Geoffrey Castillo

The attraction effect and its explanations

Keywords: attraction effect, asymmetric dominance effect, decoy effect, range effect, risky choice, individual decision-making JEL codes: C91, D11, D80, D90

The attraction effect violates choice consistency, one of the central assumptions of economics. I present a risky choice experiment to test it and disentangle some of its explanations. I find the attraction effect, but in a smaller magnitude than previously thought. I uncover a 'range effect' that shows that people weight more attributes whose range increases. I also show that the aggregate results hide considerable heterogeneity between subjects.

Introduction

The attraction effect 1 occurs when adding an option x to a menu {x, y}, where x is dominated by x but not by y, increases the probability of choosing x. One of the * Vienna Centre for Experimental Economics, University of Vienna, Austria. geoffrey. castillo@univie.ac.at. I am grateful to Kian Howe Ong, Till Weber, Jingcheng Fu, and especially Ashira Perera, for helping me to conduct the sessions. I thank Mohammed Abdellaoui, Robin Cubitt, Alexia Gaudeul, Kaisa Herne, Joel Huber, Peter Landry, Daniel Seidmann, Chris Starmer, Robert Sugden, Gerelt Tserenjigmid, Ryan Webb, and Erik Wengström for comments on earlier versions. I also thank numerous seminar and conference participants for helpful comments. The Nottingham School of Economics Research Ethics Committee approved the experiment reported in this paper. 1 Sometimes referred to as the asymmetric dominance effect or, rather confusingly, the decoy effect.

1 most popular topics in marketing research and psychology, it has been observed with consumer products, job candidates, political candidates, political issues, medication, investment opportunities, and environmental goods. 2 The attraction effect has even been observed among birds, bees, moulds, monkeys, and frogs. 3This body of research suggests that preferences are context-dependent and so challenges the principles of consistency and utility maximisation central to economics. For this reason it has served as a motivation for many economic papers. 4 Most of the evidence, however, comes from studies using hypothetical problems: of the 52 experiments in marketing reviewed by Lichters et al. (2015b) only one, [START_REF] Doyle | The robustness of the asymmetrically dominated effect: Buying frames, phantom alternatives, and in-store purchases[END_REF], uses real incentives; and, in economics, one of the rare incentivised studies on the attraction effect is [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF]. Moreover, even in marketing where its study is widespread, there are still some debates about the replicability of the attraction effect.5 Finally, after 30 years of research we still do not know how to explain the attraction effect [START_REF] Simonson | Mission (largely) accomplished: What's next for consumer BDT-JDM researchers?[END_REF]. There are two possibilities [START_REF] Wedell | Distinguishing among models of contextually induced preference reversals[END_REF]: adding an asymmetrically dominated option could change the weighting of the different attributes, or it could change the choice process. These explanations have so far not been teased apart, which renders precarious any attempt to explain the attraction effect.

To solve these problems I setup a controlled, incentivised laboratory experiment aimed at testing and explaining the attraction effect in the context of choice under risk. My first main finding is that, despite higher incentives, stricter controls, and a more refined design, I replicate the attraction effect. It is, however, smaller than previously reported. The second main finding is that none of the existing explanations is able to fully explain the attraction effect. In fact I observe another effect, the 'range effect', that runs against the attraction effect.

What previous studies have referred to as 'the attraction effect' actually encompasses two effects, each with a different implication: one violates the Weak Axiom of Stochastic Revealed Preference while the other violates the Regularity Condition. 6 Instead of focusing on one or the other, I study both in the same experiment; and, since the Weak Axiom of Stochastic Revealed Preference imposes more requirements on consistency I predict that we should observe its violations more often.

The results confirm that the attraction effect exists, and, as predicted, violations of the Weak Axiom of Stochastic Revealed Preference are observed more often.

The effect I observe is, however, roughly half of what was previously reported. Yet, it cannot be explained by random errors and so is a systematic deviation from consistency.

One way of explaining the attraction effect is to assume that adding decoys changes people's relative weighting of the attributes constituting the options. This weights explanation supposedly stems from a negative range effect: increasing the range of an attribute makes people weight this attribute less. If this interpretation is correct then increasing the attribute ranges even more should generate more negative range effect and thus more attraction effect. To test this prediction I create new decoys which double the attribute ranges compared to the typical decoys.

The results show that these new decoys do not cause more attraction effect and so reject the weights explanation.

Another way of explaining the attraction effect is to say that a decoy changes how people choose. As its name suggests, an asymmetrically dominated decoy is dominated by only one option and so, if people have weak or imprecise preferences, they might feel compelled to choose the dominating option. Thus this process explanation hinges on the decoys being asymmetrically dominated and removing the asymmetric dominance should eliminate the attraction effect. I test this by introducing a second class of new decoys which also double the attribute ranges but are symmetrically dominated. Because they double the ranges the weights explanation predicts more attraction effect, just like the previous new decoys. But because they are symmetrically dominated, the process explanation predicts no attraction effect.

In fact they trigger, not more, not none, but a negative attraction effect. This negative attraction effect results from a positive range effect: increasing the range of an attribute makes people weight it, not less, but more. It further invalidates the weights explanation and runs against the attraction effect.

This range effect shows that people weight more attributes whose range increases.

A similar idea recently surfaced in economics with [START_REF] Kőszegi | A model of focusing in economic choice[END_REF] focusing model, but it has a long tradition in psychology. 7 In the focusing model increasing the range of payoffs, say at a particular date in an intertemporal choice context, makes people weight more this date. [START_REF] Andersson | Differences attract: An experimental study of focusing in economic choice[END_REF] and [START_REF] Dertwinkel-Kalt | Concentration bias in intertemporal choice[END_REF] have verified this intuition experimentally and the range effect documented in the present experiment demonstrates that it can be extended to risky choice.

Apart from their implications for choice, these results confirm that firms can exploit the attraction effect to influence consumers 8 when risk matters, for example when selling financial products or insurance. Firms can similarly exploit the range effect to direct the attention of consumers, something that has not yet been considered. The results also demonstrate that we need to take into account the range effect when trying to explain the attraction effect.

Compared to [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF] the present experiment uses higher incentives, options with different expected values, and a more concrete and transparent procedure, so it constitutes a more robust test of the attraction effect. Crucially, it also introduces new manipulations to test its explanations. More details on the similarities and the differences are provided in the main text. Another close study is [START_REF] Soltani | A rangenormalization model of context-dependent choice: A new model and evidence[END_REF] who use context effects to test a new theory of context-dependent choice.

They also find the attraction effect using simple binary gambles, but they do not differentiate between the two definitions of the attraction effect and between the weights and process explanations. [START_REF] Kroll | The relevance of irrelevant alternatives[END_REF] also study the attraction effect but they focus on its impact on certainty equivalents.

7 See the weight-change literature [START_REF] Fischer | Range sensitivity of attribute weight in multiattribute value models[END_REF][START_REF] Goldstein | Judgments of relative importance in decision making: Global vs local interpretations of subjective weight[END_REF][START_REF] Mellers | Trade-offs depend on attribute range[END_REF][START_REF] Von Nitzsch | The effect of attribute ranges on weights in multiattribute utility measurements[END_REF][START_REF] Wedell | Testing models of trade-off contrast in pairwise choice[END_REF][START_REF] Wedell | Using judgments to understand decoy effects in choice[END_REF] and the similarity literature [START_REF] Mellers | Similarity and choice[END_REF]Mellers et al., 1992a,b). 8 [START_REF] Eliaz | On the strategic use of attention grabbers[END_REF] and [START_REF] Ok | Theory of product differentiation in the presence of the attraction effect[END_REF] show in their models how this might happen.

Definitions of the attraction effect

The attraction effect is a type of context effect whereby the introduction of a supposedly irrelevant option, the 'decoy', causes a choice reversal. In the attraction effect this decoy is asymmetrically dominated: it is dominated by only one of two options. Throughout this paper I will use superscripts to denote the decoys.

For example x , that we have already encountered in the introduction, is the asymmetrically dominated decoy of x: it is dominated by x but not by y; similarly, y is dominated by y but not by x.

There are two ways to define the attraction effect. The first one, used for example by [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF], looks at the combined effect of x and y and so keeps constant the number of options: Attraction Effect WASRP. The probability of choosing x is greater in {x, y, x } than in {x, y, y }, and the probability of choosing y is greater in {x, y, y } than in {x, y, x }.

The second way, more common in marketing research, focuses on the effect of one decoy at a time and so varies the number of options:

Attraction Effect Regularity.
1. The probability of choosing x is greater in {x, y, x } than in {x, y}.

2. The probability of choosing y is greater in {x, y, y } than in {x, y}.

As their names imply, Attraction Effect WASRP violates the Weak Axiom of Stochastic Revealed Preference [START_REF] Bandyopadhyay | Stochastic revealed preference and the theory of demand[END_REF] while Attraction Effect Regularity violates the Regularity Condition. These are stochastic versions of the Weak Axiom of Revealed Preference and of the Chernoff condition. 9 The Regularity Condition is the weakest consistency requirement of stochastic choice and is satisfied by all random utility models (Luce and Suppes, 1965, Theorem 41, p. 346). WASRP is a stronger requirement and necessarily implies the Regularity Condition (Dasgupta and Pattanaik, 2007).

Previous studies have used interchangeably the two definitions of the attraction effect, but since the Regularity Condition is a weaker consistency requirement, 9 See Appendix A for details on WASRP and the Regularity Condition.

Attraction Effect Regularity is a more serious violation of consistency. We can therefore expect to observe more Attraction Effect WASRP than Attraction Effect Regularity. Now that we know what is the attraction effect and what it implies, we can look at how it can be explained.

Explanations of the attraction effect

I follow [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF] and study the attraction effect with simple binary gambles. 10Figure 1, which I will use throughout, depicts the gambles and the different classes of decoys used in the experiment. The gambles offer a probability p of winning £x and a probability 1p of winning £0, denoted by (x, p). In the Figure probabilities p are on the x-axis, and winning amounts x on the y-axis. Following the previous literature I focus on two types of gambles: ω p = (x p , p p ) and ω £ = (x £ , p £ ). As can be seen in the Figure ,ω p is better in the probability attribute, p p > p £ , while ω £ is better in the £x attribute, x £ > x p . As we have already seen the asymmetrically dominated decoys testing the classical attraction effect are denoted by a single prime: these are ω p and ω £ , also depicted in the Figure . All decoys will follow the same naming convention: the superscript denotes the class of decoy while the subscript denotes the option to which the decoy is attached to.

Over the years, researchers in psychology and marketing research have proposed many explanations to the attraction effect, which can be grouped into two categories [START_REF] Herne | The role of decoys in choice: a review of research on context dependent preferences[END_REF][START_REF] Köhler | Why does context matter?: Attraction effects and binary comparisons[END_REF][START_REF] Wedell | Distinguishing among models of contextually induced preference reversals[END_REF].

Weights explanation

The first is a preference-based explanation: it argues that adding decoys changes the weights people attach to the attributes [START_REF] Huber | Adding asymmetrically dominated alternatives: Violations of regularity and the similarity hypothesis[END_REF][START_REF] Simonson | Choice in context: Tradeoff contrast and extremeness aversion[END_REF][START_REF] Tversky | Context-dependent preferences[END_REF]. For example, according to this weights explanation people choose ω p following the introduction of ω p because they weight less the money attribute x. They weight it less because, as the explanation goes, the attribute range changes: Figure 1 shows that ω p increases the range of money from ∆ x to ∆ x . Thus the attraction effect as explained by the weights explanation arises from a negative range effect: increasing an attribute range makes people weight less this particular attribute.

x p ω £ ω p p £ p p x £ x p ∆ x ∆ p ω * ω * p ω * £ ω p ω £ ω p ω £ 2 • ∆ x 2 • ∆ p ∆ x ∆ p ω x
If this weights explanation is correct then increasing even more the ranges should cause more negative range effect and so more attraction effect. To test this prediction I created the decoys ω p and ω £ . As can be seen in the Figure, these new decoys are also asymmetrically dominated (hence the prime symbols), but compared to ω p and ω £ they double the ranges, from ∆ x and ∆ p to 2 • ∆ x and 2 • ∆ p (hence the double primes). So, ω p and ω £ should cause more attraction effect due to an increased negative range effect. They also test one of the findings of [START_REF] Heath | Asymmetric decoy effects on lower-quality versus higher-quality brands: Meta-analytic and experimental evidence[END_REF] meta-analysis: the greater the range extension, the more pronounced the attraction effect.

Process explanation

The second category of explanations is a heuristic-based explanation: it argues that adding a decoy changes how people make a choice [START_REF] Ariely | Seeking subjective dominance in multidimensional space: An explanation of the asymmetric dominance effect[END_REF][START_REF] Hedgcock | Trade-off aversion as an explanation for the attraction effect: A functional magnetic resonance imaging study[END_REF][START_REF] Simonson | Choice based on reasons: The case of attraction and compromise effects[END_REF]. This process explanation hinges on the decoys being asymmetrically dominated. For example, ω p dominates ω p but ω £ does not; one could then feel that ω p gives good reason to choose ω p . So, the process explanation explains the attraction effect as a change of decision process triggered by the introduction of an asymmetrically dominated option.

To test for this explanation, I remove the asymmetric dominance from ω p and ω £ to create ω * p and ω * £ (the star symbol replacing the prime symbol should help the reader remember that the star decoys are symmetrically dominated). As Figure 1 shows, ω * p and ω * £ have the same range extension as ω p and ω £ , at 2 • ∆ x and 2 • ∆ p , so the weights explanation predicts the same attraction effect. On the other hand, since ω * p and ω * £ are symmetrically dominated the process explanation predicts that the attraction effect will disappear. In other words, if the attraction effect disappears the process explanation can be ruled out; if it holds it has to come from the weights explanation alone. For this reason ω * p and ω * £ are the important decoys that allow me to discriminate between the two explanations. They are intentionally close to the double prime decoys, the only real difference being the removal of the asymmetric dominance.

Wedell (1991, Experiment 2) followed a similar approach and used the decoy ω (also represented in Figure 1) which removes the asymmetric dominance from ω p . He found that adding ω has no effect and so concluded against the weights explanation, which subsequent studies confirmed [START_REF] Wedell | Testing models of trade-off contrast in pairwise choice[END_REF][START_REF] Wedell | Using judgments to understand decoy effects in choice[END_REF].

But he used, for example, ω p = ($20, 0.5), ω £ = ($33, 0.3) and ω = ($18, 0.3).

With these, the increase in the range of winning amounts is minimal, from ∆ x = 33 -20 = $13 to ∆ x = 33 -18 = $15. The weights explanation might work but only for greater range extensions, hence why I am doubling the ranges.

Relation to choice models

Most models have followed the weights explanation. For example, the negative range effect is the backbone of Bushong et al.'s (2017) 'model of relative thinking', which thus predicts that we should see more attraction effect at ω p and ω £ than at ω p and ω £ . [START_REF] Kőszegi | A model of focusing in economic choice[END_REF], by contrast, build their 'model of focusing' on the positive range effect, according to which one puts more weight on an attribute when this attribute range increases. They thus predict that, when introducing ω p , we should observe more choice of ω £ , which is in contradiction to the attraction effect.

Tserenjigmid's (2018) 'choosing with the worst in mind' model explains the attraction effect using reference dependence where the reference point is determined by the minimum of the attributes. For example, in Figure 1, ω * is the reference point when facing the menu {ω p , ω £ } since it is composed of the minimal probability p £ and of the minimal amount of money x p . Introducing ω p changes the reference point to ω , which decreases the reference of x from x p to x . Because of decreasing sensitivity the marginal value of x decreases, which increases the relative marginal value of p, leading to more choice of ω p and thus causing the attraction effect.

Introducing ω p decreases further down the reference of x and creates even more attraction effect.

Landry and Webb's (2019) 'pairwise normalization model' is also related to the weights explanation. In their model, inspired by the neuroscience of perception, attribute levels are first compared in a pairwise fashion before being normalised.

But since it does not rely on ranges or on a reference point the model is more general.

Since all of these models are closely tied to the weights explanation, they predict that the removal of the asymmetric dominance, when we move from ω p and ω £ to ω * [START_REF] Gerasimou | Partially dominant choice[END_REF] is the only economic model explicitly built on the process explanation. In this model, preferences are incomplete and the asymmetric dominance of the decoy helps the decision-maker break ties between alternatives that are otherwise incomparable. It could thus be seen as the theoretical underpinning of [START_REF] Simonson | Choice based on reasons: The case of attraction and compromise effects[END_REF] 'choice based on reason' argument. To be applied, however, one needs to elicit indifference between ω p and ω £ , which is uncommon in attraction effect research.

Other explanations

In [START_REF] Bordalo | Salience and consumer choice[END_REF] salience emerges from the comparison of the payoff of a lottery in a given state of the world to the payoffs of other lotteries in the same state of the world. Given that in the present experiment each and every gamble will give a different amount of money with a different probability, and given the incentive mechanism which picks a gamble pair or triple at random, it can be argued that the payoffs are all in different states of the world, so salience theory would be silent.

Other models, for example Eliaz et al. (2011), de Clippel and[START_REF] De Clippel | Reason-based choice: A bargaining rationale for the attraction and compromise effects[END_REF], Manzini and Mariotti (2012b), [START_REF] Masatlioglu | Revealed attention[END_REF] and [START_REF] Manzini | Two-stage threshold representations[END_REF], introduce weakenings of the consistency requirements of rational choice to capture context effects such as the attraction effect. These models do not try to pin down a particular mechanism behind the attraction effect and so could fit in either category.

Finally, ω * in Figure 1 tests for the possibility that simply introducing a third option in itself causes an effect, even if it is neither asymmetrically dominated nor increasing the attribute ranges.

Summary

To summarise, I use four classes of decoys, each asking a different question:

• The prime decoys ω p and ω £ test for the classical attraction effect.

• The double prime decoys ω p and ω £ keep the asymmetric dominance but double the attribute ranges. The weights explanation predicts that we will observe a stronger attraction effect due to the negative range effect.

• The star decoys ω * p and ω * £ also double the attribute ranges but remove the asymmetric dominance. The weights explanation predicts that we will observe the same attraction effect compared to the previous decoys, while the process explanation predicts that the attraction effect will disappear. These star decoys also allow me to test most existing models.

• ω * does not increase attribute ranges and is symmetrically dominated, so both explanations predict that it will have no effect.

The next Section presents the experimental design implementing these decoys in the laboratory.

Experimental design

Parameter sets, between-and within-subject comparisons

I started by creating 14 sets of gambles. To account for risk aversion and test the robustness of the attraction effect, in each set the expected value of ω £ is 20% higher than the one of ω p .11 For the first 11 sets, the expected values of ω p and ω £ are approximately £5.8 and £7; for the remaining 3 sets, they are £10 and £12.12 By contrast, [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF] gambles were all of the same expected value of 30 Finnmarks, which was approximately £3.3 in 1999.

Each set accommodates all decoys previously mentioned to ensure that all sets have the same underlying structure and so can be compared. Accommodating ω * p and ω * £ imposes the most restrictions because the probabilities of ω p and ω £ should be such that the range of probabilities can be doubled. For example, ω p = (£7.2, 0.8) and ω £ = (£23, 0.3) cannot be part of a valid set because they cannot accommodate ω * £ : the probability range is ∆ p = 0.8 -0.3 = 0.5 and it is impossible to introduce a ω * £ that doubles it. Table 1 presents the resulting 14 sets.

Then, of the first 11 low-stake sets (sets a to k), I randomly selected 3 to study the decoys {ω p , ω £ }, 3 to study {ω p , ω £ } and, since these are the most important ones, 5 to study {ω * p , ω * £ }, making sure that the sets assigned to a class of decoys were not too similar. I also randomly assigned one of the 3 remaining high-stake sets to each class.

The experiment is setup to study Attraction Effect WASRP within-and betweensubject; and Attraction Effect Regularity, due to concerns about experimenter demand effects,13 only between-subject. Note that [START_REF] Wedell | Distinguishing among models of contextually induced preference reversals[END_REF] and Herne Set

ω p ω £ ω * ω p ω £ ω p ω £ ω * p ω * £ p p x p p £ x £ p * x * p p x p p £ x £ p p x p p £ x £ p * p x * p p * £
x * £ a 0.8 7 0.55 12.5 0.55 7 0.75 6 0.5 11.5 0.6 1.5 0.3 8 0.55 1.5 0.3 7 b 0.75 7.5 0.55 12.5 0.55 7.5 0.7 6.5 0.5 11.5 0.6 2.5 0.35 8.5 0.55 2.5 0.35 7.5 c 0.75 7.5 0.5 14 0.5 7.5 0.7 6.5 0.45 13 0.55 1 0.25 8.5 0.5 1 0.25 7.5 d 0.7 8 0.5 14 0.5 8 0.65 7 0.45 13 0.55 2 0.3 9 0.5 2 0.3 8 e 0.7 8 0.45 15.5 0.45 8 0.65 7 0.4 14.5 0.5 0.5 0.2 9 0.45 0.5 0.2 8 f 0.65 8.5 0.5 14 0.5 8.5 0.6 7.5 0.45 13 0.55 3 0.35 9.5 0.5 3 0.35 8.5 g 0.65 8.5 0.45 15.5 0.45 8.5 0.6 7.5 0.4 14.5 0.5 1.5 0.25 9.5 0.45 1.5 0.25 8.5 h 0.6 9.5 0.5 14 0.5 9.5 0.55 8.5 0.45 13 0.55 5 0.4 10.5 0.5 5 0.4 9.5 i 0.6 9.5 0.45 15.5 0.45 9.5 0.55 8.5 0.4 14.5 0.5 3.5 0.3 10.5 0.45 3.5 0.3 9.5 j 0.6 9.5 0.4 17 0.4 9.5 0.55 8.5 0.35 16 0.45 2 0.2 10.5 0.4 2 0.2 9.5 k 0.5 11.5 0.3 22.5 0.3 11.5 0.45 10.5 0.25 21.5 0.35 0.5 0.1 12.5 0.3 0.5 0.1 11.5 l 0.75 13.5 0.5 25 0.5 13.5 0.7 12.5 0.45 24 0.55 2 0.25 14.5 0.5 2 0.25 13.5 m 0.7 14.5 0.45 27.5 0.45 14.5 0.65 13.5 0.4 26.5 0.5 1.5 0.2 15.5 0.45 1.5 0.2 14.5 n 0.65 15.5 0.5 24.5 0.5 15.5 0.6 14.5 0.45 23.5 0.55 6.5 0.35 16.5 0.5 6.5 0.35 15.5

Note. The gambles not used in the experiment have been greyed out.

(1999) only studied Attraction Effect WASRP within-subject, while most studies in psychology and marketing research only studied Attraction Effect Regularity between-subject. The combination of the two approaches allows me to study the effect from both angles. And, if the effect were to appear only in one context, I would be suspicious of the reality of the effect.

Table 2 describes how I combined the two types of attraction effect and withinand between-subject comparisons in the same experiment. As can be seen in the [START_REF] Greenwald | Within-subjects designs: To use or not to use?[END_REF] or even experimenter demand [START_REF] Zizzo | Experimenter demand effects in economic experiments[END_REF]

Procedure and incentives

To achieve concreteness the experiment made use of pairs of 10-sided dice to describe and play the gambles. All subjects had the dice on their desk throughout the experiment, they were encouraged to examine them and they knew that these would be the dice used to play their chosen gamble at the end of the experiment.

The choice tasks themselves also referred to the dice (see Appendix B.2 for a sample). By contrast, previous experiments on the attraction effect using gambles, such as [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF], relied on a random number generator on the computer and described the probabilities in abstract terms.

The experiment was incentivised using the PRINCE mechanism [START_REF] Johnson | PRINCE: An improved method for measuring incentivized preferences[END_REF]. It adds transparency to the traditional random incentive system by asking subjects entering the laboratory to draw a sealed envelope that contains a piece of paper describing the entire choice task (of the 28 they face in the experiment) that will matter to determine their earnings. At the end of the experiment the subject

and the experimenter open the envelope and flip through the booklets to find the task described on the piece of paper. The subject then plays the gamble she has chosen in this particular choice task and is paid accordingly, plus a show-up fee.

Implementation

The experiment took place across five sessions between the end of April and the beginning of June 2016 at the CeDEx laboratory in Nottingham. 207 subjects were recruited randomly using ORSEE [START_REF] Greiner | Subject pool recruitment procedures: organizing experiments with ORSEE[END_REF]. A session lasted about 1 hour for an average payment of £11.37 (SD = £7.43). Each subject made 28 choices (14 in each booklet) and two subjects left a task blank, which leaves 5794 choices to exploit.

Results

Denote by ωi the decoy associated with ω i , i, j ∈ {p, £} i = j, c(•) the observed choice in a menu, and Pr(•) the proportion of subjects exhibiting a particular pattern. Within-subject Attraction Effect WASRP is characterised by

Pr ω i = c {ω p , ω £ , ωi } and ω j = c {ω p , ω £ , ωj } -Pr ω j = c {ω p , ω £ , ωi } and ω i = c {ω p , ω £ , ωj } > 0, (1)
which I will test using a one-sided McNemar test. Note that this test rules out explanations of the attraction effect based on random errors: If random errors were present, they would affect both choice patterns in (1) equally, and there is no reason to believe that one pattern would be affected more than the other. Looking at the difference between the two patterns thus rules out random errors and reveals truly anomalous behaviour.14

Between-subject Attraction Effect WASRP is characterised by

Pr

ω i = c {ω p , ω £ , ωi } -Pr ω i = c {ω p , ω £ , ωj } > 0, (2)
and between-subject Attraction Effect Regularity by

Pr

ω i = c {ω p , ω £ , ωi } -Pr ω i = c {ω p , ω £ } > 0, (3)
both tested using a one-sided χ 2 test.

The neutral decoy ω * is in principle associated with neither ω p nor ω £ . I choose to look at its impact on ω p and define the within-subject effect of ω * by

Pr

ω p ∈ c {ω p , ω £ } and ω £ ∈ c {ω p , ω £ , ω * } -Pr ω £ ∈ c {ω p , ω £ } and ω p ∈ c {ω p , ω £ , ω * } = 0, (4)
tested using a two-sided McNemar test; and its between-subject effect, by

Pr

ω p ∈ c {ω p , ω £ , ω * } -Pr ω p ∈ c {ω p , ω £ } = 0. (5)
tested using a two-sided χ 2 test.

There is a (small) attraction effect

I start with the classical attraction effect, using the decoys ω p and ω £ . The top-left graph of Figure 2 focuses on within-subject Attraction Effect WASRP, when ω p is seen first (ω p → ω £ , first row) or when ω £ is seen first (ω £ → ω p , second row).

We see that the attraction effect is significant in both cases so the experiment replicates the results from [START_REF] Wedell | Distinguishing among models of contextually induced preference reversals[END_REF] and [START_REF] Herne | The effects of decoy gambles on individual choice[END_REF]; but note that when ω £ is seen first the effect is in the right direction for all parameter sets but significant in only one. The top-right graph of Figure 2 shows that Attraction Effect WASRP carries-over to between-subject comparisons. Moving to Attraction Effect Regularity, the bottom-left graph of Figure 2 shows that the effect appears with ω p but not with ω £ . The effect ω p causes, however, is small, and a closer inspection shows that this effect stems primarily from only one parameter set.

ω p → ω £ ω £ → ω p ω p → ω £ ω £ → ω p ω * p → ω * £ ω * £ → ω * p 4.8 † 9.7 † † † † 1.4 7.7 † † † -9.1 ‡ ‡ ‡ ‡ 1.
So, I replicate the two types of attraction effects, but with two new observations.

As predicted, Attraction Effect Regularity appears to be weaker than Attraction Effect WASRP. Second, the attraction effect I observe is considerably smaller than previously reported. For example, the within-subject Attraction Effect WASRP observed by Herne (1999, Experiment 1) varies, depending on the parameter set, between 16% and 35.2%, averaging at about 24.1%. Mine, by contrast, varies between 2% and 23.5%, averaging at about 7.2%. Still, it is significant, and it survives to the higher incentives, the gambles with different expected values, the more transparent incentive mechanism, and the increased concreteness of the choice situation.

There is no clear support for the weights explanation

Consider then ω p and ω £ . As we saw, these decoys are also asymmetrically dominated but they double the range of the weakest attribute of their associated option. The weights explanation predicts that they will cause a stronger attraction effect, while the process explanation predicts no change at all.

The results provide mixed evidence. Within-subject Attraction Effect WASRP now appears in one direction only, when subjects are first exposed to ω £ , but at a higher significance level (top-left, third and fourth row); and so for 3 out of 4 parameter sets, compared to only one when subjects were first exposed to ω £ . Between-subject, Attraction Effect WASRP is less pronounced than before (top-right, second row).

For Attraction Effect Regularity (bottom-left, fourth row), ω p has more incidence than ω p : introducing ω p increases the proportion of subjects choosing ω p by almost 8 percentage points, compared to 5 for ω p (bottom-left, third row). ω £ , just as ω £ , has no effect.

Therefore, asymmetrically dominated decoys that double the attribute range do not lead to a significantly changes in the strength of the attraction effect. This result fails to confirm a prediction of the weights explanation and its associated models.15

A positive range effect operates against the attraction effect

The real test, however, comes from ω * p and ω * £ . Remember that according to the weights explanation, we should observe the same attraction effect; but according to the process explanation, the effect should vanish.

Neither is correct: Looking at within-subject Attraction Effect WASRP (top-left, rows 5 and 6) introducing ω * p in the first booklet and ω * £ in the second causes a negative attraction effect whereby subjects switch from ω £ to ω p and not from ω p to ω £ . This effect is highly significant at the aggregate level, significant for 3 out of 6 parameter sets, and always in this direction. Introducing ω * £ first has no effect.

This negative attraction effect carries over to between-subject Attraction Effect WASRP (top-right, last row).

Between-subject Attraction Effect Regularity resulting from the introduction of ω * £ is also negative, which makes the effect clearer: introducing ω * £ decreases the choice of ω £ and so increases the choice of ω p (bottom-left, last row). So, instead of favouring ω £ as would predict the weights explanation, ω * £ actually favours ω p .

Recall from Figure 1 that ω * £ increases the range of the probability attribute. When the probability range is increased, subjects tend to chose ω p , the superior option in terms of the probability attribute: it is a positive range effect. Note that ω * p causes no between-subject Attraction Effect Regularity at the aggregate level.

Therefore, symmetrically dominated decoys do not lead to an attraction effect, and in some cases generate the opposite effect. This observation is in the opposite direction of that predicted by the weights explanation, and is also a puzzle for the process explanation.

Order effects

The first two rows of the bottom-right graph in Figure 2 report the within-subject effect of introducing the neutral decoy ω * . As explained in equation ( 4), here a positive within-subject effect means that subjects switched more from ω £ to ω p than the opposite. Figure 2 shows that this happened when ω * was introduced only in the second booklet (first row), but not when it was added in the first booklet then removed (second row). The last row of the bottom-right graph shows that ω * had no between-subject effect.

It is likely that this pattern reveals a simple order effect by which subjects chose more ω p as the experiment progressed. Since ω p is less risky than ω £ , this result can be explained by subjects becoming more risk averse over the course of the experiment.16

This order effect partially explains why introducing ω £ , ω £ or ω * p in the second booklet caused less or no effects. These decoys are supposed to favour ω £ , but the order effect pushes subjects to pick the less risky option ω p . Without the order effect, we might have observed a similar reactivity to decoys.

The order effect, however, cannot explain everything. The between-subject results are immune to it because subjects faced the parameter sets in the same

order; yet we saw that, even between-subject, decoys had less grip on ω £ than on ω p . It might be that subjects had imprecise preferences [START_REF] Butler | Imprecision as an account of the preference reversal phenomenon[END_REF] for ω £ and thus had difficulty to interpret a decoy that supposedly favoured it.17 It might also be that they perceived ω £ to be particularly unattractive and tried to avoid it even in the presence of a decoy. Meta-analyses (e.g. [START_REF] Heath | Asymmetric decoy effects on lower-quality versus higher-quality brands: Meta-analytic and experimental evidence[END_REF][START_REF] Milberg | Synthesis of attraction effect research[END_REF] have consistently found that low-quality options are less responsive to decoys than high-quality ones. If we assimilate a probability to a quality attribute, this would explain why ω p and ω £ responded differently to decoys. [START_REF] Malkoc | Between a rock and a hard place: The failure of the attraction effect among unattractive alternatives[END_REF] have also shown that attraction effect is attenuated when the options are unattractive. In their setting, however, both target options are unattractive-they are for example losses-whereas in the present experiment only ω £ can be said to be unattractive.

Reduced-form regression results

Table 3 reports the results of a logistic regression looking at the impact of the different classes of decoy on the probability of choosing ω p or ω £ . This regression reproduces the analysis leading to the between-subject Attraction Effect Regularity in Figure 2, but further controls for subject fixed-effects, option order within a task, and task order.

We see that the results presented above carry through: ω p and ω p increase the probability of choosing ω p while ω * p does not, which lends support to the process explanation. On the other hand, decoys do not affect ω £ except for ω * £ which actually decreases the probability of choosing it and so favours ω p , which is the positive range effect. ω * does not affect choice in this between-subject setting.

Finally, the order of the task affects positively the probability of choosing ω p , which is the order effect previously mentioned.

Heterogeneity of choice behaviour

Up to now we have looked at the effects only at the aggregate; we can get a better sense of subjects' heterogeneity by grouping them into different classes. To do so, I say that for a given subject a decoy ωi affects ω i if it does so in at least half of the parameter sets for which it is involved. For example, ω * i affects ω j , thus creating a positive range effect, if a subject switches to ω j following the introduction of the decoy in at least 3 of the 6 parameter sets where this type of decoy is present.

I then define classes of choice behaviour. Subjects can

• fit in the process explanation class if ω i and ω i affect ω i but not ω * i ;

• fit in the weights explanation class if ω i , ω i and also ω * i affect ω i ;

• fit in the 'process then positive range effect' class if ω i and ω i affect ω i , but ω * i affects ω j ;

• and finally, fit in the consistent class is they choose the same, or almost always the same, option between the two booklets.

Figure 3 shows the results from this classification. We see that almost half of the subjects, 100, can be classified as consistent. Of these 100 subjects, 46 are perfectly consistent-they always choose the same option between the booklets; 9 are affected only by ω * but are otherwise perfectly consistent; and 45 choose the same option more than 85% of the time and never exhibit one of the other effects.

The attraction effect, while present, thus leaves many subjects unaffected.

The second most popular class, with 68 subjects, is the process class. Of them, 30 exhibit more attraction effect at ω i than at ω i : for these subjects the decoy benefits from being close to its target. In contrast, 28 subjects exhibit more effect at ω i than at ω i : for them the decoy needs to be asymmetrically dominated, but within the asymmetric dominance decoys that increase the range create more effect;

this is a 'process + negative range effect' class. There remains 10 subjects who exhibit as much effect at ω i than at ω i .

These two classes encompass more than 80% of the subjects. Then, 10 subjects fit in the 'process then positive range effect' class: they exhibit the classical attraction effect as long as the asymmetric dominance is maintained, and then switch to the positive range effect.

Only 3 subjects fit in the weight class,. This was expected since, as Figure 2 showed us, the star decoys never created an attraction effect. Finally, 26 subjects do not fit in any of these classes.

These results suggest a different interpretation of the shift observed between

B.1 Gambles in attraction effect experiments

I relied on simple binary gambles to test the attraction effect for mainly two reasons. The first one is practical: gambles maximise internal validity. To understand why, note that in principle the attraction effect holds with any option, so the only criteria to consider when choosing an option is that it renders the experiment internally valid. A recent debate21 has clarified which options do so. The options should allow for economic consequences (Lichters et al., 2015a), which rules out hypothetical options as well as options that are costly and difficult to implement, such as the cars, TV sets and apartments routinely used in the consumer research literature.

The researcher should also have good reason to believe that subjects perceive the options as intended. For example, not everyone knows that the T-bone steaks used by Yang and Lynn (2014) are dominated by Porterhouse steaks (Simonson, 2014); some might even prefer the supposedly dominated T-bone decoys. The problem here stems from the use of complex options with many attributes, which requires the researcher to arbitrarily select two attributes out of all potential ones (an approach criticised by Frederick et al., 2014). People, however, might have preferences for the attributes not selected (Huber et al., 2014). Simple binary gambles easily solve these issues: they can be incentivised, since subjects can play their chosen gamble at the end of the experiment and be paid accordingly; and there is no need to arbitrarily select two attributes because simple binary gambles have two natural attributes, probability and money.

The second reason to use gambles is that they hold a special place for economists. They have proven to be a popular proving ground for theories [START_REF] Starmer | Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk[END_REF] and knowing whether the attraction effect might manifest itself when one tries to, say, elicit risk preferences, is important for the design of experiments and the interpretation of results. Further, testing the explanations to the attraction effect might give us hints on what to incorporate in future theories of risky choice.

As for external validity, gambles readily appear in domains such as financial and medical decision-making. The results obtained with gambles can thus be directly applied to these domains. It is true that the abstractness of gambles might dampen our ability to generalise to other domains; nonetheless, more concrete options, such as medical coverage or loans, would in turn hinder internal validity, since the requirements outlined above would be hard to satisfy; and for any selected option one could always ask how much exactly the results generalise. All in all, in the tension between internal and external validity [START_REF] Schram | Artificiality: The tension between internal and external validity in economic experiments[END_REF][START_REF] Starmer | Experiments in economics: should we trust the dismal scientists in white coats?[END_REF] I have favoured the former.

B.2 Sample choice task

In one of the envelopes, the options are: • writing P means that I will receive £7 if the dice throw yields a number from 1 to 80, and £0 otherwise; • writing Q means that I will receive £12.5 if the dice throw yields a number from 1 to 55, and £0 otherwise.

B.3 Procedure

Subjects waited outside the laboratory in line. An experimenter controlled their student card against the list generated by ORSEE and let them enter. Once inside the laboratory (Picture 4a) they were greeted by a second experimenter who asked them to draw a number from a pouch. This number determined their seat number as well as their group, as defined in Table 2 of the main text. They were then directed to take an envelope from the box corresponding to their group (Picture 4b). In each box were approximately 190 envelopes (Picture 4c). Inside each envelope was a piece of paper describing one of the 28 choice tasks that a member of the group assigned to this box could encounter in the experiment. Subjects were instructed to take this envelope with them but to not open it -all subjects obeyed this instruction. The draw of the envelope was without replacement. Subjects then went to their assigned desk. There they found the instructions, a pen and two 10-sided dice (Picture 4d). Once everyone was seated the experimenter started reading the instructions, which are found in Appendix B.4. The experimenters also controlled subjects' answers to the control questions. Then, the experimenters distributed the first decision booklet and the subjects started completing the 14 tasks contained inside. Depending on their seat number and so their group, subjects received different booklets. The booklets differed in the decoys seen, again as described in Table 2 of the main text. Appendix B.2 provides a sample of a choice task found in the booklets.

In the instructions subjects were instructed to raise their hand when they would finish the first booklet. As soon as a subject did the experimenters went to see her, collected the first booklet and gave her the second booklet containing 14 additional choice tasks. On average subjects took 15 minutes to complete each booklet.

Subjects were further instructed to raise their hand when they would finish the second booklet. Once everyone had finished the experimenters started the payment phase. When an experimenter came to a subject the experimenter gave her her first booklet. Then, the experimenter asked the subject to open the envelope and they flipped through the two booklets to find the choice task described on the piece of paper. Together they looked at the gamble the subject chose in this particular choice task and the experimenter read out loud the text at the bottom of the choice task (Appendix B.2) describing what would happen depending on the result of the dice. The experimenter asked the subject to draw her dice and, depending on the result, the subject won or lost. In any case the subject also received a show-up fee, of which they were not aware before this moment. The experimenter wrote the final payoff of the subject on a piece of paper, which the subject took to the centre of the room where a third experimenter collected it and paid the subject accordingly. Finally the subject exited the room In total the experiment took about 1 hour.

B.4 Instructions

The next pages reproduces the instructions as they were seen by the subjectsthey are here reproduced two-pages-on-one to save space.

Instructions

Welcome to the experiment. Please switch off your electronic devices and remain silent. If you have a question at any time, raise your hand and an experimenter will come to your desk to answer it. * * * In this experiment, you will face options that give you chances to win amounts of money. An option is for example a 20%-chance of winning £35.

To represent the chances of winning, we will refer to the dice placed on your desk, which you are welcome to examine as much as you want throughout the experiment. One die is used for the tens and the other for the units. Throwing the dice together and adding the results yields a number from 1 to 100. For example, the throw below is 31:

The next one is 8: 100 is obtained by getting zeros on both dice. Each ten and each unit is equally likely, so each number from 1 to 100 is also equally likely.

1

In terms of the dice, the 20%-chance of winning £35 is equivalent to £35 for numbers from 1 to 20, and £0 for numbers from 21 to 100. We will represent it as follows: All tasks in this experiment will ask you to select one from a set of two or three such options. The options in these tasks are always gambles which will feature different chances of winning various amounts of money. You will record your selected option in each of these tasks in decision booklets, which we will distribute soon. There will be a different task of this form on each page.

You will play one of the gambles that you select for real at the end of the experiment, meaning that you might win some money in this real task. The task that is real for you has already been selected in the following way: When you entered the room, you picked one from a set of envelopes. Each envelope contains a piece of paper describing one of the tasks that you will face in your booklet. Any one of the tasks that you face could be for real, but you will not know which one is for real until the end of the experiment.

At the end of the experiment, an experimenter will come to your desk and open your envelope. The piece of paper in the envelope will show the task that will be real for you. The experimenter will then look in your decision booklet to see which option you picked in this task. That option will be a gamble which will specify an amount of money that you can win depending on the throw of the dice. You will then roll the dice to see whether you earn money from the real task or not. If you win, you will be paid in cash the amount specified in your chosen option.

So, as you respond to the tasks remember that for each task, the option that you pick could turn out to be the one that you play for real at the end of the experiment. Because of this, we suggest that you treat each task as if it is for real and as if it is the only task you face since at the end of the experiment you will only face one task for real (i.e. the one contained in the envelope you now have).

Let us illustrate this with an example.

In one of the envelopes, the options are:

P

• writing P means that I will receive £12 if the dice throw yields a number from 1 to 60, and £0 otherwise;

• writing Q means that I will receive £25 if the dice throw yields a number from 1 to 40, and £0 otherwise.

This example assumes that you chose option P. Then, imagine that another page of the decision booklet is:

3

Page B

In one of the envelopes, the options are:

R

• writing P means that I will receive £20 if the dice throw yields a number from 1 to 30, and £0 otherwise;

• writing Q means that I will receive £10 if the dice throw yields a number from 1 to 50, and £0 otherwise;

• writing R means that I will receive £40 if the dice throw yields a number from 1 to 20, and £0 otherwise.

Here, we assume that you chose option R. At the end of the experiment, the experimenter comes to your desk and opens your envelope. Suppose that your envelope contains: The experimenter will look for these options in your decision booklet. They were in Page A above and we assumed that you chose option P. The experimenter will then throw the dice. In accordance with what is written in the decision booklet, you would get £12 if the throw yields a number from 1 to 60, and £0 if it yields a number from 61 to 100.

5

Questions

We want to make sure you understand the procedure fully, so we have designed two questions to test your understanding. These questions have no bearing on the rest of the experiment. Please answer them and raise your hand when you have finished; an experimenter will come to verify your responses.

Question 1

Imagine that the options in your envelope are:

Q

• writing P means that I will receive £14 if the dice throw yields a number from 1 to 70, and £0 otherwise;

• writing Q means that I will receive £35 if the dice throw yields a number from 1 to 20, and £0 otherwise.

9

Page B

In one of the envelopes, the options are:

R

• writing P means that I will receive £50 if the dice throw yields a number from 1 to 20, and £0 otherwise;

• writing Q means that I will receive £12 if the dice throw yields a number from 1 to 50, and £0 otherwise;

• writing R means that I will receive £23 if the dice throw yields a number from 1 to 30, and £0 otherwise.

Figure 1 :

 1 Figure 1: Decoys used to test the attraction effect and its explanations.

p

  and ω * £ , should not change choice. In fact the D-WARP condition in Tserenjigmid (2018) explicitly prevents it. Therefore, ω * p and ω * £ further allow me to test this class of models.

Figure 2

 2 Figure 2 reports the aggregate results of the experiment. I will comment on parameter-set irregularities when appropriate. Disaggregated and detailed results can be found in Appendix C. Across the whole experiments subjects chose decoys only 10 times, so in what follows I will not mention decoy choices.

Figure 2 :

 2 Figure 2: Attraction effects (in percent) at the aggregate level. One, two, three and four symbols indicate significance at α = 0.1, 0.05, 0.01 and 0.001.

Figure 3 :

 3 Figure 3: Number of subjects classified into the different patterns of choice.

ω

  £ and ω * £ . At the aggregate ω £ favours ω £ and ω * £ favours ω p , not because the introduction of ω * £ makes subjects shift from ω £ to ω p , but because some subjects-the ones in the process class-stop choosing ω £ while others-the ones in the process then positive range class-start choosing ω p . Such heterogeneity should thus be taken into account when studying the attraction effect, something that was lacking in previous studies. Appendix B.3 describes how the experiment was conducted and how the PRINCE mechanism was implemented. Appendix B.4 reproduces the instructions.

  are the ones in my envelope, give me option: (write P or Q in the box)

( a )

 a Entering the laboratory. (b) Boxes containing the envelopes. (c) Inside a box. (d) A subject's desk.

Figure 4 :

 4 Figure 4: Pictures of the experiment.

  , if the dice throw yields a number from 1 to 20, you will receive £35. If it yields a number from 21 to 100, you will receive £0. * * *

  are the ones in my envelope, give me option: (write P or Q in the box)

  are the ones in my envelope, give me option: (write P, Q or R in the box)

  are the ones on the following page of the decision booklet: are the ones in my envelope, give me option: (write P or Q in the box)

  are the ones in my envelope, give me option: (write P, Q or R in the box)

Table 1 :

 1 Parameter sets.

Table ,

 ,

subjects are split into 4 groups. All subjects encounter the 14 parameter sets in the same order in the first booklet but they see different decoys depending on their group. For example, the first line of the Table corresponds to parameter set g. Subjects in the first group choose between ω p and ω £ ; while subjects in the second, third and fourth groups have the decoys ω * , ω * p and ω * £ added to their menu. Comparing Group 1 and 3, and Group 1 and 4, addresses between-subject Attraction Effect Regularity, while comparing Group 3 and 4 addresses betweensubject Attraction Effect WASRP. The menus {ω p , ω £ } and {ω p , ω £ , ω * } act as fillers between parameter sets so that subjects: (a) do not face the same class of spurious effects due to sensitisation

  . Experimenter demand effects are especially a problem for Attraction Effect Regularity: since it requires to add a single, asymmetrically dominated option to the choice between two original options, it signals what the experiment is about and subjects can easily find out what is expected from them. decoy twice in a row; (b) do not face decoys favouring the same option twice in a row. Subjects face again the 14 parameter sets in the second decision booklet (second part of Table 2) but with different decoys. Keeping the example of parameter set g, subjects in Group 3 who faced the menu {ω p , ω £ , ω * p } in the first booklet face {ω p , ω £ , ω * £ } in the second booklet, and the the other way round for subjects in Group 4. Comparing the choices across booklets for a given parameter set addresses within-subject Attraction Effect WASRP. This way subjects see a given parameter set only twice, once in each booklet with different decoys, which minimises experimenter demand and sensitisation effects. A consequence of this design is that half of the subjects see the decoy related to ω p first (ω * p in the example) while the others see the decoy related to ω £ first (ω * £ ), so the design also allows me to study the directionality of within-subject Attraction Effect WASRP. Previous studies, on the other hand, studied the effect only in one direction for a given parameter set.

Table 2 :

 2 Decoys added to {ω p , ω £ } and explanation tested for each parameter set, in order of presentation.

	Set		Group 1		Group 2		Group 3	Group 4
					Booklet 1
	g	-No decoy	ω * Neutral decoy	ω * p	Weights/Process ω * £	Weights/Process
	c	ω * p	Weights/Process ω * £	Weights/Process -No decoy	ω * Neutral decoy
	f	ω * Neutral decoy	-No decoy	ω £ Standard decoy ω p Standard decoy
	k	ω £ Standard decoy ω p Standard decoy ω * Neutral decoy	-No decoy
	j	-No decoy	ω * Neutral decoy	ω p Weights	ω £ Weights
	m	ω p Weights	ω £ Weights	-No decoy	ω * Neutral decoy
	b	ω * Neutral decoy	-No decoy	ω * £	Weights/Process ω * p	Weights/Process
	h	ω * £	Weights/Process ω * p	Weights/Process ω * Neutral decoy	-No decoy
	a	-No decoy	ω Neutral decoy
	i	ω * Neutral decoy	-No decoy	ω £ Weights	ω p Weights
	e	ω £ Weights	ω p Weights	ω * Neutral decoy	-No decoy
	d	-No decoy	ω * Neutral decoy	ω * p	Weights/Process ω * £	Weights/Process
	l	ω *
					Booklet 2
	g	ω * Neutral decoy	-No decoy	ω * £	Weights/Process ω * p	Weights/Process
	c	ω * £	Weights/Process ω * p	Weights/Process ω * Neutral decoy	-No decoy
	f	-No decoy	ω Neutral decoy
	j	ω * Neutral decoy	-No decoy	ω £ Weights	ω p Weights
	m	ω £ Weights	ω p Weights	ω * Neutral decoy	-No decoy
	b	-No decoy	ω * Neutral decoy	ω * p	Weights/Process ω * £	Weights/Process
	h	ω * p	Weights/Process ω * £	Weights/Process -No decoy	ω * Neutral decoy
	a	ω * Neutral decoy	-No decoy	ω £ Standard decoy ω p Standard decoy
	n	ω £ Standard decoy ω p Standard decoy ω * Neutral decoy	-No decoy
	i	-No decoy	ω * Neutral decoy	ω p Weights	ω £ Weights
	e	ω p Weights	ω £ Weights	-No decoy	ω * Neutral decoy
	d	ω * Neutral decoy	-No decoy	ω * £	Weights/Process ω * p	Weights/Process
	l	ω * £	Weights/Process ω * p	Weights/Process ω * Neutral decoy	-No decoy

* Neutral decoy ω p Standard decoy ω £ Standard decoy n ω p Standard decoy ω £ Standard decoy -No decoy ω * p Weights/Process ω * £ Weights/Process -No decoy ω * Neutral decoy * Neutral decoy ω p Standard decoy ω £ Standard decoy k ω p Standard decoy ω £ Standard decoy -No decoy ω *

  3 Notes. ω p → ω £ : ω p added to {ωp, ω £ } in the first booklet and ω £ in the second. McNemar tests, † : Ha: (1) p. 15; ‡ : inequality reversed.

	Within-subject	Between-subject
	Attraction Effect WASRP	Attraction Effect WASRP
	ω p ↔ ω £	6.8 † †
	ω p ↔ ω £	4.6 †
	ω * p ↔ ω * £	-3.8 ‡

Notes. χ 2 tests, † : Ha: (2) p. 16; ‡ : inequality reversed.

Table 3 :

 3 Impact of the different classes of decoy on the probability of choosing ω p or ω £ . Logit model coefficients, standard errors in parentheses. Constant omitted. Option order: the position of the gamble on the page: first, second or third. Parameter set group: a dummy identifies the parameter set used to test a class of decoy: sets {a, f, k, n} for ω i , {e, i, j, m} for ω i , and {b, c, d, g, h, l} for ω i * . p < 0.1, ** p < 0.05, *** p < 0.01.

	ω p	ω £

Note. *

For consumer products, see[START_REF] Huber | Adding asymmetrically dominated alternatives: Violations of regularity and the similarity hypothesis[END_REF],[START_REF] Heath | Asymmetric decoy effects on lower-quality versus higher-quality brands: Meta-analytic and experimental evidence[END_REF] and[START_REF] Milberg | Synthesis of attraction effect research[END_REF]. For job candidates, see, among others,[START_REF] Highhouse | Context-dependent selection: The effects of decoy and phantom job candidates[END_REF],[START_REF] Slaughter | Decoy effects and attribute-level inferences[END_REF] and[START_REF] Slaughter | Effects of two selection batteries on decoy effects in job-finalist choice[END_REF]. For the other cases, see respectively[START_REF] Pan | The attraction effect and political choice in two elections[END_REF],[START_REF] Herne | Decoy alternatives in policy choices: Asymmetric domination and compromise effects[END_REF],[START_REF] Schwartz | Are more options always better? the attraction effect in physicians' decisions about medications[END_REF],[START_REF] Schwarzkopf | The effect of attraction on investment decisions[END_REF], and[START_REF] Bateman | Decoy effects in choice experiments and contingent valuation: asymmetric dominance[END_REF]. The attraction effect has also been observed in perceptual[START_REF] Crosetto | A monetary measure of the strength and robustness of the attraction effect[END_REF][START_REF] Trueblood | Not just for consumers: Context effects are fundamental to decision making[END_REF] and inferential[START_REF] Trueblood | Multialternative context effects obtained using an inference task[END_REF] tasks.

[START_REF] Latty | Irrational decision-making in an amoeboid organism: transitivity and context-dependent preferences[END_REF];[START_REF] Lea | Irrationality in mate choice revealed by tungara frogs[END_REF];[START_REF] Parrish | Rhesus macaques (macaca mulatta) exhibit the decoy effect in a perceptual discrimination task[END_REF];[START_REF] Shafir | Context-dependent violations of rational choice in honeybees (apis mellifera) and gray jays (perisoreus canadensis)[END_REF].

See, among many others,[START_REF] Barbos | Context effects: A representation of choices from categories[END_REF],[START_REF] Cherepanov | Rationalization[END_REF], de Clippel and Eliaz (2012), Gerasimou (2016a),Manzini and Mariotti (2012a),[START_REF] Masatlioglu | Revealed attention[END_REF][START_REF] Ok | Revealed (p)reference theory[END_REF].

Frederick et al. (2014);Huber et al. (2014);Lichters et al. (2015a,b);Simonson (2014);Yang and Lynn (2014).

Both will be defined precisely in Section 2.

See Appendix B.1 for details on the use of gambles in attraction effect research.

Most studies on the attraction effect have used options with the same expected value, an approach criticised byFrederick et al. (2014) and[START_REF] Crosetto | A monetary measure of the strength and robustness of the attraction effect[END_REF].

At the time of the experiment, £1 $1.42.

Within-subject designs have more statistical power, are less noisy, and are more natural when one wants to study preferences[START_REF] Charness | Experimental methods: Between-subject and within-subject design[END_REF]. But they are also more prone to elicit

[START_REF] Cubitt | Testing explanations of preference reversal[END_REF] used the same argument in the context of preference reversals.

Mann-Whitney U tests further show that ω p does not cause more between-subject attraction effect than ω p (U = 6, p = 0.686), and ω £ , than ω £ (U = 7, p = 0.886). Given that these tests compare different parameters sets, however, they should be interpreted with caution.

[START_REF] Loomes | A microeconometric test of alternative stochastic theories of risky choice[END_REF] make a similar observation.

I thank Robert Sugden for this observation.

On the WASRP, see also[START_REF] Bandyopadhyay | Demand aggregation and the weak axiom of stochastic revealed preference[END_REF],[START_REF] Bandyopadhyay | A general revealed preference theorem for stochastic demand behavior[END_REF] andDasgupta and Pattanaik (2007).

From Chernoff (1954); also called Sen's α[START_REF] Sen | Quasi-transitivity, rational choice and collective decisions[END_REF], basic contraction consistency[START_REF] Sen | Social choice theory: A re-examination[END_REF] or the independence of irrelevant alternatives[START_REF] Nash | The bargaining problem[END_REF].

On Chernoff, see Sen (1969, Lemma 2, p. 384). On WARP, seeMas-Colell et al. (1995, Proposition 1.D.1 and 1.D.2, pp. 12-13). On WARP implying Chernoff, seeSen (1971, T.8 and Corollary, p. 314).

Frederick et al. (2014) andYang and Lynn (2014) failed to replicate the attraction effect, which prompted replies byHuber et al. (2014) andSimonson (2014) in the same issue of the Journal of Marketing Research.Lichters et al. (2015a,b) added to the debate by providing guidelines for future attraction-effect experiments.

Appendices Appendix A Weak Axiom of Stochastic Revealed Preference and Regularity Condition

To see how Attraction Effect WASRP violates the Weak Axiom of Revealed Preference, denote by P A (B) the probability that the choice from the set A lies in B. WASRP holds if and only if, for all A and B, P B (C) -P A (C) ≤ P A (A -B) for all C ⊆ A ∩ B. For us, A = {x, y, x } and B = {x, y, y }. P A (A -B) boils down to P {x,y,x } (x ), which is always 0 in properly setup attention effect experiments.

With C = {x}, WARSP becomes P {x,y,y } (x) ≤ P {x,y,x } (x), and with C = {y}, P {x,y,y } (y) ≤ P {x,y,x } (y), a condition that Attraction Effect WASRP violates. 18 Then, for the Attraction Effect Regularity and with the same notation, we know that the Regularity Condition is satisfied if and only if, for all A, B, C such that C ⊆ B ⊆ A, P B (C) ≥ P A (C). For us, B = {x, y}, and, for example, A = {x, y, x } and C = {x}. The Condition becomes P {x,y} (x) ≥ P {x,y,x } (x), which Attraction Effect Regularity 1 violates; and similarly for Attraction Effect Regularity 2 with A = {x, y, y } and C = {y}.

As said in the main text, WASRP and the Regularity Condition are stochastic generalisations of the the Weak Axiom of Revealed Preference (WARP) and of the Chernoff Condition 19 (Dasgupta and Pattanaik, 2007). If we were to use these, we would use a similar reasoning: We know that the Chernoff Condition is necessary for the existence of any preferences, while WARP is necessary and sufficient for the existence of rational preferences, so WARP necessarily implies the Chernoff Condition. 20 Thus, violations of WARP should be observed more often, since there should be more people with non-rational preferences than people without preferences at all.

Appendix B Details of the experiment

This Appendix provides additional details on the experiment. Appendix B.1 explains more in depth why I used simple binary gambles to test the attraction effect. Appendix B.2 gives a sample of a choice task found in a decision booklet. If these options are the ones in my envelope, give me option: (write P or Q in the box)

• writing P means that I will receive £8 if the dice throw yields a number from 1 to 85, and £0 otherwise;

• writing Q means that I will receive £13 if the dice throw yields a number from 1 to

Appendix C Result tables

This Appendix provides disaggregated and detailed results corresponding to all aggregated results encountered in the main text. The Tables are as follows:

• Table 4 • Within-subject Attraction Effect WASRP: from Table 4, -when the decoy of ω p is seen first (e.g. ω p → ω £ ), it is the percentage of subjects who switch from ω p to ω £ (column ω p then ω £ ) minus the percentage of subjects who switch from ω £ from ω p (column ω £ then ω p ); -when the decoy of ω £ is seen first (e.g. ω £ → ω p ), it is the percentage of subjects who switch from ω £ to ω p (column ω £ then ω p ) minus the percentage of subjects who switch from ω p from ω £ (column ω p then ω £ ). Table 6 uses a similar definition but the test is two-sided since there is no expected effect.

• Between-subject Attraction Effect WASRP: from Table 5, -it is the percentage of subjects who choose ω p when the decoy is related to ω p (column Decoy under ω p ) minus the percentage of subjects who choose ω p when the decoy is related to ω £ . Note that this last percentage is not displayed directly on the Table : it is roughly 100 minus the percentage of subjects who choose ω £ when the decoy is related to ω £ (column Decoy under ω £ ) but not exactly since some subjects chose the Decoy and these subjects are not displayed on the Table . However since the Decoy was very rarely chosen (less than 10 times across the whole experiment) this has a small impact on the numbers. The Figures in the main text report exact percentages, that is they take care of these few subjects who chose the Decoy. -It is also the percentage of subjects who choose ω £ when the decoy is related to ω £ (column Decoy under ω £ ) minus the percentage of subjects who choose ω £ when the decoy is related to ω p . The same remark applies. The result of the χ 2 test is reported in the rightmost column.

• Between-subject Attraction Effect Regularity: from Table 5, -when the decoy is related to ω p (e.g. ω p ), it is the percentage of subjects who choose ω p when there is a decoy (column Decoy under ω p ) minus the percentage of subjects who choose ω p when there is no decoy (column No decoy under ω p ); -when the decoy is related to ω £ (e.g. ω £ ), it is the percentage of subjects who choose ω £ when there is a decoy (column Decoy under ω £ ) minus the percentage of subjects who choose ω £ when there is no decoy (column No decoy under ω £ ). Table 7 uses a similar definition but here the test is two-sided since there is no expected effect.

Table 4: Percentage (n) of choice patterns by class of decoy and presentation order (within-subject).

Set

51.9 (28) 25.9 (14) 13.0 (7) 9.3 (5) 0.33

56.0 ( 28

31.4 (16) 47.1 (24) 7.8 (4) 13.7 (7) 0.82

51.0 (53) 53.4 (55) 0.18

64.1 (66) 68.3 (71) 0.41

35.9 (37) 38.5 ( 40

41.4 (43) 49.0 (50) 1.22 Set

36

(18) 38 ( 19) 10 ( 5) 16 (8) 0.69


R Ravaud

G Lemarquand
email: guy.lemarquand@univ-lemans.fr

T Roussel

Experimental measurement of the nonlinearities of electrodynamic microphones for reciprocal calibration

Keywords: Microphone, Electrodynamic, Electrical Impedance

This paper presents an experimental way of characterizing the nonlinearities of electrodynamic microphones used as acoustical sources. This functioning occurs for reciprocal calibration techniques. For this purpose, its electrical impedance is measured with a Wayne Kerr wedge which has an excellent precision. Moreover, it can be noted that the Thiele and Small model is used to characterize its electrical impedance. Furthermore, an experimental method based on Simplex algorithm allows us to construct polynomial laws which describe the dependence of the Thiele and Small parameters with the input voltage. The nonlinear variations obtained allow us to determine the nonlinear differential equation of the electrodynamic microphone. Then, this equation is solved numerically in order to confirm the accuracy of the polynomial laws obtained by the Simplex algorithm. The distortions are measured with a laser Doppler velocimeter and compared with the ones obtained by the numerical solving of the nonlinear differential equation. The experimental displacement spectrum is consistent with the theoretical one.

Introduction

Electrodynamic microphones are generally used either for recording voice and instruments or for reciprocal calibration techniques. They are often characterized by their directivity (omnidirectional, cardiod, supercardiod, etc...).

Moreover, most of the microphones are designed as pressure microphones or pressure gradient microphones which usually leads to sound coloration. Microphone directivity is the most important property since it allows to select the sound produced by only one instrument among other instruments. However, it is not the only property which has to be taken into account. Microphone linearity is an important characteristic which is strongly linked to sound fidelity.Distortions produced by electrodynamic microphone nonlinearities is a scientific topic which is studied little. However, the most interesting studies on the microphone characterization were done by Abuelma'atti with various technologies of microphones [START_REF] Abuelma'atti | Improved analysis of the electrically manisfested distortions of condenser microphones[END_REF]- [START_REF] Abuelma'atti | Large signal performance of micromachinened silicon condenser microphones[END_REF] and Niewiarowicz [START_REF] Niewiarowicz | Investigations into the transduction properties of dynamic microphone membranes subjected to transients[END_REF] [START_REF] Niewiarowicz | Determination of active compliance of dome type microphone membranes by using the indicator diagrams method[END_REF]. Experimentally, a lot of parameters have to be taken into account and vary together according to input level. For this reason, the accurate estimation of the electrodynamic microphone main nonlinearities is difficult. Moreover, time-varying effects are also present and can modify the recording quality by amplifying or reducing distortions. The knowledge of these nonlinearities can really help designing new microphones with improved sound quality.

to recent demands for miniaturization and high sound quality [START_REF] Hwang | New develpment of combined permanent-magnet type microspeakers used for cellular phones[END_REF]- [START_REF] Musha | Measurement of sound intensity using a single moving microphone[END_REF]. These new developments are based on the traditional technology. Moreover, the nonlinearities observed in these new microphones have the same physical origins as the nonlinearities observed in electrodynamic loudspeakers even if their functioning is different. Therefore, the studies carried out with electrodynamic loudspeakers [START_REF] Olsen | Nonlinear modeling of low frequency loudspeakers-a more complete model[END_REF]- [START_REF] Lemarquand | Ironless loudspeakers[END_REF] can be useful for the electrodynamic microphone ones.

However, electrodynamic microphones are damping controlled whereas the electrodynamic loudspeakers are mainly designed to be mass controlled. Consequently, electrodynamic microphones have a poor transient response which is the most important defect. It can be noted that it is one of the main problems of electrodynamic microphones but this is not the only one. This paper presents an experimental way of characterizing the nonlinearities of electrodynamic microphones. This experimental method is based on a very accurate measurement of the electrical impedance of the electrodynamic microphone.

We can say that that the electrical impedance measurement of such a transducer is the most accurate measurement we can generally realize in a laboratory. Moreover, such a measurement is simple to perform. Consequently, the experimental method presented in this paper allows us to guess what must change in an electrodynamic microphone in order to improve its fidelity. In addition, the electrodynamic microphone is used as an acoustical source in this paper. This allows us to use important input voltages to show the nonlinear effects of such transducers. Furthermore, it can be noted that the Thiele and Small model [START_REF] Small | Closed-box loudspeaker systems, part1: Analysis[END_REF] is used to characterize the electrical impedance of the electrodynamic microphone. We will show that the Thiele and Small parameters depend on the input voltage and consequently, some distortions are created.

Such distortions are measured with a laser Doppler velocimeter and predicted theoretically by solving numerically the nonlinear differential equation of the electrodynamic microphone. We can say that the experimental displacement spectrum is consistent with the theoretical spectrum. The first section presents the analytical classical model of an electrodynamic microphone and its limits.

The second section presents an experimental method based on the electrical impedance measurement to characterize the variations of the nonlinear parameters that describe the electrodynamic microphone. This way of characterizing a nonlinear system has been used in a previous paper for studying the electrodynamic loudspeaker nonlinearities [START_REF] Ravaud | Time-varying nonlinear modeling of electrodynamic loudspeakers[END_REF]. The third section presents both the theoretical and the experimental spectrums.

Classical model of electrodynamic microphones and its limits

An electrodynamic microphone is a transducer which transforms acoustic signals into electrical signals. Such an electrodynamic transducer generally includes a magnet motor, a rim and a diaphragm. The diaphragm vibration due to the acoustical excitation (the voice for example) engenders the movement of a coil which moves between two yoke pieces. Moving coil microphones use the same dynamic principle as in a loudspeaker, only reversed. When sound enters through the windscreen of the microphone, the sound wave moves the diaphragm. When the diaphragm vibrates, the coil moves in the magnetic field, producing a varying current in the coil through electromagnetic induction. However, it must be emphasized here that the parameter values are extremely different between an electrodynamic microphone and an electrodynamic loudspeaker. The apparent internal resistance R e of an electrodynamic microphone can reach 800Ω whereas it varies approximately from 2Ω to 10Ω for an electrodynamic loudspeaker. Such a difference has a great influence on the dynamic of these two transducers. In addition, the equivalent damping parameter R ms is rather weak for electrodynamic microphones: we can also say that its variation with input voltage generates distortions that are less important than the other Thiele and Small parameters when an electrodynamic microphone is used as an acoustical source. In fact, we can say that R ms represents the measurement of the losses, or damping, in a driver's suspension and moving system. Consequently, as the voice-coil displacement is greater for electrodynamic loudspeakers, the losses are generally greater. This is why this parameter does not have the same influence on the acoustical response between electrodynamic microphones and electrodynamic loudspeakers. Furthermore, the eddy currents, commonly represented by R µ , do not appear at the same frequency between an electrodynamic microphone and an electrodynamic loudspeaker. The reason lies in the fact that the magnet dimensions and the magnetic circuit dimensions is smaller in electrodynamic microphones.

Two differential equations can be used to describe the electrodynamic microphone. Such equations are also used for modeling electrodynamic loudspeakers [START_REF] Thiele | Loudspeakers in vented boxes, part i[END_REF]- [START_REF] Small | Direct-radiator loudspeaker system analysis[END_REF]. The first one is given by [START_REF] Abuelma'atti | Improved analysis of the electrically manisfested distortions of condenser microphones[END_REF].

u(t) = R e i(t) + L e di(t) dt + Bl dx(t) dt (1)
where x(t) is the position of the coil, l is the length of the coil, L e is the coil inductance, i(t) is the coil current, Bl is the force factor, R e is the electric resistor of the coil and u(t) is the input voltage. The second differential equation is given by Eq.( 2).

M ms d 2 x(t) dt 2 -Bli(t) = -kx(t) -R ms dx(t) dt (2)
where M ms is the mass of the diaphragm, Bl is the force factor, k is the equivalent stiffness of the suspensions and R ms is the equivalent damping parameter.

Inserting Eq.( 1) in Eq.( 2) leads to the complex electrical impedance given by given by Eq. [START_REF] Abuelma'atti | Large signal performance of micromachinened silicon condenser microphones[END_REF].

Z e = R e + jL e w + Bl 2 R ms + jM ms w + k jw (3)
By taking into account the eddy currents which occur at high frequencies [START_REF] Vanderkooy | A model of loudspeaker driver impedance incorporating eddy currents in the pole structure[END_REF],

Eq.( 3) is expressed as follows (Eq.4):

Z e = R e + jR µ L e w jL e w + R µ + Bl 2 R ms + jM ms w + k jw (4)
All the parameters in Eq.( 3) could be called the electrodynamic microphone parameters. As the parameters that describe the electrodynamic loudspeakers are the same, the parameters in Eq.(3) can also be called the Thiele and Small parameters. However, it must be emphasized that the parameter values are not comparable and thus, the acoustical response is very different. The main assumption of this classical model is that it is a linear model. In the next section, it is shown that a linear model is not sufficient for describing accurately the electrodynamic microphone behavior. Moreover, the nonlinearities are also different between electrodynamic loudspeakers and electrodynamic microphones.

For example, the voice-coil excursion of an electrodynamic loudspeaker is important and generate important sound pressure levels compared to the ones produced by electrodynamic microphones used as acoustical sources. Consequently, the nonlinear effects that are often predominant at low frequencies for electrodynamic loudspeakers are different for electrodynamic microphones.

Limits of a linear electro-acoustical model

This section presents the limits of the linear model for characterizing electrodynamic microphones. To do so, an electrodynamic microphone is placed in an anechoic chamber. An electrical impedance measurement is realized by using a Wayne Kerr wedge that has an excellent precision (10 -4 Ω). A voltage measurement is carried out with levels varying from 100mV to 4V. During our experiment, the electrodynamic microphone is used as an acoustical source.

Even though this situation is rather rare, the nonlinearities determined with such an approach represent very well the main defects in electrodynamic microphones. This is in fact the main aim of this paper: an accurate electrical impedance measurement can be used to estimate electrodynamic microphone nonlinearities. The electrical impedance magnitude is represented versus the input voltage and the frequency in ear phenomena of the two previous representations (Figs. 3 and4). Figures 3 and4 shows that the electrical impedance of the electrodynamic microphone depends also on input voltage. It is noted that the resonance frequency varies with respect to the input voltage; this implies that the stiffness of the suspensions or the equivalent mass depend on input voltage. In conclusion, Eq.( 4) which is generally used to describe the electrodynamic microphone is not sufficient to correctly describe its nonlinear effects. Strictly speaking, all the parameters which define the electrical impedance (Eq.4) are a function of both input level and time. Obtaining the variation laws of these parameters is necessary in order to improve the design of electrodynamic microphones and predict the distortions created by themselves. As a consequence, a general method should be found in order to determine which parameters vary a lot with the input voltage and produce some distortions. Such a general experimental method is discussed in the next section.

3 Experimental method to derive the nonlinear variations of the Thiele and Small parameters

Introduction

Our experimental method to derive the dependence of the Thiele and Small parameters with the input voltage is based on the electrical impedance measurement of the electrodynamic microphone. A real-time algorithm has been put forward to measure this impedance with a Wayne Kerr wedge that has an excellent precision (10 -4 Ω). It is noted that this wedge is especially dedicated to the electrical impedance measurement. Consequently, we can say that such a measurement device allows us to have a great confidence in the experimental measurements. Our way of characterizing the electrodynamic microphone nonlinearities allows us to predict precisely the distortions created by such transducers. Our measurement algorithm is used in order to determine at which frequencies impedance must be measured. Basically, points must be measured when electrical impedance reaches a maximum or when impedance variation with frequency is important. In short, the electrodynamic microphone is characterized by its electrical impedance which, precisely measured, allows us to construct polynomial functions for each electrodynamic microphone parameter. The polynomial functions are determined by using Simplex algorithm and their coefficients are established by using the least mean square method. The Simplex algorithm is used to determine the coefficients of each polynomial function describing the nonlinear variations of the Thiele and Small parameters. The principle of this algorithm is to minimize the difference ∆Z e between the experimental impedance and the theoretical impedance. The theoretical impedance is in fact the electrical impedance with the Thiele and Small model whose parameters are assumed to depend on input voltage. For example, the equivalent mass can be written :

M ms (u) = M ms + m n=1 μn M ms u n (5)
Each Thiele and Small parameter is represented like the previous form. Consequently, the difference ∆Z e is expressed as follows:

∆Z = n=2 n=0 Z (exp) (u) -Z (theo) (u) 2 (6)
where

Z (theo) (u) = R e (u) + jR µ (u)L e (u)w jL e (u)w + R µ (u) + Bl(u) 2 R ms (u) + jM ms (u)w + 1 jCms(u)w (7)
When the algorithm converges, all the values describing the nonlinear parameters obtained are used to solve numerically the nonlinear differential equation of the electrodynamic microphone. Figure 5 represents the error sheet between the experimental results and the theoretical ones when the Thiele and Small parameters are constant. The mean difference between the experimental and the theoretical values is 6.0Ω. In this case, we did not take into account the nonlinear variations of the Thiele and Small parameters determined by the Simplex algorithm. Figure [START_REF] Hwang | New develpment of combined permanent-magnet type microspeakers used for cellular phones[END_REF] represents the error sheet between the experimental resuts and the theoretical one when the variations of the Thiele and Small parameters are taken into account. The mean difference between the experimental and the theoretical values is 2.9Ω. As a consequence, the improvement of the electrodynamic microphone model is only possible if the nonlinear variations of the Thiele and Small parameters are taken into account.

Variations of the Thiele and Small parameters

This section discusses the sensitivity of the Thiele and Small parameters to the least mean square method. To do so, we assume that only one parameter varies at a time (though the other Thiele and Small parameters are constant). By using our least square method based on the simplex method, we determine the difference of the impedance (magnitude and phase) between the model with constant parameters and the model with one varying parameter. This Laws of variations of the Thiele and Small parameters difference allows us to determine the sensitivity of each Thiele and Small parameter. Table 1 presents the laws of variations of Thiele and Small parameters determined with our three-dimensional least mean square method.

It can be noted that the parameter that is the most sensitive to the least mean square algorithm is the force factor Bl. In addition, we see that the equivalent inductance L e is also sensitive. This implies that the magnetic circuit could be improved. In fact, it is well-known that the iron in magnetic circuits generates nonlinearities because of its saturation and its hysteresis losses. This is the reason why it can be interesting to design ironless magnetic loudspeakers [START_REF] Lemarquand | Ironless loudspeakers[END_REF].

d = kR e Bl (14)
We can also write the previous relations in the frequency domain so that [START_REF] Musha | Measurement of sound intensity using a single moving microphone[END_REF] becomes :

U = a(jw) 3 X + b(jw) 2 X + c(jw)X + dX (15)
Thus, we deduct that there is a bijective relation between U and X:

U = X A(jw) 3 + B(jw) 2 + C(jw) + D (16)
Thus

U = χX (17)
where χ = (A(jw) 3 + B(jw) 2 + C(jw) + D). In the previous section, we studied the fact that the five Small signal parameters depended on input voltage.

We deduct that these parameters can also be written as a function of the voice coil position X. Therefore, the parameters a, b, c and d in 10 become a(x), b(x), c(x) and d(x) in the nonlinear differential equation of the electrodynamic microphone. It is to be noted that solving this nonlinear differential equation is rather difficult because the denominator is not constant. It can be noted that this equation must be solved numerically in order to determine the distortions created by an electrodynamic microphone. In fact, the distortions created by a nonlinear system can be determined either analytically by using for example a Taylor series expansion or numerically. In the case of the electrodynamic microphone, we have chosen to solve numerically its nonlinear differential equation with Mathematica. This allows us to confirm the experimental displacement spectrum measured with the laser Doppler velocimeter.

Comparison between the theoretical displacement spectrum and the experimental displacement spectrum

A way of obtaining the theoretical displacement spectrum is to solve numerically the nonlinear differential equation of the electrodynamic microphone.

This can be done for example in the time-domain by assuming that the electrodynamic microphone generates only harmonics that are multiple of the fundamental harmonic (w, 2w, 3w ). This is a simplifying assumption because input voltage owns in reality many terms so that other typical nonlinear phenomena appear (intermodulations). In short, we assume the solution of the nonlinear differential equation of the electrodynamic microphone to be as the following form:

x(t) = a 1 cos(wt) + a 2 sin(wt) + a 3 cos(2wt) + a 4 sin(2wt) +a 5 cos(3wt) + a 6 sin(3wt)

The parameters a 1 , a 2 , a 3 , a 4 , a 5 and a 6 are determined numerically and are given in Table 2. Values of the coefficients given in Eq. ( 18) : these coefficients have been determined

with the explicit Runge Kutta method (numerical solving of the nonlinear differential equation of the electrodynamic microphone)

Experimental and theoretical displacement spectrums

This section presents a comparison between the experimental displacement spectrum of the electrodynamic microphone which has been obtained by using a laser Doppler velocimeter and the theoretical displacement spectrum obtained by using the solution given in Eq. ( 18). The experimental and theoretical values are given in table 3. Moreover, the results obtained are plotted in Fig. 7. The theoretical displacement spectrum is consistent with the experimental displacement spectrum. Consequently, we deduct that the experimental way of characterizing the electrodynamic microphone with its electrical impedance allows us to precisely estimate the nonlinear variations of the Small signal parameters with the input voltage. Values of the harmonics created by the electrodynamic microphone ; H1 corresponds to the fundamental, H2 is the harmonic two and H3 is the harmonic three

Fig. 1 .

 1 Fig. 1. Experimental three-dimensional representation of the electrical impedance magnitude of the electrodynamic microphone (voltage: 0 V;4 V)(frequency: 0 Hz;1000 Hz)(|Z|: 400Ω;900Ω)

  Fig. (2) A two-dimensional view allows us to see more precisely the nonlin-

Fig. 2 .Fig. 3 .

 23 Fig. 2. Experimental three-dimensional representation of the electrical impedance phase of the electrodynamic microphone (voltage: 0 V;4 V)(frequency: 0 Hz;1000 Hz)(phase: -20 deg ;+20 deg)

Fig. 4 .

 4 Fig. 4. Two-dimensional representation of the electrical impedance phase of the electrodynamic microphone (frequency: 60 Hz;240 Hz)(phase: -0.3 rad;+0.3 rad)

Fig. 5 .

 5 Fig. 5. Three-dimensional representation of the difference between the experimental impedance and the theoretical impedance ; the theoretical impedance is based on the Thiele and Small model with constant parameters (voltage: 0 V;4 V)(frequency: 0 Hz;1000 Hz)(|Z|: -200Ω;+200Ω)

Fig. 6 .

 6 Fig. 6. Three-dimensional representation of the difference between the experimental impedance and the theoretical impedance ; the theoretical impedance is based on the Thiele and Small model with variable parameters (voltage: 0 V;4 V)(frequency: 0 Hz;1000 Hz)(|Z|: -200Ω;+200Ω)

  R e + R ms L e ) Bl (12) c = (R e R ms + Bl 2 + kL e ) Bl(13)

  exp ] -5.17 -11.89 -14.1 log[x theo ] -5.24 -12.08 -15.3 Table3

Fig. 7 .

 7 Fig. 7. Experimental and Theoretical spectrums of the electrodynamic microphone

Table 1

 1

		.0023 + 0.002u + 0.06u 2	15.1%
	Bl	13.2 -15.1u + 8.09u 2	23%
	Rms	0.25 + 0.81u -0.021u 2	4.7%
	M ms	0.00025 -0.0014u + 0.0036u 2	18.1%
	k	171.28 -50.2u + 1018u 2	2.1%
	R µ	48.1

Table 2

 2

solving numerically the nonlinear differential equation of the electrodynamic microphone. The comparison between the theoretical displacement spectrum and the experimental displacement spectrum shows a very good agreement at low frequencies.

crophone

This section presents a method to obtain the nonlinear differential equation of the electrodynamic microphone. In fact, this nonlinear differential equation is the same as the one of the electrodynamic loudspeaker because the electrodynamic microphone is used as an acoustical source. In this paper, the nonlinear differential equation of the electrodynamic microphone is obtained by taking into account the variations of the Thiele and Small parameters.

These variations are obtained in the previous section by using both the Simplex algorithm with the least mean square criteria. Furthermore, we neglect here the unstationary effects (R e increases in time due to the Joule effect).

The first step for obtaining this nonlinear differential equation is to drop the parameter i(t) from the two equations ( 1) and [START_REF] Abuelma'atti | Harmonic and intermodulation distortion in electret microphones[END_REF]. From (2), i(t) can also be written as follows:

By using ( 8) and 1, we deduct :

The previous equation can also be written in the following form :