Whose Opinion Counts? Political Processes and the Implementation Problem

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1 Whose Opinion Counts? Political Processes and the Implementation Problem Rene Saran Norovsambuu Tumennasan August 13, 2011 Abstract We augment the mechanism used in Nash implementation with a political process that collects the opinions of a subset of individuals with a fixed probability distribution. The outcome is a function of only the collected opinions. We show that the necessary and sometimes sufficient condition for implementation by a specific political process can be either weaker or stronger than Maskin monotonicity. We study three such processes: oligarchy, oligarchic democracy, and random sampling. Oligarchy collects only the opinions of the oligarchs (a strict subset of the individuals). We present a Nash implementable social choice rule (SCR) that cannot be implemented by any oligarchy. Oligarchic democracy almost always collects the opinions of the oligarchs but sometimes there is a referendum (i.e., everyone s opinions are collected). We show that in economic environments, every Nash implementable SCR can be implemented by oligarchic democracy in which any three individuals act as oligarchs. In random sampling, a sample of opinions are collected randomly. We show that in economic environments, every Nash implementable SCR can be implemented by randomly sampling opinions of 4 individuals. We also provide necessary and sufficient conditions for implementation when the planner has the flexibility to choose any political process. Keywords: Nash Implementation; Political Process; p Implementation; Direct Democracy; Oligarchy; Oligarchic Democracy; Random Sampling JEL: C72; D78 We thank Subir Bose, Antonio Cabrales, Bo Chen, Kfir Eliaz, Alexander Koch, David Levine, Rudolf Müller, Julia Nafziger, Ronald Peeters, Ludovic Renou, William Sandholm, Roberto Serrano, Ricardo Serrano-Padial, William Thomson and seminar audience at Aarhus, Bielefeld, Brown, Copenhagen, Leicester, Nottingham, 2011 Spring Midwest Theory Meeting (Notre Dame), and 22nd International Conference on Game Theory (Stony Brook) for useful comments and suggestions. Department of Economics, Maastricht University, PO Box 616, 6200 MD Maastricht, The Netherlands. r.saran@maastrichtuniversity.nl; Tel: ; Fax: School of Economics and Management, Aarhus University, Bartholins Allé 10, Aarhus C 8000, Denmark. ntumennasan@econ.au.dk; Tel:

2 1 Introduction Consider a community of n individuals having a social choice rule (SCR) that specifies the socially desirable alternatives conditional on the state of the economy (preferences, technology etc.). The community would like to design a mechanism such that individuals incentives in the mechanism are aligned with its SCR, i.e., given the state, any Nash equilibrium of the mechanism must lead to a socially desirable alternative and every socially desirable alternative must be obtained under some Nash equilibrium of the mechanism. This is the implementation problem under complete information. 1 The standard mechanism used in this literature has two components: a message space for each individual and a single outcome function that maps everyone s messages to an alternative. Thus, the planner collects everyone s messages and then decides on the outcome. This is analogous to direct democracy since everyone s opinions are considered while making the decision. However, direct democracy is seldom used at national, state or even local levels. Indeed, when n is large, it will be too costly and time consuming to collect everyone s opinions while making social decisions. By the time a data collector gathers everyone s opinions, the state of the economy might change, rendering the whole exercise meaningless. In this paper, we augment the standard mechanism with a political process that selects a senate (i.e., a subset of the individuals) with an exogenous probability distribution. 2 Thus, a mechanism now has three components: a message space for each individual, a political process, and a set of outcome functions, one for each possible senate, that map messages of the senators to alternatives. This mechanism defines a strategic-form game in which all individuals simultaneously announce their messages. The political process then selects the senate and transmits the messages of only the selected senators to the planner; the messages of other individuals are ignored. 3 Finally, the planner implements an alternative using the transmitted messages and the outcome function of the corresponding senate. We are interested in the relationship between a political process and the SCRs that a community can implement if it were to adopt a mechanism which uses that political process. In designing the mechanism, the planner could face two alternative scenarios. First, the planner could be restricted to use a specific political process. In this case, we say that an SCR is implementable by the political process p or p implementable if there exists a mechanism that uses p as the political process and the set of Nash equilibrium outcomes 1 See Maskin and Sjöström (2002) and Serrano (2004) for surveys of implementation theory. 2 Alternatively, one can consider political processes in which the senate is selected endogenously, e.g., through elections. This is an interesting area of research that we hope will be pursued in future. 3 It is possible to alter the mechanism so that an individual announces a message if and only if she is selected as the senator (see Remark 2.10). There is no change in our results under this participation requirement. 2

3 of the mechanism coincides with the SCR in each state. Although lotteries over the set of alternatives can be generated in the mechanism (since the selection of the senate under p could be stochastic), only off-the-equilibrium outcomes can be lotteries. Nash equilibrium outcomes are deterministic with the same socially desirable alternative being implemented for all possible senates. Second, the planner could have the flexibility to use any political process. In this case, we say that an SCR is implementable by a political process if there exists some political process p such that the SCR is p implementable. We study both these scenarios in this paper. First, consider the scenario when the planner must use a specific political process. In his seminal contribution, Maskin (1999) studied Nash implementation which is equivalent to implementation by the process of direct democracy. In direct democracy, the probability of referendum (i.e., selecting the set of all individuals as the senate) is 1 and hence, messages of all individuals determine the outcome. Maskin (1999) showed that any Nash implementable SCR must be Maskin monotonoic. He further proved that if n 3, then Maskin monotonicity and no-veto power are sufficient conditions for an SCR to be Nash implementable. 4 It follows that in economic environments (i.e., when at most n 2 individuals have a common mostpreferred alternative) with at least 3 individuals, Maskin monotonicity is sufficient for Nash implementation. We use these results for Nash implementation as benchmarks. In particular, for each political process p, we are interested in knowing whether every SCR that is Nash implementable is also p implementable. If the answer is yes, then replacing direct democracy with such a political process that considers the messages of a smaller number of individuals can reduce the costs and time involved in collecting everyone s opinions. 5 We prove that a necessary condition for an SCR to be p implementable is that it is p monotonic. Maskin monotonicity is equivalent to p monotonicity when p is direct democracy. More generally, if p is such that every individual has a positive chance of being selected as a senator, then p monotonicity is weaker than Maskin monotonicity. This is because the planner can now implement lotteries over alternatives on off-the-equilibrium paths whenever p is stochastic, which provides greater flexibility than in Nash implementation where she is restricted to implement only alternatives. 6 On the other hand, if p is such that some 4 For necessary and sufficient conditions for Nash implementation see Moore and Repullo (1990), Sjöström (1991), Danilov (1992) and Yamato (1992). 5 We do not explicitly model such costs of collecting messages. One possibility is to assume that there is a fixed cost of collecting each message equal to c. In Section 8, we briefly discuss the implication of our results under this assumption. We must also point out that we do not consider the issue of communication and processing burdens imposed on, respectively, the individuals and planner due to large size of the message space (see, for e.g., Saijo (1988) and McKelvey (1989), who tackle this issue in the context of Nash implementation). 6 Benoît and Ok (2008, Remark 3) provide an example to show that Maskin monotonicity is not necessary for Nash implementation if the planner can implement lotteries and use her knowledge about individuals preferences over lotteries. 3

4 individuals have no chance of being selected as senators, then p monotonicity can be stronger than Maskin monotonicity. Restricting attention to those p that always select at least three senators and have a positive chance of referendum, we prove that in economic environments, p monotonicity is also sufficient for p implementation. Since p monotonicity is weaker than Maskin monotonicity for such p, we obtain the result that in economic environments, any Maskin monotonic SCR can be implemented by any political process that always selects at least three senators and has a positive no matter how small chance of referendum. We then study three specific political processes: oligarchy, oligarchic democracy, and random sampling. In oligarchy, the senate comprises of a particular strict subset of the individuals (the oligarchs) with probability 1. Thus, only the messages of the oligarchs determine the implemented alternative. Oligarchic democracy is a perturbation of the oligarchy in which the senate almost always comprises of the oligarchs but there is a small positive chance of referendum. That is, in an oligarchic democracy, the messages of only the oligarchs determine the implemented alternative with probability arbitrarily close to 1 while with the rest of the probability, the messages of all individuals determine the implemented alternative. Finally, in random sampling, a fixed number of individuals, which is strictly less than n, are selected at random to form the senate and only the messages of these randomly selected individuals determine the implemented alternative. We show that any SCR that is implementable by oligarchy is also Nash implementable by making the outcome function under the direct democracy unresponsive to the messages of the commoners (individuals who are not in the oligarchy). However, the converse is not true: we present an SCR that is Nash implementable but not implementable by any oligarchy. The reason is that p monotonicity when p is oligarchy is stronger than Maskin monotonicity since messages of the commoners are ignored in the oligarchy. The perturbed process of oligarchic democracy, on the other hand, generates dramatically opposite conclusions. Our sufficiency result for political processes that always select at least three senators and have a positive chance of referendum gives us the following corollary: in economic environments, any Nash implementable and sometimes even non-maskin monotonic SCR is implementable by oligarchic democracy that has at least three oligarchs. The identities of the oligarchs do not matter for this result; any three individuals can be designated as the oligarchs. This success of oligarchic democracy, however, does not carry over to noneconomic environments. We give an example of a noneconomic environment and an SCR that is Nash implementable but not implementable by oligarchic democracy with any subset of the individuals designated as the oligarchs. The reason for this failure of oligarchic democracy is the lack of sufficient diversity in the most-preferred alternatives of 4

5 the oligarchs. As we show, any SCR satisfying Maskin monotonicity and no-veto power can be implemented by oligarchic democracy that has at least three oligarchs who never have a common most-preferred alternative. In random sampling, every individual has a positive chance of being a senator. Hence, p monotonicity when p is random sampling, or equivalently, the necessary condition for implementation by random sampling is weaker than Maskin monotonicity. However, there is no chance of referendum under random sampling. Still, we show that in those economic environments in which each individual has at most a single most-preferred alternative, the necessary p monotonicity condition for implementation by random sampling is also sufficient if the sample size is at least 4. In arbitrary economic and some noneconomic environments, we show that any Maskin monotonic SCR is implementable by random sampling if the sample size is at least 4. Thus, in economic environments, instead of collecting everyone s messages, the planner can collect messages of only 4 randomly selected individuals in order to implement any Nash implementable SCR. Four is the minimal sample size that guarantees this result. We present an SCR that is Nash implementable in an economic environment but not implementable by random sampling when the sample size is smaller than 4. Second, consider the scenario when the planner has the flexibility to use any political process. Our results for the first scenario immediately imply that an SCR is implementable by a political process only if it is p monotonic for some political process p. The set of SCRs that are p monotonic for some p of course includes the set of all Maskin monotonic SCRs. However, there are SCRs that are not p monotonic for any p and hence, not implementable by a political process. Under a stronger assumption on preferences over lotteries, we show that in economic environments, any SCR that is p monotonic for some p that always selects at least three senators is implementable by a political process. Under mild domain restrictions, Bochet (2007) and Benoît and Ok (2008) show that Maskin monotonicity is both necessary and sufficient for implementation using lottery mechanisms. In a lottery mechanism, off-the-equilibrium outcomes can be lotteries over alternatives but equilibrium outcomes are deterministic. Maskin monotonicity remains necessary because the planner has limited information about individuals preferences over lotteries; she only knows that these are monotonic. Hence, she must design a mechanism that works for all possible monotonic preferences over lotteries that agree on the ordering of alternatives. We instead assume that the planner knows individuals preferences over lotteries and uses this information in designing the mechanism. Therefore, for some political processes, we obtain weaker necessary conditions than Maskin monotonicity. Another difference between theirs and our setup is that in lottery mechanisms, arbitrary lotteries can be used off the equilibrium whereas the structure of off-the-equilibrium lotteries in our model is defined 5

6 by the specific stochastic selection of senates. 7 The rest of the paper is organized as follows. We outline the model and provide main definitions in Section 2. In Section 3, we study the scenario when the planner is restricted use a specific political process. In Sections 4, 5 and 6, we present the results for, respectively, implementation by oligarchy, oligarchic democracy and random sampling. In Section 7, we study the scenario when the planner has the flexibility to use any political process. We provide a brief conclusion in Section 8. Longer proofs are collected in Appendix. 2 Preliminaries There is a finite set of players N = {1,..., n} with n 3. The set of social alternatives is A, which can be infinite but not singleton. A lottery l is a probability distribution with a finite support in A. For any lottery l, let l(a) denote the probability assigned by l to a A and A(l) denote the support of l. We write a for both the alternative a A and the degenerate lottery that puts probability 1 on a. Let A denotes the set of lotteries. Let Θ be set of states with at least two elements. A typical state is denoted by θ. We assume that the players have complete information about the realized state. Each player has a state dependent preference ordering (i.e., complete and transitive relation) θ i over A. 8 Let θ i and θ i denote, respectively, the strict and indifference relations derived from θ i. For any i and θ, let A i (θ) = {a A : a θ i a, a A} be the set of most-preferred alternatives for i in state θ. Since A could be infinite, without additional assumptions, A i (θ) could be empty. For any i, θ, finite A A, and for all natural numbers k 1, recursively define: P 1 (i, θ, A ) = {a A : a θ i a, a A } P k+1 (i, θ, A ) = {a A \ P k (i, θ, A ) : a θ i a, a A \ P k (i, θ, A )}. Thus, P 1 (i, θ, A ) is the set of most-preferred alternatives in A for player i in state θ, P 2 (i, θ, A ) is the set of second preferred alternatives in A for player i in state θ etc. 7 Nevertheless, in their proofs, Bochet (2007) and Benoît and Ok (2008) use simple lotteries with at most two alternatives in their respective supports. This is similar to the lottery generated under oligarchic democracy (because the senate is either the oligarchs or all individuals). In contrast to Bochet (2007) and Benoît and Ok (2008), however, commoners cannot induce a lottery in oligarchic democracy since their messages are considered only in one senate. 8 We take these state-dependent preference orderings over A as primitives of the model instead of the state-dependent preference orderings over the set of probability measures on A (set of probability measures on A is in general a superset of A). This is because every outcome in the mechanisms that we study lies in A. 6

7 Unless stated otherwise, we maintain the following assumption on preferences: Assumption 2.1. Preferences over Lotteries are Monotone: For all i N, θ Θ, and l, l A, we have K l(a) k=1 a P k (i,θ,a(l) A(l )) K l (a), K 1 = l θ i l k=1 a P k (i,θ,a(l) A(l )) and whenever at least one inequality is strict, then l θ i l. This assumption means that shifting probability weights from alternatives lower in the preference ordering to alternatives higher in the preference ordering generates a preferred lottery. It is weaker than imposing the independence axiom (Definition 7.4) on players preferences. 2.1 Environment The environment is E = N, A, ( ) ( θi ) i N θ Θ. We consider various classes of environments, the most important being economic environment as defined by Bergemann and Morris (2008). Definition 2.2. An environment E is economic in state θ if for any a A, there exist i j and alternatives a i and a j such that a i θ i a and a j θ j a. An environment E is economic if it is economic in every state θ Θ. Observe that E is an economic environment if and only if in any state θ, an alternative is most-preferred by at most n 2 players, i.e., i S A i (θ) = for all S N such that S n 1. Benoît and Ok (2008) define the following class of environments: Definition 2.3. The environment E satisfies top-coincidence condition if for any θ Θ and S N such that S = n 1, i S A i (θ) is at most a singleton. Thus, an environment that satisfies top-coincidence condition is such that in any state, any subset of n 1 players have at most one unanimously most-preferred alternative. Clearly, the class of environments identified by top-coincidence condition is weaker than the class of economic environments. We define two other classes of environments. 7

8 Definition 2.4. The environment E satisfies unique-top condition if for all i N and θ Θ, A i (θ) = A i (θ) is singleton. Thus, unique-top condition requires that whenever a player has a most-preferred alternative in some state, then that alternative is unique. An example of such an environment is when all players have strict preferences. Another example is when all players have singlepeaked preferences over A. Clearly, any environment that satisfies unique-top condition must also satisfy top-coincidence condition. However, there is no logical relation between the class of economic environments and the class of environments satisfying the unique-top condition, i.e., there can exist both economic environments that do not satisfy the unique-top condition and noneconomic environments that satisfy the unique top condition. Definition 2.5. Let S N. The environment E satisfies diversity of top alternatives for S (DTA-S) if for all θ Θ, i S A i (θ) =, i.e., there does not exist an alternative that is unanimously most-preferred by every player in S. Given S N, there can exist both economic and noneconomic environments that satisfy DTA-S. Moreover, if S n 1, then every economic environment satisfies DTA-S. However, if S n 2, then there can exist economic environments that do not satisfy DTA-S. 2.2 Social Choice Rules Social goals are embodied in a social choice rule (SCR), which is a nonempty-valued correspondence F : Θ A. The following two properties of SCRs are prominent in the literature on Nash implementation. Definition 2.6. F is Maskin monotonic if whenever an alternative a F (θ) and a / F (θ ) for some θ and θ, there exist player i N and a A such that a θ i a and a θ i a. Definition 2.7. F satisfies no-veto power if for any θ Θ and S N such that S n 1, we have A i (θ) F (θ). i S We will later introduce other notions of monotonicity and no-veto power among a subset of the players. 8

9 2.3 Mechanism A social planner, who does not know the realized state, designs a mechanism in order to implement an SCR. The standard definition of a mechanism has a message space for each player and an outcome function that maps the messages of all players into the set of alternatives. We augment this definition with a political process that selects a senate, i.e., a subset of the players. Ultimately, the outcome is a function of only the messages of the selected senators. Formally, let N = 2 N \ be the set of all possible senates. By referendum we mean the event in which N is selected as the senate. Definition 2.8. A political process p is a lottery on N, i.e., p = (p(s)) S N such that p(s) 0, S N and S N p(s) = 1. Let P be the set of all political processes on N. A mechanism is a triplet Γ = ((M i ) i N, p, (g S ) S N ), where M i is the set of opinions or messages that player i can announce. p P is the political process for selecting a senate. If S N is the selected senate, then the messages of the players in S, (m i ) i S, are transmitted to the planner and the messages of all players j / S are ignored. g S : i S M i A is the outcome function conditional on the selection of the senate S N. Note that the outcome function g S is deterministic. Let m denote a typical element of i N M i. For any m i N M i and S N, let m S be the projection of m into i S M i (note that if S = N, then m N = m). That is, given any profile m of messages of all players, m S is the profile of messages of only the players in S. For any mechanism Γ = ((M i ) i N, p, (g S ) S N ) and message profile m, let l[γ, m] denote the lottery that assigns to each a {g S (m S ) : S N }, the probability S:g S (m S )=a p(s). For any realization θ, a mechanism Γ = ((M i ) i N, p, (g S ) S N ) defines a strategic-form game Γ, θ. In this game, each player i N announces a message m i M i. Let m is be generated message profile. Then a senate S N is selected according to the political process p. The messages of the selected senators m S are transmitted to the planner who then implements the alternative g S (m S ). 9 Thus, the outcome of message profile m is lottery 9 Thus, players cannot change their messages after their selection as senators. This constraint is important for our results. If after their selection, the senators are informed about each others identities and allowed to change their messages, then the necessary condition for implementation by any political process will be stronger than the necessary condition for implementation by oligarchy. Also see Remark

10 l[γ, m]. The players preferences are given by the profile ( θ i ) i N. Let NE(Γ, θ) denote the set of Nash equilibria of Γ, θ. 10 Definition 2.9. F is p implementable if there exists a mechanism Γ = ((M i ) i N, p, (g S ) S N ) such that { } l[γ, m] : m NE(Γ, θ) = F (θ), θ Θ. F is implementable by a political process if it is p implementable for some p P. Thus, if mechanism Γ = ((M i ) i N, p, (g S ) S N ) p implements F, then (i) for any m NE(Γ, θ), we must have l[γ, m] F (θ), i.e., l[γ, m] is deterministic with g S (m S ) = a F (θ) for all S with p(s) > 0 and (ii) for any a F (θ), there exists m NE(Γ, θ) such that l[γ, m] = a. Hence, although lottery p is used in the selection of the senators, p implementation requires exact implementation of the SCR. Remark In the strategic-form game defined by a mechanism, every player announces a message even though a player s message is ignored when she is not selected in the senate. A more reasonable assumption is that a player announces a message if and only if she knows that she is a senator. This alternative assumption can be easily incorporated into the model. Suppose mechanism Γ = ((M i ) i N, p, (g S ) S N ) p implements F. Consider the following extensive-form game: first a senate S N is chosen using the political process p. This selection procedure is common knowledge. However, the selected senators in S are not informed about each other s identities. Thus, each senator only knows p and that she is selected. Then the senators in S announce their messages (m i ) i S i S M i and alternative g S ((m i ) i S ) is implemented. In this extensive-form game, every player moves at exactly one information set and thus, it has the same strategic form as Γ, θ in each state θ. Hence, the extensive-form game also implements F in Nash equilibrium. 11 This alternative game form resembles an opinion poll (e.g. a telephone survey) since participants in such polls usually do not know each others identities. Our definition of a political process is broad enough to include several procedures like direct democracy, oligarchy, oligarchic democracy and random sampling that are used in practice. We define these political processes next. 10 Like most literature on implementation, we restrict ourselves to pure strategies. 11 Under the stronger assumption that players preferences over lotteries satisfy the independence axiom, we can argue that in each state θ, the set of Perfect Bayesian equilibria of the extensive-form game coincides with the set of Nash equilibria of Γ, θ. When a player is asked to play, she finds out that she is in the senate but she does not know who else is in the senate. Due to the independence axiom, the player s incentives in the extensive-form game are the same as in Γ, θ. 10

11 2.4 Direct Democracy We define direct democracy, denoted by p D, as the political process in which the probability of referendum is 1, i.e., p D P such that p D (N) = 1. Hence, in direct democracy, every player s message is transmitted to the planner. A direct-democratic mechanism, Γ D = ((M i ) i N, p D, (g S ) S N ), is a mechanism that uses p D to select the senate. 12 Note that for any m i N M i, we have l[γ D, m] = g N (m). The literature has exclusively focused on direct-democratic mechanisms. The corresponding notion of implementation is called Nash implementation. Definition F is Nash implementable if it is p D implementable, i.e., there exists a direct-democratic mechanism Γ D = ((M i ) i N, p D, (g S ) S N ) such that { } g N (m) : m NE(Γ D, θ) = F (θ), θ Θ. Maskin (1999) proves that Maskin monotonicity is necessary for Nash implementation. Theorem 2.12 (Maskin (1999)). If F is Nash implementable, then F is Maskin monotonic. In general environments with at least three players, Maskin (1999) shows that Maskin monotonicity and no-veto power are sufficient for Nash implementation. Theorem 2.13 (Maskin (1999)). If n 3, then any Maskin monotonic SCR that satisfies no-veto power is Nash implementable. Since no-veto power is vacuously satisfied in economic environments, we have the following corollary: Corollary F is Nash implementable in an economic environment with n 3 if and only if it is Maskin monotonic. 2.5 Oligarchy Let S be any proper subset of N. We define S oligarchy, denoted by p S, as the political process that selects S as the senate with probability 1, i.e., p S P such that p S (S) = 1. Hence, in S oligarchy, the messages of all players in S the oligarchs are transmitted to 12 Since messages of all players are transmitted to the planner, we only need to specify the outcome function g N as none of the other outcome functions are ever used in a direct democracy. However, we continue to list these functions in the definition of direct-democratic mechanisms just to make it consistent with the general definition of mechanisms given earlier. Similar remarks apply to other political processes. 11

12 the planner whereas the messages of all players in N \ S the commoners are ignored. An S-oligarchic mechanism, Γ S = ((M i ) i N, p S, (g S ) S N ), is a mechanism that uses p S to select the senate. Note that for any m i N M i, we have l[γ S, m] = g S (m S ). Definition Let S N. F is implementable by S oligarchy if it is p S implementable, i.e., there exists an S-oligarchic mechanism Γ S = ((M i ) i N, p S, (g S ) S N ) such that { } g S (m S ) : m NE(Γ S, θ) = F (θ), θ Θ. 2.6 Oligarchic Democracy Let S N. By S-oligarchic democracy, we mean the political process in which either the senate is S or there is a referendum but the probability of referendum is arbitrarily close to 0. To capture the idea of arbitrarily close to 0, we define a sequence of political processes in which the probability of referendum is converging to 0. Formally, for any S N and ɛ (0, 1), let (S, ɛ)-oligarchic democracy, denoted by p (S,ɛ), be the political process that selects S as the senate with probability 1 ɛ and N as the senate with probability ɛ. Hence, in (S, ɛ)-oligarchic democracy, either the messages of only the players in S the oligarchs are transmitted to the planner with probability 1 ɛ or the messages of all players in N are transmitted to the planner with probability ɛ. An (S, ɛ)- oligarchic-democratic mechanism, Γ (S,ɛ) = ((M i ) i N, p (S,ɛ), (g S ) S N ), is a mechanism that uses p (S,ɛ) to select the senate. Note that for any m i N M i, lottery l[γ (S,ɛ), m] assigns probability 1 ɛ to alternative g S (m S ) and probability ɛ to alternative g N (m). Definition Let S N. F is implementable by S-oligarchic democracy if there exists a sequence of (S, ɛ k )-oligarchic democracies ( p (S,ɛk )) such that k=1 (i) lim k ɛ k = 0 and (ii) for all k, F is p (S,ɛk ) implementable, i.e., there exists an (S, ɛ k )-oligarchic-democratic mechanism Γ (S,ɛk ) = ((M i ) i N, p (S,ɛk ), (g S ) S N ) such that { } l[γ (S,ɛk ), m] : m NE(Γ (S,ɛk ), θ) = F (θ), θ Θ. Condition (i) in the above definition formalizes the idea that the probability of referendum is arbitrarily close to 0. Condition (ii) requires exact implementation by every political process p (S,ɛk ). In particular, if m NE(Γ (S,ɛk ), θ), then l[γ (S,ɛk ), m] is deterministic with g S (m S ) = g N (m) F (θ). 12

13 2.7 Random Sampling The final political process that we consider is n-random sampling. In this process, denoted by p n, a senate of fixed positive size n < n is randomly selected from the set of all players. Let N n denote the set of all senates S N such that S = n. Thus, in random sampling, the senate is equally likely to be any S N n and the messages of such a randomly chosen sample of players are transmitted to the planner. The n-sampling mechanism, Γ n = ((M i ) i N, p n, (g S ) S N ), is a mechanism that uses p n to select the senate. Note that for any m i N M i, lottery l[γ n, m] is such that for all S N n, it assigns probability 1/ N n to the alternative g S (m S ). Definition F is implementable by n-random sampling if it is p n implementable, i.e., there exists an n-sampling mechanism Γ n = ((M i ) i N, p n, (g S ) S N ) such that { } l[γ n, m] : m NE(Γ n, θ) = F (θ), θ Θ. Thus, implementation by n-random sampling requires exact implementation of the SCR even though a lottery is used in the sampling of the messages. 3 p Implementation In this section, we consider the situation when the planner is restricted to use a given political process p. Thus, the relevant notion of implementation is p implementation. We now provide the necessary and sufficient conditions for p implementation. 3.1 Necessary Condition For any p P and function h : N A, define the lottery induced by [p, h], denoted by l[p, h], as the lottery that assigns to each a {h(s) : S N }, the probability S N : h(s)=a p(s). Also, let N (i) = {S N : i S} be the set of those senates in which player i is a senator. Definition 3.1. F is p monotonic for a given p P if whenever a F (θ) and a / F (θ) for some a, θ and θ, then there exist player i N and function h i : N A such that h i (S) = a, S N \ N (i) with p(s) > 0, a θ i l[p, h i ] and l[p, h i ] θ i a. Thus, p monotonicity requires that whenever an alternative a drops out of the SCR in going from θ to θ, then there must exist a player i for whom there is reversal of her preferences over lotteries around a (i.e., there is a lottery that she weakly prefers to a in θ 13

14 but this is not true in θ ), where we consider only those lotteries that are induced by [p, h i ] for any h i : N A with the property that h i (S) = a for all S N \ N (i) with p(s) > 0. The following theorem shows that p monotonicity is necessary for p implementation. Theorem 3.2. If F is p implementable, then F is p monotonic. Proof. Let Γ = ((M i ) i N, p, (g S ) S N ) be the mechanism that p implements F. Suppose there exist a, θ and θ such that a F (θ) but a / F (θ ). Then there must exist m NE(Γ, θ) with l[γ, m] = a, i.e., g S (m S ) = a for all S N such that p(s) > 0. Since a / F (θ ), it must be that m / NE(Γ, θ ). Hence, there must exist i N and m i M i such that a θ i l[γ, (m i, m i )] and l[γ, (m i, m i )] θ i a. Define function h i : N A such that h i (S) = g S ((m i, m i ) S ) for all S N. Then h i (S) = a for all S N \ N (i) such that p(s) > 0. Moreover, lottery l[p, h i ] = l[γ, (m i, m i )]. Hence, player i and function h i satisfy the required condition in p monotonicity. The proof clarifies why p monotonicity requires the particular reversal of preferences over lotteries. Under the political process p, the message of player i is considered only if she is selected as the senator. Hence, by changing her message, player i can in principle change only those alternatives that are implemented under senates S N (i). As a result, player i is only able to generate a lottery that is induced by [p, h i ], where h i : N A is constant over all S N \ N (i) that can be selected with positive probability. If the political process is p D, then for any h i : N A, the lottery l[p D, h i ] assigns probability 1 to the alternative h i (N). Hence, if player i has a reversal of her preferences over lottery l[p D, h i ] around a (i.e., a θ i l[p D, h i ] and l[p D, h i ] θ i a), then it is equivalent to player i having a reversal of her preferences over alternative h i (N) around a (i.e., a θ i h i (N) and h i (N) θ i a). This observation implies that Maskin monotonicity is equivalent to p D monotonicity. We note this result below (proof is omitted): Proposition 3.3. F is Maskin monotonic if and only if it is p D monotonic. On the other hand, there are political processes p such that p monotonicity is weaker than Maskin monotonicity. In particular, if p is such that every player has a positive probability of being selected in some senate, then any Maskin monotonic SCR is also p monotonic. Intuitively, whenever a drops out of the SCR, Maskin monotonicity requires that some player has a reversal of her preferences over alternatives around a, which is stronger than that player having a reversal of her preferences over lotteries around a. Formally, we have: 14

15 Lemma 3.4. If F is Maskin monotonic, then F is p monotonic for any p P such that for all i N, there exists an S N (i) with p(s) > 0. Proof. Suppose a F (θ) but a / F (θ ). By Maskin monotonicity, there exist player i and alternative a such that a θ i a and a θ i a. Define the mapping h i : N A such that h i (S) = a, S N (i), and h i (S) = a, S N \ N (i). By assumption, there exists an S N (i) such that p(s) > 0. Therefore, lottery l[p, h i ] puts a positive probability on a. Moreover, a is the only other alternative that could be in the support of l[p, h i ]. Since preferences over lotteries are monotone, Hence, F is p monotonic. a θ i a = a θ i l[p, h i ] while a θ i a = l[p, h i ] θ i a. In fact, there are p implementable and hence, p monotonic SCRs that are not Maskin monotonic. Consider the following example: Example 3.5 (King Solomon s Dilemma). Two women, Ann (1) and Beth (2), approach King Solomon both claiming to be the mother of a baby. There are two states, {θ, θ }, with Ann (Beth) being the true mother in state θ (θ ). There are four alternatives, {a, b, c, d} where a is to give the baby to Ann, b is to give the baby to Beth, c is to cut the baby in half, and d is to kill both women and child. The women s preferences over alternatives are: θ θ a θ 1 b θ 1 c θ 1 d a θ 1 c θ 1 b θ 1 d b θ 2 c θ 2 a θ 2 d b θ 2 a θ 2 c θ 2 d King Solomon s SCR is to give the baby to the true mother, i.e., F (θ) = a and F (θ ) = b. However, this SCR is not Maskin monotonic and hence, not Nash implementable. Now, suppose there exists a lottery l A such that: l(c) = 0 and for Ann, l θ 1 b but b θ 1 l, and symmetrically for Beth, l θ 2 a but a θ 2 l. Since preferences over lotteries are monotone and l θ 1 b, we have l(a) > 0; similarly, since l 2 θ a, we have l(b) > 0; and finally, since b θ 1 l and a θ 2 l, it must be that l(d) > 0. Intuitively, when a woman is the true mother, she is willing to risk death for all to obtain the baby for herself rather than give the baby to the other woman but when she is not the true mother, she is not willing to take the same risk. As an example, suppose 15

16 both Ann and Beth have expected-utility preferences with the following Bernoulli utilities: u θ 1(a) = u θ 2 (b) = 3000, u θ 1(b) = u θ 2 (a) = 50, u θ 1(c) = u θ 2 (c) = 1, u θ 1(d) = u θ 2 (d) = 0, u θ 1 (a) = u θ 2(b) = 50, u θ 1 (c) = u θ 2(c) = 2, u θ 1 (b) = u θ 2(a) = 1 and u θ 1 (d) = u θ 2(d) = 0. Then the lottery l such that l(a) = l(b) = 1 50, l(c) = 0 and l(d) = satisfies the above requirement Now, consider the following mechanism Γ: M i = {θ, θ } for all i {1, 2}. p P is such that p({1}) = l(a), p({2}) = l(b) and p({1, 2}) = l(d). g {1} (θ) = a and g {1} (θ ) = b; g {2} (θ) = a and g {2} (θ ) = b; and g {1,2} (θ, θ) = a, g {1,2} (θ, θ ) = b and g {1,2} (θ, θ ) = g {1,2} (θ, θ) = d. In this mechanism, each woman announces whether the state is θ or θ. According to p, King Solomon collects only Ann s message with probability l(a), only Beth s message with probability l(b) and both women s messages with probability l(d). If only one woman s message is collected, then King Solomon gives the baby to that woman who is the true mother in the announced state. If both women s messages are collected, then King Solomon faces two possibilities: if both women agree on the state, then he gives the baby to that woman who is true mother in agreed state whereas if they disagree, then he kills everyone. We show that (θ, θ) is the only Nash equilibrium of Γ in state θ. Fix state θ and consider the message profile (θ, θ). In this profile, a is implemented in every senate. Ann would not deviate to θ because then she does not obtain her baby with a positive probability. On the other hand, if Beth deviates to θ, then the outcome is lottery l. But a θ 2 l. Hence, (θ, θ) is an equilibrium in state θ and l[γ, (θ, θ)] = a. Now, let us show that any other message profile is not an equilibrium in state θ. First, (θ, θ ) with outcome b in every senate is not an equilibrium because if Ann deviates to θ, then the outcome will be lottery l and l θ 1 b. Second, (θ, θ ) with outcome l is not an equilibrium because if Beth deviates to θ, then a is implemented in every senate and a θ 2 l. Finally, (θ, θ) is not an equilibrium because Ann does not obtain her baby with a positive probability but by deviating to θ, Ann would obtain her baby for sure. l[γ, (θ, θ )] = b. Hence, F is p implementable. Similarly, (θ, θ ) is the only Nash equilibrium in state θ and Here we note that for certain other preferences of the women, King Solomon s SCR is not implementable by any political process p. Finally, there are also political processes p such that p monotonicity is stronger than Maskin monotonicity. This is the case for S oligarchy p S (see Example 4.7) since some players are never selected as senators. 16

17 3.2 Sufficient Conditions Theorem 3.6. Let n 3 and E be an economic environment. If F is p monotonic for p P such that p(n) > 0 and p(s) > 0 only if S 3, then F is p implementable. Sketch of the proof : Consider mechanism Γ = ((M i ) i N, p, (g S ) S N ) in which for all i N, M i = Θ A } {{ A } Z +, N (i) +1 where Z + is the set of nonnegative integers. Note that since N (i) = N (j) for all i, j, the numbers of components in the players messages are equal. Let a typical message m i be of the form (θ i, a 1 i, (a S i ) S N (i), z i ). 13 For each S N, the outcome function g S is as follows: (i) If for every player i S, m i = (θ, a 1 i, a,..., a, 0) and a F (θ), then g S( (m i ) i S ) = a. (ii) If for S 1 players i j in S, m i = (θ, a 1 i, a,..., a, 0) and a F (θ), but m j = (θ j, a 1 j, (a S j ) S N (j), z j ) (θ, a 1 j, a,..., a, 0), then g S( (m i ) i S ) = { a if l[p, h j ] θ j a a S j if a θ j l[p, h j ], where h j : N A is such that h j (S ) = { a S j a if S N (j) if S N \ N (j). (iii) In all other cases, g S( (m i ) i S ) = a 1 j where j S is the player with the lowest index among those who announce the highest integer in (m i ) i S The size of the message space for each player can be reduced by replacing (a S i ) S N (i), i.e., one alternative for each senate in N (i), with (a S i ) S N (i):p(s )>0, i.e., one alternative for only those senates S N (i) with p(s ) > 0. However, this will make the exposition cumbersome. 14 The advantage of the integer game in our framework is that by announcing a high enough integer, a player can win the integer game in every senate. Notice that this cannot be achieved by using the modulo game in which each player announces a single integer. A particular integer might make the player win the modulo game in one senate but at the same time she could lose the modulo game in some other senate. Hence, in order to replace the integer game with the modulo game, we need each player to announce as many integers between 1 and n as the number of senates she can be a member of, N (i). All proofs, except that of Theorem 6.5, can be altered in this fashion without changing the results. In contrast, the proof of Theorem 6.5 relies heavily on the existence of one player who has the lowest index among those who announce the highest integer in the whole population. Thus, replacing the integer game with the modulo game seems difficult in that proof. However, the use of both these games in implementation has been criticized; see Jackson (1992). 17

18 Interpret a S i as the alternative player i wants to implement under senate S N (i). Thus, each player announces a state θ i, an alternative a 1 i, alternatives a S i that she wants to implement under each senate S N (i) and an integer z i. For any senate S, the outcome function g S is defined like in the canonical mechanism of Maskin (1999) with three rules. The first rule is used when all the messages received by the planner agree on the state, want a unique socially desirable alternative to be implemented under each senate and announce integer 0 (notice that the planner ignores the second component a 1 i in this rule). In this case, the planner implements the commonly agreed alternative. Thus, if each player announces the true state θ, a fixed a F (θ) for each senate and integer 0, then the first rule ensures that a is implemented under each senate. The second rule guarantees that such a message profile is an equilibrium. If player j unilaterally deviates in this situation (unilateral deviations can be identified since p(s) > 0 only if S 3), then in each S N (j), the planner implements the alternative a S j wanted by player j under senate S if and only if by doing so in every such senate (which generates lottery l[p, h j ]) would not make player j better-off. The second rule also allows the planner to use p monotonicity of the SCR to eliminate the possibility that when the true state is θ, players agree in their messages but on a bad alternative, i.e., all the players announce that the state is θ, alternative a F (θ ) for each senate and integer 0 but a / F (θ). In such a situation, the second rule gives any player j the opportunity to unilaterally deviate and implement a strictly preferred lottery l[p, h j ] by using the reversal of her preferences over such lotteries around the bad alternative a. This is done by implementing whatever player j wants under each senate S N (j). Finally, the third rule eliminates the possibility of any equilibria other than those in which all players agree in their messages. In these situations, at least n 1 players are such that each of them can implement her most-preferred alternative (proof takes care of the case when this does not exist) under senate N by announcing such an alternative in the second component of her message and a high enough integer. However, such a deviation will be improving if it does not cause a worse alternative to be implemented under other senates S N. This is obviously the case if the senate does not include the deviating player. On the other hand, if the senate includes the deviating player and the third rule is used in that senate, then the deviating player still gets her most-preferred alternative. Finally, even in those senates that include the deviating player but in which the second rule is used, the deviating player ensures that a worse alternative is not implemented since the second rule does not depend on the altered components (i.e., the second and last) of her message. Thus, in such equilibria, the alternative implemented under senate N must be most-preferred by at least n 1 players, which contradicts with the assumption of economic environment. These arguments are formalized in the proof presented in the Appendix. 18

19 It is worth emphasizing the role of each assumption in the above result. Firstly, p monotonicity ensures that any agreement on a bad alternative is avoided by giving each player the option to announce one alternative for each senate in which she could be a senator. Secondly, p(s) > 0 only if S 3 ensures that the planner can identify unilateral deviations in each senate. Finally, a positive chance of referendum, i.e., p(n) > 0, in conjunction with economic environment ensure that any disagreement is avoided. Since any Maskin monotonic SCR is also p monotonic for any p with p(n) > 0 (see Lemma 3.4), we obtain the following corollary: Corollary 3.7. Let n 3 and E be an economic environment. If F is Maskin monotonic, then F is p implementable for any p such that p(n) > 0 and p(s) > 0 only if S 3. Hence, in economic environments, any Nash implementable SCR can be implemented using any political process that always selects at least three senators and has a positive chance no matter how small of referendum. However, as we will show later in Example 5.8, this result is not true in noneconomic environments. 4 Implementation by Oligarchy Consider any set of oligarchs S N. Since p S puts probability 1 on the senate S, for any h i : N A, the lottery l[p S, h i ] assigns probability 1 to the alternative h i (S). Hence, if player i has a reversal of her preferences over lottery l[p S, h i ] around a, then it is equivalent to player i having a reversal of her preferences over alternative h i (S) around a. Furthermore, since messages of the commoners are ignored, only the preference reversals of the oligarchs matter. These observations imply the following result (proof is omitted): Proposition 4.1. Let S N. F is p S monotonic if and only if whenever a F (θ) and a / F (θ ) for some a, θ and θ, there exist player i S and a A such that a θ i a and a θ i a. Theorem 3.2 implies the following necessary condition for implementation by S oligarchy: Corollary 4.2. If F is implementable by S oligarchy, then F is p S monotonic. Clearly, p S monotonicity is stronger than Maskin monotonicity. Definition 4.3. Let S N. F satisfies S-no-veto power if for any θ Θ and S S such that S S 1, we have i S A i (θ) F (θ). 19

20 Notice that an S oligarchy in environment E = N, A, ( ) ( θi ) i N is equivalent to θ Θ a direct democracy in the restricted environment E(S) = S, A, ( ) ( θi ) i S. Hence, the sufficient conditions for implementation by S oligarchy in environment E will be equivalent to the corresponding conditions for Nash implementation in environment E(S). We thus obtain the following corollary from Theorem 2.13: Corollary 4.4. If 3 S < n, then any SCR that satisfies p S monotonicity and S-no-veto power is implementable by S oligarchy. Since S-no-veto power is vacuously true if E(S) is an economic environment, we also have the following result: Corollary 4.5. Suppose 3 S < n and E(S) is an economic environment. F is implementable by S oligarchy if and only if it is p S monotonic. θ Θ 4.1 Comparison with Nash Implementation Proposition 4.6. If F is implementable by S oligarchy for some S N, then F is Nash implementable. Proof. Let Γ S = ((M i ) i N, p S, (g S ) S N ) be the S-oligarchic mechanism that p S implements F. Define the direct democracy Γ D = ((M i ) i N, p D, (ĝ S ) S N ) with ĝ N (m) = g S (m S ), m M, and ĝ S = g S, S N. It is easy to show that for all θ, we have m NE(Γ S, θ) m NE(Γ D, θ), and hence, the result follows. Thus, in any environment, any SCR that is implementable by an oligarchy is also implementable by direct democracy (i.e., Nash implementable). However, the converse is not true. That is, there are SCRs that are Nash implementable but not implementable by S oligarchy for any S N. Consider the following example: Example 4.7. Let N = {1, 2, 3, 4}, A = {a, b, c, d} and Θ = {θ, θ }. The players preferences over alternatives are: θ θ a θ 1 b θ 1 c θ 1 d b θ 1 a θ 1 c θ 1 d b θ 2 c θ 2 d θ 2 a c θ 2 b θ 2 d θ 2 a c θ 3 d θ 3 a θ 3 b d θ 3 c θ 3 a θ 3 b a θ 4 d θ 4 b θ 4 c d θ 4 a θ 4 b θ 4 c Let F be such that F (θ) = {a, b, c} and F (θ ) = {d}. Player 1 is the only player i for whom there exists an alternative â such that a θ i â and â θ i 20 a. Player 2 is the only player i for