion on this subject). According to this hypothesis, the larger the value of M, the lower the cost. Another possible explanation for the dependency of costs with the networks could be the structural differences between Table 2. Numerical results in experimental networks. Efficiency (fraction of rounds in which the cheapest path cost was equal to or less than the threshold), and mean values of the price, price in the cheapest path, cost of the cheapest path, and cheapest path length. Results obtained from numerical simulations with each one of the two studied networks with their corresponding source and destinations. ), and therefore the observed differences in cost could be tied to variations in those properties. In order to verify the previous hypotheses, we executed a version of the model without the maximum cost threshold. With this setup, we can study long-term effects after a sufficiently large number of rounds and uncover the cost tendency. In this regime, we cannot analyze network efficiency, however, networks yielding higher cost should be more inefficient. Note that the proposed model allows extrapolating the observed behavior to larger networks with a large range of values of M. Then, we can generalize the observed experimental results to larger networks, which allow us to find the (theoretically conjectured) influence of M on prices. We ran the algorithm for 10 4 rounds and then we considered the final cost of the trade for each configuration. Results for networks of size 26, 50 and 1,000 nodes are shown in Fig. 6A-C, respectively. Simulations of trading dynamics on the aforementioned networks indicate that the number of node-disjoint paths (M) between S and D is the best indicator of final cost. Figure 6D shows that as M grows, the costs are reduced so drastically that they go to 0 for M > 3 . Moreover, the numerical results also reveal that for networks with the same value of M, the cost grows with the average path length. Indeed, this dependency explains why costs on small-world networks tend to be larger: these networks have a larger average path-length. To show that this finding is not a consequence of differences in the length of the cheapest paths, Fig. S5 of the SI displays, for the same simulations, the costs of the cheapest path normalized by the number of nodes on it versus the average path length of the network. It can be seen that the mean price of nodes in the cheapest path also correlates with the average path length. Interestingly, even though in this regime the difference in the clustering is larger than the difference in average path length, the former is not a good indicator of costs ( R 2 = 0.57 vs. R 2 = 0.79 , see Section 2.C.2, table S3, and Figure S6 in the SI). In summary, these results provide two stylized facts that may guide future inquiries in this line, namely, trading costs will be null in setups with a relatively large number of node-disjoint paths and costs should be larger in networks with larger average path length. Note that, although the influence of the type of network on efficiency, costs, and prices is an experimental result, the effect of both the number of disjoint paths and average path length on costs is a result obtained from the proposed model. ) of the cheapest path after a period of 10 4 rounds as a function of the average path length of the network. Different panels correspond to different network sizes: 26 (A), 50 (B), and 1,000 (C) nodes; colors correspond to different network models: random (blue) and small-world (magenta); and different shapes correspond to different values of the number M of disjoint paths. For each configuration, there were generated 10,000 networks of size 26, 50, and 1,000, according to the Watts-Strogatz algorithm [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] with p = 0.1, 1 and average degree from 2 to 10. The initial cost was set to 0 and the increment/decrement ratio was fixed to the experimental value ( σ/ρ = 2.4 ). Results for M > 5 are not shown as costs converge fast to 0. D: Mean value of the cost of the cheapest path versus M for the same networks. See the main text for further details.

GFigure 2 .

 2 Figure 2. SD-betweenness determines payoffs but not posted prices. (A-F) Accumulated price (A,D), accumulated payoff (B,E) and frequency in the cheapest path (C,F) of participants during a series of 15 rounds as a function of the node criticallity sd 0 (A-C) and of the SD-betweenness sd ∞ (D-F). (G) R 2 of the regression of participants accumulated payoff on sd α versus α , where α modulates the weight of the length of the paths in the S-D centrality measure. Dashed (points) line show the value of R 2 for correlation of payoffs on the betwenness (SD-betweenness). Data is pooled across any series of 15 rounds in any experimental session. For similar analyses within each experimental network, see SI Fig. S2. Figure created using ggplot2 (v.3.2.1) 21 .

Figure 3 .

 3 Figure 3. Being or not in the cheapest path determines the intermediaries price increases. (A) Mean changes in the posted price conditioned to have been (right) or not (left) in the selected cheapest path in the previous round for the random networks of 26 nodes (R26), and for the small-world network of 26 nodes (SW26). (B) Probability to increase (blue) and to decrease (pink) the posted price conditioned to have been (Y) or not (N) in the selected cheapest path, for each one of the studied networks. The error bars represent the 95% C.I. An extension of these results including the 50-nodes Random Network is displayed in SI Fig. S3.Figure created using ggplot2 (v.3.2.1) 21 .

  Figure 3. Being or not in the cheapest path determines the intermediaries price increases. (A) Mean changes in the posted price conditioned to have been (right) or not (left) in the selected cheapest path in the previous round for the random networks of 26 nodes (R26), and for the small-world network of 26 nodes (SW26). (B) Probability to increase (blue) and to decrease (pink) the posted price conditioned to have been (Y) or not (N) in the selected cheapest path, for each one of the studied networks. The error bars represent the 95% C.I. An extension of these results including the 50-nodes Random Network is displayed in SI Fig. S3.Figure created using ggplot2 (v.3.2.1) 21 .

Figure 4 .

 4 Figure 4. Numerical results of the model executed over the networks, source and destination from the experiments. Results shown are for 100 executions with 15 rounds for each network and source-destination pair, excluding the first round. Initial prices are bootstrapped from the experimental values. Values of σ and ρ are fixed and correspond to mean values of the experiment, respectively, 2.60 and 1.2.Figure created using ggplot2 (v.3.2.1) 21 .

Figure 5 .

 5 Figure 5. Numerical results of the model for networks with 50 and 100 nodes. Results shown are for 100 executions with 15 rounds for each network and source-destination pair, excluding the first round. Initial prices are bootstrapped from the experimental values. Values of σ and ρ are fixed and correspond to mean values of the experiment, respectively, 2.60 and 1.2. For similar analyses with random initial prices, see SI Fig. S7. Figure created using ggplot2 (v.3.2.1) 21 .

Figure 6 .

 6 Figure 6. Numerical results of the model. (A-C) Average final cost (in 10 4) of the cheapest path after a period of 10 4 rounds as a function of the average path length of the network. Different panels correspond to different network sizes: 26 (A), 50 (B), and 1,000 (C) nodes; colors correspond to different network models: random (blue) and small-world (magenta); and different shapes correspond to different values of the number M of disjoint paths. For each configuration, there were generated 10,000 networks of size 26, 50, and 1,000, according to the Watts-Strogatz algorithm[START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] with p = 0.1, 1 and average degree from 2 to 10. The initial cost was set to 0 and the increment/decrement ratio was fixed to the experimental value ( σ/ρ = 2.4 ). Results for M > 5 are not shown as costs converge fast to 0. D: Mean value of the cost of the cheapest path versus M for the same networks. See the main text for further details. Figure created using ggplot2 (v.3.2.1) 21 .

  Figure 6. Numerical results of the model. (A-C) Average final cost (in 10 4) of the cheapest path after a period of 10 4 rounds as a function of the average path length of the network. Different panels correspond to different network sizes: 26 (A), 50 (B), and 1,000 (C) nodes; colors correspond to different network models: random (blue) and small-world (magenta); and different shapes correspond to different values of the number M of disjoint paths. For each configuration, there were generated 10,000 networks of size 26, 50, and 1,000, according to the Watts-Strogatz algorithm[START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] with p = 0.1, 1 and average degree from 2 to 10. The initial cost was set to 0 and the increment/decrement ratio was fixed to the experimental value ( σ/ρ = 2.4 ). Results for M > 5 are not shown as costs converge fast to 0. D: Mean value of the cost of the cheapest path versus M for the same networks. See the main text for further details. Figure created using ggplot2 (v.3.2.1) 21 .

Table 1 .

 1 1. Network topology affects trading costs and prices. (A) Cost of the cheapest path for the random network of 26 nodes (R26), and for the small-world network of 26 nodes (SW26). (B) Mean price of participants in cheapest path for the same networks. Lines in the boxes denote the medians, whereas boxes extend to the lower-and upper-quartile values. Whiskers extend to the most extreme values within 1.5 interquartile range (IQR). Figure created using ggplot2 (v. 3.2.1)[START_REF] Wickham | ggplot2: Elegant Graphics for Data Analysis[END_REF] . Experimental results. Efficiency (fraction of rounds in which the cheapest path cost was equal to or less than the threshold), and mean values of the price, price in the cheapest path, cost of the cheapest path, profit, and cheapest path length for the random network with 26 nodes (R 26) and for the small-world network with 26 nodes (SW 26).

	Network	Efficiency	Price	Price in CP	Cost	Profit	Length
	R 26	0.97	11.34	5.49	28.33	1.10	6.26
	SW 26	0.68	18.10	13.16	76.52	2.38	7.00

Table 3 .

 3 Numerical results for larger networks. Efficiency (fraction of rounds in which the cheapest path cost was equal to or less than the threshold), and mean values of the price, price in the cheapest path, cost of the cheapest path, and cheapest path length. Results obtained from numerical simulations with random networks with 50 and 100 nodes (R 50, R 100) for the small-world network with 50 and 100 nodes (R50, R 100).

	Network	Efficiency	Price	Price in CP	Cost	Length
	R 50	0.98	13.12	7.03	44.76	7.68
	SW 50	0.91	14.32	8.28	77.44	10.74
	R 100	0.97	12.65	5.53	46.50	10.05
	SW 100	0.82	13.20	6.56	88.56	15.41

the latter. It is well known that clustering coefficients and average path lengths differ for the SW and the random networks considered in our experiments ( p ∈ {0.1, 1}

[START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]

Scientific RepoRtS | (2020) 10:11113 | https://doi.org/10.1038/s41598-020-68094-z

Acknowledgements

All authors have been partially supported by the EU through FET Open Project IBSEN (Contract No. 662725). A.S. acknowledges partial support by FEDER/MICINN-AEI (Spain) through Grant PGC2018-098186-B-I00 (BASIC). Y.M. acknowledges partial support from the Government of Aragón, Spain through a grant to the group FENOL (E36-17R), by FEDER/MICINN-AEI funds (Grant FIS2017-87519-P), the University of Zaragoza (Project 2015/022PIP) and from Intesa Sanpaolo Innovation Center. The funders had no role in study design, data collection and analysis, or preparation of the manuscript.

Data availability

The datasets analysed during the current study are available from the corresponding author on reasonable request. Received: 30 October 2019; Accepted: 21 April 2020

Conclusions

Our experimental results indicate that the trading network has a powerful effect on both the pricing behavior of intermediaries and the overall efficiency of the system, random networks being more efficient and showing significantly lower prices than small-world networks. However, within a network, prices are relatively insensitive to node location, but intermediaries with greater betweenness make larger profits. Informed by the experimental results, we introduced an ABM of pricing behavior to understand traders' pricing. The key input of the model is the experimental observation that intermediaries raise prices when they lie on the cheapest path and lower their prices otherwise. The model successfully reproduced qualitatively the experimental results and allowed us to extrapolate and anticipate outcomes of pricing and efficiency to scenarios involving larger networks and longer timescales. Important enough, the model also enabled the discovery of what are the key determinants of cost, namely, the number of node-disjoint paths from source to destination and the network average path length. Ultimately, this explained the differences in our experimental results: in a small-world network, the average path length tends to be larger and this leads to higher costs and lower efficiency of trading in these networks as compared to random networks.

Overall, our work reveals that the topology of trading networks is key to determine their efficiency and cost. It would be interesting to further test our conclusions using real data on trading, in particular, the finding that the availability of node-disjoint paths takes trading costs down. On the other hand, our insights may be useful for the design of competition-improved networks for goods currently overpriced due to intermediation. Further research on the role of information provided to intermediaries and on other network topologies will be also relevant to address these issues.

Methods

We carried out 5 experimental sessions, each of which is composed of 4 consecutive series of 15 rounds each. All the networks have been generated through the Watts-Strogatz algorithm [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] with different probabilities p of rewiring (p = 0.1 for small-world-like networks and p = 1 for random networks). Mean degree was �k� = 3 for the 26 nodes networks and �k� = 4 for the network of 50 nodes. A representation of each network as viewed by the participants is shown in SI Fig. S1. The experiment was conducted with 144 volunteers recruited from the volunteer pool of the project (http://www.ibsen .eu). The first experimental session was performed on 29 June 2017, jointly at Experimental Economics Labs of University of Zaragoza (UZ) and University Carlos III of Madrid (UC3M), with a random network of 50 nodes. A second experimental session was performed on 31 October 2017, at UC3M, with a SW network network of 26 nodes. Subsequent three sessions were performed at UZ on 14 November 2018 (RN, 26 nodes), 24 April 2018 (RN, 26 nodes), and 25 April 2018 (SW, 26 nodes). Table S1 of the SI shows the demographic data of the sessions.

All the participants played through a web interface after reading a tutorial (both included in SI Materials and Methods) on the screen. When everybody had gone through the tutorial, the experiment began, lasting for approximately 90 minutes. At the end of the experiments, all participants received their earnings and their showup fee. Total earnings in the experiment ranged from 5 to 82 Euros, average earning was 18.4 Euros.

All participants signed an informed consent to participate. Besides, their anonymity was always preserved (in agreement with the Spanish Law for Personal Data Protection) by assigning them randomly a username which identified them in the system. No association was ever made between their real names and the results. This procedure was checked and approved by the Clinical Research Ethical Committee of IACS, Aragon Government: accord 12/2017. All methods were performed in accordaxnce with the relevant guidelines and regulations.

Author contributions

Competing interests

The authors declare no competing interests.


Frederic Moisan

Robert Ten Brincke

Ryan O Murphy

Cleotilde Gonzalez

Running head: Incentives, Social Preferences, and Cooperation

Keywords: Incentives, Social Preferences, and Cooperation prisoner's dilemma, cooperation, social preferences, social value orientation

The Prisoner's Dilemma (PD) is a classic decision problem where two players simultaneously must decide whether to cooperate or to act in their own narrow self-interest. The PD game has been used to model many naturally occurring interactive situations, at the personal, organizational and social levels, in which there exists a tension between individual material gain and the common good. At least two factors may influence the emergence of cooperative behavior in this well known collective action problem: the incentive structure of the game itself, and the intrinsic social preferences of each of the players. We present a framework that integrates these two factors in an effort to account for patterns of high or low cooperation from repeated choice interactions. In an experiment using a collection of different PD games, and a measure of individual social preferences, we identify regions of PD games in which: (1) cooperation is independent of social preferences; (2) nice people can be exploited; and (3) being nice is consistently rewarded.

Introduction

Cooperation is an essential component in efficient social interactions. However, it requires that decision makers forgo some of their own potential gains in order to achieve better communal outcomes. This tension creates a social dilemma, a situation that pits an individual's self-interest against collective interests [START_REF] Hardin | The tragedy of the commons[END_REF][START_REF] Dawes | Social dilemmas[END_REF][START_REF] Ostrom | Revisiting the commons: local lessons, global challenges[END_REF][START_REF] Van Lange | The psychology of social dilemmas: A review[END_REF]. The Prisoner's Dilemma (PD) is a social dilemma that archetypically represents this tension. Its simple structure makes clear that each player has a strictly dominant option, namely to choose defect (i.e., not to cooperate); and the logic of this -rational‖ choice, fueled by narrow self-interest, yields unambiguous but lamentable results: people may act in their own self-interest and this negates more efficient communal outcomes. This is the tension at the heart of collective action problems [START_REF] Ostrom | Collective action and the evolution of social norms[END_REF].

Fortunately however, universal defection is not observed when people make choices in social dilemmas and a sizable proportion of decision makers choose to cooperate instead (e.g., [START_REF] Axelrod | The evolution of cooperation[END_REF][START_REF] Dawes | Anomalies: Cooperation[END_REF][START_REF] Camerer | When does -economic man‖ dominate social behavior?[END_REF][START_REF] Rand | Positive interactions promote public cooperation[END_REF][START_REF] Martin | A Description-Experience Gap in Social Interactions: Information about Interdependence and Its Effects on Cooperation[END_REF]. Two factors may influence the emergence of this cooperative behavior: (1) the particular payoffs in the PD game may reward cooperation in different ways (i.e., the "PD game" is not just one game but it is rather a constellation of potential 2-player games with particular properties) and (2) the social preferences that people hold may intrinsically compel them to care about the other's outcomes as well as their own, hence making cooperation more likely. Here, we present a coherent and integrated framework that brings together the incentive structures in the PD social dilemmas, and the intrinsic individual social preferences people have, to explain the emergence of cooperation in repeated social interactions. The results of our empirical investigation reveal that not all PD games are equal in terms of elicited behavior. We identify patterns of high and low cooperation rates that emerge from different incentive structures in PD games, and also from individual social preferences (i.e., Social Value Orientation (SVO)) among decision makers. In particular we identify incentive structures in which decision makers with a high degree of SVO are exploited; and those incentives where high SVO players can thrive and earn greater payoffs through repeated cooperation.

Social preferences and cooperation

Social preferences are perhaps best explained when contrasted against Homoeconomicus, an idealized decision maker posited to be narrowly self-interested and further believing that other decision makers are also narrowly self-interested [START_REF] Aumann | Agreeing to disagree[END_REF]. A Homoeconomicus' sole motivation is to maximize his/her own payoffs, indifferent to the payoffs of others, and further he/she expects the same from all other decision agents. Many economic models use Homo-economicus as a foundation, and game theoretic normative solutions are founded in part on this construct. In contrast, we consider here social decision agents who derive some personal pleasure from the positive payoffs that other decision makers receive. This -nicer‖ kind of decision maker is thus willing to make sacrifices for the benefit of others and hence promote the common good. A general utility framework for this kind of social decision maker can be expressed as:

( ) (1)
Here a decision maker garners positive utility for the payoff to themselves as well as some utility for the other player's payoff . Homo-economicus is simply the special case where . One can consider as a representation a decision maker's degree of -niceness‖ or the regard for other's outcomes as a result of his own actions [START_REF] Gonzalez | A Cognitive Model of Dynamic Cooperation with Varied Interdependency Information[END_REF]. This construct is known as Social Value Orientation (SVO) in social psychology [START_REF] Van Lange | The pursuit of joint outcomes and equality in outcomes: An integrative model of social value orientation[END_REF], other regarding preferences in economics [START_REF] Cooper | Other regarding preferences: a selective survey of experimental results[END_REF], welfare tradeoff ratios in evolutionary psychology [START_REF] Sell | Formidability and the logic of human anger[END_REF], and altruism in biology [START_REF] Hamilton | The genetical evolution of social behaviour I and II[END_REF] as well as in economics [START_REF] Becker | Altruism, egoism, and genetic fitness: Economics and sociobiology[END_REF][START_REF] Simon | Altruism and economics[END_REF][START_REF] Fehr | The economics of fairness, reciprocity and altruismexperimental evidence and new theories[END_REF]. Regardless of the moniker, widespread empirical work has established that the majority of human decision makers have some significant degree of positive social preferences and that typically ranges between zero and one 1 [START_REF] Messick | Motivational bases of choice in experimental games[END_REF][START_REF] Van Lange | Self-interest and beyond: Basic principles of social interaction[END_REF][START_REF] Murphy | Social value orientation theoretical and measurement issues in the study of social preferences[END_REF]. SVO can be measured as a reliable individual difference that captures these intrinsic social preferences [START_REF] Murphy | Measuring social value orientation[END_REF]Murphy & Ackerman, 2015).

The relationship between prosocial preferences (i.e., high SVO) and cooperation in PD games is sometimes [START_REF] Balliet | Social value orientation and cooperation in social dilemmas: A meta-analysis[END_REF], but not always (Kummerli, 2010;[START_REF] Kümmerli | Resistance to extreme strategies, rather than prosocial preferences, can explain human cooperation in public goods games[END_REF][START_REF] Burton-Chellew | Prosocial preferences do not explain human cooperation in public-goods games[END_REF][START_REF] Burton-Chellew | Payoff-based learning explains the decline in cooperation in public goods games[END_REF], found to be positive. A complicating factor that may help explain the conflicting results is that -the‖ PD game actually represents an infinite set of potential payoff structures that may influence cooperation in different ways. Different PD games have different relative payoffs and different games yield different behavior from decision makers holding heterogeneous social preferences. In order to better understand how individuals with diverse intrinsic social preferences may influence each other when interacting repeatedly, 1 In the literature SVO scores are sometimes reported in the metric of degrees, which is a consequence of legacy from the Ring Measure [START_REF] Liebrand | The effect of social motives, communication and group-size on behavior in an n-person multi-stage mixed-motive game[END_REF], which yielded output in terms of an angle. Transforming between the index and an SVO angle score in degrees is straightforward as ( °). The linear metric of may be easier to understand than a trigonometric scaling of social preferences and that is in part why we use it here. empirical investigation is needed and hence a major motivation for this paper. Beyond the theoretical view, it is pragmatically important and collectively beneficial to be able to create incentive structures that allow genuinely nice decision makers (i.e., those with high SVO) to promote the emergence and maintenance of cooperation. Similarly, it is important to identify the features of environments that preclude nice individuals from being exploited, and lastly to find ways in which those less inclined to cooperate may be encouraged to do so.

In the next section we define a comprehensive normalized space of PD games motivated by early theoretical work [START_REF] Rapoport | Prisoner's dilemma: A study in conflict and cooperation[END_REF][START_REF] Axelrod | Conflict of interest: An axiomatic approach[END_REF]. We also develop expectations regarding the relationship between SVO and the particular payoff structures of the games, and explicate how these factors would influence the emergence and sustainability of ongoing cooperation.

The normalized Prisoner's Dilemma space

In a standard 2x2 PD game (Figure 1), each player has two options and the game's payoffs must conform to the strict inequalities T > R > P > S. For each player, option D strictly dominates option C, but the outcome of both players choosing D leads to the payoff P for each player. This outcome is inefficient as both players could do better and receive R by each choosing C rather than D. A secondary condition of a PD game requires that 2R > T+S [START_REF] Rapoport | Prisoner's dilemma: A study in conflict and cooperation[END_REF], which ensures that the payoffs obtained from continued joint cooperation (CC) are greater than those obtained from alternating cooperation and defection (where T is earned on one round, then S on the next, both players alternate, and so on).

Early theoretical work [START_REF] Rapoport | Prisoner's dilemma: A study in conflict and cooperation[END_REF][START_REF] Axelrod | Conflict of interest: An axiomatic approach[END_REF] proposed indices of anticipated cooperation that correspond to different payoffs in PD games (see [START_REF] Stivers | Indexing Prisoner's Dilemma games: Quantifying psychological factors underling cooperative behavior[END_REF] for an overview). This theoretical approach is appealing because it reflects the reality that people often actually choose to cooperate in PD games; and it is simple, as a prediction of cooperation is possible only from the relative differences in payoffs. Other approaches to rationalize cooperation could be more complicated. For example Homoeconomicus would require a refinement of the assumption of -common knowledge of rationality‖ in order to cooperate in any finite PD game [START_REF] Kreps | Rational cooperation in the repeated prisoner's dilemma[END_REF]. Also, specifying precisely what preferences and beliefs would replace narrow self-interest and the assumption of common knowledge in order to justify cooperation is tractable, but non-trivial (e.g., [START_REF] Murphy | Social Preferences, Positive Expectations, and Trust Based Cooperation[END_REF].

Following a theoretical perspective of anticipated cooperation from normalized indices in the PD games, Figure 1 presents a configuration of all possible PD games using a straightforward transformation of PD games with scaled (normalized) payoffs as follows:

( 2 )

( 3 ) ( 4 ) ( 5 )
The best-known measure of anticipated cooperation is Rapoport's K-index [START_REF] Rapoport | A note on the -index of cooperation‖ for Prisoner's Dilemma1[END_REF], and in this notation it is defined as:

( 6 )
The K-index is designed to capture the severity of a game as it reflects the intuition that some PD games are -easier‖ than others, and hence cooperation is more likely to emerge in games with higher K values [START_REF] Rapoport | Prisoner's dilemma: A study in conflict and cooperation[END_REF][START_REF] Roth | Equilibrium behavior and repeated play of the prisoner's dilemma[END_REF] contrasted with games that have relatively low K values. such that payoffs T = 1, S = 0, and R' and P' are the free parameters on the x and y axes respectively. Any point within the displayed triangle is a particular PD game. Games to the right of R' = 0.5, which correspond to any point within the shaded right trapezoid, conform to the additional payoff requirement that 2R > (T+S), an important restriction for iterated PD games. In any case, all PD games have the same defining characteristic of a Pareto-deficient pure strategy Nash Equilibrium. However some games are -easier‖ dilemmas than others, and indices like Rapoport's K-index reflect this. The four games used in the current experiment are indicated in the PD space and are labeled I-IV. Sustainable exploitation within heterogeneous pairs of players is more likely in games from the dark grey area (R+P < T+S) than in games from the light area (R+P > T+S). Behavior is expected to be constant over time in games located on the border between the dark and light gray areas (R+P = T+S).

One advantage of using these normalized payoffs is that any PD game can be defined with just two parameters (R' and P'), each bounded between 0 and 1, but not inclusive. A second advantage of this standardization is that it allows one to make theoretical predictions regarding the level of cooperation that are isomorphic to other well know indexes. From this theoretical development we pose two questions. First, how is the exogenous payoff structure of the PD game related to emergent cooperation rates among players? Games with the same K-index (diagonals in Figure 1) are expected to result in the same level of cooperation between two players, and as the K-index increases, cooperation should also increase. Further, we can explore the separate potential effects of R' (which corresponds to the r 2 index from [START_REF] Rapoport | Prisoner's dilemma: A study in conflict and cooperation[END_REF], on cooperation which, according to Rapoport & Chammah, is conjectured to be positive. Second, how are endogenous social preferences related to emergent cooperation rates among players? Previous work has produced mixed results and this could be in part due to the different payoff structures that were used. Through a laboratory experiment, we can isolate the independent effects of different incentive structures by randomly assigning dyads to different PD games. We can also examine what sort of incentive structures allow genuinely nice players to spread their prosocial influence and induce other players they interact with to cooperate more.

The dynamics related to this question raise issues about reciprocity and learning that we address in the results section.

To address these questions we conducted an experiment that allows us to identify patterns of low and high cooperation rates across different PD games while concordantly measuring the players' social preferences. We selected four games across the normalized PD space, marked as Game I, II, III, and IV in Figure 1. These games allow us wide coverage of possible strategic interactions and do so in a way that allows us to disentangle the effects of the K-index from the effects of levels on cooperation. Game I and II share the same high K-index (K=0.6), while Games III and IV share the same low K-index (K=0.2). The expectation is that there would be more cooperation in Games I and II than in Games III and IV; and there would be similar levels of cooperation in Games I and II as well as in Games III and IV. Furthermore, Game III has the lowest level ( =0.6), followed by Game I ( =0.7) and Games II and IV ( =0.9). The expectation is that Games II and IV should yield the highest cooperation levels, followed by Game I and then Game III.

As highlighted above, incentives are not the sole driver of behavior and we expect intrinsic social preferences to matter as well. SVO reflects the extent to which a decision maker cares about the material welfare of others, which can be formally expressed through the index in the utility function from equation (1). For any PD game, there exists a threshold ̅ (

) such that, for any ̅, choosing option C is always strictly preferred to option D, regardless of whatever beliefs a person has about the other's anticipated behavior 2 . Similarly, there exists a threshold ( ) such that, for any , choosing D is always strictly preferred to C, regardless of the beliefs about the other's anticipated behavior. However, whenever ̅, SVO alone is not sufficient to determine the anticipated choice, and the behavioral prediction is contingent on a decision maker's expectations about the other player's behavior. Analytical work by [START_REF] Murphy | Social Preferences, Positive Expectations, and Trust Based Cooperation[END_REF] provides detailed predictions of such combined effects on cooperative choice in a number of PD games, including the ones selected for the current study. In particular, we have ̅ and in Games I and II (K=0.6), ̅ in Game III (K=0.2), and ̅ and in Game IV (K=0.2).

Assuming social preferences as in equation ( 1), three types of PD games can be distinguished:

2 Option C is strictly preferred to option D whenever outcome CD is strictly preferred to outcome DD and outcome CC is strictly preferred to outcome DC. In these cases, preferences alone are sufficient to account for behavior as beliefs are inconsequential, assuming decision makers have social utilities consistent with equation 1.

If

(the border line in Figure 1), then no dynamic behavior is expected. (prosocials and individualists behaving alike since ̅ is likely for both players) 4 . We test this hypothesis through games II and IV (both with a large ) in our experiment. The main difference between those two games is that Game IV is expected to be more belief dependent than Game II because of its larger gap between and ̅, which makes it more likely that ̅ for both players than in Game II. One may therefore expect more convergence to outcome DD in game IV than in game II, the latter being less restrictive on unconditional cooperators.

3. If (dark grey area in Figure 1), then social preferences are expected to promote exploitation within heterogeneous pairs. As in the previous case, whenever ̅, one's preferred option directly depends on one's expectations of cooperation from the other player. However, the difference with the previous case lies in that choosing to cooperate here requires believing that the other will defect 5 . Such a counterintuitive observation has significant consequences on the expected emergence of mutual cooperation within a pair: mutual cooperation indeed requires that ̅ for both players here 6 . If instead ̅

for at least one player, then exploitation outcomes CD and DC can be stable outcome ( , if ̅ for Player 1 a ̅ for Player 2, then Player 1 will learn over time that Player 2 always plays ill lead Player 1 ly select D). As a result, whenever is not too large (such that ̅ for at least one player in the pair), as in Game I, we expect a 4 When instead are small (under the assumption that ), it becomes more likely that or ̅ for both players, and therefore constant exploitation can be expected in heterogeneous pairs. 5 For more details, we refer to Table 3 in [START_REF] Murphy | Social Preferences, Positive Expectations, and Trust Based Cooperation[END_REF]. 6 Similarly, mutual defection requires that for both players.

significant influence of SVO on cooperation to occur where prosocial players would be exploited by individualistic players.

In summary, (1) no effect of SVO is expected whenever is sufficiently large and (e.g., Games II and IV), and (2) exploitation within heterogeneous pairs is predicted whenever is sufficiently small7 and (e.g., Games I and III). Note that the above theoretical analysis is consistent with the K-index conjecture previously described.

Indeed, in all PD games, increasing K implies a decrease of both and ̅, thereby making unconditional cooperation more likely to emerge. Furthermore, our analysis reveals that a priori there is no PD game where heterogeneous pairs can benefit prosocial players more than they benefit individualistic players. Instead, the only predicted effect of SVO in this case is the exploitation of prosocials by individualists in some games. In other words, in theory, it never pays off to be a nice player when paired with an individualistic player. We therefore aim at testing these hypotheses in the following experiment.

Methods

A total of 220 American individuals were recruited through Amazon Mechanical Turk to voluntarily participate in an online study. A total of 172 people completed all parts of the study (78%) (Mean age= 33 years old, SD=11), and of those 74 participants were female (43%). The study was incentive compatible, with a minimum guaranteed payoff of $0.92 and a maximum attainable payoff of $3.60. This includes a show-up fee of $0.60. The actual average payoff was $2.17. Because the approximate duration to complete the study in full is approximately 20 minutes, these incentives are in line with the norm for online studies. Partial payment was awarded only to subjects whose matching partner left prematurely, and then only for those parts of the study that these subjects had completed in full. All payoffs were displayed in points, where 100 points corresponds to 20 cents (a 1-to-5 ratio).

Procedure

An introductory screen informed participants of the general flow of the study.

Participants were explicitly told that their choices have monetary consequences for both themselves as well the other participant with whom they would be matched. Basic demographic information was collected before the actual experiment began. Then participants completed the SVO Slider Measure that is comprised of 15 monetary allocations between the participant and an anonymous other party, an example shown in Figure 2. The measure yields an individual score that corresponds to in Equation 18 , and provides some insight into a decision maker's willingness to make costly tradeoffs between themselves and another person. Scores near zero indicate narrow self-interest whereas values closer to one indicate greater prosociality (i.e., niceness). As shown in the Appendix (Figure 6), the distribution of SVO scores in this experiment reveals a clear bimodal pattern that is routinely observed for this construct.

Consistently with previous studies conducted in the laboratory, more extreme motivations such as competitiveness and altruism, which can be measured by the SVO Slider Measure, are rarely observed in our data (competitiveness corresponds to whereas altruism corresponds to

). We refer to [START_REF] Murphy | Measuring social value orientation[END_REF] for further details regarding the SVO slider measure.

8 As in [START_REF] Murphy | Measuring social value orientation[END_REF], the SVO index can be computed based on the 6 primary money allocation tasks (the other 9 secondary slider items, which allow disentangling prosocial motivations of reducing inequality and maximizing joint payoff, have not been used in our analysis):

where and represent the average allocations to the self and other respectively across the 6 primary slider items. change the joint allocation of resources between herself and some other person, and then push a submit button to register the choice. In this decision task, the participant acts as a -dictator‖ as the other person has no choice in the matter of resource allocations. This technique provides a clean and reliable measure of how a decision maker weighs tradeoffs between her own payoffs and the payoffs of some other mutually anonymous person.

In part two of the experiment, participants were randomly matched in fixed dyads and each dyad was randomly assigned to one of the 4 previously specified PD games, repeatedly for 60 rounds. Payoffs were scaled such that the maximum range (the difference between T and S) is 100, where T=110 and S=10 in order to explicitly avoid a potential payoff of zero. These four games are shown in the matrices in Table 1.

Table 1. The four payoff scaled games used in the experiment, corresponding to the four games in Figure 1.

Participants were not informed of the total number of rounds (60) to avoid end effects.

However, participants knew the research session would last about 30 minutes in total. On each round, participants chose between options labeled A or B (the choice options were not explicitly named -cooperate‖ or -defect‖). The payoffs were displayed in matrix format, where the row of the matrix that corresponds to the respective option was highlighted when the mouse pointer hovered over the choice button. Once both participants had made a choice, the resulting payoffs, as well as the choices for both players, were displayed. However, no historical overview or summary of choices for earlier rounds was provided. Participants' earnings were allocated based on the sum of ten randomly selected rounds that were determined at the end of the experimental session.

Results

Rapoport's K-index predicts cooperation to be higher for Games I and II (each game as a K-index of 0.6) than for Games III and IV (each with a K-index of 0.2). Our results are consistent with this general prediction. The linear regression between the K-index and the mutual cooperation level is positive and significant (r=0.10, p=0.003)9 . However, the K-index does not account for the more nuanced pattern of results. Table 2 shows the mean fraction of rounds in which players unilaterally choose to cooperate (C rate), and the mean fraction of rounds in which both players in a dyad cooperate bilaterally (CC rate). Note for example that Games I and IV have different K-values, but have similar levels of mutual cooperation (CC rate: t=0.17, p=0.86). Furthermore, games that have the same K-index (I and II; III and IV) have different cooperation levels (CC rate: t=2.07 and p=0.04; t=2.26 and p=0.03).

Table 2: Mean fraction of rounds in which players unilaterally choose to cooperate (C) and the mean fraction of rounds in which both players cooperate simultaneously (CC).

Using each player's choice in every round as the unit of analysis, we ran multilevel logistic regressions models that confirm the significant effect of K and R' on cooperative behaviour. See Table 3 for more details. There exists another organizing principle that helps explain the patterns of cooperation.

Figure 3 shows that the correlation between SVO and the individual cooperation rate (over the whole 60 rounds) is positive for games with a lower R' and close to zero for games with a higher R' 10 . The normalized index R' moderates the regimes of the PD game where SVO is a useful explanatory variable. The interpretation is that for low R' games, the payoff for unilateral defection (T) is relatively larger than the payoff for mutual cooperation (R), whereas for high R'

10 Using each player's choice in every round as the unit of analysis, running some multilevel logistic regressions similarly indicates that the probability of choosing to cooperate in any given round significantly increases with the SVO index only in games I and III. See Tables 11 and12 in the appendix for more details.

PD K R' C rate CC rate

I 0.6 0.7 0.55 0.43 II 0.6 0.9 0.75 0.68 III 0.2 0.6 0.30 0.17 IV 0.2 0.9 0.50 0.41 games the payoff difference between mutual cooperation and unilateral defection is much smaller. A low R' game therefore offers a strong incentive to defect for self-regarding individuals, but this incentive is weaker for prosocial players. Meanwhile, even for almost perfectly self-regarding players, a high R' game offers a weak incentive to defect. The effect of this mechanism may be amplified by players considering the shadow of the future [START_REF] Axelrod | The Evolution of Cooperation[END_REF], knowing that their own defection may spawn the other player's defection (e.g., negative reciprocity) in subsequent rounds.

Figure 3: Effect of R' on the strength of SVO as an explanatory variable for cooperation. Scatter plots show SVO α versus individual cooperation rates (average over 60 rounds per participant), where each dot is a subject and the least squares fit is drawn in as a line. Note that only for Games I and III is there a significant relationship between SVO and cooperation. For the other games from the right side of the PD space there exists no such relationship between social preferences and bilateral cooperative choices. Detailed statistics are provided in Table 8 from the Appendix.

The above results show a relationship between SVO and cooperation for PD games with a lower R' but they do not shed light on whether cooperation is an effective strategy for prosocial players. When is it beneficial for prosocial players to act consistently with their intrinsic preferences? Figure 4 addresses this question by showing the correlations between SVO and earnings, showing that it is only in Game I where prosocial players actually fare well. The average payoffs are higher for prosocials in Game I but not in Game III.

Figure 4:

Relation between mean payoff (over 60 rounds per participant) and SVO. For Game I, subjects with higher SVO receive higher mean payoffs. For Game III, no such correspondence emerges. Detailed statistics are provided in Table 9 from the Appendix.

The combined analysis from Figures 3 and4 suggests that prosocials can thrive in Game I-this is a case where the payoff for mutual defection (P) is relatively smaller than the payoff for unilateral defection (T). If the motives for prosocials lean towards cooperation, what does this tell us about the partner's social preferences? Do prosocials cooperate despite being paired to self-regarding players? Examining the combination of social preferences within pairs provides insights into this question. Table 4 reveals that in Game I the partner's SVO is as influential as one's own SVO (the β coefficient is similar for both variables), while in Game III cooperative behavior is only driven by each individual's own SVO (cooperation is largely independent of the partner's type). Figure 5 shows the relationship between payoffs and the relative difference in SVO within pairs. The relative difference between own and partner's SVO has no relationship to payoffs in Game I but this relationship is negative in Game III. Payoffs in Game III are lower for prosocials when the difference between a player's SVO and the other player's SVO is large (r=0.36, p=0.024). This result indicates that participants with higher relative SVO compared to their partner tend to receive lower payoffs in Game III (i.e., a more prosocial member can be exploited by a more individualistic member of the pair), and the absence of this phenomenon in Game I is what allows prosocials to thrive (i.e., a more prosocial member in a dyad can induce an individualistic partner to cooperate). The above results suggest that in Games I and III prosocial people tend to be more generally cooperative than others. However, one may wonder whether prosociality improves all types of conditional cooperation. To answer this question, we consider two simple measures of 5 indeed indicates that positive reciprocity significantly increases with higher SVO in both Games I and III. However, Table 6 reveals that SVO has no significant effect on negative reciprocity.

Instead, it suggests that people are more likely to negatively reciprocate in later rounds, regardless of their social preferences. This result simply means that although prosocial individuals appear willing to strive for mutual cooperation, they do not tolerate being exploited any more than individualistic people do.

Discussion

In this study, we show how a combination of the exogenous incentive structure of PD games and the intrinsic social preferences of decision makers can facilitate or hinder the emergence of cooperation. Specifically, our results reveal the existence of three distinct phenomena: (1) some incentive structures can prevent prosocial individuals from promoting cooperation--when R+P is sufficiently large; some structures can lead prosocial individuals to be exploited by others--when R is sufficiently small and R+P T+S; and some structures can instead promote cooperation which rewards prosocial players--when R+P << T+S. These findings complement previous studies that have provided evidence that some specific personality traits (e.g., honesty, humility, agreeableness) can account for prosocial behavior as it is observed in some economic contexts such as dictator games, ultimatum games [START_REF] Hilbig | Pillars of cooperation: Honesty-Humility, social value orientations, and economic behavior[END_REF][START_REF] Hilbig | It takes two: Honesty-Humility and Agreeableness differentially predict active versus reactive cooperation[END_REF], and public goods games [START_REF] Hilbig | Personality, punishment and public goods: Strategic shifts towards cooperation as a matter of dispositional honestyhumility[END_REF]. More specifically, such personality features appear to play an important role in explaining people's general tendency for acting fairly and/or forgiving (see, e.g., [START_REF] Hilbig | The two faces of cooperation: On the unique role of HEXACO Agreeableness for forgiveness versus retaliation[END_REF] transgressions, both of which are crucial ingredients for the emergence and sustaining of cooperation. The current study highlights how endogenous factors like SVO can be contingent on different payoff structures that may accentuate or mitigate their effects and in some cases lead to joint cooperation.

Up to this point in the existing literature, the general relationship between prosocial preferences and cooperation has produced mixed results (e.g., [START_REF] Balliet | Social value orientation and cooperation in social dilemmas: A meta-analysis[END_REF][START_REF] Kümmerli | Resistance to extreme strategies, rather than prosocial preferences, can explain human cooperation in public goods games[END_REF], perhaps due to the overconcentration of research attention to particular regions of possible PD games. For example, Axelrod's PD game (R' = 0.6 and P' = 0.2), one of the most commonly used PD settings, has a relatively low R' (R' has a lower bound of 0.5 assuming that 2R > (T+S))

and is closest to our PD game I. As our results show, the unique characteristics of this incentive structure create a context where prosocials thrive, whereas other regimes of the PD space do not have these particular properties and do not afford prosocials fertile ground for engendering cooperation. In addition, our results demonstrate a positive relationship between people's own social preferences and their choice to cooperate only in games with a low R'. In the high R' region of PD games, cooperation is not influenced by social preferences but can homogeneously emerge in games with low P'. In the low R' region, we show that games with high P' (resulting in a low K-index) can result in exploitation of prosocial individuals. Such exploitation can be accounted for by the irrelevance of beliefs whenever R+P is similar to T+S: in this case, individualists (prosocials) unconditionally prefer to defect (cooperate). On the other hand, games with low R' and P' are a fruitful domain for decision makers with high social preferences to establish outbreaks of cooperation. It is in this environment where nice decision makers can influence the self-interested decision makers to cooperate, allowing cooperative behavior to thrive. This conclusion is consistent with the fact that beliefs become more relevant to determine behavior whenever R+P is lower than T+S: an individualist may actually have highly contingent social preferences and thus only be nice and choose to cooperate if he strongly believes that the other player is likely to cooperate as well.

This research also offers important theoretical implications for the study of SVO and preferences in dynamic interactions. It is often assumed that prosociality can be captured by simply incorporating the welfare of others into one's own utility function (Equation 1). However, our results suggest more complex relationships that depend on the incentive structure of the PD and on the dynamics of the interaction between two players. For example, Game III offers some sharp predictions about what people would do depending on their social preferences (either always play C or always play D, regardless of their beliefs about the other player's behavior).

But our results suggest that people with higher SVOs in Game III do not cooperate unconditionally and we observe significant variability in individual behavior (Figure 3). While prosocials positively reciprocate more than individualists do, we infer that both types of players have a similar attitude towards negative reciprocity, regardless of their social preferences. In other words, nice people are willing to elicit cooperative behavior, but not do so naively. We find that prosocials do not tolerate being exploited more than other types of players. Note however, that such a self-protection mechanism is not sufficient to prevent nice people from being exploited, although it can reduce the negative consequences that interacting with self-regarding individuals can have (see Figure 5).

Our results also suggest that beliefs about the other player's actions are likely to play a major role in the emergence of cooperation, as it is predicted by a recent model [START_REF] Murphy | Social Preferences, Positive Expectations, and Trust Based Cooperation[END_REF]. Moreover, the belief dynamics may depend on the type of social preferences (e.g., prosocials may have more variability in their beliefs about the other's behavior than individualists) and the model from Murphy and Ackermann is static. In future studies, it could be worthwhile to investigate how the dynamics of the interaction gives rise to beliefs and how those beliefs are updated over sequential iterations of the game (this would also shed light on reciprocity and its emerges). For example, future studies may intermittently assess the beliefs a player has in anticipation of the other player's behavior and updated social preferences, contingent on their changing beliefs. Recent research showed that a cognitive model that dynamically adjusts the SVO index in a joint utility function was able to account for the observed behavior in a PD game quite accurately [START_REF] Gonzalez | A Cognitive Model of Dynamic Cooperation with Varied Interdependency Information[END_REF].

In a world with heterogeneous decision makers with varied social preferences, researchers need to pay attention to the incentive structures that may promote selfish behavior or cooperative behavior in social dilemmas as neither intrinsic preferences nor exogenous incentives tell the whole story. When interacting with others repeatedly, such as in our workplace, it is important to facilitate the structural conditions that would promote civility and foster signals of unselfishness, thus allowing nice people to coordinate and reap the efficiency gains from sustained cooperation. Increasing the consequences of unilaterally deviating from mutual cooperation is a necessary but not sufficient condition to these ends. Genuinely nice people may be exploited in such environments where mutual cooperation is not appropriately rewarding. However, it becomes possible for them to -contaminate‖ those less inclined to cooperate when the cost of mutual defection is significant. It is heartening to see evidence that prosocial decision makers can ascend and do well even in an austere social context where the temptation to defect is persistent.

Figure 1 :

 1 Figure 1: The normalized PD space. The set of PD games can be feature scaled (normalized)

Figure 2 .

 2 Figure 2. Example of one of the SVO Slider Measure items. A participant can move the slider to

  reciprocal behavior: a person positively reciprocates by cooperating (choosing C) in the round following a cooperative move by the partner. Similarly, a person negatively reciprocates by defecting (choosing D) in the round following a defective move by the partner. If social preferences make people unconditionally more cooperative, then SVO should positively correlate with positive reciprocity and negatively correlate with negative reciprocity. Table

  In fact, since ̅ in this case, people are expected to almost 3 always either strictly prefer C to D, or strictly prefer D to C across the repeated iterations of the game. In this case, mutual

	cooperation clearly requires that	̅ for both players in the pair. If this condition is not
	satisfied, it may then lead to exploitation (CD or DC) in heterogeneous pairs (where players
	have different social preferences): the most prosocial player (	̅) would keep cooperating
	while the most individualistic one (	) would keep defecting. We test these predictions in
	our experiment via the Game III condition.
	2. If	(light grey area in Figure 1), social preferences cannot promote
	exploitation within a pair whenever
		for both players (e.g., if	̅ for Player 1 and	̅ for Player 2,
	then Player 1 will l	er time that Player 2 always p	ill lead Player 1
	ect C). Similarly, mutual defection can be reached as long as is sufficiently
	large such that	̅	for both players (e.g., if	̅ for Player 1 and
	for Player 2, then Player 1 will l	er time that Player 2 always p	ill
	lead Player 1		ect D). While mutual cooperation and mutual defection can be
	equally attractive in these games, we expect little effects of SVO whenever	is large

̅ for at least one player whose preferred option directly depends on one's expectations of cooperation from the other player. More precisely, preferring to cooperate requires believing that the other player will also cooperate. As a result, mutual cooperation can be reached as long as is sufficiently large such that

Table 3 .

 3 Mixed-effect logistic regression models were fit to predict binary cooperation choices in terms of round, K and R as fixed effects (subject is the random effect) with likelihood ratio tests comparing models (M B and M C with M A , M D with M C ).

					Models
		M A		M B		M C		M D
		Estimate Std Err Estimate Std Err Estimate Std Err Estimate Std Err
	Intercept	0.546	0.273	0.643	0.278	-1.736	0.568	-3.697	0.950
	Round			-0.003	0.002
	K					5.600	1.259	35.907	1.247
	R							1.284	0.505
	df	2		3		3		4
	AIC	6860		6859		6844		6840
	BIC	6874		6880		6865		6868
	Log-likelihood	-3428	-3426	-3419	-3416
	χ 2			3.045	17.927	6.086
	df χ 2			1		1		1
	p			0.081	<0.0001	0.014

Table 4 .

 4 Multiple regressions predicting individual cooperation rate (over 60 rounds per participant) in terms of one's own SVO (SVO o ) and the partner's SVO (SVO p ) in Games I and III (p<0.01( ** ), p<0.001( *** ))

  Relation between mean payoff (over 60 rounds per participant) and relative difference in SVO of Own and Partner's SVO in Games I and III. Detailed statistics are provided in Table 10 from the Appendix.

	Figure 5.
		Game I		Game III
	Variable	SE β	β	SE β	β
	SVO o SVO p	0.118 0.118	0.429 *** 0.419 ***	0.142 0.142	0.441 ** 0.210
	R 2	0.349		0.263
	F	12.6 ***		6.61 **

Table 5 .

 5 Mixed-effect logistic regression models fitted to predict positive reciprocity choices in Games I and III in terms of round and own SVO (SVO o ) as fixed effects (subject is the random effect) with likelihood ratio tests comparing models (PR Bx and PR Cx with PR Ax ).

Table 6 .

 6 Mixed-effect logistic regression models fitted to predict negative reciprocity choices in Games I and III in terms of round and own SVO (SVO o ) as fixed effects (subject is the random effect) with likelihood ratio tests comparing models (NR Bx with NR Ax, NR Cx with NR Bx ).

In the particular case where ̅, one is indifferent between choosing either actions and may therefore play differently in different rounds.

Note that the lower bound for is 0.5 because of the assumption that .

Here the unit of analysis is the ratio of mutual cooperation over 60 rounds for each pair of matched subjects.

This research was supported by the National Science Foundation Award number: 1530479 and by the Army Research Laboratory under Cooperative Agreement Number W911NF-13-2-0045 (ARL Cyber Security CRA) to Cleotilde Gonzalez. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.

Appendix

Figure 6. Distribution of social value orientation index in the experiment. This bimodal pattern is commonly found and what is shown here is very similar to other empirical distributions reported in the literature (see, e.g., [START_REF] Murphy | Measuring social value orientation[END_REF] or the baseline condition from [START_REF] Ackermann | Reciprocity as an individual difference[END_REF].