Repeated Play and Gender in the Ultimatum Game Peter McGee and Stelios Constantinides∗† March 2012

Abstract This paper examines whether a subject’s decisions in the ultimatum game played for multiple periods using the strategy method are affected by his or her partner’s gender. We find that although there are some initial gender influences on behavior, these do not persist as subject behavior is systematically different in early periods than it is in later periods. In particular, subjects are significantly more likely to offer an even division of the pie in the first period, clustering on a focal distribution before exploring other offers. The differences between our results and the extant results in the literature underscore a need for careful replication of results dealing with gender differences.

∗ †

JEL Classifications: C9, C7 Keywords: gender in experiments, ultimatum game

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1

Introduction

The potential for systematic differences between the behavior of men and women in strategic environments has generated a great deal of research in economics (e.g. Croson and Gneezy (2009), Eckel and Grossman (2008)). There are two papers that specifically focus on gender differences in the ultimatum game: Eckel and Grossman (2001, henceforth EG) and Solnick (2001). The results of the two papers agree on some findings: there is little overall difference in the mean offers of men and women, and offers to women are lower on average than those made to men, irrespective of the gender of the proposer. There are also some differences: despite similar overall rejection rates (12.8% in EG and 12.4% in Solnick), EG report that offers by men were significantly more likely to be rejected than those made by women, while Solnick finds exactly the opposite, and the fraction of the pie offered by EG’s subjects was 37.5%, while it was 46.8% in Solnick. EG conclude by noting that, “Clearly, some puzzles remain.” This paper seeks to address the source of some of these puzzles. EG point out four significant differences in the experiments that might contribute to differences in the findings. First, EG’s subjects play eight rounds while Solnick’s subjects play a one-shot game. Second, EG use the game method while Solnick uses the strategy method. Third, the introduction of gender is different across the experiments: EG’s proposers and responders sit face-to-face while Solnick’s subjects know the first name of the subject with whom they were paired. Fourth, there may be important differences in the subject pools. The particular concern of our experiment is the effect of repeated play. Using the game method, EG find limited evidence that behavior changes across rounds, and that only behavior in the first two rounds varies much from the remainder. EG suggest that results of the one-shot design using the strategy method may reflect noisy subject behavior as subjects make decisions in an unfamiliar environment, particularly that subjects may not know what constitutes an appropriate “Minimum Acceptable Offer,” or MAO. Subjects in our experiment participate in eight rounds of an ultimatum game using exactly the same procedures as Solnick’s one-shot game. The averages from all eight periods of our data for offers, MAOs, and even splits of the pie are very different from those found in Solnick; the averages are qualitatively similar when comparing only our first round of data to Solnick, but we fail to replicate the significant results in Solnick. We also find that EG’s hypothesis is essentially correct, namely that behavior in the first round of play is systematically different than in subsequent rounds. While this is true for MAOs, the change is more often significant among offers and the results suggest that this stems from how frequently proposers offer an equal split of the pie. In the first round, the highest fraction of offers that are even splits are in same gender pairings. The prevalence of even splits declines in almost every round for three of the gender pairings (female proposer-male responder pairings are the exception and have far fewer even splits to begin with), and the difference is significant in almost every round for the same gender pairings. Subjects initially cluster their offers on the focal division of the pie— an even split —before experimenting with other offers. This manifests as gender solidarity between the proposer and responder in the first round, but such solidarity is fleeting as proposers move away from an even split and exhibit no solidarity thereafter.

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2

Experimental Design

We conducted seven sessions of the ultimatum game with sixteen subjects and eight rounds each. The sessions are summarized in Table 1. In every session, half of the sixteen subjects were randomly assigned the role of the proposer, while the other half were assigned the role of the responder.1 Throughout the experiment, proposers were referred to as Player A, and responders were referred to as Player B. Subjects remained in the same role for the duration of the session.2 Of the 112 subjects, 45% were male and 55% were female. In order to make subjects aware of their partner’s gender, we follow Solnick (2001) and tell subjects the first name of the person with whom they are paired, but they remain otherwise anonymous to each other.3 Subjects were paired with each other exactly once, and all subjects were in the same room but unaware of with whom they were bargaining. The proposer’s task was to propose a split of $20 between himself/herself and the responder. At the same time, the responder provided his/her MAO. If the responder’s MAO was less than or equal to the proposer’s offer, both received the amount allocated by the proposer. If the responder’s MAO was greater than the amount offered by the proposer, both received a payoff of zero for that round. After the proposal and MAO were submitted, proposers were told their partner’s MAO, responders were told the proposed division of the pie, and both learned the outcome for that period. The first session was conducted using paper and pencil, while the remaining sessions were programmed and conducted with the software z-Tree (Fischbacher (2007)). There were no significant differences between the data from the paper and pencil session and the computer sessions, so the data have been pooled. Subjects were paid on the basis of one randomly selected round, which was announced at the end of the session. In addition, all subjects received an $8 participation fee.4 Average earnings were $16.54 for a session lasting approximately 50 minutes.

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Results

Table 2 presents the averages for offers, MAOs, rejection rates, and fraction of offers that are even splits in Solnick’s data and for all eight rounds of our data. A cursory glance shows that there are some substantial differences in the data. The average offers across all pairings in the two experiments are very similar— 46.8% of the pie in Solnick and 46.2% of the pie in our data, but the average MAO is almost six percentage points higher in our data, the rejection rate almost nine percentage points higher, and the fraction of offers that are an even split is almost sixteen percentage points lower. Solnick finds several specific, statistically significant differences: 1

Technically there is no “responder” because we are using the strategy method and moves are made simultaneously. We adopt the term responder for ease of exposition. 2 EG’s subjects played in both roles, introducing yet another design difference. Solnick’s subjects, by construction, only played in one role, so we have decided to keep the role constant. 3 Some of the subjects’ names were not gender revealing in Solnick’s experiment. All of the names in this experiment were gender revealing. A list of first names is available upon request. 4 In Solnick’s experiment, the pie was $10 and the participation fee $2. The size of the pie and the participation fee were increased because subjects participated in more rounds.

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1. Women offer men more than they offer women 2. Offers to men are greater than offers to women 3. Men are more likely to be offered an even split than women 4. Male responders have significantly lower MAOs than female responders 5. Male responders ask for more from female proposers than male proposers 6. Female responders also ask for more from female proposers than male proposers 7. Rejection rates for offers made by female proposers are significantly higher than for male proposers. Of these seven results, however, our results go in the opposite direction for (1), (2), (3), (5), and (7), while the results for (4) and (6) go in the same direction. None of the differences in (1)-(7) is statistically significant in our data when all eight periods are used. A comparison of all periods of our data against Solnick’s one-shot game is not necessarily the best way to see how our results compare. Table 3 presents the averages for offers, MAOs, rejection rates, and fraction of offers that are even splits in Solnick’s data and for just the first round of our data. The data from just the first period of our data are much closer, on average, to those from Solnick. The average offer is only 1.2 percentage points higher, the rejection rate is 3 percentage points higher, and the fraction of offers that are an even split is 0.9 percentage points higher. MAOs, on the other hand, are almost nine percentage points higher in our data.5 The data are closer on the specific predictions, too. Only three of the seven predictions listed above—(1), (2), and (4) are in the opposite direction of what we find in the first period of our data, and only (1) is significant in our data (p=0.0770). Of the remaining four specific predictions, three are in the same direction in our data as in Solnick’s— (3), (5), and (7) —while we find no difference in the MAOs female responders paired with male proposers and the MAOs of female responders paired with male proposers, finding (6). The comparison between our first period of data and Solnick’s yields very similar averages across all pairs, and many of the results are in the same direction, but for the most part we fail to replicate the significance. It is clear from Tables 2 and 3 that behavior across all eight periods may not resemble the behavior in the first period. In order to examine how behavior changes across periods, we estimated the following equations to facilitate testing of means by period and category 5

We have no exact explanation for why MAOs are consistently higher in our data, but one possibility that occurred to us after the fact is that when we scaled up the pie and show-up fee, the show-up fee became large enough that subjects felt they could risk a higher chance of rejection with a higher MAO.

4

Of f erit = β0 ∗ M ale − M aleit + β1 ∗ M ale − F emaleit + β2 ∗ F emale − M aleit + β3 ∗ F emale − F emaleit +

j=4 k=8 X X

P airT ypej ∗ P eriodk + it

j=1 k=2

(1)

M AOit = β0 ∗ M ale − M aleit + β1 ∗ M ale − F emaleit + β2 ∗ F emale − M aleit + β3 ∗ F emale − F emaleit +

j=4 k=8 X X

P airT ypej ∗ P eriodk + it

j=1 k=2

(2)

EvenSplitit = β0 ∗ M ale − M aleit + β1 ∗ M ale − F emaleit + β2 ∗ F emale − M aleit + β3 ∗ F emale − F emaleit +

j=4 k=8 X X

P airT ypej ∗ P eriodk + it

j=1 k=2

(3) Of f erit is the offer made by the proposer in group i during period t. M AOit is the MAO submitted by the responder in group i during period t. EvenSplitit is a dummy variable equal to 1 if the offer made by the proposer in group i during period t is an even division of the pie and 0 otherwise. M ale − M ale is a dummy variable equal to 1 if both proposer and responder are male, F emale − F emale is a dummy variable equal to 1 if both proposer and responder are female, M ale − F emale is a dummy variable equal to 1 if the proposer is male and the responder is female, and F emale − M ale is a dummy variable equal to 1 if the proposer is female and the responder is male. The pair types are each interacted with a full set of period dummies, where P eriod 2 is a dummy equal to 1 if it is the second period and 0 otherwise, P eriod 3 is a dummy equal to 1 if it is the third period and 0 otherwise, and so on. it is the econometric error term with mean zero. I correct for the possible non-independence of observations at the subject level by reporting heteroskedasticity robust standard errors clustered at the proposer level for offers and even splits and at the responder level for MAOs. The results can be found in Table 4. EG’s hypothesis is correct: subject behavior in the first two periods is very noisy. In column 1 we can see that average offers in male-male, male-female, and female-female pairs are never as high as they are in the first period in any subsequent period; average offers in female-male pairs are never as low as they are in the first 5

period in any subsequent period. That female-male pairs do not change in the same way as the other pairings reflects where the offers start: offers in female-male pairs are significantly lower than those in all other gender pairings at the 10% level, while there are no statistically significany differences among the other gender pairings. These findings mirror the pattern in column 3: the fraction of offers that are even splits is lower in male-male, male-female, and female-female pairs in every period after the first, while the fraction in female-male pairs is not uniformly higher or lower in subsequent periods. Again, the percentage of even splits is lowest in female-male pairs, but unlike the average offers, it is only significantly different from the percentage of even splits in female-female pairs. Except female-female pairings in period 6, MAOs are lower in every gender pairing in every period after the second period, as can be seen in column 2 of Table 4. Table 4 shows an apparent initial solidarity in same gender pairings: in columns 1 and 3, offers and the percentage of offers that are even splits are higher and MAOs are lower in malemale and female-female pairs than in mixed gender pairs. The differences in MAOs is not significant, nor is the solidarity in offers and even splits significant in male-male pairs relative to male-female pairs. There are significant solidarity effects in female-female pairs relative to female-male pairs in both the average offer and even splits in the first period. These differences are no longer significant after the first period. As already mentioned, behavior in the first period does appear to reflect noise, but do the data support an explanation such as the MAO conjecture in EG? There are two pieces of evidence that suggest that the answer lies not in the MAO but in offers that are an even split of the pie. From Table 3 we can see that the average offer in the first period is 48% of the pie and 73.2% of offers are even splits. When comparing these same numbers to the session averages in Table 2, we can see that the average offer declined by just $0.36 to 46.2% of the pie, but the fraction of offers that are even splits declines from 73.2% of all offers to 56.5% of all offers. It seems that subjects in the first period are more likely to cluster on the arguably focal distribution of a 50-50 split of the pie. The average profit in every period is greater when one offers an even split than when one does not and the rejection rate is lower, so it is difficult to attribute this change to either expected profitability or risk-aversion. To examine this in another way, Table 5 reproduces columns 1 and 3 of Table 4 but with a slight change in the data. There are 16 offers (3.6% of the total) that are greater than half the pie, eight of which are made by the same male proposer who offers more than half the pie in every period. It is unclear what motivates such generous offers, but they are unusually self-less in the data.6 Excluding these offers, a pattern does emerge: offers and even splits in male-male pairs are significantly lower at the 10% level in ever period after the first period except period 4 and offers and even splits in female-female pairs are significantly lower at the 10% level in every period after the first except periods 2 and 4. 6 Solnick reported that, when explaining her decision to offer the entire pie to the repsonder, a proposer said, “I want at least one of us to get something,” so clearly such an offer is sometimes intentionally selfless and unprofitable. Only the one proposer allocated more than half the pie in more than one period.

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4

Conclusion

We have taken up the challenge laid down in the last paragraph of EG to engage in “systematic experimentation” to examine the puzzles they identify. The results of our experiment support EG’s hypothesis that some of the differences between their results and Solnick’s are due to noise in subject behavior. Our results, however, do not exactly match those of EG, either, so factors other than just repeated play are still at work. The differences and similarities of the three papers— EG, Solnick, and this paper — illuminate a couple important points. First, for any consensus to emerge regarding gender differences in the ultimatum game or in any other arena, replication is necessary. Croson and Gneezy conclude their survey of gender differences on preferences with a methodological note that journals appear to be biased toward publishing research that finds gender differences rather than research that does not. Researchers might be concerned that such a pall would extend to work that may simply confirm previous results and shy away from such replications. Given that the three laboratory investigations of gender and the ultimatum game come to three sets of results that agree in some areas but directly disagree in others, resolving these differences through careful replication is essential even if the primary contribution of is often just a replication of previous results. The second point is that replication must be careful. That is, one must be able to determine whether results of future studies reflect the influence of gender or features of the experimental design. EG and Solnick undertook studies that published simultaneously and their designs differed along several dimensions. This paper examined the impact of just one of these differences— repeated play —while not taking up the other two primary differences, the method used to introduce gender and the game versus strategy method. Croson and Gneezy conclude that, “A number of studies also indicate that women’s social preferences are more sensitive to subtle cues than are men’s,” a sentiment that echoes the last paragraph of EG. In light of this sensitivity to cues, design differences are particularly important when interpreting and comparing gender differences across laboratory studies.

References Croson, Rachel and Uri Gneezy, “Gender Differences in Preferences,” Journal of Economic Literature, 2009, 47, 448–474. Eckel, Catherine and Philip Grossman, “Chivalry and Solidarity in Ultimatum Games,” Economic Inquiry, 2001, 39 (2), 171–188. and , “Differences in the Economic Decisions of Men and Women: Experimental Evidence,” Handbook of Experimental Economics Results, Volume 1, eds. C. Plott and V. Smith, 2008, Chapter 113 (Elsevier, New York), 1061–1073. Fischbacher, U., “z-Tree: Zurich Toolbox for Ready-made Economic Experiments,” Experimental Economics, 2007, 10(2), 171–178. Solnick, Sara, “Gender Differences in the Ultimatum Game,” Economic Inquiry, 2001, 39 (2), 189–200. 7

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Session 1 2 3 4 5 6 7

Males (Proposers/Responders) 8 (4/4) 8 (4/4) 8 (4/4) 6 (4/2) 6 (4/2) 7 (4/3) 7 (3/4)

Females (Proposers/Responders) 8 (4/4) 8 (4/4) 8 (4/4) 10 (4/6) 10 (4/6) 9 (4/5) 9 (5/4)

Rounds 8 8 8 8 8 8 8

Table 1: Summary of Experimental Sessions Money-to-Split $20 $20 $20 $20 $20 $20 $20

Show-up Fee $8 $8 $8 $8 $8 $8 $8

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All Pairings Male Proposers-All Responders Female Proposers-All Responders All Proposers-Male Responders All Proposers-Female Responders Male Proposer-Male Responder Male Proposer-Female Responder Female Proposer-Male Responder Female Proposer-Female Responder

Observations (Solnick-MC) 65-448 36-216 29-232 38-184 27-264 22-88 14-128 16-96 13-136

Offer (Solnick-MC) 46.8-46.2 46.7-45.8 46.8-46.6 48.9-45.8 43.7-46.5 47.3-45.8 44.3-45.8 51.3-45.9 43.1-47.2

MAO Rejection Rate (Solnick-MC) (Solnick-MC) 31.0-36.8 7.7-16.3 25.9-36.8 4.2-17.6 37.3-36.8 14.6-15.1 28.1-36.4 5.8-16.8 34.2-37.1 13.5-15.9 24.5-36.6 4.5-18.2 28.2-37.0 0.0-17.2 33.9-36.2 6.3-15.6 41.5-37.2 23.1-14.7

Offers of an Even Split (%) (Solnick-MC) 72.3-56.5 72.9-52.3 68.3-60.3 81.6-55.4 59.2-57.2 81.8-52.3 64.3-52.3 81.2-58.3 53.8-61.8

Table 2: Summary Statistics for Solnick (2001) and All Periods of McGee and Constantinides (MC)

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All Pairings Male Proposers-All Responders Female Proposers-All Responders All Proposers-Male Responders All Proposers-Female Responders Male Proposer-Male Responder Male Proposer-Female Responder Female Proposer-Male Responder Female Proposer-Female Responder

Observations (Solnick-MC) 65-56 36-27 29-29 38-23 27-33 22-10 14-17 16-13 13-16

Offer (Solnick-MC) 46.8-48.0 46.7-50.0 46.8-46.1 48.9-46.5 43.7-49.0 47.3-52.0 44.3-48.8 51.3-42.3 43.1-49.0

MAO Rejection Rate (Solnick-MC) (Solnick-MC) 31.0-39.9 7.7-10.7 25.9-39.2 4.2-7.4 37.3-40.1 14.6-13.8 28.1-41.5 5.8-17.4 34.2-38.8 13.5-6.1 24.5-40.0 4.5-10.0 28.2-38.8 0.0-5.9 33.9-42.7 6.3-23.1 41.5-38.8 23.1-6.2

Offers of an Even Split (%) (Solnick-MC) 72.3-73.2 72.9-74.1 68.3-72.4 81.6-65.2 59.2-72.4 81.8-80.0 64.3-70.6 81.2-53.8 53.8-87.5

Table 3: Summary Statistics for Solnick (2001) and the First Period of McGee and Constantinides (MC)

Table 4: Marginal Effects for Offer, Minimum Acceptable Offer (MAO), and the Probability of Even Splits Regressions Dependent Variable Offer MAO Even Split 10.400*** 8.000*** 0.800*** (0.517) (1.674) (0.123) 9.765*** 7.765*** 0.706*** (0.222) (0.741) (0.115) 8.462*** 8.538*** 0.538*** (0.743) (0.420) (0.145) 9.812*** 7.750*** 0.875*** (0.138) (0.754) (0.086) -1.673 0.182 -0.436** (1.009) (1.777) (0.174) -1.150 -0.083 -0.300* (0.845) (1.562) (0.166) -0.673 -0.454 -0.073 (0.584) (1.177) (0.193) -1.218** -0.091 -0.254 (0.536) (1.546) (0.227) -1.400** -1.273 -0.345 (0.635) (1.383) (0.210) -2.700*** -2.600* -0.600*** (0.838) (1.405) (0.189) -1.233** -1.250 -0.217 (0.590) (1.985) (0.171) -0.577 0.048 -0.081 (0.389) (1.139) (0.136) -0.565 -0.698 -0.172 (0.382) (1.167) (0.122) -1.577** -0.577 -0.331** (0.649) (0.350) (0.145) -0.827*** -0.390 -0.393*** (0.272) (1.046) (0.138) -0.702 -0.452 -0.331** (0.538) (0.317) (0.148) -0.235 -0.353 -0.059 (0.195) (0.288) (0.097) -0.298 -0.565 -0.106 (0.218) (1.065) (0.115) 0.372 -1.205 0.128 (0.964) (1.548) (0.191) 0.811 -0.811 0.007 (0.739) (1.599) (0.154) 0.372 -0.705 -0.205 (0.787) (0.811) (0.197) 1.288 -1.788 0.045 (0.791) (1.609) (0.191) 1.372* -0.622 0.211 (0.769) (0.679) (0.167) 0.692 -2.769** 0.000 (0.847) (0.1.170) (0.203) 0.993 -2.629** 0.189 (0.757) (1.189) (0.183) -0.107 -0.279 -0.051 (0.303) (1.038) (0.132) -0.424 -0.250 -0.375*** (0.506) (0.999) (0.121) -0.342 -0.397 -0.169 (0.291) (0.919) (0.158) -0.401 -0.456 -0.463*** (0.428) (0.990) (0.138) -0.695*** 0.074 -0.404*** (0.247) (0.809) (0.139) -0.562* -0.438 -0.312* (0.300) (0.837) (0.162) -0.535** -0.694 -0.264** (0.260) (0.880) (0.130) 448 448 448 0.977 0.833

Variable Male-Male Male-Female Female-Male Female-Female Male-Male × Period 2 Male-Male × Period 3 Male-Male × Period 4 Male-Male × Period 5 Male-Male × Period 6 Male-Male × Period 7 Male-Male × Period 8 Male-Female × Period 2 Male-Female × Period 3 Male-Female × Period 4 Male-Female × Period 5 Male-Female × Period 6 Male-Female × Period 7 Male-Female × Period 8 Female-Male × Period 2 Female-Male × Period 3 Female-Male × Period 4 Female-Male × Period 5 Female-Male × Period 6 Female-Male × Period 7 Female-Male × Period 8 Female-Female × Period 2 Female-Female × Period 3 Female-Female × Period 4 Female-Female × Period 5 Female-Female × Period 6 Female-Female × Period 7 Female-Female × Period 8 N R2

* p<0.10, ** p<0.05, *** p<0.01 Note: The standard errors reported in parentheses are robust to clustering at proposer level for offers and even splits and the responder level for MAOs.

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Table 5: Marginal Effects for Offer and the Probability of Even Splits Regressions Excluding Offer Greater than Half the Pie Dependent Variable Variable Male-Male

Offer 9.889*** (0.110) 9.625*** (0.182) 8.462*** (0.744) 9.812*** (0.138) -1.589*** (0.561) -1.162** (0.531) -0.289 (0.287) -0.707** (0.347) -0.889** (0.400) -2.189*** (0.646) -0.722** (0.305) -0.438 (0.358) -0.554 (0.398) -1.438** (0.634) -0.825*** (0.292) -1.125*** (0.424) -0.188 (0.202) -0.268 (0.237) 0.372 (0.965) 0.811 (0.740) 0.372 (0.788) 1.084 (0.791) 1.266 (0.756) 0.692 (0.848) 0.993 (0.758) -0.188 (0.310) -0.812*** (0.262) -0.342 (0.291) -0.750*** (0.267) -0.695*** (0.248) -0.562* (0.300) -0.535** (0.260) 432 0.982

Male-Female Female-Male Female-Female Male-Male × Period 2 Male-Male × Period 3 Male-Male × Period 4 Male-Male × Period 5 Male-Male × Period 6 Male-Male × Period 7 Male-Male × Period 8 Male-Female × Period 2 Male-Female × Period 3 Male-Female × Period 4 Male-Female × Period 5 Male-Female × Period 6 Male-Female × Period 7 Male-Female × Period 8 Female-Male × Period 2 Female-Male × Period 3 Female-Male × Period 4 Female-Male × Period 5 Female-Male × Period 6 Female-Male × Period 7 Female-Male × Period 8 Female-Female × Period 2 Female-Female × Period 3 Female-Female × Period 4 Female-Female × Period 5 Female-Female × Period 6 Female-Female × Period 7 Female-Female × Period 8 N R2

Even Split 0.889*** (0.110) 0.750*** (0.113) 0.538*** (0.145) 0.875*** (0.087) -0.489*** (0.175) -0.343** (0.151) -0.089 (0.172) -0.343* (0.195) -0.434** (0.195) -0.689*** (0.180) -0.306** (0.150) -0.125 (0.132) -0.178 (0.133) -0.375** (0.141) -0.417*** (0.144) -0.321** (0.155) -0.062 (0.102) -0.107 (0.120) 0.128 (0.191) 0.007 (0.154) -0.205 (0.197) 0.098 (0.191) 0.280 (0.180) 0.000 (0.204) 0.189 (0.183) 0.000 (0.122) -0.346** (0.136) -0.169 (0.159) -0.438*** (0.140) -0.404*** (0.139) -0.312* (0.162) -0.264** (0.131) 432

* p<0.10, ** p<0.05, *** p<0.01 Note: The standard errors reported in parentheses are robust to clustering at proposer level.

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