Voting Power in Weighted Voting Games: A Lobbying Approach by Maria Montero, Alex Possajennikov and Martin Sefton 1 April 2011

Transcription

1 [Very preliminary please do not quote without permission] Voting Power in Weighted Voting Games: A Lobbying Approach by Maria Montero, Alex Possajennikov and Martin Sefton 1 April 2011 Abstract We report experiments on 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 a prize. Taking the share of the budget distributed to a voter as a measure of the voter s voting power we investigate how voting power varies with voting weights. We focus on simple apex games where there is one large voter and several small voters and find that observed behavior is not perfectly in line with equilibrium predictions. However, we find support for two qualitative features of the equilibrium. First, the share of lobbyists expenditures on the large voter exceeds her relative voting weight. Second lobbyists distribute their budgets over supermajorities. Keywords: lobbying, weighted voting, vote buying, power measures, experiment JEL Codes: D72, C72, C91 1 Contact details: maria.montero@nottingham.ac.uk, alex.possajennikov@nottingham.ac.uk, martin.sefton@nottingham.ac.uk. All authors: School of Economics, University of Nottingham. 1

2 1. INTRODUCTION In this paper we examine voting power in weighted voting games using an experimental approach. In contrast to previous studies that adopt a bargaining framework and relate voting power to realized shares of a pie, we use a lobbying framework in which lobbyists allocate a budget across voters and we relate a voter s voting power to her share of lobbying expenditures. Studies of experimental bargaining show that realized shares of the pie are sensitive to both strategic features of bargaining games and uncontrolled preferences over distributions of earnings. The lobbying approach pits subjects against each other in strictly competitive games, where experimental studies have found the impact of other-regarding preferences to be less pronounced. The basic situation we consider is a majority rule voting body where there are n 1 small voters with one vote each and one large voter with n 2 votes. This is a simple apex game with two types of minimal winning coalition. The small voters together can (just) outvote the large voter and so a coalition of all the small voters is minimal winning; the large voter and any one of the small voters together command n 1 votes and so is also minimal winning. A fundamental question is whether the large voter with n 2 votes has more, less, or the same amount of power as n 2 small voters with a single vote each. Cooperative game theory provides a number of alternative measures of voting power in apex games. Most power indices give disproportionate power to the large voter. In particular, 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, i.e. her voting power is more than the combined power of n 2 small voters. On the other hand, the Nucleolus awards power in proportion to votes. Non-cooperative game theory can also be used to measure voting power, given a model of the voting process. Suppose that voters do not care about how they vote, but instead can sell their vote to an interested agent. Because the large voter can replace n 2 small voters, it seems natural that its market value would be n 2 times the value of a small voter (see e.g. Owen et al., 2006). In fact, equilibrium market values may not be proportional to voting weights. Young (1978) studies a model in which two lobbyists with identical budgets compete for the voters, and each voter does the bidding of whoever pays them most. Measuring the worth of a voter by the expected amount they are offered in equilibrium, he finds that in the situation above a voter with 2 votes is worth more than twice as much as a voter with a single vote. 2

3 Our experiment implements Young s model under several treatments that manipulate the number of voters and the coarseness of the lobbyists budget. When lobbyists distribute 120 indivisible units we observe that the large voter gets a more than proportional average payoff for 4-player apex games and 5-player apex games, though the departure from proportionality is small and insignificant for the 4-player case. We also run experiments using a coarse budget (5 indivisible units). The advantage of the small budget is that we are able to calculate the lobbyists equilibrium strategies. 2 With a small budget, the equilibrium expected payoff for a large voter is disproportionately high and close to Young s prediction for the large budget. Experimental results for the small budget confirm that the large voter's share is higher than proportional, and the departure from proportionality is significant for both the 4- voter and the 5-voter cases. Our lobbying approach also allows us to see who of the voters lobbyists target. Does a lobbyist concentrate funds on all the small voters, or 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 (Riker's size principle). 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 lobbying game: lobbyists randomize between bribing several coalitions, some 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 design and procedures. Results are presented in Section 4 and Section 5 concludes. 2 Little is known about the equilibria of simultaneous lobbying games. These games are related to Colonel Blotto games, which are notoriously difficult to solve. In view of this difficulty, most papers in the literature have opted for considering sequential moves (Groseclose and Snyder (1996), Diermeier and Myerson (1999), Banks (2000), Dekel et al. (2009), Le Breton and Zaporozhets (2010)). Except for Young s (1978) computations (which only report expected payoffs without equilibrium strategies), nothing is known about the equilibrium of the game with asymmetric voters. Thomas (2009) analyses asymmetric objects, but in the original Blotto game in which the bidders payoff is the total value of objects won. 3

4 2. BACKGROUND Apex games Apex games are the simplest asymmetric voting games and have one large voter and n 1 small voters. There are two types of minimal winning coalitions: the large voter together with one of the small voters, and all the small voters together. Apex games 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. 3 There is an extensive literature in cooperative game theory on voting power in Apex games. The Shapley value is perhaps the most commonly used measure of voting power and can be interpreted as a bargainer s share when voting over allocations of a pie. The Shapley value gives the large player 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 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 characterization of the game given above, and not on any specific features of the voting process. Early experiments with apex games used relatively unstructured bargaining 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 collations 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 3/7 ( 42.9%) of the pie. 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. (2005) examine apex games where five subjects divide a pie of $60, using two non-cooperative voting games. Their demand-bargaining treatment uses a 3 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 (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 exists and it 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 variant of Morelli (1999) and has a unique equilibrium where each voter within a winning coalition receives a share proportional to her voting weight, but large players are much more likely to feature in 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 (Montero, 2006). The experiment shows that strategic considerations play 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). A recurrent finding from these experiments is that voters are often willing to exclude other voters and allocate noting 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 of selfish preferences. In summary, experiments using a bargaining framework find that realized shares reflect both strategic features of the bargaining process and uncontrolled preferences over distributions of earnings. In the next sub-section we describe an alternative framework. The lobbying game Our lobbying game is based on the model of Young (1978). The voters in the apex game above must vote on a bill. There are two lobbyists, one of which favours the proposed bill whereas the other wants the status quo to prevail. The two lobbyists have identical budgets, and distribute them simultaneously across the voters. 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) winning or losing is assumed to be of incomparably greater value than the prices paid. Young assumes the lobbyists have perfectly divisible budgets, whereas in our experiment we necessarily have a smallest money unit. In two of our treatments the smallest money unit is relatively small compared with the total budget (there are 120 indivisible 5

6 units); in the other two treatments the smallest money unit is rather large (there are 5 indivisible units). The first case is perhaps more realistic, but the second case allows us to compute equilibrium strategies. Table 1 summarizes the equilibrium expected share for the large player depending on the number of players and on the fineness of the budget. The three-player case is a degenerate apex game in which all voters are symmetric and all have equal voting power. The predictions for the fine budget (B = 120) are taken from Young (1978). The predictions for the coarse budget (B = 5) are derived in Appendix A; in the five-player case there is a small interval of equilibrium payoffs and the table reports the smallest value. 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 very similar regardless of whether a fine or coarse budget is used, and they are clearly above the proportional prediction. Table 1. Voting powers of large voter number of voters Proportionate Shapley Value Equilibrium Budget = 5 Budget = The lobbying game provides a quite different approach to measuring power compared to bargaining. Moreover, a large experimental literature on strictly competitive games shows little evidence of the importance of other-regarding preferences. Indeed, in experimental zero-sum games outcomes often closely approximate equilibrium outcomes (see Camerer, 2003, for a review). 3. THE EXPERIMENT The experiment was conducted at the University of Nottingham with 146 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). 6

7 All sessions used an identical protocol. Upon arrival, subjects were given a written set of instructions that the experimenter read aloud (instructions for one of the treatments are appended in Appendix B). Subjects were not allowed to communicate with one another throughout the session. At the beginning of the session subjects were randomly paired to play the lobbying game 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 tokens among objects, each of which was worth a given number of points. An object is won if a subject allocates more 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 the 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 (3, 4 or 5) and in the coarseness of the budget (5 or 120 tokens) and varied across sessions, as summarized in Table 2. Each session took approximately 1.5 hours and subjects earned on average (about $17 at the time of the experiment). Table 2. Experimental Sessions Session Number of voters Budget Number of Subjects

8 4. RESULTS For all treatments the lobbying game is a symmetric constant-sum game and so in equilibrium each lobbyist wins with probability ½ 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. Figure 1 displays a histogram of the observed difference in wins, i.e., if in a pair one subject wins in 11 rounds and the other wins in 34 rounds, the difference in wins is 23 (there is one such pair in the experiment). For comparison the theoretical distribution is also shown Fraction.2.25 Figure 1. Theoretical and Observed Distributions of Difference in Wins (all treatments) Difference 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 6.84, significantly higher (p = 0.003). The obvious interpretation is that some subjects are better than others at playing the lobbying game. We next take a closer look at individual treatments, beginning with the threevoter game. 4.1 The three-voter game In the three-voter game voters are symmetric, and the lobbying game is similar to a Colonel Blotto game, where, with continuously divisible budgets, equilibrium requires players to use 8

9 0 Fraction mixed strategies where the marginal distribution of tokens on each voter is uniform (Roberson, 2006). 4 Thus in the equilibrium of our large budget treatment the distribution of tokens on each object should be approximately uniform on {0, 1,, 80}. Figure 2 displays the empirical distribution of allocations to Object A. Figure 2. Observed distribution of budget to Object A (three-player fine budget) allocation to Object A The distribution has pronounced concentrations around 0-5 and and is far from uniform. Similar findings are reported for Colonel Blotto game experiments (Arad and Rubinstein, 2009). The distributions for objects B and C are similar. Importantly, Object A does not get any positional advantage. Out of seven pairs three place less than a third and four place more than a third of their tokens on Object A, and on average subjects place 35% of their tokens on Object A. For the coarse budget, the three-voter game has a continuum of object-symmetric equilibria involving strategies that place 3 tokens on one voter and 2 tokens on another ( 320 ) and strategies that place 3 tokens on one voter and 1 token on each of the other two voters ( 311 ) (see Appendix A). The proportions of strategies, averaged over all 45 periods, are shown in Table 3 below. Note that the equilibrium strategy 320 has the highest proportion in all pairs. In fact this strategy is the only one that survives iterative elimination of weakly 4 In a Colonel Blotto game a player s payoff is the sum of the values of objects won. 9

10 dominated strategies in the reduced object-symmetric game. However, most pairs played non-equilibrium strategies more than 20% of the time. Table 3. Strategies used in three-voter coarse budget game Type/Pair All Again, it is important to note that Object A has no positional advantage. Of eight pairs three allocate less than, four allocate more than, and one allocates exactly one third of their tokens to Object A. Across all pairs 35% of tokens were allocated to Object A. 4.2 The four-voter game In the four-voter games Object A corresponds to the large voter, and is worth 2 points, whereas objects B, C and D are worth one point each. Table 4 compares the frequencies of strategies with equilibrium for the coarse budget and shows that behavior is quite far from equilibrium. For example, the most commonly observed strategy allocates no tokens to the large voter, whereas this strategy should be played less than 5% of the time. Table 4. Strategies in four-voter coarse budget game Strategy other Predicted frequency Observed frequency If we focus on supermajority strategies, these are predicted with probability 32/77 (about 42%). The proportion observed is considerably lower than this, 31% overall, although 10

11 there is an increasing trend as shown in Figure 3, and the proportion of supermajority strategies increases to 35% in the last 10 periods. Figure 3. Proportion of supermajority strategies (four-voter coarse budget) The only two supermajority strategies predicted in equilibrium are 2111 (with 10% probability) and 1211 (with 31% probability). They are observed only 4 and 5% of the time respectively in the experiment. The most popular supermajority strategy is 3110 (10%) and 1220 are observed as well (5 and 4% of the time respectively). Thus, although we observe supermajority strategies being played quite frequently, they are not played as frequently as in equilibrium, and they are not the supermajority strategies that should be played in equilibrium. Another important property of equilibrium is that large voters should get a disproportionate share of lobbying expenditures. Do we observe this in our data? Figure 4 shows how the share of budget allocated to the large voter evolves over the 45 rounds of the game. The red line shows the proportional share (2/5). With a coarse budget the large voter s share of the budget usually exceeds her proportional share, but not by much. For the case of a fine budget it is even less clear whether the large voter s share is higher than proportionate. We provide a formal statistical comparison after looking at the five-voter game treatments. 11

12 Figure 4. Share of budget allocated to large voter in four-voter treatments 1 Coarse Budget Fine Budget Round 4.3. The five-voter game Subject decisions also differ markedly from equilibrium predictions in the five-voter game with a coarse budget. The equilibrium prediction in this case is that only two strategy types are played: the minimal winning coalition strategy and the supermajority strategy Such strategies were played only 37% of the time in the experiment; the minority strategy was rather popular as well as the minimal winning coalition strategy and the supermajority strategy 31100: see Table 5. Table 5. Strategies in five-voter coarse budget game Strategy Other Predicted frequency Observed frequency If we focus on the frequency of supermajorities, the supermajority strategy has a predicted frequency between 43% and 45%. As in the four-voter game, the proportion of 12

13 supermajorities observed is lower than predicted, 34% overall, and these often correspond to strategies that should not be played in equilibrium. Notice that the difference between equilibrium and proportional shares increases with the number of voters. Thus, in a sense the five-voter game is most favorable for finding support for the super-proportionality prediction. Figure 5 presents the evolution of the average share of budget allocated to the large voter over the 45 rounds of the five-voter treatments. The proportional division is indicated by a red line. With a coarse budget the large voter gets a disproportionate share in every round, and with a fine budget in most rounds. Figure 5. Share of budget allocated to large voter in five-voter treatments 1 Coarse Budget Fine Budget Round 4.5 Super-proportionality? Table 4 presents the share of the budget allocated to the large voter in each treatment, averaging across all rounds and pairs. For statistical tests of proportionality we use simple 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. 13

14 Fine Budget (120 tokens) Coarse Budget (5 tokens) Table 4. Budget share allocated to Object A (large voter) Number of voters Proportional Share (%) Equilibrium Share (%) Number of pairs Observed Share (%) ** * *** *, **, *** significantly different from proportional at the 10%, 5%, 1% level Consistent with Figure 4, in the four-voter games the large voters share is only slightly higher than her proportional share. The effect is significant at the 10% level in the case of the coarse budget (p = 0.054), but not in the case of the fine budget (p = 0.166). Consistent with Figure 5, in the five-voter games the large voter gets a significantly higher than proportional share in both the fine budget (p = 0.011) and coarse budget cases (p = 0.000). Thus, although equilibrium does not give an exact account of behavior, and in many respects there are marked differences (for example, the large voter s share is significantly lower than her equilibrium share in three of the four apex games), some of the qualitative properties of equilibrium are observed in our data. More specifically, the large voter receives more than a proportional share of lobbying expenditures, especially in our five-voter game treatments where super-proportional shares are significant. 5. CONCLUDING REMARKS 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 of 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. Although large voters do not get the share of lobbying expenditures predicted by 14

15 equilibrium, we do find that they get super-proportional shares, i.e., a voter with two votes is worth more than twice as much as a voter with one vote. This observed effect is particularly clear in our five-voter treatments where the predicted effect is stronger. These results may have implications for the analysis of voting blocs. In a one-man-one-vote legislature, voters may benefit if they can commit to voting together. We also observe lobbyists bribing supermajorities in our experiment. Indeed, the possibility of supermajorities may be related to differences between relative voting powers and relative voting weights. Power indices like the Shapley value, which assigns a disproportionately high payoff to larger voters, take into account all coalitions in which a player is pivotal, irrespective of whether they are minimal winning. The large voter is pivotal in many coalitions that are not minimal winning (these are coalitions that contain the large voter and at least two but not all minor voters); on the other hand, the two cases in which a minor voter is pivotal both involve minimal winning coalitions. Felsenthal and Machover (1998, p. 174) also refer to buying and selling votes in their discussion of P-power. They point out that, if an outsider stands to gain one unit if a board were to pass a certain bill and lose one unit if the board were to defeat that bill, the price it would be willing to pay to an individual voter (having no knowledge of any of the voters intentions) would equal the Banzhaf measure. This argument is not conclusive in our framework since once there are two lobbyists trying to bribe multiple voters it is not the case that voters vote independently yes or no with equal probability. Indeed Felsenthal and Machover go on to rebut this argument and dismiss the Banzhaf measure as a measure of P- power. Nevertheless, it is worth mentioning that the Banzhaf index also makes a superproportional prediction. 15

16 REFERENCES Arad, A and A. Rubinstein (2009) Colonel Blotto s Top Secret Files: Multi-Dimensional Iterative Reasoning in Action. Typescript. Banks, J. (2000) Buying Supermajorities in Finite Legislatures. American Political Science Review 94, Baron, D. and J. Ferejohn (1989) Bargaining in legislatures. American Political Science Review 83, Camerer, C.F. (2003). Behavioral Game Theory: Experiments in Strategic Interaction. Princeton: Princeton University Press. Dekel, E., Jackson, M. O., Wolinsky, A. (2009) Vote Buying: Legislatures and Lobbying. Quarterly Journal of Political Science 4, Diermeier, D. and R. Morton (2004) Proportionality versus Perfectness: Experiments in Majoritarian Bargaining, in Social Choice and Strategic Behavior: Essays in the Honor of Jeffrey S. Banks, ed. by D. Austen-Smith and J. Duggan. Berlin: Springer- Verlag, Diermeier, D. and R.B Myerson (1999) Bicameralism and its Consequences for the Internal Organization of Legislatures. American Economic Review 89, Drouvelis, M., Montero, M. and M. Sefton (2010) Gaining Power through Enlargement: Strategic Foundations and Experimental Evidence, Games and Economic Behavior 69, pp Felsenthal, D. S. and M. Machover (1998) The Measurement of Voting Power. Theory and Practice, Problems and Paradoxes. Cheltenham: Edward Elgar. Fischbacher, U. (2007) "z-tree: Zurich Toolbox for Ready-Made Economic Experiments", Experimental Economics 10, Fréchette, G., Kagel, J. and M. Morelli (2005a) Behavioral Identification in Coalitional Bargaining: An Experimental Analysis of Demand Bargaining and Alternating Offers Econometrica 73, Fréchette, G., Kagel, J. and M. Morelli (2005b) Gamson s Law versus non-cooperative bargaining theory Games and Economic Behavior 51, Fréchette, G., Kagel, J. and M. Morelli (2005c) Nominal bargaining power, selection protocol, and discounting in legislative bargaining Journal of Public Economics 89,

17 Greiner, B. (2004) "An Online Recruitment System for Economic Experiments", in: Kremer, K., Macho, V. (Eds.), Forschung und Wissenschaftliches Rechnen GWDG Bericht 63, Göttingen: Ges. für Wiss. Datenverarbeitung, pp Groseclose, T. and J. Snyder (1996) Buying Supermajorities. American Political Science Review 90, Kagel, J., Sung H. and E. Winter (2010) Veto Power in Committees: An Experimental Study, Experimental Economics 13, Le Breton, M. and V. Zaporozhets (2010) Legislative Lobbying under Political Certainty, Economic Journal 120, McKelvey, R. (1991) An Experimental Test of a Stochastic Game Model of Committee Bargaining in Contemporary Laboratory Research in Political Economy, University of Michigan Press, Ann Arbor. McKelvey, R., McLennan, A. and T. Turocy (2006) Gambit: Software Tools for Game Theory, Version , Montero, M, Sefton, M. and P. Zhang (2008) Enlargement and the Balance of Power: An Experimental Study, Social Choice and Welfare 30, Montero, M. (2006) Noncooperative foundations of the nucleolus in majority games, Games and Economic Behavior 54, Morreli, M. (1999) Demand competition and policy compromise in legislative bargaining, American Political Science Review 93, Owen, G., Lindner, I., Feld, S. L., Grofman, B. and L. Ray (2006) A Simple Market Value Bargaining Model for Weighted Voting Games: Characterization and Limit Theorems. International Journal of Game Theory 35, Selten, R. and K. Schuster (1968) Psychological Variable and Coalition-Forming Behavior in Risk and Uncertainty, ed. by K. B. Borch and J. Mossin. London: Macmillan, Thomas, C. (2009) N-Dimensional Colonel Blotto Game with Asymmetric Battlefield Values. Typescript. Wooders, J. and J. Shachat (2001) On the Irrelevance of Risk Attitudes in Repeated Two- Outcome Games, Games and Economic Behavior 34, Young, H. P. (1978) A Tactical Lobbying Game in Game Theory and Political Science (edited by Peter Ordeshook), New York University Press. 17

18 The three-voter game. APPENDIX A: EQUILIBRIA FOR COARSE BUDGET We restrict attention to object-symmetric strategies. An object-symmetric strategy is such that all objects with the same value are treated in the same way, i.e. allocations to them are symmetric. For the three-voter game this means that strategies ( (x 1 ), (x 2 ), (x 3 )) where () is a permutation of (x 1, x 2, x 3 ) have the same probability. There are 5 possible types of strategies (the rest are their permutations). These types are given in the following table /2 1/3 1/ /3 1/2 1/2 1/3 1/ /3 ½ 1/2 1/2 2/ /3 1/2 1/2 1/ /6 1/3 2/3 1/2 Here, 500 means the set of three strategies (5, 0, 0), (0, 5, 0) (0, 0, 5) that are permutations of one another. 410 represents the set of 6 strategies (4, 1, 0), (4, 0, 1), (1, 4, 0), (0, 4, 1), (1, 0, 4), (0, 1, 4). The numbers in the matrix give the probability that a strategy in the row wins against the uniform distribution of strategies in the column: thus (4, 1, 0) wins against (0, 0, 5) and ties against (5, 0, 0) and (0, 5, 0), leading to the overall probability of winning of 410 against 500 being 2/3. Because there are only two possible outcomes (winning and losing), risk attitudes are irrelevant under expected utility theory and a player s payoff can be identified with the probability of winning. Thus, the matrix represents a normal form of the game for which strategies are the uniform distributions within the given types. There is a continuum of equilibria in the game, described by strategy combinations ( (1 1) 311, (1 2) 311) for ½ 1, 2 1. This includes a pure strategy equilibrium (320,320) but recall that pure here means the uniform mixing within the set of strategies described as 320, i.e. it is mixed in the overall allocation game. In fact, 320 is the only strategy that survives the iterated elimination of weakly dominated strategies. 18

19 The four-voter game. We assume lobbyists only play strategies that treat all small objects equally. For example, strategy 4100 denotes a mixed strategy in which the lobbyist allocates 4 tokens to the large object and 1 token to one of the three small objects, chosen at random. Taking this into account, there are 16 strategy types. Four of them (5000, 0500, 0410 and 0320) are eliminated because they allocate the budget to a losing subset of players. Indeed, three of those strategies (0500, 0410 and 0320) cannot win with a probability above 0.5 against any other strategy. 5 The table below is the resulting normal form game between the two lobbyists (entries on the table correspond to the probability that the row lobbyist wins) /12 11/12 11/12 5/6 ¾ 3/4 2/ /3 1/3 1/ / /12 11/ /6 11/12 5/6 1 5/6 2/3 2/ /12 1/ / / / /12 1/12 1/ / /12 11/12 5/6 3/4 7/12 7/12 1/ ¼ 1/6 1/ / / / ¼ 1/ /12 23/ / /12 5/6 3/ /3 1/ / / / / /4 3/ /3 1/ / / / /12 5/ /3 1/3 1/6 0 5/12 1/12 1/6 0 1/4 1/ /6 1/ /3 0 1/4 1/ / Using the Gambit software (McKelvey et al., 2006) we found a unique equilibrium of the payoff matrix above, with probabilities 30/77 on 4100, 12/77 on 3200, 8/77 on 2111, 5 If two strategies tie on the apex voter, they tie overall (either they tie on all minor voters as well, or each of the lobbyists must win at least one minor voter for sure). If they do not tie on the apex voter, they would need to tie on any voter that is allocated 0 for sure. This only happens if the other lobbyist is playing 5000 (and, even in this case, 0500 would win only with probability 0.25); in all other cases the probability of winning is under

20 24/77 on 1211 and 3/77 on In this equilibrium, the share of tokens allocated to the large object is The share of each small object is Some of the strategies used in equilibrium (2111 and 1211) involve lobbyists trying to bribe a supermajority. Supermajority strategies can be optimal because of the uncertainty about the strategy of the other lobbyist. For example, suppose the other lobbyist randomizes between 4100 and Strategy 1211 does quite well against both of these strategies: it wins 2/3 of the time against 4100 and 5/6 of the time against If a lobbyist is sure that the other lobbyist is playing 4100 it would do better by playing 0221 rather than 1211 (giving up on the large voter since it can never be won by allocating just one unit, and increasing the probability of winning all three minor voters); similarly if the other lobbyist is playing 0221 is it best to play a strategy such as 1400 (this strategy wins for sure by ensuring that the apex voter and one minor voter are bribed). It turns out however that if 4100 and 0221 are played with equal probability the unique best response is In the equilibrium we have computed, supermajority strategies are played with probability 32/ One of the strategies we eliminated, 5000, wins with probability ¾ against 4100 and 3200, but loses with certainty against the other three strategies. Because ¾ x 42/77 < 0.5, we indeed have an equilibrium. [we may include this strategy in the table]. It is interesting to point out that equilibrium frequencies are not restricted by the availability of other strategies. In the reduced form of the game with only five strategies, Gambit produces a unique equilibrium. 20

21 The five-voter game Again we assume that lobbyists only play strategies that treat all small objects equally and we discard some strategies that look implausible (obtained equilibria are later checked against invasion by those strategies). We discard strategy and strategies that allocate the budget to a losing subset of small objects. This leaves 12 strategy types: /8 7/8 7/8 3/4 3/4 3/4 ¾ / /32 31/32 15/16 29/32 29/32 7/8 13/16 ¾ 0.5 3/ /8 1/ / /32 15/16 15/ /16 1 7/ /8 1/32 1/ / ¼ 1/ /48 47/48 23/24 15/16 43/48 3/4 5/ ¼ 3/32 1/ / /96 47/ / / ¼ 3/32 1/16 1/32 1/48 1/ ¼ 1/8 1/16 0 1/24 1/ / / /32 7/8 25/ /4 1/16 0 5/48 1/ / / / / / /8 1/8 0 3/8 1/ /32 1/32 1/8 0.5 Using the Gambit software, we find a continuum of equilibria with support (with weight from 4/ to 16/ ) and (with weight from 13/ to 3/7 0.43). The expected share allocated to the large object is between 19/ and 77/ Strategies and tie against each other, so equilibrium frequencies of this mix must be restricted by the availability of other strategies. Strategy is not played in equilibrium but plays an important role in determining the equilibrium mix. This strategy loses against but wins with probability 7/8 against (it gets the apex voter for sure and ties on three minor voters; winning one tie is enough to win overall). The availability of this strategy restricts the frequency of to be at most 4/7. Analogously, the availability of the supermajority strategy restricts the frequency of to be at least 16/29. This strategy wins for sure against 11111, but wins only with probability 3/32 against

22 APPENDIX B: INSTRUCTIONS General rules 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 5 tokens. You must use these to bid on 3 objects labelled A, B and C. You get points for winning objects all three 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 5 tokens. You bid by entering numbers in the boxes, and then clicking on the Submit 22

23 button. If the bids you submit do not add up to 5 the 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. 23

24 Quiz 1. Suppose your bids and your competitor s bids were as follows: Object Points Your Bid Opponent s Bid A B C How many points would you receive?. How many points would your opponent receive?. What would your earnings from this round be (in pence)?. What would your opponent s earnings from this round be (in pence)?. 2. Suppose your bids and your competitor s bids were as follows: Object Points Your Bid Opponent s Bid A B C Who wins object A? Me / My Opponent / Randomly Determined (Circle One) Who wins object B? Me / My Opponent / Randomly Determined (Circle One) For the remaining questions suppose the computer awards object B to your opponent: How many points would you receive?. How many points would your opponent receive?. What would your earnings from this round be (in pence)?. What would your opponent s earnings from this round be (in pence)?. 24