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An algorithm for verifying double implementability in Nash
and strong Nash equilibria
* Sang-Chul Suh
Department of Economics, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Received 1 January 1999; received in revised form 1 November 1999; accepted 1 November 1999
Abstract
Suh (Suh, S., 1997. Double implementation in Nash and strong Nash. Social Choice and Welfare 14, 4392447.) considered a decision making problem where there are a set of alternatives and a finite number of agents with preferences defined over the set of alternatives, and provided a necessary and sufficient condition for double implementation in Nash and strong Nash equilibria. One problem we encounter in Suh’s paper is that it is difficult to apply the condition directly to a given (social choice) correspondence. Here we provide an algorithm which helps us to verify whether a correspondence satisfies the condition or not. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Double implementation; Nash and strong Nash equilibria; Algorithm
JEL classification: C72; D78
1. Introduction
Consider an abstract model of a decision making problem where there are a set of alternatives and a finite number of agents with preferences defined over the set. A (social
choice) correspondence assigns each preference profile a set of alternatives. To achieve
the goal of selecting alternatives which are recommended by the correspondence, information about agents’ preferences is needed. But the information may not be publicly known and agents may behave strategically. We are interested in investigating the possibility of constructing a decentralized institution (or mechanism) to achieve the goal in such a situation. A mechanism consists of a set of strategies for each agent and a
*Corresponding author. Tel.:11-519-253-3000, ext. 2384; fax:11-519-973-7096. E-mail address: scsuh@uwindsor.ca (S.-C. Suh).
function which assigns each strategy profile an alternative. Suppose that an equilibrium concept, Z, describes agents’ strategic behaviors. If the set of Z equilibrium outcomes of the game given by the mechanism coincides with the set of alternatives selected by the correspondence for all possible preference profiles, then we say that the mechanism
implements the correspondence in Z equilibrium.
Maskin (1977) provided a characterization result for Nash implementation and Dutta 1 and Sen (1991a) provided a full characterization result for strong Nash implementation. While the Nash equilibrium concept applies to environments where agents cannot form coalitions, hence agents can make only unilateral deviations, the strong Nash equilibrium concept applies to environments where any possible group of agents can form coalitions. Suh (1996b) provided a full characterization result for the equilibrium concept which applies to environments where general coalition formation is possible. One important assumption implicit in those papers mentioned is that the planner has information about who can form coalitions with whom.
What correspondences are implementable if the planner does not have the information about coalition formation possibilities among agents? This question in effect asks which correspondences are (doubly) implementable both in Nash and strong Nash equilibria. Suh (1997) provided an answer to this question: a necessary and sufficient condition for
2
double implementation is provided. Although the condition completely characterizes doubly implementable correspondences, it is difficult to apply the condition directly to a correspondence in question.
We provide an algorithm which allows us to verify whether a correspondence satisfies ¨ ¨
the condition for double implementation. Sjostrom (1991) gave an algorithm for checking necessary and sufficient conditions for Nash implementation. We extend his idea in constructing the algorithm for double implementation.
We organize the remainder of the paper as follows. After introducing notation and definitions in Section 2. We provide an algorithm which allows us to verify whether a rule satisfy the condition for double implementation in Section 3.
2. Notation and definitions
Let A be an arbitrary set of alternatives. Let N5h1, . . . ,nj be a finite set of agents. Each agent i[N has preferences R , a binary relation on A which is complete, transitivei and reflexive. Let P be the strict preference relation and I be the indifference relation.i i Let R be the set of agent i’s admissible preferences and R5R 3 ? ? ? 3R . A
i 1 n
preference profile is a list R5(R , . . . ,R )[R. A (social choice) correspondence is a 1 n
mapping F which associates with each preference profile R[R a non-empty subset of
A.
1
For Nash implementation, refer to Williams (1986), Saijo (1988), Danilov (1992), Moore and Repullo (1990) and Dutta and Sen (1991b), and for strong Nash implementation, refer to Maskin (1979), Moulin and Peleg (1982) and Suh (1996a).
2
A mechanismG is a pair (S, g) of a list of strategy sets S5S13 ? ? ? 3S , where S isn i the strategy set for agent i, and a function g:S→A which associates with each strategy a
unique alternative in A. For a mechanismG 5(S, g), the outcome of the strategy profile
s5(s , . . . ,s )[S is alternative g(s). Given a preference profile R[R, a mechanism 1 n
G 5(S, g) and a strategy profile s[S, the outcome g(s) is evaluated by the profile R.
Hence the preference profile R[Rand the mechanismG define a game (G,R) in normal form.
¯ ¯
Given T#N, let sT5(s )i i[T[ST5 3i[TS and si 2T5sN \T. Given S#S, let g(S )5
< ¯g(s). Let 1 be the class of all subsets of N including the empty set. s[S
Given a preference profile R[Rand a mechanismG 5(S, g), a strategy profile s[S ¯ (G,R) and SA(G,R)5g(S(G,R)). Hence a mechanismGimplements the correspondence F in Nash equilibrium, if NA(G,R)5F(R) for all R[R. And a mechanismG implements
the correspondence F in strong Nash equilibrium, if SA(G,R)5F(R) for all R[R. A mechanismGdoubly implements a correspondence F if NA(G,R)5SA(G,R)5F(R)
for all R[R. A correspondence F is doubly implementable, if there is a mechanism which doubly implements the correspondence.
For all (a,R)[A3R and for all i[N, let L(i,a,R); hb[AuaR bi j be the set of all alternatives which are less preferred to a by agent i under profile R. For all T[1 and
T±f, and for all (a,R)[A3R, let L(T,a,R);<i[T L(i,a,R). For a given
corre-j[N\t(u,T,i ) does not strictly prefer alternative a to alternative a under profile R . i i
Particularly, if (a ,R )5(a*,R*) for all i[N\T, then for all a[B (Tu) at least one agent in T does not strictly prefer alternative a to alternative a* under profile R*.
F
Definition. A correspondence F satisfies Conditionhif for allu[Q and for all T[1
1. C (Tu)#B (Tu), C*5C (Nu), Im(F )#C (Nu) and T#T * implies C (Tu)#C (T *u) simply say that the collection satisfies Conditionh. Note that for T5f and T5N,
Condition h is read as follows: if a[C (fu) and C (iu)#L(i,a,R) for all i[N, then a[F(R); and if a[C (Nu)#L(i,a,R) for all i[N, then a[F(R).
Suh (1997) showed that Condition h is necessary and sufficient for double im-plementation.
Theorem (Suh, 1997). A correspondence F is doubly implementable if and only if F
satisfies Conditionh.
3. Algorithm
In this section we provide an algorithm which helps us to verify whether a ¨ ¨
correspondence satisfies Conditionh. This type of algorithm appears in Sjostrom (1991)
¨ ¨
and Suh (1996a,b). Sjostrom (1991) originally provided an algorithm for checking whether a correspondence satisfies the necessary and sufficient conditions for Nash implementation proposed by Moore and Repullo (1990). And Suh (1996a,b) extended his results to the implementation problem where agents may form coalitions.
u
Now we present an algorithm for constructing the collection hC (Tu)jT[1 for all
F F u
u[Q . Since for all u[Q the collection hC (Tu)jT[1 satisfies Condition h as we
u F
have shown in Lemma 1, showing the existence ofhC (Tu)jT[1 for allu[Q by using the algorithm is enough for the purpose of checking whether F is doubly implementable or not.
F `
Given F andu[Q , we define the collectionhC (T u)jT[1 as follows. The definition is iterative. Let
0
C (Nu);
>
(a,R )[A(F )L(N,a,R) where A(F ); h(a,R)[A3Rua[F(R)j1 0 0
C (Nu); ha[C (Nu)ufor all R[R, if a[C (Nu)#L(i,a,R) for all i[N, then a
[F(R)j
? ? ?
m11 m m
CN (u); ha[C (Nu)ufor all R[R, if a[C (Nu)#L(i,a,R) for all i
[N, then a[F(R)j
? ? ?
` m
C (Nu);C (Nu) as m→`.
m
Note that for all m[I, C (Nu) is independent ofu. Only for notational convenience,
m m `
we use C (u) instead of C . Suppose that given T[1 and T±N, C (u) is defined
N N T<i
for all i[N\T. Let
0 `
C (Tu);
>
i[N \TCT<i(u)>
B (Tu)1 0 0
C (Tu); ha[C (Tu)ufor all R[R, if a[C (Tu)#L(i,a,R) for all i
`
[T and CT<i(u)#L(i,a,R) for all i[N\T, then a[F(R)j
m11 m m
Theorem 1. For a correspondence F satisfying Conditionh, for allu[Q and for all
m m11 ` u `
T[1, if for some m[I, C (T u)5CT (u)5 ? ? ? 5C (T u), then C (Tu)5C (T u) for
all T[1.
Proof. The statement is proved by the following two Lemmas.
F
Lemma 2. For a correspondence F satisfying Conditionh, for allu[Q and for all
u `
T[1, C (Tu)#C (T u).
F
Proof. Consider a profileu[Q . By using an inductive argument, we will show that
u m u `
Lemma 3. For a correspondence F satisfying Conditionh, for allu[Q , if for some
m m11 u `
m[I, C (u)5C (u)5 ? ? ?, then C (u)8C (u) for all T[1.
T T T T
F
Proof. Consider a profile u[Q . To prove the statement, we will show that
`
Financial support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.
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¨ ¨
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