NASH IMPLEMENTING SOCIAL CHOICE RULES WITH RESTRICTED RANGES

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1 NASH IMPLEMENTING SOCIAL CHOICE RULES WITH RESTRICTED RANGES by M.Remzi Sanver * ABSTRACT 1 We consider Nash implementation of social choice rules with restricted ranges and show that the appropriate adaptation of Maskin monotonicity to this context depends on the range of the mechanisms: The wider is this range, the weaker is the monotonicity condition to be used. As a result, mechanisms employing outcome functions which allow for out-of-range alternatives at off-equilibrium messages can Nash implement social choice rules which fail to be Nash implementable by mechanisms with restricted ranges. The Walrasian social choice correspondence is a particular instance of this. Moreover, social choice rules which are not Maskin monotonic can be monotonized by the addition of artificial out-of-range alternatives a point we illustrate through Solomon s Dilemma. Keywords: Maskin Monotonicity, Nash Implementation, Range Restrictions, Walrasian Solution, Solomon s Dilemma * Department of Economics, İstanbul Bilgi University, 80310, Kuştepe, İstanbul, Turkey 1 I am particularly indebted to Goksel Asan, Matthew Jackson, Efe Ok, Ipek Ozkal-Sanver, Arunava Sen, Tayfun Sönmez and William Thomson for their comments as well as their encouragement. A preliminary version of this paper has been presented at the Murat Sertel Israeli-Turkish Conference on Economic Theory, May 2004, Bilgi Unversity, Istanbul. I thank Eddie Dekel and all the participants. 1

2 1. INTRODUCTION It is well-known since Maskin (1977, 1999) that every Nash implementable social choice rule must satisfy a particular monotonicity condition. Moreover Maskin monotonicity, combined with a no veto power condition, ensures the Nash implementability of social choice rules. We explore the world of social choice rules with restricted ranges. So we consider a social choice rule F whose domain consists of orderings over some set A while not every element of A is in its range. In other words, there are elements in A which are chosen at no preference profile within the domain of F. The question we pose is about the shape that Maskin monotonicity takes in this world. To illustrate our point, let us look at the Walrasian allocation rule which can be seen as a social choice rule with restricted range. Consider a pure exchange economy where the fixed initial endowments are public information. Under appropriate assumptions ensuring the existence of Walrasian equilibria, it makes sense to speak about the Walrasian social choice correspondence (WSCC) which assigns to every preference profile the corresponding Walrasian allocations of the economy. The WSCC is of restricted range. For its domain consists of individuals preferences over possible allocations within their consumption spaces while its range is restricted to those which are feasible. For example, in a two-agent economy, the range of the WSCC will be limited to the allocations within the Edgeworth box while its domain will include orderings over all possible allocations. 2

3 Hurwicz et al. (1995) note the non-monotonicity of the WSCC 2 and define the (monotonic) concept of a constrained WSCC. In fact, Thomson (1999) shows that the constrained WSCC is the minimal monotonic extension of the WSCC. 3 As the WSCC is of restricted range, the claims about its non-monotonicity are based on a specific adaptation of Maskin monotonicity to this framework. To see this adaptation, let us refer to the standard claim about the non-monotonicity of the WSCC which is of the following taste: Consider a pure trade economy with two agents and two goods. Fix the consumption spaces and initial endowments of the agents. Take a preference profile (R 1, R 2 ) where R i stands for the preference of agent i {1, 2}. Let x be a Walrasian allocation of this economy at (R 1, R 2 ). In particular, the allocation x happens to be at the boundary of the Edgewoth box. Now consider a preference profile (R 1, R 2 ) where the preference of agent 2 remains intact. The preference of agent 1 is changed in such a way that every feasible allocation which was in the lower contour set of x at R 1 remains in its lower contour set at R 1. On the other hand some non-feasible allocations (outside of the Edgeworth box) which used to be in the lower contour set of x at R 1 are not in the lower contour set of x at R 1. Interestingly, the allocation x is no more Walrasian at (R 1, R 2 ) while it should be, as the shrinking of the lower contour set of x for agent 1 happens outside the Edgewoth box- hence the non-monotonicity argument. This argument adapts Maskin monotonicity to the world of social choice rules with restricted ranges in a particular manner where out-of-range alternatives are not considered. This is appropriate when one has to use mechanisms employing outcome functions which do not allow for out-of-range alternatives even at off-equilibrium messages. On the other hand, we show that the appropriate adaptation and strenght of Maskin monotonicity for the world of social choice rules with restricted ranges depends on the range of the mechanism used to implement it. As a result, the WSCC is Nash implementable when we use mechanisms that allow for out-of-range 2 which is also mentioned by Jackson (2001) 3 The minimal monotonic extension is a concept proposed by Sen (1995) to evaluate the extent of nonmonotonicity of social choice functions, by extending them minimally to social choice correspondences which are Maskin monotonic. As particular applications, we have computations of the minimal monotonic extensions of matching rules by Kara and Sönmez (1996), economic allocation rules by Thomson (1999) and scoring rules by Erdem and Sanver (forthcoming). 3

4 alternatives at off-equilibrium messages. Moreover, social choice rules which are not Maskin monotonic can be monotonized by the addition of artificial out-of-range alternatives a point we illustrate through Solomon s Dilemma. Section 2 introduces the basic notions. Section 3 states the main results. Section 4 gives two examples, namely the WSCC and the Solomon s Dilemma, to illustrate that the results of the previous section can be fruitfully applied. Section 5 makes some closing remarks. 2. BASIC NOTIONS Taking any integer n 3, we consider a society N = {1,, n} confronting a nonempty set of alternatives A. We let R stand for the set of all complete and transitive binary relations over A. We assume that every agent i N has a preference R i R over A. 4 A typical preference profile over A is denoted by R = (R 1,..., R n ) R N. Taking any non-empty set D R N, we conceive a social choice correspondence (SCC) as a mapping F: D A where A = 2 A \ { } is the set of all non-empty subsets of A. Given any SCC F: D A, we write r(f) = R D F(R) for the range of F. We say that F is of restricted range if and only if r(f) A. 5 A mechanism is an (n+1)-tuple µ = ({M i } i N, h) where M i is the non-empty message space of agent i and h: M A is the outcome function which assigns an element of A to each message profile m M = the mechanism µ. n i= 1 Mi. We write r(µ) = m M h(m) for the range of 4 with the usual following interpretation: For any a, b A, a R i b means agent i finds a at least as good as b. We write P i and I i for the respective strict and indifference counterparts of R i. 4

5 3. NECESSARY AND SUFFICIENT CONDITIONS At each R R N, a mechanism µ induces a normal form game Γ(µ, R) = {(M i, R i )} i N where M i is the strategy space of agent i and R i, by a slight abuse of notation, is his preference over M such that for any m, m M, we have m R i m if and only if h(m) R i h(m ). We write ν(γ(µ, R)) for the set of Nash equilibria of the game Γ(µ, R). We say that a mechanism µ Nash implements a SCC F: D A if and only if given any R D we have F(R) = m ν(γ(µ, R)) h(m). A SCC F is said to be Nash implementable whenever there exists a mechanism which Nash implements F. Proposition 3.1: A mechanism µ Nash implements a SCC F: D A only if r(f) r(µ) A. Proof: Let µ be a mechanism which Nash implements F. The inclusion r(µ) A holds trivially. To establish r(f) r(µ), take any x r(f). So there exists some R D where x F(R). As µ Nash implements F, there exists m ν(γ(µ, R)) with h(m) = x. Thus, x r(µ). Q.E.D. Let L(a, R i ) = {x A a R i x} be the lower contour set of an alternative a A for an agent i N with a preference R i R. Given any B A, a SCC F: D A is said to be (Maskin) monotonic relative to B if and only if for any R, R D and any a A, we have a F(R) a F(R ) whenever L(a; R i ) B L(a; R i ) for every i N. Note that given any B, B A with B B, monotonicity relative to B implies monotonicity relative to B. Hence the condition takes it weakest form when B = A, in which case it coincides with the standard definition of Maskin monotonicity. 6 5 The symbol is used for strict set inclusion, so there exists some x A such that x r(f). 6 Koray (2003) and Koray and Pasin (2003) use this and various other generalizations of Maskin monotonicity for a different analysis of Nash implementable social choice rules. 5

6 Theorem 3.1: A SCC F: D A is Nash implementable by a mechanism µ only if F is monotonic relative to r(µ). Proof: Let µ be a mechanism which Nash implements F. Take any R D and any a F(R). As µ Nash implements F, there exists m ν(γ(µ, R)) with h(m) = a. Now take any R D with L(a; R i ) r(µ) L(a; R i ) for all i N. It is straightforward to check that m ν(γ(µ, R )). Thus a F(R ). Q.E.D. A SCC F: D A is said to satisfy the no veto power condition if and only if given any R D and any a A we have #{i N : L(a, R i ) = A} n 1 a F(R). Theorem 3.2: A SCC F: D A is Nash implementable if F is monotonic relative to some B A with B r(f) and satisfies the no veto power condition. Proof: Take F and B as in the statement of the theorem. To show the Nash implementability of F, we use a Maskin-type mechanism µ = ({M i } i N, h) where M i = D B IΝ. So a typical message m i = (R i, b i, n i ) of an agent i N, consists of a preference profile R i within the domain of F, an alternative b i not necessarily in the range of F and a positive integer n i. The outcome function h is defined as follows: In case all agents agree in (R, b) while b F(R), the outcome is b. Suppose everybody but one agrees on (R, b) with b F(R). Then the outcome is still b, unless the deviator j N announces some a A with b R j a, in which case a is implemented. In all other cases, the agent announcing the highest integer (ties can be broken arbitrarily) is the dictator. One can check that this mechanism Nash implements F. 7 7 The range of the mechanism we use in the proof is B. So it employs an outcome function where outof-range alternatives belonging to the set B \ r(f) may show up in off-equilibrium messages while they never appear at equilibria. We can get rid of these by requiring agents to announce outcomes in r(f) hence restricting the range of the mechanism from B to r(f). We know by Theorem 3.1 that this will be at the expense of strenghtening the monotonicity condition from being relative to B to being relative to r(f). 6

7 Our results generalize those of Maskin (1999) which are about SCCs of full range. 8 To see this, remark that, by Proposition 3.1, any mechanism µ which Nash implements a SCC F of full range must also be of full range, i.e r(µ) = A must hold. Thus, given a SCC F of full range, Theorem 3.1 announces that F is Nash implementable only if F is monotonic relative to A which is equivalent to the standard definition of Maskin monotonicity. Similarly, we know by Theorem 3.2 that every SCC F of full range which is monotonic relative to A and satisfies the no veto power condition is Nash implementable. On the other hand, our results make also a point for SCCs of restricted range. Maskin monotonicity, in its standard form, is defined for SCCs of full range. What would its appropriate adaptation be, when the SCC F in question is of restricted range r(f)? Would it be monotonicity relative to r(f)? Theorems 3.1 and 3.2 announce that this need not be the case. In fact, monotonicity relative to any B with r(f) B A will work. In particular, it is possible to disregard the range restriction of F and take B = A, which weakens the monotonicity condition as much as possible. So there may exist a SCC F which is not monotonic relative to its (restricted) range r(f) while F is monotonic relative to some B r(f). As a result, although F cannot be Nash implemented by a mechanism µ with r(µ) = r(f) 9, there may be some µ with r(f) r(µ) B which Nash implements F. As mentioned in Footnote 7, this is at the expense of using a mechanism where out-of-range alternatives may appear at nonequilibrium message profiles. We will discuss the plausibility of this in the last section. But before, we give two applications of our results. 8 We say that a SCC F is of full range if and only if r(f) = A. 7

8 4. APPLICATIONS As a first application, we analyze the Walrasian SCC. Consider a pure trade economy with at least two goods. The parameters C and E stand for the respective consumption spaces and initial endowments of agents. Let A(C) be the set of all possible allocations within the consumption spaces of the agents and B(C, E) A be the set of feasible allocations which is a function of both the consumption spaces and initial endowments. Let D be the set of admissible preference profiles over A(C). The profiles in D consist of individual orderings satisfying standard assumptions. We assume that given any R D, the economy (C, E, R) has a Walrasian equilibrium. We can hence define the WSCC W C,E : D A which assigns to every R D, the set W C,E (R) B(C, E) of Walrasian allocations of the economy (C, E, R). It is straightforward from the definiton of an Walrasian allocation that W C,E is monotonic relative to A(C) while it trivially satisfies the no veto power condition. Hence, by Theorem 3.2, we obtain the following result: Proposition 4.1: The WSCC is Nash implementable, i.e., there exists a mechanism µ with r(µ) = A(C) such that µ Nash implements W C,E. As a second application, we consider Solomon s Dilemma. 10 The original story is well-known: Two women, Anna and Beth, come to Solomon claiming to be the mother of a child. Solomon can give the child to Anna, give the child to Beth or cut the child into two. We refer to these three alternatives as a, b and c respectively. The preferences of these women in the two possible states- which are State α and State β where the child belongs to Anna and Beth respectively are as follows: Anna Beth State α State β State α State β a a b b b c c a c b a c 9 i.e., µ uses no out-of-range alternatives 10 An excellent treatment of the problem can be found in Moore (1992). 8

9 Solomon s justice requires that F(α) = {a} and F(β) = {b}. Clearly, F is not Maskin monotonic. Now suppose Solomon announces two additional alternatives: He will give the child to Anna and furthermore marry Anna (to which we refer as a*) He will give the child to Beth and furthermore marry Beth (to which we refer as b*). Naturally, a* is always the best outcome for Anna and b* is always the best outcome for Beth. On the other hand, Anna prefers b* to b when she is the true mother (as she wishes her child to grow up under the protection of Solomon) but she prefers b to b* when she is not the true mother (because of obvious envy reasons). Similarly, Beth prefers a* to a when she is the true mother and a to a* when she is not. Hence the preferences of Anna and Beth in the two states are now as follows: Anna Beth State α State β State α State β a* a* b* b* a a b b b* c c a* b b a a c b* a* c The SCC is again F(α) = {a} and F(β) = {b}. So Solomon will never marry these two women. On the other hand F is monotonic relative to the set {a, b, c, a*, b*} while it fails to be monotonic relative to {a, b, c}. This example does not claim that Solomon s Dilemma is Nash implementable, as there are only two agents. Nevertheless, it gives an instance where a non-monotonic social choice rule can be monotonized by the artificial addition of out-of-range alternatives This is a perspective which can give fruitful results in the theory of Nash implementation. For example, Benoit and Ok (2004a) and Sanver (2004) weaken the necessary and sufficient conditions for Nash implementability by using mechanisms with awards. These mechanisms have the added element of a transferable good, say money, at their disposal. So they create artificial alternatives (with awards) while these only appear at off-equilibrium messages. 9

10 5. CLOSING REMARKS Our results rely on using mechanisms which employ outcome functions where out-ofrange alternatives show up in off-equilibrium messages - but never appearing at equilibria. The plausibility of this depends on how out-of-range outcomes are conceived by players. In fact, there is an infeasibility argument that one can bring to this approach: If players believe in the infeasibility of out-of-range alternatives and think that these can never arise, then they would not behave as anticipated by the central planner. In that case, we are constrained to use mechanisms which have ranges coinciding with the range of the SCC to be implemented. As a result Maskin monotonicity has to be adapted in its strongest form. 12 The infeasibility argument could especially be meaningful for the WSCC, as the Edgeworth box not only defines the range but also the feasible set. 13 Nevertheless, one can answer this by conceiving the out-of-range alternatives as allocations which are made feasible by an external planner who wishes to implement the social choice rule in question. This brings no cost at the end, as these out-of-range promises never appear at equilibrium. 14 After all, there are also more natural cases as is the Solomon s Dilemma- where players are not supposed to make any infeasibility claim about out-of-range alternatives. To sum up, we wish to give a formal analysis of Nash implementation when social choice rules are of restricted range. In this world, it is possible to expand the set of Nash implementable SCCs by widening the range of the mechanism we use, i.e., by allowing out-of-range alternatives at non-equilibrium messages. This result is obtained by a Maskin (1999) type of mechanism, which should make us alerted towards the fact that Maskin's mechanism can give positive results which are not 12 As Hurwicz et al. (1995) and Thomson (1999) do in order to establish that the WSCC is not Nash implementable. 13 A similar critique could apply to the mechanisms with awards (see Footnote 11) used by Benoit and Ok (2004a) and Sanver (2004). Although awards only appear at off-equilibrium messages, their approach is always open to the infeasibility argument. 14 A classical example of this can be found in Schmeidler (1980) who implements the WSCC by Nash and strong Nash equilibria. Needless to say, the corresponding mechanism uses out-of-range outcomes at non-equilibrium messages. This perspective should not be underestimated, as it establishes a general 10

11 implied by Maskin s theorem or there is more to Maskin's mechanism than Maskin's Theorem as Benoit and Ok (2004b) say. 15 We wish to close noting that our example about Solomon s Dilemma gives an instance where adding artifical out-of-range alternatives may monotonize SCCs which are originally non-monotonic. Of course this requires some control on how these artificial alternatives are ranked by the agents. Although there are certain positive results in this direction (such as Benoit and Ok (2004a) and Sanver (2004)) 16, the possibility of further using this perspective to implement social choice rules which otherwise fail to be Nash implementable remains as an open question. compatibility between the nature of market equilibria and Nash equilibria in normal form games which otherwise fails to exist. 15 One can see Benoit and Ok (2004a, 2004b) for interesting instances of this. 16 See Footnote 11 11

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13 Moore J. (1992), Implementation, Contracts and Renegotiation, In: Laffont, J.J. (ed.) Advances in Economic Theory, Econometric Society Monographs, Cambridge University Press, Sanver, M. R. (2004), Nash Implementing Non-Monotonic Social Choice Rules by Awards, mimeo. Schmeidler, D. (1980), Walrasian Analysis via Strategic Outcome Functions, Econometrica, 48, Sen, A. (1995), The Implementation of Social Choice Functions via Social Choice Correspondences; a General Formulation and a Limit Result, Social Choice and Welfare, 12, Thomson, W. (1999), Monotonic Extensions on Economic Domains, Review of Economic Design, 4,