Welfare-maximizing scoring rules VERY PRELIMINARY DRAFT Adam Giles and Peter Postl 27 February 2012 Abstract We study a cardinal model of voting with three alternatives and three agents whose vnm utilities are private information. We restrict ourselves to voting protocols given by two-parameter scoring rules, as studied by Myerson (2002). We show that a good two-parameter scoring rule reduces welfare losses relative to other, more common, voting rules such as Plurality, Approval voting, and the Borda Count. Oxera, Oxford. Department of Economics, University of Birmingham, Birmingham B15 2TT, United Kingdom, p.postl@bham.ac.uk. 1
Any theory of voting which does not allow for intensity of preference is certainly incomplete, and any voting system which does not permit its expression cannot be wholly satisfactory. (from Meek (1975), A transferrable voting system including intensity of preference, Mathematique et science humaines, Vol 50, p. 23-29) 1 Introduction Within the large literature on Social Choice, there are two theorems that every economist knows: Arrows impossibility theorem, and the Gibbard- Satterthwaite theorem. These important negative results tell us what we cannot do, not what we can do. In this paper, we look at how far we can get with voting rules in a restricted setting without transferable utility. To do so we use a simple model of voting with three alternatives and three agents who s vnm utilities for the alternatives are private information. Much of the social choice literature proceeds in an axiomatic fashion, imposing desirable properties such as independence of irrelevant alternatives, the Pareto criterion, strategy proofness and so on, whilst dealing with agent s ordinal preferences. By comparison, surprisingly little has been covered in the way of analyzing voting systems by cardinal standards of welfare. To make the problem tractable we restrict attention to a two-parameter family of voting rules known as (A, B)-scoring rules, as studied Myerson (2002). This family of voting rules contains many of the more common voting rules. With out setting we ask: What are equilibrium voting-strategies under (A, B)- scoring rules? After characterizing symmetric equilibrium voting-strategies, we adopt a utilitarian welfare criterion and ask which of the (A, B)-scoring rules generate the highest level ex ante expected utilitarian welfare? The results we present in this paper show that in our simple model, Plurality rule and Negative voting cause high welfare-losses relative to first best. Whilst Approval voting performs much better than either of these rules, it does not perform as well as (A, B)-scoring rules that allow agents to express their relative strength. Also, rules which use agents full ordinal rankings in making a collective choice perform better than those which do not. However, such voting systems do not seem to be particularly widely used. Whilst approval voting does allow for the communication of relative strength of preference, it does not convey unambiguously the agents ordinal preferences. Simple scoring rules (such as the Borda count) suffer the opposite problem, 2
in that they make use of agents full ordinal rankings, but neglect information on relative strength of preference. An alternative approach to studying welfare-maximizing voting-rules (and a much more general one) would be to view the problem as one of mechanism design, employing the revelation principle and deriving the optimal incentive compatible direct revelation mechanism. Adopting such an approach in a setting with a continuum of voter-types is very diffi cult. General results do not exist for settings with multi-dimensional private information and nontransferable utility. A simpler approach (also not adopted here for computational reasons) would be to restrict the problem to discrete type spaces, in which case it becomes a linear programming problem. In this case, one could characterize numerically the welfare-maximizing direct mechanism. This approach has been adopted for the case of two agents in a related (but simpler setting) by Börgers and Postl (2009). They find that welfare-losses relative to the first best are very small in their setting. There is a small literature that has studied voting rules and the issue of the communication of intensity of preference. For example Sonnenschein and Jackson (2007) show that in the limit incentive constraints become negligible and the first best can be implemented. Hortala-Vallve (2009) shows that in a case with a finite number of collective decisions the first best cannot be implemented. Our work is related also to that of Apesteguia et al. (2010). They look at which decision rules maximize welfare under various different welfare criteria. While they assume that agents have cardinal utilities, these can only be understood by the agents on an ordinal level. Furthermore, they assume that agents reveal their ordinal rankings truthfully, and find the welfaremaximizing decision rules. In general, they find optimal rules to be some form of scoring rule. For utilitarian welfare, with a uniform distribution of intensity of preference they find that the optimal decision rule corresponds to the Borda count. In our paper, where agents need not vote in line with their true ordinal ranking, we find (A, B)-scoring rules that dominate in welfareterms the Borda count. The remainder of this paper is structured as follows: The model is presented in Section 2. In Section 3 we characterize the equilibrium voting strategies for (A, B)-scoring rules. Section 4 is dedicated to welfare comparisons of the different (A, B)-scoring rules. Section 5 provides concluding remarks. Appendix 1 contains details of numerical results about the welfarelevels associated with different (A, B)-scoring rules. Appendix 2 contains all 3
proofs. 2 The Model 2.1 Basic set-up There are n + 1 agents who must collectively choose one alternative from the set {X, Y, Z}. Each agent i I {1,..., n + 1} has a von Neumann Morgenstern utility function u i : {X, Y, Z} R representing his preferences over lotteries on {X, Y, Z}. We denote by u i = (u i X, ui Y, ui Z ) the vector of von Neumann Morgenstern utilities that agent i assigns to outcomes X, Y and Z, respectively. We normalize utilities so that u i [0, 1] 3. These features of the von Neumann Morgenstern utility functions are common knowledge. For i = 1,..., n + 1 we refer to the vector u i as agent i s type. We assume that u i is a random variable whose realization is only observed by agent i. The agents types are stochastically independent, and they are identically distributed. We assume that for each agent i, the von Neumann Morgenstern utilities u i X, ui Y and ui Z are drawn independently from [0, 1] according to distribution function G with strictly positive density g on [0, 1]. 1 The typedistribution is common knowledge. The probability that a voter has any one of the six possible ordinal rankings over the set of alternatives is: ( 1 ( u i X ) ) u i X g(u i Y )du i Y g(u i Z)du i Z g(u i X)du i X = 1 6. 0 0 u i Z 2.2 Scoring Rules In the framework of section 2.1, we will study and compare voting mechanisms that are scoring rules as defined in Myerson (2002). Under a scoring rule, each voter is asked to submit a vote vector v i (v i X, vi Y, vi Z ), where vi k denotes the number of points given by agent i to alternative k {X, Y, Z}. 1 As we are dealing with state-dependent expected utility here (with a continuum of states of the world that determine the voters types), each agent s von Neumann Morgentstern utility function is unique up to a positive affi ne transformation with statedependent intercept and state-independent slope (see Ritzberger (2000) for details). We can therefore find, for each agent, an appropriate intercept and slope parameter so that the von Neumann Morgenstern utilities of the three alternatives are in the unit interval for all states of the world. 4
The vote vectors of all agents are added up, and the alternative with the most points is chosen. In case of a tie for the most points, an alternative is chosen randomly from among those with the most points, each with equal probability. Different scoring rules are characterized by the different sets of vote vectors that are available to the agents. In this paper we consider a family of scoring rules which are characterized by two parameters A and B such that 0 B A 1. In an (A, B)-scoring rule, each agent must choose a vote vector that is a permutation of either (1, A, 0) or (1, B, 0). That is, the voter must give a maximum of 1 point to one alternative, a minimum of 0 points to some other alternative, and either A or B points to the remaining alternative. Many well-known voting rules are special cases of (A, B)-scoring rules. (A, B)-scoring rules 5
For example: 1. Plurality Voting: A = B = 0, so that each agent must choose a vote vector that is a permutation of (1, 0, 0). 2. Borda Count: A = B = 0.5, so that each agent must choose a vote vector that is a permutation of (1, 0.5, 0). 3. Approval Voting: A = 1 and B = 0, so that each agent must choose a vote vector that is a permutation of either (1, 0, 0) or (1, 1, 0). 4. Negative Voting: A = B = 1, so that each agent must choose a vote vector that is a permutation of (1, 1, 0). Figure 1 provides an illustration of all (A, B)-scoring rules. While there is a continuum of (A, B)-scoring rules, the finite number of agents in our set-up implies that (A, B)-scoring rules can be divided into a finite number of equivalence classes. We say that any two scoring rules with parameters (Â, B) and (Ã, B) are equivalent if every pair of corresponding aggregate vote vectors under these two scoring rules generates the same probability distribution over outcomes {X, Y, Z}. 2 Equivalence classes for n + 1 = 3 voters 2 For example, in the case of three agents, if two scoring rules with parameters (Â, B) and (Ã, B) are equivalent, then the two corresponding aggregate vote vectors ( 2 i=1 vi X, 2 i=1 vi Y, 2 i=1 vi Z ) = (2 + B, 1 + 2Â, 0) and ( 2 i=1 ṽi X, 2 i=1 ṽi Y, 2 i=1 ṽi Z ) = (2 + B, 1 + 2Ã, 0) generate the same distribution over {X, Y, Z}. 6
For the special case of n+1 = 3 voters, Fig. 2 displays the 70 equivalence classes into which the various (A, B)-scoring rules can be divided. Each black line segment, each black dot, and each white polygon represents a different equivalence class. 3 Equilibrium voting strategies Each scoring rule defines a Bayesian game with n + 1 players/agents. A pure strategy for each agent i is a function that assigns to every type u i of agent i one of the vote vectors available to i under the given scoring rule (we denote by V A,B the set of all vote-vectors available to an agent under a given (A, B)-scoring rule). In this paper, we look for symmetric Bayes Nash Equilibria (BNE) in which the strategy used by all agents displays the following properties: 3 (P1) One point is assigned to highest-ranked alternative, and zero points to the lowest-ranked alternative. More formally, if v i 1 and v i 3 are the scores that agent i assigns to his highest- and lowest-ranked alternatives, resp., then the strategy used by agent i is such that v i 1 = 1 and v i 3 = 0 for all agents i I. (P2) If A > B, the number of points assigned to the middle-ranked alternative depends on the voter s intensity of preference. More formally, let u i 1 denote the vnm utility of agent i s highest-ranked alternative, u i 2 the vnm utility of i s middle-ranked alternative, and u i 3 the vnm utility of i s lowest-ranked alternative. Also, denote by v i 2 the score that agent i assigns to this alternative. The strategy used by agent i is such that: where α [0, 1]. v i 2 = { B if u i 2 < αu i 1 + (1 α)u i 3 A if u i 2 > αu i 1 + (1 α)u i 3 Suppose agent j uses a voting strategy of the type described above. From the point of view of any other agent i j, the strategy adopted by agent j, together with the fact j s type u j is a random variable, implies that j s vote vector v j is a discrete random variable with sample space 3 There may, of course, be other equilibria. However, the level of coordination between agents required to achieve many of these equilibria seems unreasonable. 7
V A,B. Given any type u j (consisting of components u j 1 > u j 2 > u j 3), the probability Pr{(1, B, 0) u j 1 > u j 2 > u j 3 } that agent j submits the vote vector v j = (v1, j v2, j v3) j = (1, B, 0) is: ( 1 ( u1 ) ) αu1 +(1 α)u 3 0 0 u 3 g(u 2 )du 2 g(u 3 )du 3 g(u 1 )du 1 ( ( u1 ) ) u1 p(α). (1) u 3 g(u 2 )du 2 g(u 3 )du 3 g(u 1 )du 1 1 0 0 and the probability that agent j submits the vote vector v j = (v1, j v2, j v3) j = (1, A, 0) is: Pr{(1, A, 0) u j 1 > u j 2 > u j 3 } = 1 p(α) Lemma 1 For any (A, B)-scoring rule, if the n opponents of agent i adopt the same voting strategy that satisfies property P1 and, if A > B also property P2, then it is a best response for agent i to set v i 1 = 1 and v i 3 = 0. The following proposition states our main characterization result for the case of n = 2 opponents for each agent (i.e. n + 1 = 3 agents in total). Work is ongoing to establish this result for arbitrary n 2. Proposition 1 For every (A, B)-scoring rule with A > B, there exists α (0, 1) such that each agent uses the same voting strategy that satisfies properties P1 and P2. Proposition 1 establishes the existence of a symmetric BNE in voting strategies that satisfy properties P1 and P2. Note that the value α need not be unique. For the purpose of welfare comparisons of (A, B)-scoring rules below, we shall use the value of α that maximizes ex ante expected social welfare whenever there are multiple equilibria. While we have explicitly constructed the proof of Proposition 1 for n+1 = 3 agents, it can be readily generalized to the case where n + 1 is a multiple of 3. If n + 1 is not a multiple of 3, then the equilibrium value α may not be in (0, 1), but may lie in [0, 1] instead. Finally, note that the task of computing analytically fixed point(s) α is very hard. In what follows, we therefore compute numerically the fixed points for specific distribution functions G. 8
4 Welfare Comparisons Having characterized symmetric Bayes Nash Equilibria of the game induced by any (A, B)-scoring rule in the presence of n + 1 = 3 agents, we now move on to a comparison of the level of ex ante expected (utilitarian) welfare across all possible (A, B)-scoring rules. In order to have a benchmark, we first compute ex ante expected welfare in the hypothetical first best scenario in which the types u i of all three voters, once realized, become observable to the social planner. 4.1 First best welfare A social planner who aims to maximize ex ante expected social welfare must choose for each profile u (u 1, u 2, u 3 ) of the three agents types the alternative in {X, Y, Z} that maximizes the sum of the agents ex post utilities. More formally, the planner chooses a decision rule f : [0, 1] 3 [0, 1] 3 [0, 1] 3 ({X, Y, Z}), u (f X (u), f Y (u), f Z (u)), where ({X, Y, Z}) is the set of probability distributions over {X, Y, Z}, and f k (u) denotes the probability that alternative k {X, Y, Z} is chosen when the agents type-profile is u. Ex ante expected social welfare associated with a decision rule f is: [ E u k {X,Y,Z} f k(u) ( ) ] i I ui k We say that a decision rule f is first best if for every k {X, Y, Z}: i I ui k > max { i I l {X,Y,Z}\k ui l} f k (u) = 1 i I ui k < max { i I l {X,Y,Z}\k ui l} f k (u) = 0 In order to compute ex ante welfare of a first best decision rule, we define three random variables w k i I ui k (one for each k {X, Y, Z}), which has support [0, 3]. Due to the i.i.d. nature of all vnm utilities u i k, each random variable w k has the same cumulative distribution function H. We can now compute first best welfare as the expectation of the random variable max{w X, w Y, w Z }, which has the following cumulative distribution function: H(w) = Pr(max{w X, w Y, w Z } w) = (H(w)) 3 9
4.2 Welfare of (A, B)-scoring rules In order to compute welfare losses (relative to first best welfare) under the different (A, B)-scoring rules, we need to compute expected welfare in the BNE of the various (A, B)-scoring rules. To do this, due to the symmetry of our set-up, all we need to do is to compute the expected utility of one agent, and then multiply it by 3 (as we are considering here the case of n + 1 = 3 agents). Observe that the expected utility of any single agent is: U i (u i ) { Ui (B, α, u i ) if u i 2 < α u i 1 + (1 α )u i 2 U i (A, α, u i ) if u i 2 > α u i 1 + (1 α )u i 2 where U i (σ, α, u i ) (with σ {A, B}) is the expected utility of agent i of type u i when he submits the vote-vector (v1, i v2, i v3) i = (1, σ, 0) that corresponds to his true ordinal ranking and in which σ points are assigned to i s secondranked alternative. 4 Due to the symmetry of our set-up, we can compute the ex ante expected utility E[U i (u i )] of agent i by multiplying by 6 (as there are six possible ordinal rankings of the alternatives in {X, Y, Z}) agent i s expected utility for an arbitrary ordinal ranking (as implied by vnm utilities u i 1 > u i 2 > u i 3, where u i l is agent i s vnm utility from his l-th ranked alternative): ( 1 ( u1 ) ) α u 1 +(1 α )u 3 6 U i (B, α, u i )g(u 2 )du 2 g(u 3 )du 3 g(u 1 )du 1 0 0 u 3 1 ( u1 ( u1 ) ) +6 U i (A, α, u i )g(u 2 )du 2 g(u 3 )du 3 g(u 1 )du 1 0 0 α u 1 +(1 α )u 3 4.3 Bounding second best welfare - INCOMPLETE!! While first best welfare is an important benchmark, it is not entirely satisfactory for measuring the performance of different (A, B)-scoring rules. The reason is that first best decision rules do not take into account the incentive constraints of individual agents, who, when given the opportunity to express directly they vnm utilities to the social planner, may find it in their interest to misrepresent their vnm utilities. A better benchmark for the evaluation 4 Note that the BNE value of α is fully determined by the model primitives, which is why the ex ante expected utility of agent i does not depend on α but only on his type u i. 10
of (A, B)-scoring rules would therefore be the optimal decision rule among all decision rules that are incentive compatible (we refer to such decision rules as second best decision rules). Definition 1 A decision rule f is incentive compatible if for all i I and for all u i, û i [0, 1] 3 we have: k {X,Y,Z} E u i[f k(û i, u i )]u i k }{{} =U i (û i,u i ) k {X,Y,Z} E u i[f k(u i, u i )]u i k }{{} =U i (u i,u i ) µ i (u i ) The following lemma states standard results from the mechanism design literature for our set-up: Lemma 2 A DRM f is incentive compatible if and only if: 1. µ i is convex, 2. At every u i at which µ i is differentiable: µ i (ui ) u i k all k {X, Y, Z}. = E u i[f k (u i, u i )] for Convexity of µ i implies that agent i s conditional expected probability of alternative k, E u i[f k (u i, u i )], is increasing in agent i s vnm utility of u i k : E u i[f k (u i, u i )] u i k 0 for all k {X, Y, Z}. Following Hortala-Vallve (2009), we can state that if a decision rule f is incentive compatible, then µ i is homogeneous of degree one: µ i (λu i ) = λµ i (u i ) (for λ R ++ ). Homogeneity of µ i implies that the decision rule f bunches all proportional types. By definition, µ i (λu i ) = k {X,Y,Z} E u i[f k(λu i, u i )] (λu i k ). Then, µ i (λu i ) = λµ i (u i ) E u i[f k (λu i, u i )] = E u i[f k (u i, u i )] Observe that homogeneity of degree zero imposes the following restrictions on incentive compatible decision rules f: Take two types of agent i: (u i X, ui Y, ui Z ) and (û i X, ûi Y, ûi Z ), and suppose X i Y i Z. Under state-dependent expected utility, the following transformation of agent i s vnm utilities can be performed wlog: (u i X ui Z, ui Y ui Z, 0) and (ûi X ûi Z, ûi Y ûi Z, 0), which 11
obtains by subtracting the vnm utility of the lowest-ranked alternative from the vnm utilities of the other alternatives. Note that, in general, we cannot wlog divide these transformed types by different constants. However, incentive compatibility (in particular, homogeneity of degree zero) implies that the decision rule f must treat a type (u i X ui Z, ui Y ui Z, 0) the same as the type (1, ui Y ui Z, 0), and must treat a type (û i u i X ui X ûi Z, ûi Y ûi Z, 0) the same as Z the type (1, ûi Y ûi Z, 0). û i X ûi Z While incentive compatibility may impose further constraints on decision rules f beyond homogeneity of degree zero, it is clear that no second best decision rule can generate higher ex ante welfare than a decision rule that respects only the homogeneity requirement. Under such a decision rule, any two agents i and j who have the same ordinal ranking, and for whom u i 2 u i 3 u i 1 u i 3 = uj 2 u j 3 u j 1 u j 3 have the same expected utility, and therefore the same best response function. Therefore, there always exists a BNE of the game induced by the decision rule f such that agents i and j report the same message to the social planner. We now state the following conjecture (as yet unproven) which should provide a more realistic benchmark for the evaluation of the welfare-performance of different (A, B)-scoring rules: Conjecture 1 Among decision rules f for which the bunching of proportional types -condition holds, the optimal decision rule assigns to each agent i who reports his ordinal ranking i over {X, Y, Z}, as well as a number θ i = (u i 2 u i 3)/(u i 1 u i 3), an expected type-vector τ i. In this expected typevector τ i, the alternative that agent i identifies as his highest-ranked alternative is assigned a value of E[u 1 θ i ]; the alternative that he identifies as his middle-ranked alternative is assigned a value of θ i E[u 1 θ i ]+(1 θ i )E[u 3 θ i ]; and the alternative that he identifies as his lowest-ranked alternative is assigned a value of E[u 3 θ i ]. On the basis of expected types τ i for all agents, the decision rule selects the one that would be first best if the τ i were the agents true types. 4.4 Numerical results This section accompanies the diagrams and tables found in Appendix 1. Looking at the diagrams indicating welfare losses, it is clear that (A, B)- 12
scoring rules with large A and relatively small B provide the lowest welfare loss. In fact in each case, the best (A, B)-scoring rule is any one from the open set of points marked in grey in the Figure 3 below. Negative voting is consistently the worst rule, indicated in the figures by the black dot at (1,1). The plurality rule is consistently the second-worst rule, indicated in the figures by the dark grey dot at (0,0). The best simple scoring rule (i.e. rules for which A = B) is the one that is consistent with the results in Apesteguia et al. (2010). Approval voting performs better than either negative voting or plurality rule, but not as well as the best simple scoring rule, or the optimal (A, B)- scoring rules shown in Fig. 3. Almost all rules that satisfy 1 > A > B > 0 yield higher welfare than those that do not, which is due to the fact that these rules use the agents full ordinal rankings in making a collective choice. One surprising fact that is evident from Fig. 3. is that it is possible to pick an (A, B)-pair that is extremely close to (1, 0) (approval voting), but yields substantially higher welfare. This discontinuity stems from the discrete number of agents in the model. Our conjecture in the previous section sharpens the intuition for our numerical results. To highlight this, we consider (for simplicity) in the remainder of this section the case where the distribution G from which the agents individual vnm utilities are drawn is the uniform distribution on [0, 1]. In this case, E[u i 1 θ i ] = E[u i 1] = 0.75, and E[u i 3 θ i ] = E[u i 3] = 0.25 for for every θ i and any i I. Therefore, the expected type of agent i in the decision rule described in Conjecture 1 is: τ i = (τ i 1, τ i 2, τ i 3) = (0.75, 0.75θ i + 0.25(1 θ i ), 0.25) Applying an affi ne transformation where τ i l = (τ i l τ i 3)/(τ i 1 τ i 3) yields: τ i = ( τ i 1, τ i 2, τ i 3) = (1, θ i, 0) Now consider an (A, B)-scoring rule in gray shaded area of Fig. 3. For scoring rules in this equivalence class, the unique value α that characterizes the symmetric BNE voting strategy is α = 0.7926. Now consider the particular (A, B)-scoring rule where A = 0.9 and B = 0.4, which is contained in the shaded area in Fig. 3. Note that any agent i whose type u i gives rise to a number θ i > 0.7926 submits a vote-vector that is a permutation of (1, 0.9, 0), while any type u i with θ i < 0.7926 submits a vote-vector that is a permutation of (1, 0.4, 0). These two vote-vectors are approximately equal 13
to the expected types who use these vote-vectors in equilibrium: (1, 0.9, 0) E[u i θ i > α ] and (1, 0.4, 0) E[u i θ i < α ]. 5 In other words, conditional on an agent submitting the vote-vector (1, 0.9, 0), his expected value of θ i is approx. 0.9, and condition al on an agent submitting the vote-vector (1, 0.4, 0), his expected value of θ i is approx. 0.4. Under Approval voting, which performs less well than any rule from the shaded set in Fig. 3., the equilibrium value of α is α = 0, 745967. Types with a value of θ i > 0.745967 submit a vote-vector that is a permutation of (1, 1, 0), while types with a value of θ i < 0.745967 submit a vote-vector that is a permutation of (1, 0, 0). Clearly, these vote-vectors do not represent very well the average type of agent who uses them, hence the higher welfare-loss relative to the benchmark decision rule in Conjecture 1. The same can be said of Negative Voting, and the plurality rule. The Borda count, which is the optimal simple scoring rule (i.e. one where A = B), effectively assigns all agents an expected type of τ i = ( τ i 1, τ i 2, τ i 3) = (1, 0.5, 0). This is also equal to the expected type E[u i θ i ] of each agent who uses this vote-vector, but as it assigns it to every type, it is not as effective as the optimal best (A, B)-scoring rule. Numerically calculating expected welfare under the benchmark mechanism in Conjecture 1 using Monte-Carlo experiments for the case of the uniform distribution yields a welfare-loss against the First Best of 1.604%. The welfare loss of the best (A, B)-scoring rule against the benchmark mechanism in Conjecture 1 is, for the uniform case, equal to 0.458%. This value is very small, and suggests that most of the welfare-loss can be attributed to the necessity of bunching all proportional types. This observation fits well with the finding Börgers and Postl (2009), who noted that welfare-losses were almost negligible in their setting, where agents types were normalized so that each agent s most preferred alternative had a vnm utility of 1, and each agent s least preferred alternative had a vnm utility 0. Consequently, there were no proportional types that had to be bunched. Finally, looking at the tables in Appendix 1, it is clear that - although not optimal - the best simple scoring rules often come close to the best (A, B)- scoring rule in terms of expected welfare. 5 The actual expected value of θ i among those who submit the vote-vector (1, 0.9, 0) is 0.8963, and the actual expected value of θ i among those who submit the vote-vector (1, 0.4, 0) is 0.3963. Note that the pair (A, B) = (0.8963, 0.3963) constitutes a twoparameter scoring rule in the equivalence class highlighted in Fig. 3. 14
5 Conclusion The aim of this paper was to study cardinal standards of welfare in voting problems without transferable utility using (A, B)-scoring rules. Our results show that in this simple model, Plurality rule and Negative voting cause high welfare-losses relative to first best. Whilst Approval voting performs much better than either of these rules, it does not perform as well as rules with a relatively large A-value and small B-value. More generally, it seems that voting rules that allow agents to express relative strength of preference perform better than those which do not. Also, rules which use agents full ordinal rankings perform better than those which do not. However, such systems do not seem to be particularly widely used. Whilst approval voting does involve relative strength of preference, the resulting vote vectors do not convey unambiguously the agents ordinal preferences. Simple scoring rules (for which A = B) suffer the opposite problem. In future work we intend to extend our model to settings with a larger number of agents or alternatives. With a larger number of agents, one would expect the highly discontinuous nature of the welfare-levels across (A, B)-scoring rules to be smoothed out by making the equivalence classes into which (A, B)-scoring rules can be divide much finer. Another extension might look into optimality of different rules in a common value setting, as opposed to the private value setting covered here. There is some research already in this area, for example Ahn and Oliveros (2010) use (A, B)-Scoring rules to show that in a common value setting with three alternatives, the best equilibria under Approval voting are more effi cient than Plurality of Negative voting. Finally, future research might also look at the continued use of numerical optimization techniques in a discrete setting. Whilst the issue of characterizing optimal direct revelation mechanisms for more than two agents remains an open problem, improvements in computing power should allow the use of finer discrete type-sets than has been possible so far. 6 References Ahn, D., Oliveros, S., 2010. Approval voting and scoring rules with commonvalues. UC Berkeley Mimeo. Apesteguia, J., Ballester, M. A., Ferrer, R., 2010. decision rules. Review of Economic Studies 78 (1), 1-16. 15 On the justice of
Borgers, T., Postl, P., 2009. Effi cient compromising. Journal of Economic Theory 144, 2057-2076. Hortala-Vallve, R., 2009. Ineffi ciencies on linking decisions. Social Choice and Welfare 34, 471-486. Myerson, R., 2002. Comparison of scoring rules in poisson voting games. Journal of Economic Theory 103 (1), 219-251. Postl, P., 2006. A bayesian mechanism design approach to optimal voting rules when agents preferences are private information. University of Birmingham mimeo. Sonnenschein, H., Jackson, M., 2007. Overcoming incentive constraints by linking decisions. Journal of Economic Theory 75 (1), 241-257. 7 Appendix 1 The following tables and figures provide the relative welfare losses of a number of (A, B)-scoring rules for three distributions G, each given by a different Beta-distribution. The figures represent the ex ante expected welfare in the BNE of every possible (A, B)-scoring rule. In particular, the figures show the relative welfare-loss relative to first best (the welfare of which has been calculated using Monte-Carlo experiments). The lowest welfare-loss is indicated in white, while black indicates the highest welfare-loss. Intermediate welfarelosses are shown in different shades of gray. The three Beta-distributions that we have chosen in this paper are: Beta(1,1) (which corresponds to the uniform distribution), Beta(2,1) (which places more probability mass on high realizations), and Beta (1,2) (which places more probability mass on low realizations). 16
17
Relative welfare-loss for Beta(1,1) (uniform) Relative welfare-loss for Beta(1,2) 18
Relative welfare-loss for Beta(2,1) 8 Appendix 2 Proof. Step 1: Take any agent and a profile of his n opponents vote vectors such that alternative X has the highest score. I.e. the aggregate vote vector v = ( j i vj X, j i vj Y, j i vj Z ) is such that j i vj X > max{ j i vj Y, j i vj Z }. Now, in the individual vote vector of each agent j, interchange the scores assigned to alternatives X and Y : v j = (v j Y, vj X, vj Z ). This gives rise to the aggregate vote vector v = ( j i vj Y, j i vj X, j i vj Z ) under which alternative Y as the highest score. Similarly, by interchanging the scores assigned to alternatives X and Z in the individual vote vectors of the n opponents, we can obtain the aggregate vote vector ṽ = ( j i vj Z, j i vj Y, j i vj X ) such that alternative Z has the highest aggregate score. This shows that, for every aggregate vote vector v VA,B n s.t. alternative X has the highest score, there exist vectors v, ṽ VA,B n s.t. alternative Y (alternative Z, resp.) has the highest aggregate score. Therefore, the number of aggregate vote vectors for which alternative Y has the highest score equals the number of aggregate vote vectors for which alternative X has the highest score. Likewise, the number of aggregate vote vectors for which alternative Z has the highest score equals the number of aggregate vote vectors for which alternative X has the highest score. Step 2: Take any agent and a profile of his n opponents vote vectors such that alternative X has the same aggregate score as alternative Y, but higher than the aggregate score of alternative Z. I.e. the aggregate vote vector v = ( j i vj X, j i vj Y, j i vj Z ) across the n opponents is such that j i vj X = j i vj Y > j i vj Z (i.e. the social ranking of the three alternatives by the n opponents is X n Y n Z, where n denotes the social preference relation of the n opponents over outcomes in {X, Y, Z}). Now, in the individual vote vector of each agent j, interchange the scores assigned to alternatives X and Z: ṽ j = (v j Z, vj Y, vj X ). This gives rise to the aggregate vote vector ṽ = ( j i vj Z, j i vj Y, j i vj X ) under which alternatives Y and Z have the same aggregate score that exceeds the score of alternative X (i.e. the social ranking is Y S Z S X). I.e. for every aggregate vote vector in VA,B n s.t. alternatives X and Y tie while beating alternative Z, there exists an aggregate vote vector in VA,B n s.t. alternatives Y and Z tie while beating 19
alternative X. Consequently, the number m of aggregate vote vectors in VA,B n s.t. X and Y tie while beating Z is the same as the number of aggregate vote vectors m in VA,B n s.t. Y and Z tie while beating alternative X. Now consider an aggregate vote vector v in VA,B n s.t. alternatives X and Z tie while while beating alternative Y (social ranking: X n Z n Y ). By interchanging the X- and Y -scores in the individual vote vectors, we obtain an aggregate vote vector s.t. alternatives Y and Z tie while beating alternative X (social ranking: Y n Z n X). I.e. for every aggregate vote vector in VA,B n s.t. alternatives X and Z tie while beating alternative Y, there exists another aggregate vote vector in VA,B n s.t. alternatives Y and Z tie while beating alternative X. Consequently, the number m of aggregate vote vectors in VA,B n s.t. X and Z tie while beating Y is the same as the number of aggregate vote vectors m in VA,B n s.t. B and C tie while beating alternative A. Therefore: m = m = m. Step 3: For each voter j, the probability that a permutation v j of a given vote vector v j arises is the same as the probability that v j arises. E.g. Pr{(1, A, 0)} = 1 (1 p(α)), where p(α) is the probability that, conditional on 6 a given ordinal ranking, voter j assigns the lower score of B to his middleranked alternative. We have Pr{(A, 1, 0)} = 1 (1 p(α)) = Pr{(1, A, 0)}, 6 which is due to the fact that all ordinal rankings are equally likely, and that all voters, conditional on a given ordinal ranking, use the same strategy to decide whether to assign a score of B or a score of A to their middle-ranked alternative. On the basis of Steps 1-3, we can immediately conclude that the probability that alternative X has the highest score across the n opponents of agent i is the same as the probability that alternative Y has the highest score, which, in turn, is the same as the probability that alternative Z has the highest score. I.e. we have Pr{X} = Pr{Y } = Pr{Z} = 1/3. Step 4: Regardless of the value of n 2, there always exist aggregate vote vectors s.t. X n Y n Z, and consequently, there exist equally many aggregate vote vectors s.t. X n Z n Y and s.t. Y n Z n X. In addition, if there are n = 3 opponents (or any multiple of 3), there are also vote vectors s.t. X n Y n Z. However, such three-way ties do not exist for n = 2 opponents, or if n/3 is not an integer (e.g. n = 148 or n = 55. For n = 55, I can create two-way ties even though 55/2 is not an integer. To see this, suppose that three opponents have vote vectors so that there is a three-way tie. E.g. (1, A, 0), (0, 1, A) and (A, 0, 1), which gives an aggregate of (1+A, 1+A, 1+A) across these three opponents. Now let (55 3)/2 = 26 20
submit the vote vector (1, B, 0), while the remaining 26 submit (B, 1, 0). This yields an aggregate vote vector across all n = 55 voters of (1 + A + 26(1 + B), 1 + A + 26(B + 1), 1 + A) = (27 + 26B + A, 27 + 26B + A, 1 + A)). Step 5: Now consider agent i and suppose he submits a vote vector that assigns a score of 1 to alternative U, a score of σ {A, B} to alternative V, and a score of 0 to alternative W, where U, V, W {X, Y, Z}, U V W. Now consider aggregate vote vectors across the n opponents of agent i where U ties with one of the other two alternatives. If U n V n W, we now have ( j i vj U + 1, j i vj U + σ, j i vj W ) so that U n+1 V n+1 W ; and if U n W n V, now have ( j i vj U + 1, j i vj V + σ, j i vj U ) so that j i vj U + 1 max{ j i vj V + σ, j i vj U }. In each such event where agent i s vote vector generates an outright win for alternative U (and there is a positive number of such events), alternative U gains probability weight 1/2 at the expense of alternatives V and W, resp. As the probability distribution across the three alternatives prior to accounting for voter i s vote vector was uniform (Pr{U} = Pr{V } = Pr{W } = 1/3), it now holds that Pr{U} > 1 > 3 max{pr{v }, Pr{W }}. Obviously, in case that there exist three-ways ties, alternative U gains yet more probability weight at the expense of alternatives V and W. Furthermore, in all events where U narrowly loses to either V or W, alternative U will gain yet more probability weight at the expense of the other two alternatives. Step 6: Now suppose wlog that voter i s ordinal ranking is X i Y i Z, with u i X > ui Y > ui Z. As voter i s expected utility is a weighted sum of these three vnm utilities, it is immediate that as much probability mass should be concentrated on alternative on u i X, and as little as possible on alternative on u i Z. It is therefore optimal for voter i to submit the vote vector that assigns a score of 1 to alternative X, and a score of 0 to alternative Z. Proof. Wlog consider agent i. Suppose agents j i use the same strategy, and that this strategy satisfies P1 and P2, with parameter α [0, 1]. The probability that any agent j i submits the vote vector (v j X, vj Y, vj Z ) = (1, B, 0), or any permutation thereof, is given by 1 p(α), where p(α) is given in 6 (1). The probability that any agent j submits the vote vector (v j X, vj Y, vj Z ) = (1, A, 0), or any permutation thereof, is given by 1 (1 p(α)). Therefore, the 6 aggregate vote vectors in VA,B 2 will occur either with probability ( 1 6 p(α))2, with probability 1p(α)( 1(1 p(α))), or with probability ( 1 6 6 6 (1 p(α)))2. Note that the probability p(α) is a continuous function of α, with p(0) = 0 and p(1) = 1. 21
By Lemma 1, agent i s choice of vote vector reduces to the question of whether to set v i 2 = A, or v i 2 = B. Observe that relative to setting v i 2 = B, choosing v i 2 = A (i.e. submitting the vote-vector (v i 1, v i 2, v i 3) = (1, A, 0)) implies a shift in probability mass to alternative 2 from either alternative 1, or alternative 3, or both. To see this, consider the following aggregate vote-vector across all agents j i: ( j i vj 1, j i vj 2, j i vj 3) = (A, 1, 1 + A) This vote vector occurs with probability ( 1 6 (1 p(α)))2, which is positive for α < 1. If agent i submits the vote-vector (v i 1, v i 2, v i 3) = (1, B, 0), the aggregate vote-vector across all three agents is: ( j I vj 1, j I vj 2, j I vj 3) = (1 + A, 1 + B, 1 + A) This implies that alternatives 1 and 2 are both chosen with probability 1/2. If, instead, agent i submits the vote vector (v i 1, v i 2, v i 3) = (1, A, 0), the aggregate vote-vector across all three agents is: ( j I vj 1, j I vj 2, j I vj 3) = (1 + A, 1 + A, 1 + A) This implies that all three alternatives are chosen with equal probability of 1/3. Next, consider the following aggregate vote-vector across all agents j i: ( j i vj 1, j i vj 2, j i vj 3) = (B, 1, 1 + B) This vote vector occurs with probability ( 1 6 p(α))2, which is positive for α > 0. If agent i submits the vote-vector (v i 1, v i 2, v i 3) = (1, B, 0), the aggregate votevector across all three agents is: ( j I vj 1, j I vj 2, j I vj 3) = (1 + B, 1 + B, 1 + B) This implies that all three alternatives are chosen with equal probability of 1/3. If, instead, agent i submits the vote vector (v i 1, v i 2, v i 3) = (1, A, 0), the aggregate vote-vector across all three agents is: ( j I vj 1, j I vj 2, j I vj 3) = (1 + B, 1 + A, 1 + B) This implies that alternative 2 is chosen. The above two examples establish that for any α [0, 1], submitting vote-vector (v i 1, v i 2, v i 3) = (1, A, 0) implies 22
(relative to the case where vote vector (v1, i v2, i v3) i = (1, B, 0) is submitted) a strict increase in the probability that alternative 2 is chosen, at the expense of the probabilities with which alternatives 1 and 3 are chosen. Define now by π σ i,l (α) (where l = 1, 2, 3) the probability that agent i s l-th ranked alternative is chosen if i submits the vote vector (v1, i v2, i v3) i = (1, σ, 0) (where σ {A, B}), and the other two agents j i use the same strategy satisfying P1 and P2. The probability π σ i,l (α) is of the following form: π σ i,l(α) = a σ ( 1 6 p(α))2 + b σ 1 6 p(α)(1 6 (1 p(α))) + cσ ( 1 6 (1 p(α)))2 (2) where a σ denotes the ex post probability that alternative 1 is chosen when agent i submits the vote-vector (v1, i v2, i v3) i = (1, σ, 0) and the vote-vectors submitted by the other two agents j i are both permutations of (1, B, 0). Note that, given our tie-breaking rule, the ex post probability of an alternative being chosen is either 1 (if this alternative has a higher aggregate score than the other two alternatives), or 1/2 (if the aggregate score of this alternative equals the aggregate score of one other alternative, and this score exceeds that of the remaining alternative), or 1/3 (if the aggregate scores of all three alternatives are the same). Likewise, b σ denotes the ex post probability that alternative 1 is chosen when agent i submits the vote-vector (v1, i v2, i v3) i = (1, σ, 0), while one other agent submits a vote-vector that is a permutation of (1, B, 0), and the remaining agent submits a vote-vector that is a permutation of (1, A, 0). Finally, c σ denotes the ex post probability that alternative 1 is chosen when agent i submits the vote-vector (v1, i v2, i v3) i = (1, σ, 0), while the vote-vectors submitted by the other two agents j i are both permutations of (1, A, 0). We can then write agent i s expected utility from submitting either of his two vote vectors (1, A, 0) and (1, B, 0) as follows: U i (A, α, u i ) π A i,1(α)u 1 + π A i,2(α)u 2 + π A i,3(α)u 3 U i (B, α, u i ) π B i,1(α)u 1 + π B i,2(α)u 2 + π B i,3(α)u 3 The argument in the second and third paragraphs of this proof implies that: π A i,1(α) < π B i,1(α) π A i,2(α) > π B i,2(α) π A i,3(α) < π B i,3(α) 23
As π A i,1(α) + π A i,2(α) + π A i,3(α) = 1 = π B i,1(α) + π B i,2(α) + π B i,3(α), it is immediate that: π A i,2(α) π B i,2(α) = [ π B i,1(α) π A i,1(α) ] + [ π B i,3(α) π A i,3(α) ] (3) As the increase π A i,2(α) π B i,2(α) in the probability of alternative 2 under vote-vector (1, A, 0), relative to vote-vector (1, B, 0), is obtained by transferring probability from alternative 1 and/or alternative 3 to alternative 2, it follows that the left-hand side of (3) is strictly greater than either of the two bracketed terms on the right-hand side of (3). We can re-write (3) as follows: 1 = πb i,1(α) π A i,1(α) π A i,2 (α) πb i,2 (α) + πb i,3(α) π A i,3(α) π A i,2 (α) πb i,2 (α) π B i,3(α) π A i,3(α) π A i,2 (α) πb i,2 (α) = 1 πb i,1(α) π A i,1(α) π A i,2 (α) πb i,2 (α) (4) Now observe that agent i will submit vote-vector (1, A, 0) (rather than (1, B, 0)) iff: U i (A, α, u i ) U i (B, α, u i ) > 0 [ π A i,2 (α) π B i,2(α) ] u i 2 > [ π B i,1(α) π A i,1(α) ] u i 1 + [ π B i,3(α) π A i,3(α) ] u i 2 Using (4), we can re-write this inequality as follows: ( ) u i 2 > πb i,1(α) π A i,1(α) π A i,2 (α) πb i,2 (α)ui 1 + 1 πb i,1(α) π A i,1(α) π A i,2 (α) πb i,2 (α) Given the expression for the probabilities π σ i,l in (2), it holds that the coeffi cient [π B i,1(α) π A i,1(α)]/[π A i,2(α) π B i,2(α)] of u i 1 is in (0, 1) for every α [0, 1]. 6 Furthermore, the coeffi cient [π B i,1(α) π A i,1(α)]/[π A i,2(α) π B i,2(α)] is a continuous function of α. In order to obtain a symmetric BNE in which 6 We have verified this by computing for all 0 B < A 1 the chosen alternative(s) in the cases where: (i) the two agents j i submit vote-vectors that are both permutations of (1, B, 0), (ii) the two agents j i submit vote-vectors that are both permutations of (1, A, 0), and (iii) one agent submits vote-vectors that are permutations of (1, B, 0) and the other submits vote-vectors that are permutations of (1, A, 0). u i 3 24
agent i adopts the same voting strategy as the other two agents j i (and which satisfies P1 and P2, with some α [0, 1]), it must hold that: π B i,1(α) π A i,1(α) π A i,2 (α) πb i,2 (α) = α By appealing to Brouwer s Fixed Point Theorem, we can conclude that the function [π B i,1(α) π A i,1(α)]/[π A i,2(α) π B i,2(α)] has at least one fixed point α, and all fixed points lie in the open unit interval (0, 1). 25