The Value of Votes in Weighted Voting Games: An Experimental Study of Majoritarian Contests with Asymmetric Battlefields 1

Transcription

1 The Value of Votes in Weighted Voting Games: An Experimental Study of Majoritarian Contests with Asymmetric Battlefields 1 Abstract Maria Montero, Alex Possajennikov, Martin Sefton 2 University of Nottingham Theodore L. Turocy 3 University of East Anglia June 2013 We analyze the following lobbying game. Two lobbyists have identical budgets and simultaneously distribute them across voters in a legislature. Each voter votes for the lobbyist who pays them most and the lobbyist who receives most votes wins the game. Taking the share of the budget distributed to a voter as a measure of the voter s value or power, we investigate how this value varies with voting weights. We focus on apex games where there is one large voter and several small voters. We compute the full mixed equilibrium distribution of strategies for these games numerically. The lobbyists observed behavior is not perfectly in line with equilibrium predictions, but we find some support for two qualitative features of the equilibrium. First, the share of lobbyists expenditures on the large voter exceeds her relative voting weight, though this effect is not significant in all treatments. Second, lobbyists often distribute their budgets over supermajorities rather than over minimal winning coalitions. Keywords: experiment, contests, Blotto game, weighted voting, lobbying JEL Codes: D72, C72, C91 1 We would like to thank seminar participants at New York University, Keele University, University of East Anglia, and conference participants at the Voting Power in Practice Symposium at LSE 2011, M-BEES 2011, SING7 2011, Contest, Mechanisms and Experiments Conference at Exeter 2012, SAET 2012 and GAMES 2012 for their comments and the Nuffield foundation for financial support. 2 Contact details: maria.montero@nottingham.ac.uk, alex.possajennikov@nottingham.ac.uk, martin.sefton@nottingham.ac.uk. 3 Contact details: T.Turocy@uea.ac.uk 1

2 1. INTRODUCTION Consider a situation with four voters. One of the voters has 2 votes, the other three voters have 1 vote each, and 3 votes are needed for a majority. There are two types of minimal winning coalitions in this situation: the large voter together with one of the small voters, and the three small voters together. Suppose that voters do not care about how they vote, but instead can sell their vote to an interested agent. Because the voter with 2 votes can replace two voters with 1 vote, it seems natural that its market value would be twice the value of a voter with 1 vote. 4 In this paper, we study a strategic model to determine the market value of votes. We use a lobbying framework in which two lobbyists with identical budgets compete for the voters, and each voter is allocated to whoever pays them most. We then measure the worth of a voter by the expected amount they are offered in equilibrium. This framework was pioneered by Young (1978), who introduced it in the following way: An economic approach to the problem of measuring the relative power or value of members in a decision making body is to ask how much the various members would be paid by outside agents trying to manipulate the outcome. A particular case is that in which there are two competing agents, or lobbyists, on opposite sides of the issue (Young, 1978, p. iii). Young (1978) claims that it is not necessarily the case that market values are proportional to votes in equilibrium; in particular a voter with 2 votes is worth more than twice as much as a voter with 1 vote in the situation above. Similarly, most power indices give a disproportionate share to the large voter. The Shapley value (which is considered to be the most appropriate index when there is a resource to be distributed, see Felsenthal and Machover, 1998) predicts a more than proportional payoff to the large voter. Furthermore, if we consider the family of apex games to which the game in the above example belongs, the share predicted by the Shapley value converges to 1 as the number of voters increases, even 4 This idea is behind several classical concepts such as von Neumann and Morgenstern s (1944) main simple solution and the competitive solution of McKelvey et al. (1978). 2

3 though the proportion of votes controlled by the large voter is never above 0.5. Following Young (1978), we consider two games from the class of apex games: the four-voter game introduced above, and a five-voter game where one voter has 3 votes, four voters have one vote each, and 4 votes are needed for a majority. Young (1978) solved these two games assuming a finitely divisible budget (the size of which is not stated) and reported the expected amounts allocated to the large voter but not the equilibrium strategies. In both cases the large voter is predicted to receive a share of the budget above its proportion of the votes. We calculate equilibrium strategies numerically (assuming a budget of 120 indivisible units) and obtain results consistent with Young s predictions for the expected amount allocated to the large voter. We test these predictions in an experimental setup in which two lobbyists have the same budget of 120 units and must distribute them across voters simultaneously. We run three treatments that manipulate the number of voters in the apex game: a baseline treatment with three identical voters, and a 4-voter treatment and a 5-voter treatment corresponding to the examples above. We observe that the large voter gets a more than proportional average payoff for 4-voter apex games and 5-voter apex games, though the departure from proportionality is small and insignificant for the 4-voter case. The strategic lobbying approach also allows us to see who of the voters lobbyists target. Does a lobbyist concentrate funds on all the small voters, or on the large voter and one of the small voters? Or does a lobbyist hedge and spread the budget over all voters? At first sight it seems that, since a minimal winning coalition is sufficient to pass a proposal, there is no point in bribing a coalition larger than minimal winning. However, because a lobbyist does not observe the strategy of the other lobbyist, it may be optimal to spread the money over a larger coalition. Groseclose and Snyder (1996) and Banks (2000) make this point in a model where vote buyers move sequentially. We find that supermajorities also arise in the equilibria of our simultaneous-move lobbying game: lobbyists randomize between bribing several coalitions, most of which are larger than minimal winning. Supermajorities are also observed in our experiment, although not as often as equilibrium theory predicts. In the next section we review the background theoretical and experimental literature on weighted voting and strictly competitive games. Section 3 describes our experimental 3

4 design and procedures. Results are presented in Section 4 and Section 5 concludes. 2. WEIGHTED VOTING GAMES Let N = {1,2,, n} be the set of voters. Each voter i has w i votes, and a threshold q denotes the number of votes needed to pass a proposal. A subset S of N is called a winning coalition if i S w i q. If S is winning but no strict subset of S is winning, S is called a minimal winning coalition. The focus of our paper is on how the asymmetry of weights affects the value of votes. To analyze this question we focus on apex games. Apex games are the simplest asymmetric voting games and have one large (apex, major) voter and n 1 small (base, minor) voters. They can be described as weighted majority games in which the large voter controls n 2 votes, the n 1 small voters control one vote each, and n 1 votes are needed to achieve a majority. 5 There are two types of minimal winning coalitions: the large voter together with one of the small voters, and all the small voters together. There is an extensive literature in cooperative game theory on apex games. They games already appear in von Neumann and Morgenstern (1944, pp ) and are the focus of Davis and Maschler (1965). The Shapley (1953) value is one possible measure of voting power since it can be interpreted as a bargainer s share when voting over allocations of a pie. The Shapley value gives the large player in an apex game a share of (n 2)/n, more than the proportion of total votes that she controls. Alternative concepts have been proposed. For example the nucleolus (Schmeidler, 1969) gives the large player a share equal to her proportion of the votes, (n 2)/(2n 3). Note that cooperative solutions are based on the description of the game given above, and not on any specific features of the voting process. Early experiments with apex games used relatively unstructured bargaining 5 There are many equivalent representations for weighted majority games. For example, if four voters have 3, 2, 1 and 1 votes respectively and 4 votes are needed for a majority, the game is also an apex game. In what follows we use the homogeneous representation w 1 = n-2, w 2 = = w n = 1 and q = n 1 (this representation assigns votes in such a way that all minimal winning coalitions have the same number of votes). For apex games, the homogeneous representation is unique (up to rescaling). Note that even if we accept representations that are not homogeneous, the strong voter in an apex game cannot have more than half of the votes. 4

5 procedures and compared outcomes with those predicted by alternative cooperative concepts. For example, Selten and Schuster (1968) had twelve five-person groups bargain face to face over 40 DM. Minimal winning coalitions formed in 10 groups, and the large player featured in 8 of these. Subjects were instructed to try and earn as much money for themselves as possible, but even so allocations within minimal winning coalitions were more equitable than predicted. Across all groups, the large player earned on average 44.6% of the pie, only slightly more than the proportional share of the pie, 3/7 ( 42.9%). Voting power can also be analyzed via bargaining shares realized from noncooperative bargaining games, and these are perhaps particularly well-suited to laboratory experimentation. Fréchette et al. (2005a) examine apex games where five subjects divide a pie of $60, using two non-cooperative voting games. Their demand-bargaining treatment uses a variant of Morelli (1999) and has a unique equilibrium where each voter within a minimal winning coalition receives a share proportional to her voting weight, but large players are much more likely to feature in minimal winning coalitions. As a consequence the large player s expected share exceeds her nucleolus value. In another treatment they use the Baron and Ferejohn (1989) closed-rule protocol with recognition probabilities equal to relative voting weights. For this treatment equilibrium values correspond to nucleolus values (see Montero, 2006). The experiment shows that strategic considerations play an important role in determining outcomes. However, behavior does not differ so much across protocols as predicted by equilibrium. Moreover, in contrast to theoretical predictions, but in line with earlier apex game experiments, allocations within winning coalitions are more equitable than predicted. These findings are also consistent with other experiments on voting games (e.g., McKelvey, 1991, Diermeier and Morton, 2004, Fréchette et al., 2005b, Fréchette et al., 2005c, Montero et al., 2008, Kagel et al., 2010, Drouvelis et al., 2010, Miller and Vanberg, 2011). A recurrent finding from these experiments is that voters are often willing to exclude other voters and allocate nothing to excluded voters, consistent with a disregard for other voters earnings, and at the same time allocations within winning coalitions are more egalitarian than predicted by equilibrium analysis with selfish preferences. In summary, experiments using a bargaining framework find that realized shares 5

6 reflect both strategic features of the bargaining process and uncontrolled preferences over distributions of earnings. In the next section we describe an alternative framework. 3. THE LOBBYING GAME Our lobbying game is based on the model of Young (1978). In this model, the voters in the weighted voting game must vote on a bill. The bill passes if the total number of votes in favor is at least q; otherwise the status quo prevails. In addition to the n voters there are two lobbyists, one of which favors the proposed bill whereas the other wants the status quo to prevail. Each lobbyist has a budget to be allocated across voters. The size of the budget is the same for both lobbyists and is denoted by B. The lobbyists allocate their budgets simultaneously. The voters do not care about how they vote and do the bidding of whoever pays them most. The lobbyists are assumed to spend their entire budget; this would be the case if they care a lot about the outcome but are budget constrained. In the words of Young (1978, pp. 1-2) winning or losing is assumed to be of incomparably greater value than the prices paid. where lobbyist j if j j j j A pure strategy of lobbyist j is ( x 1,..., xn ) such that x i 0 for all i and x i B, j x i is the amount allocated to voter i. Voter i votes for the alternative favored by x x and for the other alternative if j i k i 6 x x j i k i ; if j i k i i N x x the voter decides at random: he votes for the bill with probability 0.5 and against it with probability 0.5. From the point of view of the lobbyists, there are only two possible outcomes of the game: either the bill passes or the status quo prevails. If we normalize utilities so that the utility of getting the preferred outcome is 1 and the utility of getting the other outcome is 0, the game between lobbyists is a constant-sum game. Each lobbyist maximizes its expected utility by maximizing the probability of getting its preferred outcome. Note that since there are only two possible outcomes this prediction is independent of the risk attitude of the lobbyists. Following Young (1978), we will use the total lobbyists expenditures on a voter as a measure of the worth or power of this voter. We can compare the value of votes held by the apex voter and a minor voter and determine whether a bundle of votes has a value over and above what one would expect if there was a uniform price per vote.

7 The lobbying game provides a quite different approach to measuring power compared to bargaining. Lobbyists move simultaneously in order to minimize any asymmetries due to the order of moves. To minimize the effect of preferences in the experiment, the lobbying approach pits subjects (as lobbyists) against each other in strictly competitive games, where experimental studies have found the impact of other-regarding preferences to be less pronounced (see Camerer, 2003, for a review). The lobbying game is a particular case of conflict with multiple battlefields (see Kovenock and Roberson (2012)). Performance in this class of conflict is measured by the players success or failure in the component conflicts or battlefields (a battlefield corresponds to a voter in our case). Conflicts with multiple battlefields are classified into groups according to the following criteria: contest success functions (CSF), objective functions and cost functions. Since the highest bidder for each voter wins the voter for sure, our game has an auction CSF. Each lobbyist aims to win a majority of the votes, hence both lobbyists have a majoritarian objective. As to the cost function, lobbyists are budget constrained and their resources are treated as use-it-or-lose-it. Our game is closely related to the Colonel Blotto game, which is notoriously difficult to solve. In the more standard formulation of the Colonel Blotto game, the bidders strategy sets are the same but the bidders payoff is the total value of battlefields won. 6 In contrast, in Young s lobbying game the bidders do not care about the number of votes won as long as it is enough for winning overall, and thus maximize the probability of winning a majority of the votes. The Colonel Blotto game and its variants have applications in many disciplines, 6 To our knowledge, the first study of a similar game is in Borel (1921), where he considers three identical battlefields and a majoritarian objective. Later studies introduced the term Colonel Blotto and adopted the value-of-battlefields-won objective. For this formulation of the Colonel Blotto game, Roberson (2006) shows that the marginal univariate distributions of the budgets must be uniform in any equilibrium when all battlefields are identical. Thomas (2009) analyses the case of asymmetric battlefields and shows that uniform marginals are a sufficient condition for equilibrium. 7

8 including political science (see the survey by Kovenock and Roberson (2012)). Many papers have studied models of lobbying, electoral campaigns and distributive politics (e.g. Groseclose and Snyder (1996), Diermeier and Myerson (1999), Banks (2000), Dekel et al. (2009), Le Breton and Zaporozhets (2010), Brams and Davis (1974), Snyder (1989), Klumpp and Polborn (2006), Laslier (2002), Laslier and Picard (2002), Roberson (2008)) but none of these models deals with the equilibrium of the simultaneous-move lobbying game with a majoritarian objective, identical budgets and asymmetric voters. 4. THE EXPERIMENTAL GAMES The description of the lobbying game above does not impose any restriction on the amounts allocated to a given voter. We are not aware of any analysis of the equilibrium of the game with infinitely divisible budget for apex games. However, it is easy to show that the equilibria with uniform marginals found by Thomas (2009) for the more standard Blotto game do not carry over to our game. 7 Young (1978) reports the expected amount allocated to each voter in equilibrium in two instances of apex games (n = 4 and n = 5) for a finitely divisible budget, the size of which is not reported in the paper. Voters are not subjects in our experiment; they are described as objects that are allocated to the highest bidder. In the experiment we have apex games with n = 4 and n = 5 (and also the degenerate apex game with n = 3 where all players are symmetric). The set of possible allocations is necessarily discrete as in Young (1978). Lobbyists in our experiment have a budget of 120 indivisible units. We calculated equilibrium strategies computationally. 7 Suppose the total size of the budget is 5 (the total size does not matter since the budget is continuously divisible). Consider a strategy such that the amount allocated to the apex voter is uniformly distributed between 0 and 4, and the amount allocated to each small voter is uniformly distributed between 0 and 2. An example of such a strategy would be (4-4ε, 2ε, ε, 1+ ε) with probability 0.5 and (4-4ε, 2ε, 1+ ε, ε) with probability 0.5, where ε is uniformly distributed between 0 and 1. Any such strategy can be bettered by a strategy that puts 2 on one of the minor voters and 3 on the apex voter. This alternative strategy wins exactly one minor voter with probability 1, and the apex voter with probability 3/4, hence it wins with probability 3/4 overall. 8

9 In doing so, we focused on voter-symmetric strategies. In the degenerate apex game with symmetric players (n = 3), a voter-symmetric strategy is a mixed strategy that puts equal weights on all possible permutations of a given allocation. For example, one possible votersymmetric strategy consists in playing each of (60,60,0), (60,0,60) and (0,60,60) with probability 1/3. In the other two apex games the permutation only affects players of the same type, i.e., the minor players. One possible voter-symmetric strategy with n = 4 consists of playing each of (80,40,0,0), (80,0,40,0) and (80,0,0,40) with probability 1/3. Note that the equilibrium of the game with voter-symmetric strategies is also an equilibrium of the original game. Even with this restriction, the number of available strategies is rather large 8. We obtained results consistent with Young s predictions for the expected amount allocated to the large voter. Table 1 summarizes the equilibrium expected share for the large voter depending on the number of voters. The table also includes the proportional prediction (i.e., the share of the total votes controlled by the large player) and the Shapley value for comparison. The equilibrium predictions are clearly above the proportional prediction. Number of voters Table 1. Voting powers of large voter Proportional Shapley Value Equilibrium There have been several experimental studies on variants of the Colonel Blotto game. 9 The most closely related to ours are Avrahami and Kareev (2009), Chowdhury et al. (2013), and Arad and Rubinstein (2012). As our experiment, those papers have an auction CSF with budget-constrained use-it-or-lose-it costs; the main difference with our experiment is that we 8 The number of voter-symmetric strategies for B = 120 is approximately 1,200 (n = 3), 52,000 (n = 4) and 430,000 (n = 5). 9 Dechenaux et al. (2012) survey the experimental literature on contests more generally. 9

10 have a majoritarian objective and asymmetric battlefields. Avrahami and Kareev (2009) run an experiment with 8 battlefields. One battlefield is selected at random for each player, and the player having more tokens in his/her selected battlefield wins. They focus on asymmetries in the two players resources (each player may have 12, 18 or 24 indivisible tokens). The equilibrium prediction is that the weaker player should give up on a certain proportion of battlefields so that he can have an equal chance against the stronger player on the remaining ones. They observe that the weak player is more likely to leave a battlefield empty and this likelihood increases with the budget available to the other player and also over time (their experiment has 8 rounds). Chowdhury et al. (2013) also look at asymmetric resources, but with a much finer grid (200 and 120 tokens, with a mesh of 0.5 tokens) and a longer time horizon (15 rounds). They also have 8 battlefields, but unlike in the previous experiment all 8 battles are played out. The main purpose of their experiment is to compare the auction CSF with a lottery CSF. In the auction treatment the stronger player should allocate a random but positive amount to all battlefields, while the weaker player should allocate zero to a randomly selected subset of battlefields; in the lottery treatment both players should divide their tokens equally between all battlefields. They find that the probabilities of winning for players 1 and 2 are as predicted by the equilibrium, and the bidding strategies differ across treatments in the direction predicted. In Arad and Rubinstein (2012), each subject allocates 120 troops to 6 battlefields. In order to encourage deep thinking, they modify the game in two ways: first, nobody wins the battle if both subjects allocate the same number of troops to a battlefield; second, the game is played as a round-robin tournament in which each subject s allocation is pitted against everybody else s. They interpret the observed choices as reflecting iterated reasoning in several dimensions. Hortala-Vallve and Llorente-Saguer (2010) consider a modification of the Colonel Blotto game in which the two players have cardinal valuations of the battlefields, and these valuations are private information. The main question in their paper is that of efficiency. They find that observed behavior is far from equilibrium, but the realized welfare is similar to the equilibrium prediction. Duffy and Matros (2013) consider contests with asymmetric 10

11 battlefields and a lottery CSF; unlike in the auction CSF, the equilibrium is in pure strategies. In some of their treatments a player s payoff is the total value of the battlefields won; in others the players have a majoritarian objective in a situation similar to our apex games. Some of the battlefields are interchangeable in terms of the majoritarian objective even though they have different nominal values. Experimental data support the equilibrium prediction and subjects largely disregard these nominal asymmetries between battlefields. All those experimental papers have use-it-or-lose-it budget constrained costs. There are also some experiments with linear costs. Mago and Sheremeta (2012) consider contests with three identical objects, a majoritarian objective and linear costs. Their focus is the comparison of the simultaneous version of the contest (based on Szentes and Rosenthal (2003)) and the sequential version (based on Konrad and Kovenock (2009)). They observe significant deviations from equilibrium in both cases and a substantial amount of overdissipation. Kovenock et al. (2010) consider weakest-link contests with linear costs, based on theoretical work by Kovenock and Roberson (2010). There are four targets and the attacker needs to win one of them, whereas the defender needs to defend all of them. They observe substantial overdissipation, but behavior is close to equilibrium in other dimensions (e.g. the attacker attacks at most one target, the defender covers either all targets or none most of the time, and the frequency with which players allocate zero to every target is close to the equilibrium prediction). Even though the best shot/weakest link contest is motivated by the optimal defense of networks against terrorism, it can also be interpreted as a game between two lobbyists, one of which favors the status quo and the other favors the alternative, if the alternative can only be approved by unanimity. Casella et al. (2012) also study a market for votes experimentally, though this market is very different from ours. There are no outside lobbyists, but voters have cardinal preferences over the outcome and can either buy or sell votes from other voters. Given that the Walrasian equilibrium does not usually exist in this setup, they define a new competitive equilibrium concept that combines price-taking behavior with less standard assumptions: traders best respond to the demands of other traders, mixed demands are allowed, and market clearing holds only in expectation so rationing is typically required ex post. They test this 11

12 theoretical prediction using a continuous double auction, and find that the predicted welfare effects of the market for votes are realized. All voters are initially endowed with one vote each, and the concept of competitive equilibrium they introduce assumes a fixed price per vote, hence the question of whether large voters can obtain a disproportionate payoff does not arise in their setup. 5. EXPERIMENTAL DESIGN AND PROCEDURES The experiment was conducted at the University of Nottingham with 148 subjects recruited from a university-wide pool of undergraduate students using ORSEE (Greiner, 2004). The experiment consisted of nine computerized sessions, with no subject participating in more than one session. The experiment was programmed in z-tree (Fischbacher, 2007). All sessions used an identical protocol. Upon arrival, subjects were given a written set of instructions that the experimenter read aloud 10. Subjects were not allowed to communicate with one another throughout the session. At the beginning of the session subjects were randomly paired to play one of the treatments for 45 rounds. Subjects were not told who of the other people in the room was paired with them, but they knew that they were playing the same subject throughout. Keeping subjects in the same pairs allows us to treat each pair as an independent observation since subjects in one pair cannot influence or be influenced by the decisions of subjects in any other pair. The subjects were required to distribute 120 tokens among objects, each of which was worth a given number of points. 11, 12 An object in the experiment corresponds to a voter in the model; in particular object A always represents the apex player. The points an object is worth correspond to the votes a voter controls. An object is won if a subject allocates more 10 Full instructions for one of the treatments can be found in Appendix A. 11 We also run some sessions with a very coarse budget of 5 indivisible units. The equilibrium predictions and experimental results for these treatments are qualitatively similar and can be found in appendix B. 12 Subjects had 90 seconds to submit an allocation. If subjects timed out, the computer made a default decision allocating zero tokens to each object. Across all sessions only 28 out of 6,660 allocation decisions resulted in a timeout. 12

13 tokens to it than the opponent, or, if both subjects allocate the same number of tokens, if the subject wins the random computer draw. The subject that wins the most points in a given round is paid At the end of each round, subjects were informed of how much they bid for each object, how much the opponent bid, who won each object and whether it was a random draw. Note that, since a subject either wins 0.50 or nothing in each round, equilibrium strategies are independent of risk preferences (Wooders and Shachat, 2001). Treatments differ in the number of voters/objects (3, 4 or 5) as summarized in Table 2. We refer to these treatments as AP3, AP4 and AP5. There were three sessions for each treatment. Each session took approximately 1.5 hours and subjects earned on average (about $17 at the time of the experiment). Table 2. Experimental treatments Treatment Number of objects Number of pairs Number of subjects AP AP AP RESULTS 6.1 Predicted and observed allocations to voters The three-voter case In the three-voter game voters are symmetric, and the lobbying game is similar to a Colonel Blotto game. 13 In equilibrium, the players use mixed strategies where the marginal distribution of tokens on each voter is approximately uniform, in our case on {0,,80} Recall that in a Colonel Blotto game a player s payoff is the sum of the values of objects won. However with three objects a lobbyist will typically win either one or two of the three objects rather than all three or none, so the game is very close to a game with only two possible outcomes like ours (the only possible exception is the case in which both lobbyists submit exactly the same bid for all three objects, in which case tie-breaking may result in one of them winning all three). 14 According to Hart (2008), this distribution has different weights on odd and even numbers, but since we report the distributions over intervals of size 10, these different weights average out. 13

14 Figure 1 displays the equilibrium and the empirical distribution of allocations (all three voters are pooled together). Figure 2 displays the empirical distribution of allocations to each of the three voters separately. Figure 1. Predicted and observed allocations in the three-voter game Figure 2. Observed allocations to each voter in the three-voter game The distribution has pronounced concentrations around 0-5 and and is far from uniform. Similar findings are reported for Colonel Blotto game and related experiments (Avrahami and Kareev (2009), Chowdhury et al. (2013), Mago and Sheremeta (2012)) and all-pay auctions (Potters et al. (1998), Gneezy and Smorodinsky (2006), Ernst and Thöni, (2009)). The distributions for all three voters are similar though subjects allocated slightly more to voter 1 than to voter 2 than to voter This is consistent with experimental findings on the Colonel Blotto game. Chowdhury et al. (2013) observe that subjects on average allocate more tokens towards the first boxes (see their table 3 and figure 3). Arad and Rubinstein (2012) also observe positional effects: subjects tend to allocate more tokens to the central battlefields 3 and 4, and particularly few tokens to battlefield 6. Interestingly, successful strategies in the tournament also allocated more tokens to the middle battlefields (recall that the objective in their game is to win as many objects as possible on average, rather than a majority of the objects). Avrahami and Kareev (2009) choose a field at random for each player separately, hence positional considerations do not arise in their experiment. 14

15 The four-voter case We computed equilibrium strategies numerically. It is difficult to describe the entire equilibrium distribution, therefore we present the marginal distributions of tokens allocated to the large voter and to a small voter in Figure 3. Unlike in the Colonel Blotto game with symmetric battlefields, the marginal distributions do not characterize the equilibrium: there may be non-equilibrium distributions that produce the same marginals. Figure 3 compares the equilibrium predictions with the marginal distributions actually observed in the experiment. Figure 3. Predicted and observed allocations (four-voter treatment) The observed distribution for the large voter is bimodal and has a substantial mass around 0-5 and 56-85, while the equilibrium distribution is broadly increasing up to around 105. Subjects in the experiment allocated a small amount to the large voter too often compared with the equilibrium prediction. The observed allocation is qualitatively similar to the one observed in the three-voter case (see Figure 2). The observed strategies either allocate little to the large voter, or they allocate an amount that is close to 2/3 of the budget. This may have appeared natural to the subjects, since 3 votes are needed to win and the large voter is 15

16 worth 2 votes. Analogously, a small voter is often allocated an amount close to 0 or an amount close to 1/3 of the budget. This again may have appeared natural since a small voter controls 1 vote and 3 votes are needed to win. The five-voter case As for the four-voter game, Figure 4 compares the equilibrium predictions (computed numerically) with the marginal distributions actually observed in the experiment. Figure 4. Predicted and observed allocations (five-voter treatment) Similarly to the four-voter treatment, the observed distribution for the large voter is bimodal and has a substantial mass around 0-5 and , while the equilibrium distribution is broadly increasing up to around 115. Experimental subjects gave the large voter a negligible amount too frequently, while giving small voters a moderately high amount too often. The observed allocations have two peaks, one close to 0 and another close to 3/4 of the budget (for the large voter) and 1/4 of the budget (for a small voter). These peaks are consistent with subjects allocating most of the budget across a minimal winning coalition and 16

17 dividing it proportionally to the number of votes within this coalition. 6.2 The value of a vote A qualitative equilibrium prediction is that the large voter receives a superproportional payoff, thus the price of a vote may differ across voters. It is clear from the observed marginal distributions that the average budget allocation to the large voter is below the equilibrium prediction in the four and five-voter treatments. The following figure shows the c.d.f. of the observed versus the theoretical distribution for the large voter in these treatments; the theoretical c.d.f. is below the observed one for most of the values. Figure 5. Predicted vs observed CDFs of allocations to the large voter Even though the allocation to the large voter is clearly below the equilibrium prediction, the question remains of whether it is superproportional to the number of votes. The following figure looks at the payoff per vote for the large voter relative to the payoff per vote for a small voter. For example, in the four-voter treatment, a proportional payoff would allocate 48 points to the large voter and 24 points to a small voter; each of the voters then receives 24 points per vote and the ratio is 1. The equilibrium prediction is that the large voter receives around 60 points and a small voter receives around 20 points; the large voter then gets 30 points per vote compared to the small voter s 20 points per vote. The equilibrium prediction is superproportional and the ratio is

18 Figure 6. Relative allocation per vote across treatments Figure 6 shows a small positional advantage for object A in AP3. The allocation ratio is essentially identical in AP4, hence the small departure from proportionality can be attributed to a positional effect rather than to strategic considerations. Allocations are clearly superproportional in AP5, though even in this case they are well below the equilibrium prediction. Figure 7 shows how the share of budget allocated to object A evolves over the 45 rounds of the game for all treatments. The red (dark) line shows the proportional share, and the green (pale) line shows the equilibrium share (when this is different from the proportional share). 18

19 Figure 7. Share of budget allocated to object A over time 0.6 Share of budget allocated to the large voter in AP Proportional Equilibrium Observed 19

20 To make sure that we concentrate on stable behavior, we test for the presence of trends in the data. Taking each matching pair as an observation, we calculate the correlation coefficient between object A s share and round. There is a marginally significant downward trend in AP3 (two-sided p-value: ), consistent with the initial positional advantage of object A disappearing in later rounds. There is no significant trend in AP4 (two-sided p- value: ), but there is a significant downward trend in AP5 (two-sided p-value: ). Hence, subjects are moving away from equilibrium over time in AP5, though as we will see in the next subsection there are other measures by which observed play is becoming closer to equilibrium in all treatments. Looking at the second half of the experiment (last 22 rounds) there are no significant trends in any of the treatments (two-sided p-values: for AP3, for AP4, 1 for AP5). Table 3 presents the share of the budget allocated to object A in each treatment, averaging across all rounds and pairs. For statistical tests of proportionality we use sign tests treating each pair as an independent observation. Our alternative hypothesis is that the large voter s voting power is greater than proportional, as predicted by equilibrium, and so we report one-sided p-values. The exception is the three-voter treatment in which object A is worth the same number of points as the other two objects, hence any difference in observed share with other objects must be due to positional effects. We also report one-sided p-values for this treatment for comparability with the other two treatments. Number of voters Table 3. Budget share allocated to Object A (large voter) Proportional Share (%) Equilibrium Share (%) 20 Number of pairs Observed Share (%) All Last 22 rounds rounds ** * *** 48*** *, **, *** significantly different from proportional at the 10%, 5%, 1% level Object A has a small but significant positional advantage in the three-voter treatment (all rounds: p = ). This advantage becomes insignificant in the last 22 rounds of the game (p = ).

21 Consistent with Figure 7, in the four-voter games the large voter s share is only slightly higher than her proportional share (all rounds: p = ; last 22 rounds: p = ). In the five-voter games the large voter gets a significantly higher than proportional share in all rounds (p = ) and in the last 22 rounds (p = ). After using this conservative approach to test our main hypothesis, we analyze individual allocations to the large voter (object A) using multivariate analysis. We use random effects regressions with robust standard errors, separately for each of our three treatments: AllocA AllocA OppAllocA Log( RTime) it 0 1 i( t 1) 2 i( t 1) 3 1/ t ) D u, 4 ( 5s s i it s where AllocA it refers to the number of tokens allocated by subject i to object A in round t, AllocA i(t-1) is the same variable lagged, OppAllocA i(t-1) is the corresponding lagged variable for the opponent of subject i, Log(RTime) it is the logarithm of the response time, and D s is a session dummy. Table 4. Determinants of allocation to the large voter (Object A) Dependent variable: AllocA it AP3 AP4 AP5 AllocA i(t-1) (0.180) (0.862) (0.671) OppAllocA i(t-1) * * (0.078) (0.371) (0.090) Log(RTime) it *** *** (0.798) (0.000) (0.000) 1/t ** *** *** (0.019) (0.001) (0.007) Constant *** *** *** (0.000) (0.000) (0.000) # Observations 2,024 2,262 2,143 # Subjects p-values in parenthesis based on robust standard errors; *significant at 10%, **significant at 5%, ***significant at 1%. 21 it

22 We excluded any observations in which a subject timed out either in the current round or in the previous one, and observations in which the subject s opponent timed out in the previous round (observations in which the opponent timed out in the current round are not affected). The regressions include session dummies, which are insignificant in all cases. Our findings indicate that that there is no autocorrelation in the amount allocated to object A. Recall that we have partner matching, hence these results are consistent with the reduced autocorrelation in partner treatments in Chowdhury et al. (2013) compared to their stranger treatments. The opponent s lagged allocation is marginally significant in some of the treatments. In equilibrium there should be no effect of this variable, but behaviorally there could be a reaction of allocating less to object A when the opponent has allocated more. The response time variable is strongly significant in the two treatments with asymmetric objects, indicating that subjects that took longer to decide allocated fewer tokens to object A. The variable 1/t is designed to model a diminishing trend, evidence of which we observed in the previous nonparametric analysis; the regression finds a significant negative trend in all treatments Supermajorities versus minimal winning coalitions Since lobbyists only need to win a majority of the points, it would appear optimal to concentrate bids on a minimal winning coalition rather than on a broader set of voters. We define a minimal winning coalition strategy as a pure strategy such that the set of voters receiving a strictly positive allocation constitutes a minimal winning coalition. If this set is a winning but not minimal winning coalition, we speak of supermajority strategies. Supermajority strategies can be optimal because of the uncertainty about the strategy of the other lobbyist. For example, consider the game with three voters and suppose the other lobbyist randomizes equally between bidding (60,60,0), (60,0,60) and (0,60,60). A lobbyist playing exactly the same strategy would win half of the time, whereas a lobbyist playing (61,58,1) would win with probability 2/3. Indeed, any two voter-symmetric strategies that 16 Restricting the regression to the last 22 rounds, we find that the trend variable is not significant anymore, whereas the significance of other variables remains broadly unchanged. 22

23 consist of permutations of minimal winning coalition strategies tie against each other, so it is the availability of supermajority strategies that makes the game nontrivial. Supermajority strategies are predicted over 90% of the time in equilibrium. For the three-voter treatment, any strategy that is not a supermajority strategy is automatically a minimal winning coalition strategy (except for timeouts and strategies that concentrate the entire budget on one voter, which happened only once in the experiment). 42% of the observed divisions in AP3 correspond to minimal winning coalitions. 17 For the other two treatments we can divide the observed minimal winning coalition strategies into MWC1 (the large voter together with one of the small voters) and MWC2 (all small voters together). The predicted frequency of MWC1 is approximately 5% in AP4 and 7% in AP5. The predicted frequency of MWC2 is less than 1% in both treatments. The observed frequencies are much greater than the predictions: about 19% of observed distributions in AP4 correspond to MWC1, and around 13% to MWC2. In AP5 the values are 20% and 11% respectively. The proportion of supermajority strategies observed in our experiment increases over time, but remains well below the equilibrium prediction. Figure 8 presents the evolution of the proportion of supermajorities and minimal winning coalitions over time in all three treatments. 17 Mago and Sheremeta (2012) find similar results in their majoritarian contest with linear costs. Subjects should make positive bids on all objects, but 35% of the time they bid only on two out of three. 23

24 Figure 8. Proportion of supermajority versus MWC strategies over time 24

25 6.4 Iterative reasoning: popular strategies and unit digits Subjects in our experiment used many strategies. In what follows we focus on popular strategy types. We consider all permutations of a particular pure strategy as one strategy type. For example, denotes the set of strategies {(80, 40, 0, 0), (80, 0, 40, 0), (80, 0, 0, 40)}. This is similar to the concept of voter-symmetric strategies, except that there is no presumption that different permutations have the same frequency. The following tables show the most popular strategy types in each treatment, pooling across all subjects and rounds. Popular strategies in the three-voter case Table 5. Popular strategy types in the three-voter treatment Rank Strategy types Frequency Average Payoff Average Response (n = 2070) (actual) (expected) Time (sec.) Standard k-level reasoning (see Stahl and Wilson, 1995) is impracticable in our setup 18, but as in Arad and Rubinstein (2012) it is possible to interpret some of the patterns 18 Calculating the best response to a uniform allocation of tokens would be very difficult, even if we agree on what constitutes a uniform allocation of tokens: each token being allocated at random, each strategy in a huge normal form game being played with equal probability, each voter-symmetric strategy being played with equal probability 25

26 observed as coming from a form of iterated reasoning. Two caveats are in order: first, we are pooling data over 45 rounds, and iterative reasoning may be more natural for games that are played for the first time; second, the complicated structure of the game does not allow for a one-to-one identification between strategies and levels of reasoning: the same strategy may be a very naïve choice or a best response found after several levels of iteration. The simple strategy was played 3 out of 46 times in round 1, and around 2% overall. This strategy can be seen as a natural starting point, analogous to level 0. It can be beaten by many others, the most salient being strategy type , which was also the most popular strategy type in round 1, played 15 out of 46 times ( appeared 7 times in round 1; it also beats and ties against ). Strategy type appears to be specifically designed to be played against strategy type , and wins with probability 2/3 against it. Note that the top 5 strategy types account for about 46% of observed play. Notice also that , and have shorter response times than other strategies such as and This is consistent with these strategies being associated with lower levels of reasoning, and in particular with being the instinctive choice. Since being predictable is a disadvantage in this game, it is not surprising that popular strategy types would have low average payoffs when pitted against the empirical distribution (similar results are found by Arad and Rubinstein (2012)). The exception is strategies requiring a higher level of reasoning ( , and ). Popular strategies in the four-voter case Table 6. Popular strategy types in the four-voter treatment Rank Strategy types Frequency Average Payoff Average Response (n = 2340) (actual) (expected) Time (sec.)

27 Interpretation of the popular choices as evidencing iterated reasoning is more difficult here. Strategy appears to be the obvious starting point, though less so than in AP3 since the four objects are asymmetric. Although this strategy was not played in round 1 at all (the proportional strategy was played once), it still has the shortest response time in the table. The next step would be to concentrate forces on a minimal winning coalition and beat Strategy can be seen as one of the simplest best responses to this strategy, and it is the most popular strategy overall and also in round 1 (6 out of 52 times); is another fairly popular strategy type that focuses on a minimal winning coalition. The next step may be to still focus on a minimal winning coalition, but trying to beat other minimal winning coalition strategies. For example, could be seen as a level 2 strategy type that does well against both and , though it is not the best possible response. Notice that strategies and , which can be seen as at least level 2, are the most successful strategies in the table in terms of expected payoffs against the empirical distribution. It is not clear how to classify strategy type It can be seen as a relatively naïve strategy type that focuses on the minimum set of objects required and allocates tokens proportional to their value, but it can also be seen as a better response to and In general, if we expect the other player to focus on the large object and one small object, a better reply would be to put more tokens on the large object (for example, type wins with probability 11/12 against type ). At some point strategy (and even ) would emerge as a better reply, hence it is not possible to say with certainty that is the result of simple level 1 reasoning, especially in later rounds. Another possible next step is that, if the other player is likely to focus on a minimal 27

28 winning coalition, a player may be better-off by putting some tokens on all objects. For example, strategy is a best response against other popular strategy types like , and (though it of course loses against seemingly more naïve strategies like and ). Supermajority strategies that are not uniform can be interpreted as the result of at least two steps of reasoning, since they appear to be designed to be played against minimal winning coalitions. The five most popular strategy types account for just 18% of observed play, so players are much less predictable than in AP3. 38% of observed play corresponds to strategies that were played only once, and even in round 1 there is very little repetition (the 52 choices in round 1 correspond to 35 different strategy types). Popular strategies in the five-voter case Table 7. Popular strategy types in the five-voter treatment Rank Strategy types Frequency Average Payoff Average Response (n = 2250) (actual) (expected) Time (sec.) Similarly to the previous treatments, we can think of the uniform distribution as the starting point; this is perhaps less natural than in other treatments since not only are the objects asymmetric but the resulting numbers are not round. This strategy was 28

29 played once in round 1 and had a frequency of overall (16 th most frequent strategy type overall) and one of the shortest response times at sec. The next step would be to divide the budget equally within minimal winning coalitions. Strategy is the most popular strategy overall was not popular, but other strategies that allocate the budget to the large object and one small object were: the second and third most popular strategy types, and , have this feature. As in the previous treatment, a better response to this type of strategy would allocate more tokens to the large voter, until becomes a better response. Strategy type turned out to be quite successful in this respect, since it is a better response to many of the popular strategy types. Another possible step is to spread the budget over the entire set of objects. Three of the popular strategy types, , and are of this sort. Strategy seems more naïve than the other two, since it is only one step away from the simple minimal winning coalition strategy The other two strategies can be seen as a combination of escalation of the amount of tokens on the large object while putting some tokens everywhere. Even though they appear to be deep, these two strategy types have the fastest response times in the table. This may be partly due to these strategy types not requiring the additional decision of which small object to favor over the other small objects. Strategies seem deeper here than in the other treatments. Besides the lack of popularity of some of the obvious strategies like and , response times are considerably longer. The average response time over the whole treatment is sec, compared to sec for AP4 and sec for AP3; these average response times are much longer than justified by the mere fact of having more objects. Similarly to AP4, play is much less predictable than in AP3. The five most popular strategy types account for 15% of play, and the 50 choices in round 1 correspond to 36 different strategy types. Unit digits over time Many of the observed strategies used round numbers (unit digit 0), but the frequency of unit 29

30 digit 0 decreases over time. The next figures show the proportion of unit digits 0, 1 and 2 over time. Figure 9 refers to treatment AP3 pooling all voters together. Figure 10 pools treatments AP4 and AP5 together (the figures for each of the treatments are very similar) but separates the large and small voters. The proportion of unit digit 0 decreases over time in all treatments, but this is more pronounced in AP4 and AP5. Subjects appear to have thought more deeply in AP4 and AP5, both in terms of strategies being more varied and in terms of digits other than 0 being used. Figure 9. Proportion of unit digits over time in AP3 Figure 10. Proportion of unit digits over time across AP4 and AP5 30

31 6.5 Measures of Deviations from Equilibrium Difference in wins For all treatments the lobbying game is a symmetric constant-sum game and so in equilibrium each lobbyist wins with probability 1/2 in any play. This means that a player expects to win 22.5 out of the 45 games. In fact, some do considerably better than this. For example, in one of the pairs one subject won 11 rounds and the other won 34 rounds, so that the difference in wins was 23. Figure 11 shows the observed frequencies of each possible value of the difference in wins. For comparison the theoretical distribution is also shown. Figure 11. Theoretical and Observed Distributions of Differences in Wins The figure shows that fewer than expected pairs have a small difference in wins and more than expected have a large difference in wins. Theoretically, the expected difference in wins is 5.38, while in the data the average difference in wins is This difference is significant (Chi square test p = 0.028). The obvious interpretation is that some subjects are better than others at playing the lobbying game. Note that this measure of deviation from equilibrium is quite conservative. It may happen that each subject wins with approximately equal probability even though strategies 31

32 are far from equilibrium. In the next subsection we consider another measure of deviation from equilibrium. Beating the empirical distribution Another way to measure how far observed play is from equilibrium is to compute the best response to the empirical distribution of strategies and its associated expected payoff. Intuitively, the greater the expected payoff one can obtain against the empirical distribution, the further the distribution is from equilibrium play. We compare the exploitability of the empirical distribution in different stages of the experiment. AP3 AP4 AP5 Strategy Payoff Strategy Payoff Strategy Payoff Table 8. Best reply against the empirical distribution Overall Rounds 1-15 Rounds Rounds ; For all treatments, it is possible to find (voter-symmetric) strategies with an expected payoff of over 0.6. The exploitability diminishes over time except for the case of AP5 between the middle and the last third of the experiment. Strategies that would have been successful in AP3 allocate a reasonably large amount to one of the objects and a moderate amount to the other two. For example, would beat and win with probability 2/3 against many of the popular strategies, such as , and In AP4, strategy beats , , and , even though it only wins with probability 1/3 against This strategy beats the modes of 70 (for the large voter) and 40 (for a small voter) and also allocates something to all small voters to do well against minimal winning coalition strategies. 32

33 The strategy that emerges as best reply in AP5 is less intuitive. A strategy like would win with probability 0.59, but apparently it is not the best possible way to beat the empirical distribution. Strategy beats the mass at 30 (for two of the small objects) and 20 (for the other two) while putting a positive amount on the large voter. In doing so, it beats some of the popular strategies like , , , and Note that, even though subjects seem to have thought more deeply and strategies are more varied in AP4 and AP5 compared to AP3, the expected payoff against the empirical distribution in AP3 is slightly lower. We believe this is due to AP4 and AP5 being more complex games in which suboptimal strategies are more exploitable. 19 Explaining success The previous subsections show that behavior is different from equilibrium and some subjects are systematically more successful than others. In what follows, we will attempt to uncover the determinants of success in our experiment by running a random effects probit regression of the following form, separately for each of the three treatments: Win ( 0 1Win i( t 1) 2 log( RTime) it 3AllocA it 4MADTempit it MADAcr BR SupMaj u ), 5 it 6 it 7 it i it where is the c.d.f. of the standard normal distribution, Win it is a binary variable that takes value 1 if subject i won at round t and 0 otherwise, MADTemp it is the mean absolute deviation of allocations across rounds (if y ijt is the allocation of subject i to object j at time t and n is the number of objects in the treatment, MADTemp n y y n ), it ijt ij( t 1) / j 1 MADAcr it measures the variability of the allocation across objects in a given round 19 For example, suppose subjects play simple minimal winning coalition strategies. In AP3, the best response against the mixed strategy that puts equal probability on (60,60,0), (60,0,60) and (0,60,60) wins with probability 2/3. In AP4, the best response against the mixed strategy with support {(80,40,0,0), (80,0,40,0), (80,0,0,40), (0,40,40,40)} gets 3/4 (if each of the four strategies is played with equal probability) or 5/6 (if (0,40,40,40) is played with probability 1/2 and each of the others with probability 1/6). For AP5, the probabilities of winning against the analogous mixed strategies on {(90,30,0,0,0), (90,0,30,0,0), (90,0,0,30,0), (90,0,0,0,30), (0,30,30,30,30)} would be 4/5 and 7/8 respectively. 33

34 n ( MADAcr it yijt 120 / n / n ), BR it is a binary variable that indicates whether subject i s j 1 allocation in round t is consistent with a myopic best response to the opponent s allocation at round t-1 (the indicator being 1 does not necessarily mean that the subject is trying to play a myopic best response) and SupMaj it is an indicator variable for subject i playing a supermajority strategy. Table 9. Determinants of winning Dependent variable: Win it AP3 AP4 AP5 Win i(t-1) (0.213) 0.095* (0.076) *** (0.000) Log(RTime) it (0.821) ** (0.017) (0.881) AllocA it * (0.053) (0.218) (0.222) MADTemp it 0.005*** (0.002) 0.009*** (0.000) 0.020*** (0.000) MADAcr it (0.162) (0.180) (0.222) BR it 0.112* (0.059) 0.192*** (0.000) 0.244*** (0.000) SupMaj it 0.120* (0.083) (0.565) (0.529) Constant ** (0.044) * (0.058) * (0.083) # Observations 2,024 2,254 2,124 # Subjects Marginal effects; p-values in parenthesis, * significant at 10%, ** significant at 5%, *** significant at 1%. Consistent with Chowdhury et al. (2013) we find that MADTemp it is highly significant, hence unpredictability of the allocation is one of the main determinants of 34

35 success; also consistent with their results, the variability of the allocation across objects does not have a significant effect on the probability of winning. The best response indicator has a positive and highly significant effect, hence allocations consistent with playing a myopic best response were more successful than other allocations. The supermajority variable is not significant CONCLUDING REMARKS Our experiment finds support for some of the qualitative equilibrium predictions. Although large voters do not get the share of lobbying expenditures predicted by equilibrium, we do find that a voter with three votes is worth more than three times as much as a voter with one vote. These results may have implications for the analysis of voting blocs. In a one-man-onevote legislature, voters may benefit if they can commit to voting together. The most popular strategies in the experiment appear to be consistent with iterated reasoning. Suppose the equal distribution of the budget among all voters is the intuitive level 0 strategy. Minimal winning coalition strategies that appear to be designed to beat this strategy would be classified as level 1 and were frequently observed in the early rounds of the experiment. Supermajority strategies are a better reply to combinations of minimal winning coalition strategies and can be seen as level 2. Bribing supermajorities can be rational because of the uncertainty about the strategy of the other player. If the strategy of the other player is known, it is always possible to beat it with a minimal winning coalition strategy and supermajorities have no added advantage. However, in our setting bribing supermajorities can act as insurance against a distribution of strategies of the other player. The proportion of supermajority strategies increases during the course of the experiment. Nevertheless, not all supermajority strategies are equally reasonable; for example the 20 We could also have included variables measuring the frequency of unit digits 1 and 2 in the allocation; we have not included them in the main regression due to their high correlation with the supermajority variable. If instead of the supermajority variable we include dummy variables for the number of times digits 1 and 2 appear in the allocation as in Arad and Rubinstein (2012), the effect on the results is negligible: these variables are not significant and the significance of the other variables is essentially unchanged. 35

36 equal allocation of tokens across all voters did not do well in the experiment. Success in the experiment was correlated with the variability of the allocation across time and with the use of allocations consistent with myopic best response. Previous studies of voting power using non-cooperative bargaining games have studied various sources of asymmetries between voters. Experiments where voters differ in voting weights (e.g. Fréchette et al., 2005a, Drouvelis et al., 2010) find that voters tend to exploit strategic advantages to some extent, but not to the extent predicted by equilibrium analysis with selfish preferences. Perhaps the willingness or the ability to exploit strategic advantages is tempered by equity considerations. An intriguing feature of our results is that the large voter gets more than small voters, but not by as much as equilibrium predicts, even in a setting where we believe equity considerations have little role. 36

37 General rules APPENDIX A: INSTRUCTIONS Welcome! This session is part of an experiment in the economics of decision making. If you follow the instructions carefully and make good decisions, you can earn a considerable amount of money. In this session you will be competing with one other person, randomly selected from the people in this room, over the course of forty-five rounds. Throughout the session your competitor will be the same but you will not learn whom of the people in this room you are competing with. The amount of money you earn will depend on your decisions and your competitor s decisions. It is important that you do not talk to any of the other people in the room until the session is over. If you have any questions raise your hand and a monitor will come to your desk to answer it. Description of a round Each of the forty-five rounds is identical. At the beginning of each round your computer screen will look like the one below. You have 120 tokens. You must use these to bid on 4 objects labelled A, B, C and D. You get points for winning objects object A is worth 2 points and the other objects are worth 1 point each. For each object you can bid any whole number of tokens (including zero), but the total bid for all objects must add up to 120 tokens. You bid by entering numbers in the boxes, and then clicking on the Submit button. If the bids you submit do not add up to 120 the 37

38 computer will indicate by how many tokens the bid needs to be corrected. If you do not submit a valid bid within 90 seconds the computer will bid for you and will place zero tokens on each object. When everyone in the room has submitted their bids, the computer will compare your bids with those of your opponent. Your computer screen will look like the one below (the bids in the figure have been chosen for illustrative purposes only): You win an object if you bid more for it than your opponent. (If you and your opponent bid the same amount the computer will randomly decide whether you or your opponent wins the object, with you and your opponent having an equal chance of winning the object. In this case the computer screen will indicate with an asterisk that the object was awarded randomly). The winner of the round is the person who gets the most points. The winner of the round earns 50 pence, the other person earns zero. Ending the Session At the end of the session you will be paid the amount you have earned from all forty-five rounds. You will be paid in private and in cash. Now, please complete the quiz. If you have any questions, please raise your hand. The session will continue when everybody in the room has completed the quiz correctly. 38