Andrzej Baranski
email: a.baranski@nyu.edu

Diogo Geraldes
email: diogogeraldes@gmail.com

Ada Kovaliukaite
email: ada.kovaliukaite@univie.ac.at

James Tremewan
email: james.tremewan@gmail.com

José Apesteguía

Katherine Baldiga

Olivier Bochet

Christine Exley

Becky Morton

Ronald Peeters

Ernesto Reuben

Arno Riedl

Aldo Rustichini

Xiaoyue Shan

Lise Vesterlund

Baranski

An Experiment on Gender Representation in Majoritarian Bargaining

Keywords: JEL Classification: C72, J16, J31 multilateral bargaining, gender gap, lab experiment, discrimination, coalitions

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Introduction

Evidence from multiple studies and surveys reveals that women are largely underrepresented in decision-making bodies worldwide across business, academic, and political domains. 1 This imbalance has led to widespread calls for policies aiming to close the gender gap of female representation in decision-making bodies (e.g., European Commission Gender Equality Strategy 2020-2025 and United Nations 2030 Sustainable Development Goals). Besides a first-order equity concern for gender parity, it is often argued that women's under-representation in committees, boards, and teams may systematically lead to less desirable overall outcomes and affect women in particular negatively.

The latter undesirable outcome is arguably more likely if decisions are reached according to the expressed views of a majority, where if women are underrepresented, males might form alliances to advance their interests, further diminishing women's actual influence on decision-making.

However, the precise way in which the gender composition of committees, boards, and teams causally affects the outcomes of negotiations is unknown and warrants careful investigation. This is especially important for understanding decision processes within modern organizations and firms where many decisions are made by teams [START_REF] Lazear | Personnel economics: The economist's view of human resources[END_REF], even at high hierarchical levels [START_REF] Menz | Functional top management team members: A review, synthesis, and research agenda[END_REF]. 2 For example, top management team members (i.e., senior executives in charge of different functional areas in their organizations), must hold regular meetings to devise business strategies but there is no guarantee that their interests are fully aligned [START_REF] Edmondson | A dynamic model of top management team effectiveness: Managing unstructured task streams[END_REF].

This may lead to the formation of alliances to secure power and influence in the organization and advance self-interested agendas. In business partnerships, such as legal firms, important managerial and profit-sharing decisions are reached via negotiations with the participation of partners. 3 Another setting where bargaining certainly takes place within firms is in self-managed teams in which members may engage in internal negotiations to assign responsibilities and tasks to achieve their objectives. Because the interests and incentives of the members within an organization or team are unlikely to fully overlap, and often can be highly divergent, alliances pursuing mutually beneficial outcomes are expected to arise [START_REF] Mithani | So what exactly is a "coalition" within an organization? a review and organizing framework[END_REF]. 4 But what role does gender, and the gender composition of an organization or team, play in determining which coalitions will form and the end result of negotiations? An important body of work has emerged to investigate gender differences in bargaining, mainly via laboratory experiments. 5 1 In the 116 th United States Congress (2019Congress ( -2021)), only 23% of members of Congress and 26% of Senators are women.

In 2019, 24.9% of all members of parliamentary bodies around the world were women (Inter-parliamentary Union, 2020). Women represent only 22% of chief executives in the United States [START_REF] Huang | Women in the workplace[END_REF].

2 See [START_REF] Gavetti | The behavioral theory of the firm: Assessment and prospects[END_REF] for a summary of research on the behavioral theory of the firm, specifically relating to the political processes within firms.

3 See [START_REF] Wesemann | 2015 global partner compensation survey[END_REF] for a survey on law firm compensation schemes, in which subjective schemes entailing negotiations are quite common in the U.S. 4 According to [START_REF] Mithani | So what exactly is a "coalition" within an organization? a review and organizing framework[END_REF], "[a] coalition, therefore, is a compromise between individuals with overlapping but nonidentical interests undertaken to improve potential gains against others with relatively divergent interests". 5 Gender differences in bargaining have also been studied in the field [START_REF] Castillo | Gender differences in bargaining outcomes: A field experiment on discrimination[END_REF][START_REF] Leibbrandt | Do women avoid salary negotiations? Evidence from a largescale natural field experiment[END_REF][START_REF] Andersen | On the cultural basis of gender differences in negotiation[END_REF][START_REF] Hernandez-Arenaz | Women ask for less (only from men): Evidence from bargaining in the field[END_REF][START_REF] Säve-Söderbergh | Gender gaps in salary negotiations: Salary requests and starting salaries in the field[END_REF]. See also Recalde and Vesterlund 1

Studies that exogenously vary gender pairing in bargaining have focused exclusively on bilateral bargaining settings [START_REF] Eckel | Chivalry and solidarity in ultimatum games[END_REF][START_REF] Solnick | Gender differences in the ultimatum game[END_REF][START_REF] Sutter | Gender pairing and bargaining-beware the same sex![END_REF][START_REF] Eriksson | Gender differences in initiation of negotiation: Does the gender of the negotiation counterpart matter?[END_REF][START_REF] Hernandez-Arenaz | Gender differences in alternating-offer bargaining: An experimental study[END_REF][START_REF] Exley | Knowing when to ask: The cost of leaning in[END_REF]. 6 Taken together, the results from the previous studies indicate that women are more generous than men [START_REF] Eckel | Gender and negotiation in the small: are women (perceived to be) more cooperative than men?[END_REF][START_REF] Engel | Dictator games: A meta study[END_REF][START_REF] Bilén | Are women more generous than men? a metaanalysis[END_REF] and women are more likely to accept lower offers than men in the ultimatum game. While bilateral settings are a natural starting point for investigating bargaining behavior, they do not capture essential elements inherent to more general bargaining settings like the ones discussed previously. In particular, if women are more willing than men to accept lower shares, they become more attractive to others (men and women alike) as partners in an alliance and, consequently, one would expect men to be left out of agreements more often. Yet, evidence from a field study suggests that males are empirically preferred as bilateral bargaining partners instead [START_REF] Hernandez-Arenaz | Women ask for less (only from men): Evidence from bargaining in the field[END_REF]. As such, it is unclear whether the lower payoffs that women earn in bilateral settings are a harbinger of a gender gap in a multilateral, majoritarian setting like the one we study.

In this paper, we conduct a multilateral bargaining experiment where individual interests are clear and divergent as induced by the experimental incentives. Monetary payoffs stand in for all interests of parties (e.g., being assigned preferred tasks in a self-managed team) trading off external validity for better control over preferences. Participants in our main experimental treatment were randomly placed in triads composed of one female and two male subjects (hereafter labeled MMF).

We also conducted three other treatments varying the gender composition: all female (FFF), female majority (FFM), and all male (MMM). The treatment MMF will be our main treatment of interest as it emanates from our motivation and research question, while the other treatments will help us answer ancillary questions.

The subjects' task was to divide a sum of money through a free-form, majoritarian bargaining protocol taking place via computer terminals. Gender revealing silhouettes representing each subject were displayed during the bargaining. 7 In the game played by subjects, an agreement is temporarily reached when at least two members of the triad support a division of the money. For the agreement to be ratified and binding, it must be continuously supported for a pre-determined ratification time period. During the ratification period, a member who is not a party to the pre-agreement may engage in making offers in order to lure partners into a new alliance, thus impeding the agreement from becoming binding. Also, parties to the preliminary agreement are free to reconsider and propose alternatives. 8

(2020) who thoroughly review the field experiments focusing on wage bargaining and prices. 6 Laroze et al. (2020) study gender, along with race and political orientation, in an experimental three-player Baron-Ferejon game [START_REF] Baron | Bargaining in legislatures[END_REF], but do not exogenously manipulate gender, making it difficult to isolate its role.

7 Each displayed silhouette was based on the participant's revealed gender when they registered in our recruiting system. In all cases, the subject's self-reported gender in a questionnaire matched their registered gender. 8 Importantly, our experiment builds on previous studies of unstructured bargaining games such as [START_REF] Bolton | How communication links influence coalition bargaining: A laboratory investigation[END_REF]; [START_REF] Bolton | How do coalitions get built? Evidence from an extensive form coalition game with and without communication[END_REF]; [START_REF] Camerer | Dynamic unstructured bargaining with private information: theory, experiment, and outcome prediction via machine learning[END_REF]; [START_REF] Tremewan | Voting rules in multilateral bargaining: using an experiment to relax procedural assumptions[END_REF], and Siegenthaler and Kamm (forth-Our experimental setting is different from previous studies of gender differences in bargaining games in the following ways. First, by studying a multilateral setting, we naturally allow for agreements to include the smallest feasible alliance (i.e., a distribution of benefits including only a simple majority of supporting members) or the all-inclusive alliance (i.e., a distribution of benefits including every member of the group). Borrowing terminology from the theoretical and experimental literature on multilateral bargaining [START_REF] Eraslan | Legislative and multilateral bargaining[END_REF], we refer in the analysis section to these types of agreements as minimum winning coalitions (MWCs) and grand coalitions (GCs), respectively. Second, the largely unstructured nature of our bargaining protocol allows us to address novel questions such as whether there are gender differences in the propensity to make the first offer, spontaneously leave temporary coalitions, or choose to make aggressive counteroffers.

Our unstructured protocol is arguably more appropriate for the study of corporate boards, teams, and other organizational settings where the formation and dissolution of alliances is central to the back-and-forth negotiation process often described. Even in bodies with formal procedures in place (i.e., binding rules for proposal making and voting) such as legislatures of judicial panels, informal negotiations typically precede the formal procedures.

In our first set of experimental sessions-conducted at Maastricht University-we uncover a gender gap in earnings: Men earn 21% more than women in MMF. Most bargaining agreements are two-way splits of the money, with women being excluded 42% of the time (i.e., receiving a share of zero). When mixed-gender alliances form, there are minimal differences in payoffs between the male and female coalition partners, meaning that the gender gap is mainly driven by female's overwhelming exclusion from alliances and not by gender differences in earnings within them. In order to probe the robustness of the gender gap, we conducted additional experimental sessions with participants from the University of Valencia in Spain. 9 We are able to replicate the gender gap in MMF, with male subjects earning 15% percent more than females.

There are two channels, which are common to both samples in our study, that help explain the gender gap in earnings. The first behavioral regularity we identify is that men are more active negotiators. We find that they are more prone to making opening offers than women. This behavior makes men more likely to be part of an alliance, which gives rise to a payoff advantage. Intriguingly, men benefit from proposing first more than women. Hence, even if females attempted to mimic the active negotiation patterns characterizing male behavior, there is no reason to expect that this will aid in substantially closing the gender gap in earnings. A second behavioral regularity contributing to the gender gap in earnings has to do with the bargaining process. We find that mixed-gender coalitions are less stable than male-male coalitions. Thus, the dissolution of initial mixed-gender coming). For a meta-analysis on structured multilateral bargaining experiments see [START_REF] Baranski | The determinants of multilateral bargaining: A comprehensive analysis of Baron and Ferejohn majoritarian bargaining experiments[END_REF].

coalitions contributes to the over-exclusion (relative to pure chance) of females from final agreements, and thus, not receiving any money from the fund to divide.

In light of these two behavioral regularities, it is natural to ask whether majorities discriminate against minorities, and if such behavior could explain the gender gap in earnings. In our first experiment (conducted in Maastricht), only 58% of coalitions are mixed gender while a random partner choice would result in 67% being mixed gender. While this may appear first hand as the result of direct discrimination, a detailed look into the bargaining process reveals this is not the case. When males make the opening offer, they invite the only female in the triad more often (56% of the time).

When a man is excluded from a MWC, they are more likely to favor the female partner with any counteroffer. Another behavioral pattern we find is that left-out males are quite aggressive and insistent in their attempts to break the mixed-gender MWCs, and included males are less loyal to their initial female partner as we find that they are more likely to break the interim agreement. Put together, these findings are not in line with a taste-based explanation for the gender gap in earnings in which men directly or intentionally discriminate against women.

However, the bargaining dynamics observed in the second experiment (Valencia) differ in some important aspects. First, we do find evidence of taste-based discrimination in Spain, as men there display a preference for partnering with each other. Second, we do not find that women are less aggressive than men in attempting to break coalitions.

Is the gender gap driven solely by gender differences in bargaining behavior or does the gender composition matter? To this end, we explore behavior in the female majority treatment (FFM). In this treatment, we also find that males earn more than females on average: 2.1% in Maastricht (not statistically significant) and 21% in Valencia (statistically significant at conventional levels). As in MMF, males are more likely than females to make opening offers, and derive higher payoffs from doing so. Importantly, females do not display a bias in their first offers when choosing a coalition partner in both samples, and division of the surplus within a mixed gender MWC are close to equal splits.

Is the gender gap driven solely by gender differences in bargaining behavior or does the gender composition matter? To this end, we explore behavior in the female majority treatment (FFM). In this treatment, we also find that males earn more than females on average: 2.1% in Maastricht (not statistically significant) and 21% in Valencia (statistically significant at conventional levels). Thus, while replacing the male majority with a female majority all but eliminates the earnings gap in Maastricht, it has minimal effect in Valencia. As in MMF, males are more likely than females to make opening offers, and derive higher payoffs from doing so. Importantly, females do not display a bias in their first offers when choosing a coalition partner in both samples, and division of the surplus within a mixed gender MWC are close to equal splits.

To answer our final question on the role of the gender composition on bargaining outcomes, we find that increasing female representation leads to more inclusive sharing of the money. Grand coali-tions are most likely to be observed in FFF and least likely in MMM. Hence, our experiment provides a causal link between female representation and outcome fairness in majoritarian bargaining.

Collectively, the results from our experiments highlight that the existence of a gender gap in majoritarian negotiations may arise through several channels. While some of these channels are shared in the two samples of our study, others are not, which hints at cultural differences as a potential explanation (we discuss this aspect in Section 4.4). We conduct a counterfactual analysis which estimates the magnitude of each channel in creating a gender gap and shows that there is no "one-size-fits-all" policy solution (see section 4.1). Instead, our analysis underscores the relevance of studying in detail the endogenous bargaining dynamics that arise in a negotiation. Policymakers and those seeking to create better organizations that can foster equitable outcomes must first analyze in detail the context and forces that can sustain male-female differences.

Our study contributes to the vast experimental economics literature on gender differences, which has found that men are more inclined to enter competitions [START_REF] Niederle | Do women shy away from competition? do men compete too much?[END_REF], contribute ideas [START_REF] Coffman | Evidence on self-stereotyping and the contribution of ideas[END_REF], lead a team [START_REF] Born | Gender and willingness to lead: Does the gender composition of teams matter? Review of Economics and Statistics[END_REF], and give advice on how to play strategically [START_REF] Cooper | A failure to communicate: an experimental investigation of the effects of advice on strategic play[END_REF]. Importantly, by now, there are robust findings showing that women's decision-making is influenced by gender stereotypes, as well as by gender composition, when the domain is stereotypically perceived as disadvantageous for women [START_REF] Coffman | Evidence on self-stereotyping and the contribution of ideas[END_REF][START_REF] Bordalo | Beliefs about gender[END_REF][START_REF] Geraldes | Women dislike competing against men[END_REF][START_REF] Shan | Does minority status drive women out of male-dominated fields? Mimeo[END_REF][START_REF] Stoddard | Strength in numbers: A field experiment in gender, influence, and group dynamics[END_REF]. Thus, understanding the role of gender in multilateral bargaining outcomes and dynamics will expand our understanding of men's and women's behavioral patterns in an important novel direction.

This article proceeds as follows. In Section 2, we present the design of the experiment and present our main results in Section 3. We discuss our results, as well as similarities and differences between subject pools, in Section 4. Finally, Section 5 concludes the article.

Experimental Design

The Bargaining Game

Participants were randomly grouped in triads to divide 12 points, which corresponded to 36 euros at the beginning of the negotiations. A silhouette indicating the gender of each member of the committee was displayed. At any moment during the bargaining process, participants could: (1) make a proposal on how to divide the twelve points, (2) provisionally support (or not) an existing proposal, or (3) withdraw the proposal after having made one. Bargaining ended and a split of the points was implemented when at least two members of the triad agreed on a split for a ten-second ratification period. This design closely follows the unstructured bargaining protocol of [START_REF] Tremewan | Voting rules in multilateral bargaining: using an experiment to relax procedural assumptions[END_REF] and the layout of the bargaining interface is displayed in Figure 1.

Importantly, each second that elapsed during the game, the value of the fund decreased by 24 cents. The total fund continues to fall each second during the ratification period. Thus, if 150 seconds went by without agreement, all players earned 0. The experiment was computerized using the zTree software [START_REF] Fischbacher | z-tree: Zurich toolbox for ready-made economic experiments[END_REF]. Screenshots, experimental instructions, and details of the bargaining interface can be found in the Online Appendix. All interactions are anonymous and there is no communication allowed between participants. In each treatment, the bargaining game was repeated 12 times. In what follows, we denominate each repetition of the bargaining game as a period. When two subjects click on the same circle (i.e., make the same offer) a red dot appears in the circle and a timer indicating the "time until agreement" is displayed so that subjects are aware that a preliminary agreement is in place. On the top part of the screen, subjects can see how the exchange rate between experimental currency units (ECUs) and Euros decays as each second elapses. For further details, see the Online Appendix.

Treatments and Session Details

We conducted four treatments (between-subjects): Male majority (MMF), Female majority (FFM), All female (FFF), and All male (MMM). This means that the gender composition of the triads was fixed in each treatment even though the triads were randomly re-matched in each of the 12 periods. The number of repetitions was selected with two reasons in mind. First, we wanted to allow for ample learning opportunities and second, we wanted the experiment to remain within a reasonable time limit. Only one period was randomly selected for payment.

We recruited an equal number of men and women to participate in each session, so that upon arriving to the laboratory, participants would see a gender balanced group. 10 During each session, we conducted two treatments concurrently but subjects only participated in one of them. Thus, there are two matching groups in each session from which participants are rematched each period. Six sessions were for the treatments MMF and FFM, and the other six sessions were for the treatments FFF and MMM. In the sessions involving the two mixed-gender treatments, 1/3 of the men were matched with 2/3 of the women to run the FFM treatment and the remaining participants assigned to treatment MMF. This particular recruiting and treatment allocation procedure was done in order to avoid revealing the objective of the experiment ex-ante. 11

Prior to the start of each session, a demographic survey was conducted which asked for gender among several other questions. Importantly, participants saw the same silhouettes they would see at the bargaining stage when selecting their gender. Thus, it was common knowledge during the bargaining game that silhouettes represented their gender and the gender of their opponents. In our sample, it was always the case that the reported gender matched what the experimenters determined when assigning participants to computer terminals. Importantly, gender was never mentioned in the instructions or invitation email. We discuss the implications of our design concerning potential experimenter demand effects in Section 4.

A total of 282 subjects participated in our first experiment. They were undergraduate students from Maastricht University in the Netherlands. A show-up fee of 5 euros was offered and earnings averaged 16.50 euros. Sessions lasted about 60 minutes.

In order to test the robustness of the gender gap in earnings, as well as to gather more data for the exploratory analysis of mixed-gender coalition dynamics (section 3.3), we conducted eight additional sessions of the mixed-gender treatments (MMF and FFM). The second experiment was conducted at the Laboratory for Research in Behavioural Experimental Economics (LINEEX), located in the University of Valencia, Spain. A total of 192 subjects took part in these experiments and all the procedures were identical to those of the sessions conducted in Maastricht. 10 The order of treatments was randomized, so that participants are effectively randomly assigned. The participants' characteristics support that our procedure to randomly assign participants across treatments has been successful (see Section C of the Online Appendix for details).

11 One cannot rule out that subjects in FFF and MMM would find it unusual that the gender composition of triads remains fixed during the experiment because they had observed a gender-balanced composition in the welcome room. Nevertheless, considering the benefits and drawbacks of having only one gender per experimental session (and thus likely revealing the experiment's aim to subjects in the FFF and MMM treatments) or implementing mixed-gender sessions to conduct FFF and MMM simultaneously, we decided on the latter design choice. Importantly, answers in the debriefing questionnaire do not reveal any concern related to this issue, nor did any subject ask about this aspect at the end of the FFF or MMM sessions.

In Table 1, we summarize the structure of the data. In Maastricht, we observed 288 bargaining games in the FFF and MMM treatments, and 276 bargaining games in the FFM and MMF treatments. In Valencia, we observed 384 bargaining games in the FFM and MMF treatments. 2 In one of our mixed-gender sessions (Maastricht), not enough subjects showed up. Hence, the two matching groups in that session had 9 subjects each.

Results

In this section, we report the results of our experiments, with the focus being on MMF. We will, however, also present the results of FFM simultaneously in tables and figures for the sake of conciseness.

We begin by testing whether there are disparities in earnings by gender. Subsequently, we aim to explain the differences, first by looking at the types of final divisions that arise, then with a more detailed look at gender differences in bargaining strategies and their impact on final agreements. At each step, we compare and contrast the results from our two different subject pools. We note here that all groups eventually reached an agreement, and therefore we observe a monetary division for every game played.

Are There Gender Differences in Earnings?

Average earnings by gender are displayed in We summarize in our first result:

Result 1. In male majority groups, male subjects earn significantly more than female subjects in both samples. In female majority groups, males also earn more than females, but the difference is significant only in Valencia.

Gender Differences in Bargaining Outcomes

In this subsection, we focus on the overall split of the money. How many members receive a share of the money and how many receive nothing? Are MWCs the most common types of agreements as in related multilateral bargaining games? We then investigate the degree to which the observed gender gap in earnings in MMF can be explained by the prevalence of MWCs, the rate at which females are excluded from them, and/or the distribution of earnings within coalitions.

Final Agreements: Types of Coalitions

We start by analyzing the type of allocations that triads agreed upon. In Table 3, we report the proportion of MWCs for each treatment 13 , which are all in line with the literature [START_REF] Palfrey | Experiments in political economy[END_REF][START_REF] Baranski | The determinants of multilateral bargaining: A comprehensive analysis of Baron and Ferejohn majoritarian bargaining experiments[END_REF]. Clearly, MWCs are modal, representing approximately 80% of all agreements.

In Figure 2, we see that the proportion of MWCs is increasing with experience in all treatments. Table 3 shows a clear pattern in Maastricht: As the number of males increases in the triad, so does the proportion of MWCs. To probe the statistical significance of these observations, we conduct a probit regression of the probability of a MWC on the number of females in the group (clustering at the matching group level), using only the data from Maastricht where we have all four treatments. The marginal effects are reported in Table A1. Our regression shows that each additional female in a triad represents a 5 percentage point decrease in the probability of observing a MWC (p < 0.01). Regressing instead on treatment dummies, of the six bilateral comparisons we find significant differences only between FFF and MMM (p < 0.01) and between FFM and MMM (p = 0.039).

These results are robust to controlling for a period trend. An unreported probit regression finds that the difference in proportions of MWCs between the two treatments in Valencia is not statistically significant (p = 0.312).

Gender Composition of MWCs

Table 4 reports the proportion of mixed-gender coalitions in MMF and FFM. In Maastricht, we find that 67% of the time MWCs are mixed-gender in FFM-which is consistent with equiprobable partner choice-but only 58% in MMF (see top of Table 4). This implies that outcomes are worse for women than for men when they are in the minority, but are not better when in the majority. There is only marginal statistical evidence that mixed-gender coalitions in MMF occur significantly less than In Valencia, men are more likely to be part of a MWC in both mixed-gender treatments. Mixedgender coalitions occur 59% of the time in MMF, which is less than predicted by a random coalition formation process (p = 0.036). In FFM, these represent 77% which is higher than predicted by chance (p = 0.018).

Result 2. In the male majority groups, females are more likely to be excluded from MWCs than predicted by chance. In female majority groups, the gap persists in Valencia but males and females are included with equal chance in Maastricht.

Gender Differences in the Division of Surplus within Coalitions

We now investigate whether there are gender differences in the split of the surplus within mixedgender coalitions (see Table 4). Women take 49% of the pie in FFM and 50.4% in MMF in Maastricht, and 49.5% and 50.9% respectively in Valencia. These figures differ statistically from 50% only in MMF in Valencia (p = 0.067), but the magnitude of this difference is small, and in the opposing direction to explain lower female earnings overall. Regarding GCs, in no case does the average share of a female differ statistically from the equitable one-third.

Result 3. In male majority groups there is no gender difference in the share received between partners in a mixed gender coalition. The same is true in female majority groups.

Dynamics in Mixed-gender Treatments

In contrast to an initial expectation that women may earn less as more agreeable coalition partners, the preceding analysis demonstrated that the main reason females earn less is due to being excluded from MWCs, rather than receiving a smaller share of agreements they are part of. Why are females generally more likely to be excluded from a MWC?

In this section, we conduct an exploratory analysis to better understand the gender differences in the formation of coalitions. We focus on four initial events of the bargaining games, all of which could potentially explain the lower earnings of women.

The first three events we look at are who makes the first offer, who the first offer is made to, and the probability an initial agreement is implemented. These events could explain differences in earnings in the following ways. First, if males are more likely to make the first proposal, and this proposal is more likely to result in an initial agreement, then as initial agreements are implemented in the majority of the games, males will clearly be present in more implemented MWCs. Second, a preference for males to make offers to other males will also reduce the probability a female is in the final coalition. Third, if mixed-gender coalitions in MMF are less likely to be implemented, all-male coalitions will be more prominent.

The probability an initial agreement is implemented can be influenced by two important factors:

the type of counteroffers that are made by the excluded player during the first provisional agreement, and the probability that such a counteroffer is accepted (and thus dissolves the initial agreement). To assess the first factor, we examine the first counter-offer made after the initial temporary agreement is formed. In our unstructured game, it is not clear whether a particular offer has been "accepted" because any number of counteroffers by any of the three players can intervene before a new temporary agreement is formed. Therefore, we leave the analysis of acceptance rates for Section 4, where we propose a simple model of behavior that allows to back out the implied willingness to accept counter-offers.

We focus our analysis on these events of the bargaining process because they are straightforward to observe and quantify, and the number of comparable observations available as the game goes on decreases rapidly (most games end after the first agreement and, as time elapses, the ever-decreasing number of games is divided among an exponentially increasing number of possible histories). Also, in around 80% of the games of MMF and FFM, it is the first or second provisional agreement that is implemented. We also limit our attention to MWCs, which are more straightforward to analyze because they are the vast majority of agreements and in a GC proposal or counteroffer, it is not clear who a proposal has been made to.

Who Proposes First?

We now test if males are more likely to make opening offers than females. Note that in mixed-gender treatments, the majority gender should make the opening offer 2/3 of the time if each gender had equal likelihood of proposing first.

As shown in Table 5, we see that men are indeed more likely than women to make the opening offer. In Maastricht, when men are the majority, they move first 79% of the time, substantially greater than 2/3 (p = 0.023). When women are the majority, one of them moves first 49% of the time, substantially less than 2/3 (p = 0.080). The results in Valencia are similar: in MMF, men propose first 80% of the time and in FFM, women propose first 45% of the time, in both cases significantly different from 2/3 (p < 0.01 and p = 0.018, respectively). 15A further analysis reveals that male first movers earn significantly above the mean share of the fund (1/3) but females do not (pooling both samples). Figure 3 shows the mean share that a subject who makes the opening offer earns in the final agreement. Pooling over both samples in MMF, males that make the opening offer end up with over 36% of the money while females only with 29.7%, a statistically significant difference (p = 0.037). A similar pattern arises in FFM (38% vs 33%).

Importantly, these result also hold when controlling for the share of money that the first proposer is demanding for him or herself (see Table A2 for regression results). Notes: The y-axis measures the proportion of the fund that subject making the opening offer receives in the agreed allocation. 95% confidence intervals, clustering s.e. at the matching group level.

Result 4. Males are more likely than females to make the first offer. Conditional on making the opening offer, males earn on average more than females, and above the mean share (1/3 of the fund), but female first movers earn at or below the mean share. These results hold in male majority and female majority groups.

Is There a Gender Bias in Initial Proposals?

Here we test for a gender bias in the target of initial proposals. As can be seen in the second line of Table 5, when males are the majority in Maastricht, a male first-mover proposes a MWC to the other male in 44% of cases, so any bias is in favor of women (p = 0.052). On the other hand, in Valencia, male majority first movers make offers to the other male 57% of the time, which is significantly higher than predicted by a coin toss (p = 0.004).

In neither sample do women in FFM appear to target their offers at a particular gender. The proportion of initial proposals made by females to females does not differ statistically from 50% (49% in Maastricht and 47% in Valencia).

Result 5. In male majority groups, the target partner of a male's first proposal shows different patterns between samples. In Maastricht males are biased towards making first proposals to females but display a preference for partnering with other males in Valencia. In female majority groups, we find no statistical evidence that females show a gender bias in choosing their coalition partner.

Stability of Initial Agreement

One pattern found in both subject pools is that mixed-gender coalitions are more stable in FFM than in MMF, implemented 9.p.p. more often in Maastricht (FFM = 69%, MMF=60%; p = 0.226) and 14 p.p. more often in Valencia (FFM = 68%, MMF = 54%; p = 0.048). 16 This suggests that excluded males may be better at breaking coalitions than excluded females, for example by being more likely to make a competing offer. Further supporting this notion is the fact that all-male coalitions are consistently more stable than mixed-gender coalitions in MMF, with implementation rates differing by 12 p.p. (p = 0.141) in Maastricht and 18 p.p. in Valencia (p < 0.01). The comparison between stability of mixed-gender and all-female coalitions in FFM in Valencia is also consistent with this reasoning, with the former 20 p.p. more likely to be implemented than the latter (p = 0.086). However, in the same treatment in Maastricht, all-female coalitions 6 p.p. more stable than mixed-gender coalitions (p = 0.357). This does not rule out excluded males being more likely to make attractive offers here, in the case that females in this treatment are particularly averse to accepting outside offers. Result 6. In male majority groups, male-male coalitions are more stable than mixed-gender coalitions. In female-majority groups, mixed-gender coalitions are more stable than female-female coalitions. Taken together, this reveals a general tendency for initial coalitions to be more stable when the excluded player is a woman.

Propensity to Make Attractive Counteroffers

We consider an "attractive counteroffer" to be an offer of a MWC by the excluded player, as we found that GCs are much less likely to be implemented (see Section 3.2.1). 17 In both MMF and FFM in Maastricht, females are less likely to make such an offer, with differences of 13 p.p. and 9 p.p. respectively, although the differences are not statistically significant. This difference is driven by the fact that excluded females are more likely not to counter-offer at all (11% vs. 5%) and more likely to propose GCs (21% vs. 14%). By contrast, male and female behavior in Valencia is almost identical in these respects.

With regard to bias in the target of counteroffers, of the first counter proposals following an initial mixed-gender provisional MWC in MMF in Maastricht, 45% of those made by excluded men are MWC proposals to women, compared to 31% to the other men. In Valencia, the figures are 37% and 25% respectively. Thus, we can rule out gender discrimination in counteroffers as an explanation for women being left out of final agreements.

Result 7. Males in both male-majority and female-majority groups are more likely than females to make attractive counteroffers in order to break initial coalitions in Maastricht. Males and females behave almost identically in this respect in Valencia.

Discussion

In this section, we first seek to shed light on the role of the different mechanisms in explaining the gender gap in the male majority treatment. We do so by providing back-of-the-envelope calculations based on a simplified model of behavior. We seek to further our understanding of the instability of mixed-gender coalitions because part of this behavior cannot be directly observed. In particular, we provide an estimate of the willingness to accept external counter-offers. Next, we discuss three issues that may affect the interpretation of our results. First, we investigate whether participants noticed the gender composition indicated by silhouettes. Second, we analyze the effect of personal characteristics (i.e., risk and altruism, among others) on the share of the surplus that participants earn. Finally, we discuss how cultural aspects may explain differences in behavior between our samples.

The Role of Different Mechanisms in Explaining the Gender Gap in the Male

Majority Treatment 18

In order to identify the precise role of each channel in the emergence and magnitude of the gender gap, we developed a simplified model of behavior. The model should have the following desirable features: it needs to be sufficiently complex to evaluate the main channels discussed in Section 3.3, but should be as simple as possible to allow straightforward analysis and avoid over-fitting.

In addition, it should roughly describe a majority of games. With this in mind, we chose a model which allows for an initial offer and a counteroffer (the first or second temporary agreement was implemented in 75% of games). The model includes as variables the propensity of males to make the first proposal (denoted by p m in Table 6), the explicit bias by males (b), gender differences in the tendencies to challenge existing coalitions (c m for males' and c f for females' willingness to make counteroffers when excluded), and leave existing coalitions (a m and a f ). 19The first four parameters (p m , b, c m , c f ) are be taken directly from the data, as shown in Table 5. The parameter p m is the proportion of first offers made by males; b is the proportion of first offers made by males in which that proposal is a MWC including the other male; c m and c f are the proportion of games in which the excluded male and female, respectively, proposes a MWC to one of the initial coalition members.

Obtaining the acceptance rates of counteroffers (a m and a f ) from the data is more challenging because in the unstructured game, it is not clear whether an offer is explicitly rejected. This is because one or more counter-offers can intervene, from and to any of the three players, before the next temporary agreement forms. To address this issue, we find parameters which are consistent with both the observed proportion of mixed-gender coalitions and the proportion of initial agreements which were implemented, under the assumption that our model is correct. 20Our estimated acceptance rates support the observation that men are more prone to abandon coalitions. Specifically, the solution to our model yields a M m = 0.57 and a M f = 0.02 for Maastricht, and a V m = 0.68 and a V f = 0.25 for Valencia. These estimates are consistent with our previous analysis and observation that females display a stronger commitment to their initial coalition partner.

The estimate of a f in Maastricht is unrealistically low. As the only free parameters in the model, a f and a m capture all gender differences in dynamics that lead to fewer implemented mixed gender coalitions which are not specifically accounted for in our model. This includes gender differences in propensity to spontaneously leave a coalition, and the impact of differences in counter-offer and acceptance rates from any second or further temporary agreement that might have eventuated. There-fore, the proportion of the earnings gap we attribute in the following analysis to acceptance rates is probably exaggerated (and counter-offer rates understated). Although our estimates of the magnitudes of these individual channels may be imperfect, our estimate of the total effect should be a reliable indication of the overall importance of gender differences in bargaining strategies that affect the stability of coalitions.

In Table 6, we eliminate each bias one at a time, as well as counter-offer and acceptance rates jointly, and consider how interchanging male and female patterns of behavior impact the gender difference in earnings. Note that for consistency with the model, the baseline earnings difference is calculated based on the proportion of mixed-gender coalitions in the data, assuming only 50-50 splits. No male bias in partner selection (b = 0.5) 0.79 0.87

B. Ratio if females behave like males:

When making counter-offers (c f = c m ) 0.86 0.83

When accepting outside offers (a f = a m ) 0.94 0.92

Both (c f = c m , a f = a m ) 0.99 0.90

C. Ratio if males behave like females:

When making counter-offers (c m = c f ) 0.88 0.83

When accepting outside offers (a m = a f ) 0.98 0.83

Both (c m = c f , a m = a f ) 0.98 0.83
See the main text for the meaning of each variable. The notation a f = a m means the female acceptance rate is set to the male rate observed in the data, i.e., what happens if females act like males, whereas a m = a f means the opposite.

Interestingly, our results show that increasing the number of female first proposals or eliminating the bias males display in Valencia is relatively ineffective. In both samples, our calculations suggest that when female behavior is set at male mean levels in terms of making and accepting counterproposals the gender gap shrinks substantially, and virtually closes in Maastricht. This exercise further underscores the importance of analyzing the bargaining dynamics beyond the opening offer to understand an existing gender gap in earnings. For example, the effect of males making first offers, and any bias in those offers is diminished if counter-offer and acceptance rates are sufficiently high, as seems to be the case in our data.

We are cautious, though, about the policy implications one may derive from these findings because, ultimately, one should conduct direct experimental tests in which each channel is experimentally manipulated before suggesting any policy intervention. One concern is that the aggressiveness of female counteroffers may respond endogenously to the degree of male bias in selecting coalition partners, something which is consistent with our data (more bias and less gender difference in counteroffer behavior in Valencia than in Maastricht).

Did Participants Notice the Gender Composition?

In a debriefing question, we find that 72% and 68% of the participants reported that they did not notice the silhouettes in the Maastricht and Valencia experimental sessions, respectively. However, we are cautious about interpreting these answers. Because we also find evidence for explicit discrimination in explaining the gender gap in earnings, this indicated that using silhouettes to prime gender was not necessarily unsuccessful. That is, despite participants' (negative) answer to the manipulation check, we cannot rule out that: (1) they did not notice consciously, but the effect is subconsciously operating, and (2) they noticed the silhouettes but lied about it to avoid being labeled sexist or discriminatory.

In Table 7, we check for discrimination in first offers, controlling for whether or not the offerer correctly stated their bargaining partners' genders in the post-experimental questionnaire. In the MMF treatment of the Valencia experiment, where we observed the strongest evidence of samegender bias, only men who report that they did not notice they are interacting with one male and one female, significantly favor males. This observation can be interpreted in two ways: that those who noticed the gender composition were able to counteract their biases or, alternatively, that those who discriminated based on gender wished to conceal their motivations. One could potentially investigate if answers are closer to the actual gender composition by incentivizing subjects for accurately reporting their recollection.

Do Personal Characteristics Explain Differences in Payoffs?

Another question that naturally arises is whether the gender disparity in earnings can be explained by gender differences in personal characteristics that likely influence behavior in multilateral bargaining [START_REF] Gillen | Experiments with measurement error: Techniques with applications to the caltech cohort study[END_REF][START_REF] Van Veldhuizen | Gender differences in tournament choices: Risk preferences, overconfidence, or competitiveness[END_REF]. To examine this aspect, we conducted a linear regression to investigate the effect of gender on earnings, controlling for cognitive ability (using the three-question cognitive reflection test), self-reported risk preferences, altruism, enjoyment derived from competing, and enjoyment derived from winning. For succinctness, we pool data from both heterogeneous groups in both locations. In Table 8, we see that while many of these variables are significant and their inclusion explains part of the gender effect, the residual effect remains sizeable and statistically significant. This suggests that the gender disparities we observe cannot be fully explained by simple measures of commonly considered personal characteristics.21

The Role of Culture

The role of cultural differences in bargaining behavior has been widely recognized in economics (see [START_REF] Roth | Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo: An experimental study[END_REF] and [START_REF] Henrich | Does culture matter in economic behavior? ultimatum game bargaining among the machiguenga of the peruvian amazon[END_REF] for evidence from ultimatum games). However, establishing a correlation between external measurements of general cultural traits22 and bargaining behavior has been an unfruitful endeavor to the best of our knowledge. In a meta-analysis of ultimatum games containing data from all over the world, [START_REF] Oosterbeek | Cultural differences in ultimatum game experiments: Evidence from a meta-analysis[END_REF] Ultimately, our results are a proof of existence, that different mechanisms can underlie the same gender gap in majoritarian bargaining. We find that there are both commonalities and interesting behavioral differences between subject pools from two different countries. Because we are conducting identical procedures with identical incentives (as in [START_REF] Roth | Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo: An experimental study[END_REF]; [START_REF] Henrich | Does culture matter in economic behavior? ultimatum game bargaining among the machiguenga of the peruvian amazon[END_REF]), only two possible explanations are left: that our samples differ in their composition (i.e., majors that students are enrolled in, for example) or cultural differences. Further research may shed light on how cultural traits affect majoritarian bargaining outcomes by determining sample-selection criteria a priori and setting forth testable hypotheses.

Concluding Remarks

Collectively, the results from our experiments demonstrate that gender, and the gender composition of a bargaining committee, affect the dynamics of negotiations and agreements reached. Importantly, men and women employ different bargaining strategies. Our study documents the existence of a gender gap in earnings in majoritarian bargaining: Men earn more than women, a finding which holds in two different subject pools. We identify three underlying mechanisms explaining this gap, which differ in importance in the two groups of subjects. First, men are more likely to make the opening offer, which enhances their odds of being part of a winning coalition. Interestingly, the first-mover advantage is enjoyed mainly by men as women are rarely better off for proposing first.

Second, gender differences in the bargaining process also play a role in explaining differentials in earnings. When a preliminary alliance between a male and a female subject forms, these are quite unstable in male-majority treatments. Their dissolution leads to the subsequent formation of a male-male alliance, and as such, mixed-gender alliances are less likely to be ratified.

A third mechanism explaining the gender gap is discrimination, for which we find strong evidence in the Spanish subject sample. In particular, we find that men tend to partner with each other in male-majority triads more than expected by chance, whereas women do not display such behavior when they are in majority. Male-male MWCs generally end in agreement (close to 70% of the time) in male-majority triads.

Insofar as our results extend to other populations, they suggest that the preferences of females in male-dominated top management and self-managing teams will be under-weighted in outcomes.

External validity is less of a concern in the latter, as many members of our subject pool are likely to work in such teams in the future. Employers concerned about ensuring all views are given equal weight, or fairness of outcomes in, for example, distribution of tasks within a team, should be mindful of the gender-makeup of teams, as we find evidence in one of our samples that increasing female representation closes the gender gap. Managers and human resources specialist can consider putting in place formal protocols for decision-making that minimize the impact of the channels we have identified. Further experiments can shed light as to the effectiveness of such institutional modifications.

It remains to be studied if bargaining protocols can be engineered to counteract the effect of behavior which may systematically affect female bargainers.

Relatedly, several previous studies find that the gender composition of a group also affects outcomes in other domains. Specifically, Bourreau-Dubois et al. ( 2020) find that all-women French judge panels impose higher child support payments than all-male or mixed-gender panels. [START_REF] Boyd | Untangling the causal effects of sex on judging[END_REF] find that males in judge panels lacking women are less likely to rule in favor of plain-tiffs in cases of discrimination in the United States. [START_REF] Apesteguia | The impact of gender composition on team performance and decision making: Evidence from the field[END_REF] report results from a management training game played in groups and finds that groups of three women are more likely to invest in corporate social responsibility programs compared to other group compositions. Besides the problem of non-random allocation into groups for establishing causality, the three previous settings have one aspect in common, which differentiates them from our study: the resulting decisions have consequences over third parties. We argue, however, that generosity toward others will display the same increasing pattern in the number of women in the committee. Recent work by Cason et al.

(2021) reveals that increasing female representation in groups leads to higher pro-social outcomes in a coordination game. Further experimental work is needed to understand how exogenous variation in the gender composition of a committee affects bargaining outcomes when the decisions being made have consequences for uninvolved parties.

In closing, three decades of experimental investigations on gender differences in bargaining have focused on bilateral settings and little is known about gender differences in multilateral bargaining.

We report the existence of a gender gap and identify plausible behavioral mechanisms that can give rise to it. We view this exploratory examination of the detailed bargaining dynamics as what Alvin

Roth refers to as a "search for facts" [START_REF] Roth | Introduction to experimental economics[END_REF]. Having identified these plausible mechanisms, we leave for future work experiments specifically designed for testing related hypotheses. Also, investigating richer settings, such as those where power and status differentials between men and women exist, will help bring the laboratory environment closer to real-world settings where such gender differentials are observed [START_REF] Babcock | Women don't ask: Negotiation and the gender divide[END_REF][START_REF] Sandberg | Lean in: Women, work, and the will to lead[END_REF].

A Supporting Tables and Figures

B Stylized model for the counterfactual analysis

In this section, we describe in more detail the stylized model of bargaining behavior presented in the paper, and that we use to perform the counterfactual analysis reported in the paper. The purpose of this model is to incorporate the main forces underlying the gender gap that we have uncovered while retaining tractability and conciseness.

Consider the following timeline:

1. One player is randomly chosen to make the first proposal. The player is a male with probability p m .

2. A proposal is made. If a male is the first proposer, the proposal is made to the other male with probability b (and the offer in the proposal is immediately accepted by the recipient, which means that a temporary coalition forms with probability 1).

3. A counteroffer is made by the excluded player with probability c m (c f ) if the excluded player is male (female). Each player in the initial coalition is equally likely to be the target of the counteroffer.

4. The counteroffer is accepted (i.e., the old coalition dissolves and a new one forms) with probability a m (a f ) if the target of the counteroffer is male (female).

5. If the counteroffer is accepted, it is implemented (i.e., bargaining ends), otherwise the counteroffer passes to the remaining coalition member who accepts it with the same probabilities (a m if male or a f if female).

6. If neither coalition member accepts a counteroffer, the initial proposal is implemented.

We now proceed to derive the probability that a male-male final coalition is implemented, given the timing of the game as a function of the parameters just described. In the following table, we present each possible contingency under which a male-male MWC forms and its associated likelihood of occurrence.

p m • b • (1 -c f )
(1.b) makes an offer to the other male, female makes counteroffer, and neither male accepts

p m • b • c f • (1 -a m ) 2
(1.c) makes an offer to the female, excluded male makes counteroffer to a male, which is accepted

p m • (1 -b) • c m • 1 2 • a m
(1.d) makes an offer to the female, excluded male makes counteroffer to female who rejects, then the male accepts

p m • (1 -b) • c m • 1 2 • (1 -a f ) • a m
2. Probability that a female proposes first and...

(2.a) excluded male makes counteroffer to a male, which is accepted

(1 -p m ) • c m • 1 2 • a m
(2.b) excluded male makes counteroffer to a female who rejects, then the male accepts

(1 -p m ) • c m • 1 2 • (1 -a f ) • a m
Thus, the total probability of observing a male-male MWC is given by Note that equation (1) has two unknowns: a m and a f . All the other parameters are matched from the data as explained in the body of the paper. Here we discuss why it is not possible to match the probability of accepting a counteroffer (a m and a f ). In the actual experiment, when a counter-offer is made, another member may suddenly make a different counter-offer which then causes the initial counter-offerer to retract. It may also happen that a member of the temporary coalition proposes a new division virtually at the same time at which he/she was being the target recipient of a counteroffer. These variables, thus, are hard to identify from the data. Instead of attempting to consider all possible scenarios for these events (which would result in few observations per event), we will back out their values by considering a second independent equation: the probability that an initial coalition ends in agreement.

To derive the probability that an initial coalition is implemented we proceed as before by explaining each possible contingency and its associated likelihood of occurrence. (2)

To obtain the implied values of a m and a f , we solve these two equations numerically. In the supplementary materials, we include the code that we used to do so for replication purposes.

Figure 1 :

 1 Figure 1: Bargaining Interface

Figure 2 :

 2 Figure 2: Minimum Winning Coalitions over Periods of Play, by Sample

Figure 3 :

 3 Figure 3: Share of the Fund for First Proposers in Approved Allocations

  MWC MM = p m b(1 -c f ) + p m bc f (1 -a m ) 2 + p m (1 -b)c m (a m -a m a f 2 ) + (1 -p m )c m (a m -

  (1.a) makes an offer to the other male, and female doesn't counterofferp m • b • (1 -c f )(1.b) makes an offer to the other male, female makes counteroffer, and neither male acceptsp m • b • c f • (1 -a m ) 2(1.c) makes an offer to the female, and there is no counterofferp m • (1 -b) • (1 -c m )(1.d) makes an offer to the female, and counteroffer is rejected by bothp m •(1-b)• c m •(1-a m )•(1-a f )2. Probability that a female proposes first and...(2.a) there is no counteroffer (1 -p m ) • (1 -c m )(2.b) and counteroffer is rejected by both recipients(1 -p m ) • c m • (1 -a m ) • (1 -a f )Thus, the total probability of observing that first agreement ends in approval is given byP First = p m b(1 -c f ) + p m bc f (1 -a m ) 2 +p m (1 -b)(1 -c m )+ p m (1 -b)c m (1 -a m )(1 -a f )+(1 -p m )(1 -c m ) + (1 -p m )c m (1 -a m )(1 -a f ).

Table 1 :

 1 Description of Data Set

	Treatment Subjects Periods Agreements 1 Matching groups 2
	Maastricht
	FFF	72	12	288	6
	FFM	69	12	276	6
	MMF	69	12	276	6
	MMM	72	12	288	6
	Valencia
	FFM	96	12	384	8
	MMF	96	12	384	8

1 An agreement refers to a bargaining outcome (there are no disagreements in our sample). It is the number of periods times the number of subjects divided by 3.

Table 2 .

 2 Focusing first on our main treatment of interest, we find a significant gender gap in MMF in both samples. Males earn 22% more than females (11.34 EUR vs. 9.32 EUR, p < 0.094) in Maastricht and 15% (11.02 EUR vs 9.61 EUR, p = 0.078) in Valencia.Does the gender gap close when females are in the majority? The answer depends on the sample.The gender gap in FFM varies from 2% in Maastricht (10.90 EUR vs 10.66 EUR, p = 0.844) to 21% in Valencia (11.95 EUR vs 9.88 EUR, p = 0.016). Pooling data across location and mixed-gender treatments, men earn approximately 12% more than women (11.21 EUR vs. 10.02 EUR, p < 0.01, Wilcoxon signed rank test using gender differences of matching group averages).12

Table 2 :

 2 Average Earnings (in Euros)

		FFF	FFM	MMF	MMM
	Combined data
	Females		10.21	9.49
	Males		11.51	11.15
	p-value		0.042 * * < 0.01 * * *
	Maastricht
	Females	10.65	10.66	9.32
	Males		10.90	11.35	10.78
	p-value		0.844	0.094 *
	Valencia
	Females		9.88	9.61
	Males		11.95	11.02
	p-value		0.016 * *	0.078

*

The p-values are based on Wilcoxon signed rank tests comparing within-matching group gender differences in averages. n = 14 for combined data; n = 6 in Maastricht; n = 8 in Valencia. * p < 0.1, ** p < 0.05, *** p < 0.01.

Table 3

 3

	: Proportion of Minimum Winning Coalitions by
	Treatment
		FFF FFM MMF MMM
	Maastricht
	All MWCs	0.74 0.74	0.84	0.87
	Equal splits (50% each) 0.53 0.55	0.59	0.64
	Valencia
	All MWCs	0.79	0.74
	Equal splits	0.57	0.48

Table 4 :

 4 Bargaining Outcomes in Mixed-Gender Treatments p < 0.1, ** p < 0.05, *** p < 0.01. Null hypotheses for t-tests: mixed-gender MWCs = 2/3; Female share in MWCs = 0.5; Female share in GCs = 1/3. Standard errors clustered at the matching group level.

	FFM	MMF

*

Table 5 :

 5 Main Results of Analysis of Dynamics Test based on a linear regression with standard errors clustered at matching group level. Comparison with 2/3. 2 Test based on a linear regression with standard errors clustered at matching group level. Comparison with 1/2. 3 In parentheses we display the number of counteroffers divided by the number of relevant cases.

	FFM	MMF
	Maastricht Valencia Maastricht Valencia

* p < 0.1, ** p < 0.05, *** p < 0.01. 1

Table 6 :

 6 Ratio of Female to Male Earnings under Alternative Behavior

	Maastricht Valencia

Table 7 :

 7 Linear Regression for First Offer to Male

		Maastricht	Valencia
		FFM	MMF	FFM	MMF
		(females only) (males only) (females only) (males only)
	Noticed Correctly	-0.120	-0.106	-0.108	-0.097
		(0.078)	(0.081)	(0.062)	(0.083)
	Constant	0.548***	0.465***	0.563***	0.573***
		(0.055)	(0.039)	(0.032)	(0.023)
	H0: Constant=0.5
	F-stat	0.76	0.81	3.79*	10.56**
	H0: Constant+Noticed Correctly=0.5
	F-stat	0.56	4.80*	0.82	0.13
	N	118	194	155	248
	R 2	0.01	0.01	0.01	0.01
	Standard errors clustered at the matching group level. * p<0.1, ** p<0.05, *** p<0.01

Table 8 :

 8 Linear Regression for Share of the Surplus (in points) Earned Risk and Altruism are self-reported attitudes on a scale from 0 (low) to 10 (high). CRT refers to the number of correct answers (0-3) to the standard Cognitive Reflection Task questions. Enjoys Competing and Enjoys Winning variables are self-reported willingness to compete and win respectively on a scale 1 (strongly disagree) -7 (strongly agree).

		(1)	(2)	(3)	(4)	(5)	(6)	(7)
	Male	0.496*** 0.471***	0.475***	0.386*** 0.466*** 0.470*** 0.314**
		(0.121)	(0.116)	(0.122)	(0.130)	(0.121)	(0.116)	(0.131)
	Risk		0.0263					0.0162
			(0.0188)					(0.0182)
	Altruism			-0.0607***				-0.0430*
				(0.0193)				(0.0223)
	CRT				0.146***			0.140***
					(0.0457)			(0.0456)
	Enjoys Competing					0.0529*		0.0474*
						(0.0263)		(0.0268)
	Enjoys Winning						0.0884**	0.0643
							(0.0393) (0.0398)
	Constant	3.752*** 3.606***	4.191***	3.598*** 3.520*** 3.298*** 3.286***
		(0.0638)	(0.134)	(0.164)	(0.0624)	(0.133)	(0.241)	(0.425)
	N	3,960	3,960	3,960	3,960	3,960	3,960	3,960
	R 2	0.008	0.009	0.011	0.012	0.009	0.010	0.016
	Robust standard errors in parentheses, clustered at the Subject level. Pooling both locations. *** p<0.01, ** p<0.05, *
	p<0.1 .

  find that none of Hofstede et al.'s 2005 measures correlate with average transfers. Moreover, conducting a meaningful analysis of the effect of cultural traits on bargaining dynamics is evidently not possible with only two countries in our sample.

Table A1 :

 A1 Proportion of MWCs by Treatment (Maastricht) Robust standard errors in parentheses, clustered at the matching group level. Treatment FFM is the base level. * p < 0.1, ** p < 0.05, *** p < 0.01

			(1)	(2)	(3)	(4)
	VARIABLES	MWC	MWC	MWC	MWC
	FFM		-0.000385		0.00110
			(0.0756)		(0.0767)
	MMF	0.0937 Table A2: Linear Regression for Share of the Pie Received 0.0919
		(0.0598) by Subjects making the Opening Offer (Pooled for both (0.0601)
	MMM	Samples)	0.122***	0.125***
			(0.0452)		(0.0459)
	Number of Females		-0.0503*** (1)		(2)	-0.0518**
		Male First Mover (=1 if yes)	(0.0195) 0.050*** 0.049***	(0.0203)
	Period of Play		0.0294*** 0.0294*** (0.016) (0.016)
		MMF		-0.030	(0.00534) -0.031	(0.00541)
	p-values for Wald Tests		(0.027)		(0.027)
	FFM v MMF FFM v MMM MMF × Male	0.215 0.039**	0.014 (0.034)	0.236 0.016 0.038** (0.034)
	MMF v MMM	0.589		0.520
	Observations Share demanded (as proportion) 1,128 1,128	1,128 0.063	1,128
	Probit regressions reporting marginal effects. FFF is the base level.	(0.063)
	Standard errors clustered at the matching group level reported in parentheses. Constant. 0.329*** 0.298***
	* p < 0.1, ** p < 0.05, *** p < 0.01	(0.014)		(0.031)
		N		1320		1320
		F-stat		4.602		5.692
		R 2		0.014		0.014

Table B5 :

 B5 Contingencies under which Male-Male Coalitions Form in MMF in our Simplified Bargaining Game

	Contingency	Probability
	1. Probability that a male proposes first and...
	(1.a) makes an offer to the other male, and female
	doesn't counteroffer

Table B6 :

 B6 Contingencies under which the First Temporary Agreement is Implemented in our Simplified Bargaining Game

	Contingency	Probability
	1. Probability that a male proposes first and...

Note that the aim of this paper is not to study cultural differences. The only reason we conducted additional experimental sessions in Spain was that the BEELab in Maastricht was closed due to COVID-19. Nevertheless, we acknowledge and discuss the possible role of culture in our results in Section 4. This unplanned switch turned out to be serendipitous as it allowed us to not only show the robustness of some findings to different subject pools but also identify new mechanisms not present in the first subject pool.

The small difference in earnings between the homogenous treatments is driven by the fact that all-male groups come to an agreement marginally faster than all-female groups (15.31 seconds in MMM vs. 16.85 seconds in FFF). See Section D3 in the Online Appendix for an analysis of time to agreement.

The proportion of GCs are almost exactly the inverse of MWCs, so we do not report these separately. The difference is the small number of games where one-player took everything.

In this section and the following, p-values are obtained from linear regressions on a constant, with standard errors clustered at the matching group level.

Recall that we have restricted our analysis to MWCs. If we consider all initial proposals, we observe the same qualitative gender bias, although it is less extreme in the FFM treatments.

The probability with which initial agreements are implemented, according to gender composition, treatment, and location, can be seen in TableA3.

We are not suggesting that making GC counter-proposals cannot, in some instances, be effective in breaking MWC coalitions. However, we prefer to focus on MWC counter-proposals because they are much more often implemented, so when looking for explanations for the gender earnings gap they are more important. Other behaviors by excluded parties are detailed in TableA4, for the interested reader.

We are grateful for the remarks of an anonymous referee that encouraged us to look deeper into the causes of the gender gap.

We present the details of our model in Appendix B for the interested reader.

In order to identify two parameters we need two independent equations, so must match two features of the data. We chose these two features because they are observed in all games (so are most precisely estimated) and are fundamental to the outcomes we are trying to explain. The derivation of these equations is detailed in Appendix B.

Clearly, we cannot rule out the possibility of measurement error in our control variables as[START_REF] Gillen | Experiments with measurement error: Techniques with applications to the caltech cohort study[END_REF] suggest can be important when investigating gender differences in behavior.

See for example[START_REF] Hofstede | Cultures and organizations: Software of the mind[END_REF] for five variables that may presumably help delimit cultural boundaries.

gratefully recognizes financial support by the Center of Behavioral Institutional Design and Tamkeen under the NYU Abu Dhabi Research Institute (Award CG005). Tremewan gratefully acknowledges funding from the French Agence Nationale de la Recherche, under grant ANR-21-CE03-0018 (project ENDURA).

Notes: The y-axis measures the proportion of the opening proposals in which all members are being offered a positive share. 95% intervals, clustering standard errors at the matching group level.


David Coeurjolly
email: david.coeurjolly@liris.cnrs.fr

Isabelle Sivignon
email: isabelle.sivignon@liris.cnrs.fr

Measure of Straight Lines and its Applications in Digital Geometry ⋆

In digital geometry, objects such as digital straight lines can be considered as equivalent classes of Euclidean straight lines with respect to a digitization process. This paper investigates the analysis of the set of straight lines whose digitization is a given digital straight segment with the help of tools from integral geometry. After a definition of a measure on such sets, we illustrate several applications of it in digital geometry.

Introduction

When investigating relationships between Euclidean and Digital geometries, we are facing difficulties induced by the digitization process: Given a digital set X ⊂ Z 2 , there is an infinity of shapes Ω X ⊂ R 2 such that X = Ω X ∩ Z 2 (in this statement, we have considered the Gauss digitization process [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF] for the sake of simplicity). The set Ω X can be considered as the preimage of X for the Gauss digitization. If we want to perform geometrical analysis on digital objects we usually associate properties on X from properties of its preimage set Ω X . For example, X is digitally convex if there exists a continuous convex shape in Ω X [START_REF] Sklansky | Measuring concavity on a rectangular mosaic[END_REF]. In shape analysis, many approaches exist to decide if the set X is digitally straight (see [START_REF] Klette | Digital straightness-a review[END_REF] for a survey). Among all the definitions of Digital Straight Lines (DSL for short), we consider the following one: X is a DSL if there exist a Euclidean straight line in Ω X .

In this paper we focus on the analysis of a subset of Ω X which only corresponds to straight lines: we define L X the set of straight lines (maybe empty) such that the digitization of each line in L X is exactly X. Note that we can define a digital straight segment (DSS for short) and its associated preimage as the set of Euclidean lines whose digitization contains X. In the literature, the preimage L X has been widely used to control DSS recognition algorithms [START_REF] Dorst | Discrete representation of straight lines[END_REF][START_REF] Dorst | Decomposition of discrete curves into piecewise straight segments in linear time[END_REF][START_REF] Lindenbaum | On recursive, o(n) partitioning of a digitized curve into digital straigth segments[END_REF][START_REF] Klette | Digital straightness-a review[END_REF], or to perform geometric reconstruction from a digital contour [START_REF] Breton | Towards an invertible euclidean reconstruction of a discrete object[END_REF]. In some sense, L X can be considered as an uncertainty characterization of X with straight lines: Given a set X, the thinner the L X , the finer the DSS characterization of X.

We propose here to formalize and investigate this preimage analysis with the help of integral geometry results [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF]. More precisely, we use the mathematical measure of lines in the plane to evaluate a kind of accuracy of the DSS recognition. Similar analyses have been proposed in [START_REF] Kiryati | Digital or analog hough transform?[END_REF] to compare the continuous and the analytical Hough transforms. The authors have considered the measure of the DSS worst case in both models for recognition purposes. We provide here a closed formula to evaluate the measure of lines and we illustrate several applications in digital geometry.

Measure of Straight Lines in the Plane

Preliminaries

In this section, we first recall integral geometry results [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF]. First of all, let us consider a polar representation (ρ, φ) of a straight line l with φ ∈ [0, 2π] and ρ ≥ 0 Hence, a point p(x, y) ∈ R 2 belongs to l if

x cos φ + y sin φ -ρ = 0 .
In integral geometry and in the theory of geometrical probability, the measure of a set L of lines is defined as the integral over the set of a differential form ω = f (ρ, φ)dρ ∧ dφ, provided the fact that this integral exists in the Lebesgue sense. The differential form is defined using notations and rules of the exterior calculus for which an overview is given in [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF]. In our context, exterior calculus will coincide with classical differential calculus in most situations.

To define a measure of straight lines, the function f should be chosen such that the resulting integral is invariant under the group of motions in the plane (rotation and translation). In fact, the invariance implies that f is constant [START_REF] Crofton | On the theory of local probability[END_REF][START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF]. Hence using f (ρ, φ) = 1 we have: Theorem 1 ( [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF]). The measure of a set of lines G(ρ, φ) is defined by the integral over G of the differential form dG = dρ ∧ dφ .

(1)

Note that the measure is always taken at absolute value. In the following, since we will only consider continuous and closed domains G, the integral always exists in the Lebesgue sense. As a consequence, the measure is positive, monotonic

(if A ⊆ B, then m(A) ≤ m(B)), additive (if A ∩ B = ∅, then m(A ∩ B) = m(A) + m(B))
, and the measure of the empty set is 0.

Density in the (a, b)-linear space

As defined above, the measure definition perfectly matches with the (ρ, φ)-Hough transform for computer vision purposes [START_REF] Duda | Use of the hough transform to detect lines and curves in pictures[END_REF]. Indeed, given a continuous and closed domain G in the (ρ, φ) space, the measure of G is

m(G) = G dρ ∧ dφ , (2)
which simply corresponds to the area of G in (ρ, φ).

In digital geometry, when considering DSS preimages [START_REF] Dorst | Discrete representation of straight lines[END_REF][START_REF] Mcilroy | A note on discrete representation of lines[END_REF][START_REF] Dorst | Decomposition of discrete curves into piecewise straight segments in linear time[END_REF][START_REF] Lindenbaum | On recursive, o(n) partitioning of a digitized curve into digital straigth segments[END_REF][START_REF] Klette | Digital straightness-a review[END_REF], we usually prefer the slope-intercept linear parameter space, denoted (a, b)-space in the following, since it allows to characterize arithmetical properties embedded in DSS. This parameter space maps the line axy + b = 0 to the point (a, b). Conversely, straight lines in the (a, b)-space are mapped onto points in the primal space.

In order to obtain the measure of straight lines in this space, we need to perform a transformation in the integral calculus. From [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF], we know that if the straight lines are parametrized by their intercept u (resp. v) along the x-axis (resp. y-axis), we can change the coordinates

ρ = uv √ u 2 +v 2 φ = arctan(u/v) (3)
to obtain dG = uv

(u 2 + v 2 ) 3 2 du ∧ dv . (4)
If we consider now the (a, b) parametrization widely used in digital geometry, we define the transformation:

u = -b a v = b (5)
and obtain

ρ = b √ 1+a 2 φ = arctan(-1 a ) (6)
whose Jacobian matrix is

J = ab (1+a 2 ) 3/2 1 √ 1+a 2 1 1+a 2 0 (7) Hence, dG = dρ ∧ dφ = |det(J)|da ∧ db (8)
Finally, Lemma 1. The measure of a set of lines G(a, b) is defined by the integral over G of the differential form

dG = da ∧ db (1 + a 2 ) 3 2 . ( 9
)
Even if the (a, b)-parametrization leads to a more complex differential form, we demonstrate in the next sections that the measure of DSS preimages in the (a, b)-space can be easily computed.

In terms of geometric probability, the measure allows us to obtain interesting properties. For example given two bounded and closed domains Ω and Ω ′ with Ω ′ ⊂ Ω and Ω, Ω ′ ⊂ R 2 , the probability that a random line intersects Ω ′ with the hypothesis that it also intersects Ω is p = m(Ω ′ ) m(Ω) . For further developments on probability results from a measure, please refer to [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF].

Measure of Lines of a closed (a, b)-space domain

Let us consider a polygonal domain P in the (a, b)-space defined by its boundary vertices {p i = (a i , b i )} i=0..n-1 (counter-clockwise). Given a function f measurable on P (in the Lebesgue sense), a classical way to compute the integral of f over P is to decompose the computation into a finite summation of triangular integration processes. As illustrated in Fig. 1, let T i be the oriented triangle {(0, 0), p i , p (i+1 mod n) } for 0 ≤ i ≤ n -1. For each triangle T i , let δ i be the sign of its signed area (

δ i = sign(a i b i+1 -a i+1 b i )).
In the rest of the paper, all index shifts such as i + j must be considered modulo n. Using the above notations The next step is to compute the integration over triangles T i . To solve that problem, we also consider a linear variable transformation which maps the triangle T i to the unit triangle {(0, 0), (1, 0), (0, 1)}

P f (a, b)dadb = n-1 i=0 δ i Ti f (a, b)dadb . (10) + - p p
a = a i a ′ + a i+1 b ′ b = b i a ′ + b i+1 b ′ whose Jacobian matrix is J i = a i a i+1 b i b i+1 (11) Thus Ti f (a, b)dadb = |det(J i )| 1 0 1-b ′ 0 f (a ′ , b ′ )da ′ db ′ (12)
Note that the sign of det(J i ) and δ i coincides. Finally, we obtain the following lemma:

Lemma 2. The measure of a set of straight lines defined in a polygonal domain

P = {a i , b i )} i=0..n-1 in the (a, b)-space is m(P ) = n-1 i=0 (a i b i+1 -a i+1 b i ) • g(a i , a i+1 ) (13)
with

g(a i , a i+1 ) =                1 1+a 2 i + √ 1+a 2 i if a i = a i+1 , 1 1+ √ 1+a 2 i if a i+1 = 0, 1 1+ √ 1+a 2 i+1 if a i = 0, ai+1- √ 1+a 2 i ai+1+ai(-1+ √ 1+a 2 i+1 ) aiai+1(ai+1-ai)
otherwise.

(

) 14
Proof. With Eq. ( 9), ( 10) and ( 12), we have

m(P ) = n-1 i=0 (a i b i+1 -a i+1 b i ) 1 0 1-b ′ 0 da ′ db ′ (1 + (a i .a ′ + a i+1 .b ′ ) 2 ) 3 2 (15)
Then, thanks to the mapping of T i triangles to the unit triangle, the rightmost integral term can be evaluated to obtain closed formulas presented in Eq. ( 14). The function g(a i , a i+1 ) in the lemma statement defines the result of the integral computation obtained by a symbolic mathematical solver.

3 Measure of Digital Straight Segments and Applications

DSS Preimage and Farey Fan

In the following, we use the preimage definition proposed by Lindenbaum and Bruckstein [START_REF] Lindenbaum | On recursive, o(n) partitioning of a digitized curve into digital straigth segments[END_REF]. Consider a straight line y = a 0 x + b 0 (without loss of generality, (a 0 , b 0 ) ∈ [0, 1] × [0, 1[, i.e. straight lines are supposed to be in the first octant), the digitization of this line using the Object Boundary Quantization (see [START_REF] Jonas | Digital representation schemes for 3d curves[END_REF] for a survey on digitization schemes) on an unit grid is the set of discrete points such that X = (x, y) ∈ Z 2 | ⌊a 0 x + b 0 -y⌋ = 0 . The preimage of X is thus defined by

L X = {(a, b) ∈ [0, 1] × [0, 1[ | ∀(x, y) ∈ X, y ≤ ax + b < y + 1} . (16)
In the (a, b)-space L X is a convex polygon whose vertices are embedded in a strong arithmetical structure, the Farey Fan [START_REF] Mcilroy | A note on discrete representation of lines[END_REF]. Indeed, each pixel in X induces two linear constraints with integer coefficients. The Farey Fan of order n is the arrangement obtained as the intersection of all possible linear constraints associated to pixels in [0, n] × [0, n]. In other words, it is the arrangement of the set of lines {y/x, 0

≤ y ≤ x, 1 ≤ x ≤ n} in the domain [0, 1] × [0, 1] [12]
. The vertex coordinates of this arrangement are irreducible fractions whose denominator is less or equal to n. Finally, such an arithmetical structure allows us to design fast recognition algorithms since the cells in the Farey Fan of order n defines all DSS of length n + 1 in the first octant [START_REF] Dorst | Decomposition of discrete curves into piecewise straight segments in linear time[END_REF][START_REF] Lindenbaum | On recursive, o(n) partitioning of a digitized curve into digital straigth segments[END_REF].

A first application of the measure is to merge both arithmetical properties and uncertainty within the same framework. Indeed, each cell of a Farey Fan of order n represents a DSS X of length n + 1. With the measure associated to this preimage (and thus to X), we can evaluate the quantity (in terms of integral measure) of straight lines whose digitization contains X. Hence, Fig. 3 illustrates Farey Fans of order from 0 to 5 in which the gray value corresponds to the measure computed with Lemma 2 on each cell (the white color cell is the preimage with maximal measure). As expected, we can observe that the cells have a high variability with respect to the measure. If we consider DSS examples in Fig. 2, both DSS preimages are in the Farey Fan of order 4 (Fig. 3-(e)). However, if we consider the following probability test: given a random straight line l whose slope and intercept is in [0, 1] × [0, 1], the probability such that the digitization of l contains the left DSS is 3.7 times higher than the probability of the right DSS. Hence, the measure is a powerful tool to perform a posteriori evaluation of DSS recognitions.

Representative Euclidean Straight Line of a DSS

In applications such as the polygonal reconstruction from a digital curve, we sometimes need to associate a representative Euclidean straight line to a preimage. In the literature, such a representative straight line is usually computed as the point in the (a, b)-space defined as the mid-point of the preimage diagonal obtained by the two vertices of equal abscissa [START_REF] Vialard | Geometrical parameters extraction from discrete paths[END_REF]. The advantage of such an approach is that both the obtained straight line and the DSS slopes are identical.

We propose here a new definition of the representative Euclidean straight line based on integral geometry tools. Definition 1. Given a continuous and closed domain G(a, b), the representative Euclidean Straight Line is the center of mass of G on the density function defined in Sect. 2.2. In other words, such a point is defined as

G a dadb G dadb , G b dadb G dadb (17)
In terms of probability, the centroid definition corresponds to the expected values of the normalized probability distributions defined from the density on the a and b variables in the Lebesgue sense.

The denominators in Eq. ( 17) correspond to the measure G. To evaluate the numerators on a polygonal domain P , we can use the evaluation scheme detailed in Sect. 2.3: we decompose the integration over P into a finite sum of integrations over the unit triangle. Finally, we evaluate the integrals to obtain closed formulas. Lemma 3. The Representative Euclidean straight line of a polygonal domain

P = {(a i , b i )} i=0..n-1 in the (a, b)-space is the point ( Ca m(P ) , C b m(P ) ) with C a = n-1 i=0 (a i b i+1 -a i+1 b i ) • g a (a i , a i+1 ) C b = n-1 i=0 (a i b i+1 -a i+1 b i ) • g b (a i , a i+1 , b i , b i+1 ) and g a (a i , a i+1 ) =                arcsinh(ai)- a i √ 1+a 2 i a 2 i if a i = a i+1 , ai+1-arcsinh(ai+1) a 2 i+1 if a i = 0, ai-arcsinh(ai) a 2 i if a i+1 = 0, aiarcsinh(ai+1)-ai+1arcsinh(ai) aiai+1(ai-ai+1)
otherwise.

g b (a i , a i+1 , b i , b i+1 ) =                        (bi+bi+1)arcsinh(ai)- a i 0 @ b i +a i 0 @ -a i + s 1+ 1 a 2 i √ 1+a 2 i 1 A b i+1 1 A √ 1+a 2 i 2a 3 i if a i = a i+1 , ai+1((-2+ √ 1+a 2
i+1 )bi+2bi+1)+(bi-2bi+1)arcsinh(ai+1)

2a 3 i+1 if a i = 0, ai " 2bi+ " -2+ √ 1+a 2 i " bi+1 " +(-2bi+bi+1)arcsinh(ai) 2a 3 i if a i+1 = 0, NumApprox(a i , a i+1 , b i , b i+1 ) otherwise.
where NumApprox(a i , a i+1 , b i , b i+1 ) is the numerical approximation of

1 0 1-b ′ 0 (b i a ′ + b i+1 b ′ )da ′ db ′ (1 + (a i .a ′ + a i+1 .b ′ ) 2 ) 3 2 (18)
The proof of this lemma is similar to the proof of Lemma 2: g a and g b functions are obtained using a formal integration over the unit triangle. A specific point concerns the g b function in the general case for which no closed formula has yet been found: we failed to find a suitable parameter transformation in order to help the symbolic integrator to find a solution. However, a closed formula exists to integrate the function in Eq. (18) over a rectangular domain. Hence, the integral can be approximated using the Rectangle Approximation method (function N umApprox). Figure 3 illustrates the center of mass for each preimage in Farey Fans of order from 0 to 5. In Fig. 4, we compare our rep-

(a) (b) (c) (d) (e) (f)
Fig. 3. Farey Fan of order from 0 to 5 ((a) -(f )) with the center of mass for each preimage. The color corresponds to a linear mapping between the preimage measure and the gray level map (the white color cell is the preimage with maximal measure).

resentative line computation to the one proposed in [START_REF] Vialard | Geometrical parameters extraction from discrete paths[END_REF] on the Farey Fan of order 4 (Fig. 4-(a)). Note that even if points in the (a, b)-space associated to the representative lines in both definitions may differ, the associated lines in the primal space may be visually close. Furthermore, thanks to the definition of the preimage in Eq. ( 16), both the restriction to positive slope straight lines (a ∈ [0, 1]) and the OBQ digitization scheme make the representative line for horizontal digital segment shifted and not perfectly horizontal (see Fig. 4-(b)). Despite, the proposed representative line definition is consistent to the straight line measure theory. Furthermore, the proposed framework can be generalized to different DSS preimage defintions, including for example, lines with negative slopes or the Gauss digitization model (see Sec. 1)

Contour analysis with Digital Tangent and Preimage Measures

In this section, we provide a last illustration of the power of the measure to analyze contours. Let us denote by C = {p 0 , . . . , p n-1 } a digital closed curve. We consider N k (i) the sequence {p i-k , . . . , p i , . . . , p i+k } (again, index shifts are computed modulo n). From the measure definition detailed above, we can design the following linearity map.

Definition 2 (Linearity map). Given a closed contour C, the linearity map is a function λ : C → R that associates for each point p i ∈ C, the measure of straight lines of N k (i).

From this definition, several observations can be made:

if λ(i) = 0, it means that N k (i) is not a DSS since there is no Euclidean straight line whose digitization contains N k (i); -if λ(i) > λ(j), it means that the set of straight lines whose digitization contains N k (i) is wider than the set of straight lines whose digitization contains N k (j).

Roughly speaking, the linearity map measures the uncertainty of N k (i) with respect to the straight lines. From a computational point of view, to compute the λ function, we just have to compute the measures in the Farey Fan of order 2k which gives us the set of patterns of length 2k + 1 whose measure is not null (up to symmetry and translation). Then, we scan the sequences of 2k + 1 pixels in C and associate the corresponding measure. Figures 5 (b) and (d) illustrate the linearity map computed from local patterns of length 2k+1 = 5 from the kangaroo curve (Figure 5(a)). In Fig. 5(b) and (c) the grey map represents the measure computed for each pixel: white stands for high values while black pixels have a low measure. Cross marked pixels are those for which the local pattern of length 5 is not a digital segment. Local artefacts due to the fixed length of the local patterns can be observed on the "steps" of the curve (vertical or horizontal pixel sequences). In Fig. 5(d), the λ(i) graph computed for the kangaroo curve is depicted. Note that high values generally correspond to central pixels of the steps. Indeed, such steps are associated to the highest measure value in the Farey Fan of order 4. To perform a finer analysis of digital contours, it would be of interest to locally adapt the size of the patterns.

In the literature, to design curvature or tangent estimators [START_REF] Lachaud | Analysis and comparative evaluation of discrete tangent estimators[END_REF][START_REF] Lachaud | Fast, accurate and convergent tangent estimation on digital contours[END_REF] or to extract a minimal DSS decomposition of a digital contours [START_REF] Feschet | On the min dss problem of closed discrete curves[END_REF], the concept of discrete tangential cover has been widely used to avoid the k parameter. More precisely, given a closed digital contour C, a discrete tangent at a point p ∈ C is the longest DSS centered at p (see [START_REF] Vialard | Geometrical parameters extraction from discrete paths[END_REF] for a more formal definition). In that case, we can also define a linearity map computed on the discrete tangent (in some sense, the neighborhood N k (i) is adapted locally). Figures 5(c) and (e) illustrate preliminary results obtained using the discrete tangents to compute the measure at each pixel. Note that compared to Fig. 5(b) and (d), the behavior of the measure along the curve is smoother, and that each pixel is now assigned a measure value. The λ(i) graph using discrete tangents is depicted in Fig. 5(e). Contrary to the λ(i) graph computed from local patterns (Fig. 5(b)), the high values of the measure now seem to match some high curvature parts of the curve. As mentioned above, the linearity map seems to be a complementary tool to the curvature estimation.

Conclusion and Future Works

In this paper, we have introduced concepts from integral geometry and geometric probability in the digital geometry context. More precisely, we have focused on the computation of the measure of straight lines in the (a, b) parameter space which allows to first have a precise analysis of Farey Fans in terms of measure. Then, we have investigated a formal and consistent definition of a representative Euclidean straight line associated to a DSS. Finally, we have defined a new tool, so called the linearity map, to describe discrete contours. At this point, only a preliminary analysis of this feature map has been addressed. A formal comparison between such a tool and a classical curvature based descriptor is a challenging future work. Furthermore, it would be interesting to consider similar integral geometry analysis of other objects such as digital circles or digital polynomials.

Fig. 1 .

 1 Fig. 1. Integration scheme over a polygonal domain.

Fig. 2 .

 2 Fig. 2. Two DSS of length 5 with measure respectively 0.0738 (left) and 0.0198 (right).

Fig. 4 .

 4 Fig. 4. Comparison between the proposed representative straight line computation (circles in (a) and bold lines (b)) and the existing approach (squares in (a) and gray lines in (b)): (a) differences depicted in the Farey Fan of order 4 and (b) in the primal space.

Fig. 5 .

 5 Fig. 5. Preliminary results of the linearity map for the kangaroo curve (a). Figures (b) and (c) focus on the kangaroo leg part: the lower the measure, the darker the pixel. In (b) and (d), local patterns are used to compute the measure, while discrete tangents are used in (c) and (e).

⋆ This work was partially funded by the ANR project GeoDIB ANR-06-BLAN-0225.