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Implementation via mechanisms with transfers
Mathematical Social Sciences 61 (2011) 65–70
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Implementation via mechanisms with transfers Jianxin Yi School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
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Article history: Received 25 December 2008 Received in revised form 23 July 2010 Accepted 27 October 2010 Available online 2 November 2010 JEL classification: C72 D71 D78
abstract We consider the following problem: suppose that the social choice rule is given, but the planner has the opportunity to redistribute among agents some numeraire commodity (‘‘money’’) by compensatory transfers; can the planner find a transfer such that the social choice rule is Nash implementable by the transfer? In this paper, we establish a necessary and sufficient condition for Nash implementation by transfers. Furthermore, we construct a simplified mechanism, which Nash-implements the desired social choice rule by transfers. © 2010 Elsevier B.V. All rights reserved.
Keywords: Nash implementation Monotonicity Semi-monotonicity Transfers
1. Introduction Implementation theory attempts to answer the following important question. Consider an environment with a set of agents and a set of feasible outcomes, together with a set of possible states of the world. These agents must make a social choice from the set of feasible outcomes according to the states of the world. Then can a mechanism, or game form, be designed for the agents to play such that the equilibrium outcomes coincide with the ones specified by the social choice rule? In general, whether or not a social choice rule is implementable depends on the solution concept. Implementation theory has considered a variety of solution concepts, including equilibrium in dominant strategies, Bayesian equilibrium, and Nash equilibrium. Maskin (1999a) initiated the study of Nash implementation. He showed that a condition on social choice rules called monotonicity is necessary for Nash implementation. Furthermore, when there are at least three agents, any social choice rule which satisfies monotonicity and no veto power is Nash implementable. His work not only provides us with an understanding of what is implementable in Nash equilibrium, but also provides a blueprint for many techniques used in the literature.
E-mail address: [email protected]. 0165-4896/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2010.10.004
Maskin’s monotonicity is a fairly demanding condition, and many well-known social choice rules fail to be Nash implementable (see Examples 1–3). Example 1 (Plurality Rule. From Abreu and Sen, 1991). There are three outcomes, C = {a, b, c }, three agents, I = {1, 2, 3}, and two states, Θ = {θ , ζ }. Preferences are described as follows. u1 (·, θ ) u2 (·, θ ) u3 (·, θ )
a 3, 1, 1,
b 2, 3, 2,
c 1; 2; 3;
a 3, 1, 1,
b 2, 3, 3,
c 1; 2; 2;
u1 (·, ζ ) u2 (·, ζ ) u3 (·, ζ ).
Let F (θ ) = a and F (ζ ) = b. Then, F is not Nash implementable. Example 2 (Condorcet Rule. From Palfrey and Srivastava, 1991). There are three outcomes, C = {a, b, c }, five agents, I = {1, 2, 3, 4, 5}, and two states, Θ = {θ , ζ }. Preferences are described as follows. u1 (·, θ ) u2 (·, θ ) u3 (·, θ ) u4 (·, θ ) u5 (·, θ )
a 3, 3, 1, 1, 1,
b 2, 2, 3, 2, 2,
c 1; 1; 2; 3; 3;
a 3, 3, 2, 1, 1,
b 2, 2, 3, 2, 2,
c 2; 2; 3; 3; 3;
u1 (·, ζ ) u2 (·, ζ ) u3 (·, ζ ) u4 (·, ζ ) u5 (·, ζ ).
Let F (θ ) = b and F (ζ ) = c. Then, F is not Nash implementable.
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J. Yi / Mathematical Social Sciences 61 (2011) 65–70
Example 3 (Borda Rule. From Jackson, 2001). There are three outcomes, C = {a, b, c }, three agents, I = {1, 2, 3}, and two states, Θ = {θ, ζ }. Preferences are described as follows. u1 (·, θ) u2 (·, θ) u3 (·, θ)
a 3, 2, 1,
b 2, 1, 3,
c 1; 3; 2;
a 3, 2, 1,
b 2, 1, 2,
c 1; 3; 3;
u1 (·, ζ ) u2 (·, ζ ) u3 (·, ζ ).
Let F (θ ) = {a, b, c } and F (ζ ) = {c }. Then F is not Nash implementable. On the other hand, the Muller and Satterthwaite (1977) theorem shows the restrictiveness of monotonicity in the voting context. It states that if the outcomes are finite, agents are more than three, the set of admissible preferences includes all strict preferences, then any social choice function that is monotonic is necessarily dictatorial. For these reasons, much of Nash implementation theory has been developed and more positive results are obtained by changing the requirement of implementation. For example, Palfrey and Srivastava (1991) discuss the implementation in undominated Nash equilibrium. Jackson and Palfrey (1992) weakens the solution concept to undominated strategies, and proves that if strict value distinction is satisfied, then any social choice rule is implementable in undominated strategies. Abreu and Sen (1991) consider virtual implementation in Nash equilibrium. Bochet (2007) and Benoit and Ok (2008) consider the problem of Nash implementation with lottery mechanisms. Maskin (1999b), Jackson and Palfrey (2001) and Amorós (2004) examine Nash implementation via renegotiation. Sanver (2006) and Benoit and Ok (2008) discuss Nash implementation by awards. Other interesting works include Sanver and Sanver (2006) and Barlo and Dalkiran (2009). Recall the mechanism design problem under incomplete information. In this setting, the possible states of the world are the profiles of private type. Every agent can observe his type which is not known collectively to the other agents in the society. When agents may hold private information, implementation in dominant strategies (or incentive compatibility) is very important. An advantage of implementation in dominant strategies is that if the mechanism designer is an outsider, he does not need to know the probability density to successfully implement a social choice function. However, Gibbard (1973) and Satterthwaite (1975) show that for a very general class of problems there is no hope of implementing satisfactory social choice functions in dominant strategies. The Gibbard–Satterthwaite theorem is a quite negative result. It implies that transfer functions will be needed for dominant strategy implementation of non-dictatorial social choice functions in some settings. Vickrey (1961), Clarke (1971) and Groves (1973) prove that if the social choice function is efficient, then there exists a transfer function such that the social choice function is incentive compatible by the transfer. On the assumption of unrestricted domain, Roberts (1979) shows that a social choice function is incentive compatible by a transfer if and only if it satisfies PDA. Bikhchandani et al. (2006) establish a difference condition (WMon), which is necessary and sufficient for incentive compatibility of social choice function by transfers on order-based domains. Similarly, to the use of transfers to provide correct incentives mentioned above, we consider the following problems. Suppose that the social choice rule is given and is not Nash implementable, but the planner has the opportunity to redistribute among agents some particular commodity; then, can the planner find a transfer such that the social choice rule with the transfer is Nash implementable? If the answer is negative, what restrictions, if any, must be placed on the SCR? Furthermore, can we require that the transfer is budget balanced?
For convenience, we call a social choice rule T-Nash implementable, if we can find a transfer such that the social choice rule is Nash implementable by the transfer. In this paper, we establish a necessary and sufficient condition which characterize the social choice rules that are T-Nash implementable when there are two or more agents. Especially, a social choice rule must be T-Nash implementable if it is Nash implementable. But the converse is not true. Therefore we successfully implement a wider range of social choice rules. Moreover, for a social choice rule to be T-Nash implemented, we construct a simplified mechanism to implement it. Note that although, in many cases, we can find a transfer function such that the social choice function is incentive compatible by the transfer, but the transfer is not budget balanced. Thus, equilibrium outcomes in certain states will lead to a waste of money and a departure from ‘full’ efficiency. Here, it is shown that, if a social choice rule is T-Nash implementable, it can be T-Nash implementable by a budget-balanced transfer. Furthermore, we can see that T-Nash implementation and individual rationality are compatible. This paper is organized as follows. Section 2 provides definitions. Section 3 discusses the T-Nash implementation of social choice rules. Section 4 establishes two simplified mechanisms. The proofs are in Appendix. 2. Preliminaries Let C be the set of social alternatives or outcomes, I = {1, 2, . . . , n} is the set of agents with n ≥ 2, Θ is the set of possible states of the world. At state θ , each agent i ∈ I is assumed to have a utility function ui (·, θ ) over the set C . We shall refer to the collection (I , C , Θ , {ui }i∈I ) as an environment. A social choice rule (SCR) F is a mapping F : Θ → 2C ∼ {φ}, where F (θ ) is interpreted as the set of alternatives that are deemed socially optimal in state θ . A social choice function (SCF) is a singlevalued SCR, i.e., a function f Θ → C . F is said to be monotonic, if, for any θ , ζ ∈ Θ and any a ∈ F (θ ), we have a ∈ F (ζ ), whenever
{x ∈ C |ui (a, θ ) ≥ ui (x, θ )} ⊆ {x ∈ C |ui (a, ζ ) ≥ ui (x, ζ )} for all i ∈ I. F is said to be semi-monotonic, if, for any θ , ζ ∈ Θ and any a ∈ F (θ ), we have a ∈ F (ζ ), whenever ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . A mechanism in the environment (I , C , Θ , {ui }i∈I ) is a pair (S , g ), where S = S1 × S2 × · · · × Sn is a product of strategy spaces and g: S → C is an outcome function. Let si ∈ Si denote agent i’s strategy. A strategy profile is denoted as s = (s1 , s2 , . . . , sn ). For any s ∈ S, i ∈ I, let s−i = (s1 , · · · , si−1 , si+1 , . . . , sn ). A strategy profile s∗ ∈ S is a Nash equilibrium of mechanism (S , g ) at state θ if ui (g (s∗ ), θ ) ≥ ui (g (s∗−i , si ), θ ) for all i ∈ I and all si ∈ Si . The set of all Nash equilibria at state θ is denoted as NE ((S , g ), θ ). The mechanism (S , g ) Nash-implements F if and only if g (NE ((S , g ), θ )) = F (θ ) for all θ ∈ Θ . F is Nash implementable if there exists a mechanism (S , g ) which Nashimplements F . Given an environment (I , C , Θ , {ui }i∈I ), we consider a new environment (I , C × Rn , Θ , {vi }i∈I ), where vi : C × Rn × Θ → R takes a quasilinear form, i.e.,
vi (x, y1 , y2 , . . . , yn , θ ) = ui (x, θ ) + yi . A transfer is a function p : Θ → Rn . The transfer p is budget balanced if p1 (θ ) + · · · + pn (θ ) = 0 for all θ ∈ Ω .
J. Yi / Mathematical Social Sciences 61 (2011) 65–70
Given a SCR F : Θ → 2C ∼ {φ} and a transfer p: Θ → Rn , we define (F , p) to be a correspondence (F , p)(θ ) = {(a, p1 (θ ), p2 (θ), . . . , pn (θ )) ∈ C × Rn |a ∈ F (θ )} for all θ ∈ Θ . In particular, we rewrite (F , p) as (F , 0) if p(θ ) = (0, 0, . . . , 0) for all θ ∈ Θ . That is, (F , 0)(θ ) = {(a, 0, 0, . . . , 0) ∈ C × Rn |a ∈ F (θ )}. Given a mechanism (S , (g , t )) in the new environment (I , C × Rn , Θ , {vi }i∈I ), where S = S1 × S2 × · · · × Sn is a product of strategy spaces, g: S → C is an outcome function and t : S → Rn a transfer function. We say that a mechanism (S , (g , t )) satisfies the budgetbalanced condition if t1 (s) + · · · + tn (s) = 0 for all s ∈ S. Now, we can describe the primary question precisely for a given SCR F Θ → 2C ∼ {φ} which is not Nash implementable, does there exist a transfer p : Θ → Rn , such that (F , p) is Nash implementable in the environment (I , C × Rn , Θ , {vi }i∈I )? If the answer is affirmative, we call that F is T-Nash implementable and p is feasible with respect to F . 3. T-Nash implementation
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In many cases, it is impossible to implement a social choice rule which satisfies individual rationality constraints. However, it is in the use of transfers, we give a possible result to implement an individually rational SCR. For current purpose, we first present Theorem 2 which characterizes the feasible transfers with respect to a SCR. Theorem 2. Suppose that F is semi-monotonic, then, for any p : Θ → Rn , p is feasible with respect to F if and only if for any θ , ζ ∈ Θ , we have p(θ ) = p(ζ ) whenever either condition (a) or (b) below holds. (a) There is an a ∈ F (θ ), such that ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . (b) There is an a ∈ F (ζ ), such that ui (a, ζ ) − ui (x, ζ ) ≤ ui (a, θ ) − ui (x, θ ) for all i ∈ I and all x ∈ C . Let Mi = infx,θ ui (x, θ ) and ti (θ ) = supη {wi (η)|f (η) = f (θ )}. Moreover, suppose that ti (θ ) < +∞ for any θ ∈ Θ and Mi = infx,θ ui (x, θ ) > −∞.
In this section, we begin to explore the consequences of T-Nash implementation when there are two or more agents. It is known that the two-person problem is different from its many-person counterpart for Nash implementation. Nevertheless, we show that semi-monotonicity is a necessary and sufficient condition for TNash implementation of an n-person social choice rule with n ≥ 2. Then, we characterize completely the class of transfers which are feasible with respect to a given SCR.
Theorem 3. (1) Suppose that the social choice function f is semimonotonic. Let pi (θ ) = ti (θ ) − ui (f (θ ), θ ). Then p is a feasible transfer with respect to f and (f , p) satisfy the individual rationality constraints. (2) Suppose that the social choice rule F is semi-monotonic. Let pi (θ ) = ti (θ ) − Mi . Then, p is a feasible transfer with respect to F and (F , p) satisfy individual rationality constraints.
Theorem 1. Suppose that |I | ≥ 2. Then, there exists a transfer p : Θ → Rn , such that (F , p) is Nash implementable in the environment (I , C × Rn , Θ , {vi }i∈I ), i.e., F is T-Nash implementable if and only if F is semi-monotonic in the environment (I , C , Θ , {ui }i∈I ).
A natural question then is whether optimal transfers exist. In other words, if social choice rule F is semi-monotonic, is there a feasible transfer p, such that F and p satisfy following conditions. (1) (F , p) satisfy the individual ∑n rationality constraints. ∑n (2) p is optimal, that is | i=1 pi (θ )| ≤ | i=1 qi (θ )| for any n θ ∈ Θ and any feasible transfer q : Θ → R which satisfies the individual rationality constraints.
Examples 1–3 above give three social choice rules which are Nash implementable by transfers. In fact, we can check that any transfer p : Θ → Rn is feasible with respect to the Plurality rule F given in Example 1 and the Condorcet rule given in Example 2. Any transfer p : Θ → Rn is feasible with respect to the Borda rule given in Example 3 if p(θ ) = p(ζ ). By definition, we know that monotonicity implies almost monotonicity (Sanver, 2006) and almost monotonicity implies semi-monotonicity, so Nash implementable
⇒ Nash implementable by awards (Sanver, 2006; Benoit and Ok, 2008)
⇒ T-Nash implementable. But the converse is not true. Example 1 provides a counterexample. Up to this point, we have studied whether we can Nashimplement by transfers a social choice rule. But efficiency also requires that none of the numeraire be wasted, that is, the transfer function is budget balanced. If the transfer function is not balanced, there is either waste inside the system of n agents or source of financing from outside it. Fortunately, for a semi-monotonic F , it is possible to find a budget-balanced transfer p, such that (F , p) is Nash implementable. This point can be seen in the proof of Theorem 1. In many applications, agent’s participation in the mechanism is voluntary. As a result, the social choice rule must not only be implemented by a mechanism but also satisfy certain participation (or individual rationality) constraints if it is to be successfully implemented. Formally, Suppose that agent i ∈ I can receive a utility wi (θ ) by withdrawing from the mechanism when the social state is θ . Then, to ensure agent i’s participation, we must satisfy the individual rationality constraints ui (a, θ ) ≥ wi (θ ) for all θ ∈ Θ and all a ∈ F (θ ).
4. Simplified mechanism If (F , p) is monotonic, (F , p) is Nash implementable. Note that, the canonical mechanism which Nash-implements (F , p) uses a huge strategy space. For example, like Maskin’s mechanism (Maskin, 1999a), the agent i’s strategy space Si is of the form Θ × (C × Rn ) × N. According to the Moore and Repullo mechanism (Moore and Repullo, 1990), the agent i’s strategy space Si is more complicated. Now, we shall simplify the canonical mechanism. We propose a simple mechanism in which each agent announces a social state, a feasible social outcome, plus one number in any play. That is, Si = Θ × C × R for each agent i. Such mechanisms bring about a reduction in the communication complexity measured in dimension and has the advantage of informational efficiency of decentralized decision making. Although, semi-monotonicity is necessary and sufficient for the T-Nash implementation of n-person social choice rule with both n = 2 and n ≥ 3. There are also some interesting differences between n = 2 and n ≥ 3. For example, we can find a budgetbalanced mechanism to implement an n-person social choice rule (F , 0) with n ≥ 3. But, we do not know whether there is a budgetbalanced mechanism to implement a 2-person SCR (F , 0). 4.1. Three or more agents Consider the following mechanism (S , (g , t )), where Si = Θ × C × R for each i ∈ I and the outcome functions g : S → C and ti : S → R are defined by the following three rules. For a vector of strategies s = (s1 , s2 , . . . , sn ) ∈ S, R4.1 if si = (θ , c , m) for all i ∈ I and c ∈ F (θ ), then g (s) = c and ti (s) = pi (θ ) for all i ∈ I;
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J. Yi / Mathematical Social Sciences 61 (2011) 65–70
R4.2 if there exists j ∈ I such that si = (θ , c , m) for all i ̸= j and c ∈ F (θ ), but sj = (θj , cj , mj ) ̸= (θ , c , m), then, (a) if uj (c , θ ) + pj (θ) ≥ uj (cj , θ ) + mj , set g (s) = cj ,
tj (s) = mj
and
ti (s) = −
mj n−1
for each i ̸= j;
ui (a, ζ ) + ti (s∗ ) < ui (g (s∗−i , si ), ζ ) + ti (s∗−i , si ).
(b) if uj (c , θ ) + pj (θ ) < uj (cj , θ ) + mj , set g ( s) = c ,
tj (s) = pj (θ ) and
ti (s) = pi (θ ) for each i ̸= j;
R4.3 in all other cases, let si = (θi , ci , mi ), then g (s) = ck ,
tk (s) = mk
and
ti (s) = −
mk n−1
for each i ̸= k,
where k is the lowest indexed i such that mi ≥ mj for all j ∈ I. Theorem 4. Suppose that |I | ≥ 3. Then (S , (g , t )) Nash-implements F if (F , p) is monotonic. Furthermore, (S , (g , t )) is budget balanced if p is budget balanced. 4.2. Two agents Consider the following mechanism (S , (g , t )), where S1 = S2 = Θ × C × R and the outcome functions g : S → C and ti : S → R are defined by the following rules. Given a s = (s1 , s2 ) ∈ S, where s1 = (θ1 , a1 , λ1 ) and s2 = (θ2 , a2 , λ2 ) (R5.1) if (θ1 , a1 ) = (θ2 , a2 ) = (θ , a) and a ∈ F (θ ), then
(g (s), t1 (s), t2 (s)) = (a, p1 (θ ), p2 (θ )); (R5.2) if (θ1 , a1 ) ̸= (θ2 , a2 ) and λ1 = λ2 = 0, then
(g (s), t1 (s), t2 (s)) = (a2 , p1 (θ2 ) − 1, u2 (a1 , θ1 ) + p2 (θ1 ) − u2 (a2 , θ1 ) − 1); (R5.3) if (θ1 , a1 ) ̸= (θ2 , a2 ), then
(a , λ , λ ) 1 1 1 if u1 (a2 , θ2 ) + p1 (θ2 ) ≥ u1 (a1 , θ2 ) + λ1 (g (s), t1 (s), t2 (s)) = (a2 , p1 (θ1 ), p2 (θ2 )) if u1 (a2 , θ2 ) + p1 (θ2 ) < u1 (a1 , θ2 ) + λ1 when λ1 ̸= 0, λ2 = 0 and
(a , λ , λ ) 2 2 2 if u2 (a1 , θ1 ) + p2 (θ1 ) ≥ u2 (a2 , θ1 ) + λ2 (g (s), t1 (s), t2 (s)) = (a1 , p1 (θ1 ), p2 (θ2 )) if u2 (a1 , θ1 ) + p2 (θ1 ) < u2 (a2 , θ1 ) + λ2 when λ1 = 0, λ2 ̸= 0; (R5.4) in all other cases, let
(g (s), t1 (s), t2 (s)) =
(a1 , λ1 , λ1 ) (a2 , λ2 , λ2 )
Let (S , (g , t )) T-Nash-implements F . Then, there is a vector of strategies s∗ = (s∗1 , s∗2 , . . . , s∗n ) ∈ NE (S , (g , t ), θ ) such that g (s∗ ) = a. It suffices to prove that s∗ ∈ NE (S , (g , t ), ζ ). Suppose that the conclusion does not hold. Then there is an i ∈ I and an si ∈ Si such that
if λ1 ≥ λ2 if λ1 < λ2 .
Theorem 5. Let |I | = 2. If (F , p) is monotonic, (S , (g , t )) Nashimplements F . Acknowledgements I am grateful to two anonymous referees and an associate editor for their helpful comments and suggestions. Appendix Proof of Theorem 1. The necessity part is simple. For any θ , ζ ∈ Θ and any a ∈ F (θ ), suppose that, ui (a, θ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C .
Note that s∗ ∈ NE (S , (g , t ), θ ) implies that ui (a, θ ) + ti (s∗ ) ≥ ui (g (s∗−i , si ), θ ) + ti (s∗−i , si ). Therefore, we have ui (a, θ ) − ui (g (s∗−i , si ), θ ) > ui (a, ζ ) − ui (g (s∗−i , si ), ζ ) which is a contradiction. The sufficiency part will be obtained as a corollary following Lemmas 1 and 2. Lemma 1. If F is semi-monotonic then (F , 0) is monotonic in the environment (I , C × Rn , Θ , {vi }i∈I ). For any θ , ζ ∈ Θ and any (a, 0, . . . , 0) ∈ (F , 0)(θ ), suppose that
{(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, θ ) + 0 ≥ ui (x, θ ) + yi } ⊆ {(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, ζ ) + 0 ≥ ui (x, ζ ) + yi } for all i ∈ I. If ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) is not true for some j ∈ I and some x¯ ∈ C , i.e., uj (a, θ ) − uj (¯x, θ ) > uj (a, ζ ) − uj (¯x, ζ ). Take (z1 , . . . , zn ) ∈ Rn such that uj (a, θ ) − uj (¯x, θ ) > zj − 0 > uj (a, ζ ) − uj (¯x, ζ ). Then
(¯x, z1 , . . . , zn ) ∈ {(x, y1 , . . . , yn ) ∈ C × Rn |uj (a, θ ) + 0 ≥ uj (x, θ ) + yj } whereas
(x0 , z1 , . . . , zn ) ̸∈ {(x, y1 , . . . , yn ) ∈ C × Rn |uj (a, ζ ) + 0 ≥ uj (x, ζ ) + yj }, which is a contradiction. Therefore, a ∈ F (ζ ), which implies (a, 0, . . . , 0) ∈ (F , 0)(ζ ). Lemma 2. Suppose that |I | ≥ 2, (F , p) is Nash implementable iff (F , p) is monotonic in the environment (I , C × Rn , Θ , {vi }i∈I ). Note that, when |I | ≥ 3, the result follows immediately from Maskin’s theorem, since (F , p) automatically satisfies no veto power in the environment (I , C × Rn , Θ , {vi }i∈I ). So we only verify it when |I | = 2. Take B = C × Rn and Ci (a, θ ) = {(x, y1 , y2 ) ∈ C × R2 |ui (a, θ ) + pi (θ ) ≥ ui (x, θ ) + yi }. Moreover, for any a, θ , b, η, let e(a, θ , b, η) = (a, p1 (θ ) − 1, u2 (b, η) − u2 (a, η) + p2 (η) − 1). Then, e(a, θ , b, η) ∈ C1 (a, θ ) ∩ C2 (b, η). Note that there is no ζ , such that
{(x, z1 , z2 ) ∈ C × R2 |u1 (a, θ ) + p1 (θ ) ≥ u1 (x, θ ) + z1 } ⊆ {(x, z1 , z2 ) ∈ C × R2 |u1 (a, ζ ) + y1 ≥ u1 (x, ζ ) + z1 } and
{(x, z1 , z2 ) ∈ C × R2 |u2 (b, η) + p2 (η) ≥ u2 (x, η) + z2 } ⊆ {(x, z1 , z2 ) ∈ C × R2 |u2 (a, ζ ) + y2 ≥ u2 (x, ζ ) + z2 }. It is not difficult to check that B, Ci (a, θ ) and e(a, θ , b, η) satisfy Condition µ2 (Moore and Repullo, 1990). Thus (F , p) is Nash implementable.
J. Yi / Mathematical Social Sciences 61 (2011) 65–70
Proof of Theorem 2. Sufficiency: it suffices to prove that (F , p) is monotonic. Suppose that for any θ , ζ ∈ Θ , and any a ∈ F (θ ),
{(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, θ ) + pi (θ ) ≥ ui (x, θ ) + yi } ⊆ {(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, ζ ) + pi (θ ) ≥ ui (x, ζ ) + yi } for all i ∈ I. It is easy to check that ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . Therefore (F , p)(θ ) = (F , p)(ζ ). Necessity: Suppose that, for any θ , ζ ∈ Θ , there is an a ∈ F (θ ), such that ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . Then
{(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, θ ) + pi (θ ) ≥ ui (x, θ ) + yi } ⊆ {(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, ζ ) + pi (θ ) ≥ ui (x, ζ ) + yi } Note that (F , p) is monotonic, it follows that a ∈ F (ζ ) and p(θ) = p(ζ ). Therefore, p is feasible with respect to F . Proof of Theorem 3. Theorem 3 will be obtained as a corollary of Theorem 2.
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If s∗ falls into (R4.3), let s∗i = (θi , ci , mi ) for each i ∈ I. Then g (s∗ ) = ck , where k is the lowest indexed i such that mi ≥ mj for all j ∈ I. Note that uk (g (s∗ ), θ ) + tk (s∗ ) = uk (ck , θ ) + mk
< uk (ck , θ ) + l = uk (g (s∗−k , sk ), θ ) + tk (s∗−k , sk ) for any sk = (θk , ck , l) with l > mk . Thus, by announcing a high enough number l > mk , agent k can profit from the deviation from s∗k . This contradicts the assumption that s∗ is a Nash equilibrium. Finally, suppose s∗1 = s∗2 = · · · = s∗n = (ζ , c , m) and c ∈ F (ζ ). For each i ∈ I and each (x, λ1 , λ2 , . . . , λn ) ∈ C × Rn , suppose that ui (c , ζ ) + pi (ζ ) ≥ ui (x, ζ ) + λi . Let si = (η, x, λi ) ̸= (ζ , c , m). By rule (R4.2), we have g (s∗−i , si ) = x, ti (s∗−i , si ) = λi . Note that s∗ ∈ NE (S , (g , t ), θ ) implies ui (g (s∗ ), θ ) + ti (s∗ ) ≥ ui (g (s∗−i , si ), θ ) + ti (s∗−i , si ). Thus
Proof of Theorem 4. First, let us check that
ui (c , θ ) + pi (ζ ) ≥ ui (x, θ ) + λi .
(F (θ), p1 (θ ), . . . , pn (θ )) ⊆ (g , t )(NE (S , (g , t ), θ )) for any θ ∈ Θ .
By hypothesis, it follows that
For any c ∈ F (θ ), let s∗i = (θ , c , 1) for all i ∈ I. Then g (s∗−i , s∗i ) = c and ti (s∗−i , s∗i ) = pi (θ ) for all i ∈ I. For any j ∈ I and any sj = (θj , cj , mj ) ̸= s∗j , if uj (c , θ ) + pj (θ ) ≥ uj (cj , θ ) + mj , then
So, (S , (g , t )) Nash-implements F .
(g , t )(s∗ ) = (c , p1 (ζ ), . . . , pn (ζ )) ∈ (F , p)(θ ).
uj (g (s∗−j , s∗j ), θ ) + tj (s∗−j , s∗j ) = uj (c , θ ) + pj (θ )
≥ uj (cj , θ ) + mj ≥ uj (g (s∗−j , sj ), θ ) + tj (s∗−j , sj ). If uj (c , θ ) + pj (θ ) < uj (cj , θ ) + mj , then uj (g (s∗−j , s∗j ), θ ) + tj (s∗−j , s∗j ) = uj (c , θ ) + pj (θ )
= uj (g (s∗−j , sj ), θ ) + tj (s∗−j , sj ). Thus, (s∗−j , s∗j ) ∈ NE (S , (g , t ), θ ) and (c , p1 (θ ), . . . , pn (θ )) ∈ (g , t ) (NE (S , (g , t ), θ )). Next, let us check that
(g , t )(NE (S , (g , t ), θ )) ⊆ (F (θ ), p1 (θ ), . . . , pn (θ )) for each θ ∈ Θ . Let s∗ = (s∗1 , s∗2 , . . . , s∗n ) ∈ NE (S , (g , t ), θ ). We must show that s∗ falls into (R4.1). If not, then either rule (R4.2) or rule (R4.3) applies to (s∗1 , s∗2 , . . . , s∗n ). If s∗ = (s∗1 , s∗2 , . . . , s∗n ) falls into (R4.2), without loss of generality, we can assume that,
Proof of Theorem 5. We need to show that (F (θ ), p1 (θ ), p2 (θ )) = (g , t )(NE (S , (g , t ), θ )) for any θ ∈ Θ . The proof proceeds in three steps. Step 1: (F (θ ), p1 (θ ), p2 (θ )) ⊆ (g , t )(NE (S , (g , t ), θ )) for any θ ∈ Θ. For any a ∈ F (θ ), let s∗1 = s∗2 = (θ , a, 0), then g (s∗1 , s∗2 ) = a, t1 (s∗1 , s∗2 ) = p1 (θ ) and t2 (s∗1 , s∗2 ) = p2 (θ ). Consider any s1 = (θ1 , a1 , λ1 ). If (θ , a) ̸= (θ1 , a1 ), then when λ1 ̸= 0, u1 (g (s1 , s∗2 ), θ ) + t1 (s1 , s∗2 ) u1 (a1 , θ ) + λ1 u1 (a, θ ) + p1 (θ )
=
if u1 (a, θ ) + p1 (θ ) ≥ u1 (a1 , θ ) + λ1 if u1 (a, θ ) + p1 (θ ) < u1 (a1 , θ ) + λ1
≤ u1 (a, θ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), θ ) + t1 (s∗1 , s∗2 ) when λ1 = 0, u1 (g (s1 , s∗2 ), θ ) + t1 (s1 , s∗2 ) = u1 (a, θ ) + p1 (θ ) − 1
≤ u1 (a, θ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), θ ) + t1 (s∗1 , s∗2 ). Similarly, we can verify that u2 (g (s∗1 , s∗2 ), θ ) + p2 (θ ) ≥ u2 (g (s1 , s2 ), θ ) + t (s∗1 , s2 ) for any s2 = (θ2 , a2 , λ2 ). Therefore, (s∗1 , s∗2 ) ∈ NE (S , (g , t ), θ ) and ∗
s∗1 = s∗2 = · · · = s∗n−1 = (ζ , c , m) ̸= s∗n = (ζn , cn , mn )
and
c ∈ F (ζ ). By rule (R4.2), we have
mn u (c , θ ) − 1 n n−1 if un (c , ζ ) + pn (ζ ) ≥ un (cn , ζ ) + mn u1 (g (s∗ ), θ ) + t1 (s∗ ) = u1 (c , θ ) + p1 (ζ ) if un (c , ζ ) + pn (ζ ) < un (cn , ζ ) + mn . Let s1 = (ζ , c ′ , l) where l > max{m, mn }, then by (R4.3), we have u1 (g (s∗−1 , s1 ), θ ) + t1 (s∗−1 , s1 ) = u1 (c ′ , θ ) + l. Therefore, by announcing a high enough number l, agent 1 profits from the deviation from s∗1 , thereby contradicting the assumption that s∗ is a Nash equilibrium.
(a, p1 (θ ), p2 (θ )) = (g , t )(s∗1 , s∗2 ) ∈ (g , t )(NE (S , (g , t ), θ )). Step 2: For any ζ ∈ Θ , if (s∗1 , s∗2 ) ∈ NE (S , (g , t ), ζ ), where s∗1 = (θ1 , a1 , λ1 ) and s∗2 = (θ2 , a2 , λ2 ), then (θ1 , a1 ) = (θ2 , a2 ), a1 ∈ F (θ1 ) and λ1 = λ2 = 0. To see this, suppose the contrary, that is, following four cases applies to (s∗1 , s∗2 ). Case 1: (θ1 , a1 ) = (θ2 , a2 ) = (θ , a), a ∈ F (θ ) and λi ̸= 0 for some i ∈ {1, 2}. Suppose, without loss of generality, that λ2 ̸= 0. Let s1 = (θ1′ , a′1 , λ′1 ) and (θ1′ , a′1 ) ̸= (θ2 , a2 ), λ′1 ̸= 0, λ′1 > λ2 . Note that, g (s∗1 , s∗2 ) = a, t1 (s∗1 , s∗2 ) = p1 (θ ) and g (s1 , s∗2 ) = a′1 and t1 (s1 , s∗2 ) = λ′1 . Since u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 ) ≥ u1 (g (s1 , s∗2 ), ζ ) + t (s1 , s∗2 ),
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u1 (a, ζ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 )
we have
≥ u1 (g (s1 , s∗2 ), ζ ) + t1 (s1 , s∗2 ) = u1 (x, ζ ) + λ1 .
u1 (a, ζ ) + p1 (θ ) ≥ u1 (a′1 , ζ ) + λ′1 . Note that, this is impossible because u1 (a′1 , ζ ) + λ′1 can be arbitrarily large by choosing λ′1 large enough. Case 2: (R5.2) applies to (s∗1 , s∗2 ), that is (θ1 , a1 ) ̸= (θ2 , a2 ) and λ1 = λ2 = 0. Let s2 = (θ2 , a2 , λ′2 ), where λ′2 ̸= 0 and u2 (a1 , θ1 ) + p2 (θ1 ) ≥ u2 (a2 , θ1 ) + λ′2 > u2 (a1 , θ1 ) + p2 (θ1 ) − 1. Then
If λ1 = 0, let
s1 =
(θ , x, −ε) (ζ , a, −ε)
for any ε > 0. By rule (R5.3), we have g (s1 , s∗2 ) = x and t1 (s1 , s∗2 ) = −ε . By assumption, we have u1 (a, ζ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 )
≥ u1 (g (s1 , s∗2 ), ζ ) + t1 (s1 , s∗2 ) = u1 (x, ζ ) + λ1 − ε.
u2 (g (s∗1 , s∗2 ), ζ ) + t2 (s∗1 , s∗2 )
= u2 (a2 , ζ ) + u2 (a1 , θ1 ) + p2 (θ1 ) − u2 (a2 , θ1 ) − 1 < u2 (a2 , ζ ) + λ′2 = u2 (g (s1 , s∗2 ), ζ ) + t2 (s1 , s∗2 ). This is a contradiction. Case 3: (R5.3) applies to (s∗1 , s∗2 ). Note that, (θ1 , a1 ) ̸= (θ2 , a2 ) and, suppose, without loss of generality, that λ1 ̸= 0 and λ2 = 0. Let s2 = (θ2 , a2 , λ′2 ) such that λ′2 ̸= 0, λ1 < λ′2 . Then u2 (g (s∗1 , s∗2 ), ζ ) + t2 (s∗1 , s∗2 )
=
u2 (a1 , ζ ) + λ1 u2 (a2 , ζ ) + p2 (θ2 )
Note that the right-hand side of the above expression can be arbitrarily large by choosing λ′2 large enough. But the left-hand side is bounded. This is a contradiction. Case 4: (R5.4) applies to (s∗1 , s∗2 ). Suppose, without loss of generality, that λ1 ≥ λ2 . Then g (s∗1 , s∗2 ) = a1 and t1 (s∗1 , s∗2 ) = λ1 . Choose s1 = (θ1 , a1 , |λ1 | + 1), then , g (s1 , s∗2 ) = a1 and t1 (s1 s∗2 ) = |λ1 | + 1. Note that, u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 ) ≥ u1 (g (s1 , s∗2 ), ζ ) + t1 (s1 , s∗2 ). This means u1 (a1 , ζ ) + λ1 ≥ u1 (a1 , ζ ) + |λ1 | + 1, which is a contradiction. The combination of (1)–(4) completes the proof of step 2. Step 3: For any ζ ∈ Θ , if (s∗1 , s∗2 ) ∈ NE (S , (g , t ), ζ ), then (g , t ) (s∗1 , s∗2 ) ∈ (F , p)(ζ ). By step 2, we have s∗1 = s∗2 = (θ , a, 0), a ∈ F (θ ), g (s∗1 , s∗2 ) = a, ti (s∗1 , s∗2 ) = pi (θ ) for each i ∈ {1, 2}. For any (x, λ1 , λ2 ) ∈ C × R2 , suppose that u1 (a, θ ) + p1 (θ ) ≥ u1 (x, θ) + λ1 . We need to show that u1 (a, ζ ) + p1 (θ ) ≥ u1 (x, ζ ) + λ1 . If θ = ζ , there is nothing to prove. So consider the case of θ ̸= ζ . If λ1 ̸= 0, let
(θ , x, λ1 ) (ζ , a, λ1 )
It follows immediately that u1 (a, ζ ) + p1 (θ ) ≥ u1 (x, ζ ) + λ1 since ε can be chosen arbitrarily close to zero. Similarly, u2 (a, θ ) + p2 (θ ) ≥ u2 (x, θ ) + λ2 for any (x, λ1 , λ2 ) ∈ C × R2 implies u2 (a, ζ ) + p2 (θ ) ≥ u2 (x, ζ ) + λ2 . By monotonicity of (F , p), (g , t )(s∗1 , s∗2 ) = (a, p1 (θ ), p2 (θ )) ∈ (F , p)(ζ ). References
if u1 (a2 , θ2 ) + p1 (θ2 ) ≥ u1 (a1 , θ2 ) + λ1 if u1 (a2 , θ2 ) + p1 (θ2 ) < u1 (a1 , θ2 ) + λ1
≥ u2 (a2 , ζ ) + λ′2 = u2 (g (s∗1 , s2 ), ζ ) + t2 (s∗1 , s2 ).
s1 =
if x ̸= a if x = a
if x ̸= a if x = a. ,
By rule (R5.3), we know that g (s1 , s∗2 ) = x and t1 (s1 s∗2 ) = λ1 . Therefore, (s∗1 , s∗2 ) ∈ NE (S , (g , t ), ζ ) implies
Abreu, D., Sen, A., 1991. Virtual implementation in Nash equilibrium. Econometrica 59, 997–1021. Amorós, P., 2004. Nash implementation and uncertain renegotiation. Games and Economic Behavior 49, 424–434. Barlo, M., Dalkiran, N.A., 2009. Epsilon-Nash implementation. Economics Letters 102, 36–38. Benoit, J-P., Ok, E.A., 2008. Nash implementation without no-veto power. Games and Economic Behavior 64, 51–67. Bikhchandani, S., Chatterji, S., Lavi, R., Mu’alem, A., Nisan, N., Sen, A., 2006. Weak monotonicity characterizes deterministic dominant-strategy implementation. Econometrica 74 (4), 1109–1132. Bochet, O., 2007. Nash implementation with lottery mechanisms. Social Choice and Welfare 28, 111–125. Clarke, E.H., 1971. Multipart pricing of public goods. Public Choice 2, 19–33. Gibbard, A., 1973. Manipulation of voting schemes: a general result. Econometrica 41, 587–602. Groves, T., 1973. Incentives in teams. Econometrica 41, 617–631. Jackson, M.O., 2001. A crash course in implementation theory. Social Choice and Welfare 18, 655–708. Jackson, M.O., Palfrey, T.R., 1992. Implementation in undominated strategies: a look at bounded mechanisms. Review of Economic Studies 59, 757–775. Jackson, M.O., Palfrey, T.R., 2001. Voluntary implementation. Journal of Economic Theory 98, 1–25. Maskin, E., 1999a. Nash equilibrium and welfare optimality. Review of Economic Studies 66, 23–38. Maskin, E., 1999b. Implementation and renegotiation. Review of Economic Studies 66, 39–56. Moore, J., Repullo, R., 1990. Nash implementation: a full characterization. Econometrica 58, 1083–1100. Muller, E., Satterthwaite, M., 1977. The equivalence of strong positive association and strategy proofness. Journal of Economic Theory 14, 412–418. Palfrey, T.R., Srivastava, S., 1991. Nash implementation using undominated strategies. 59, 475–501. Roberts, K., 1979. The characterization of implementable choice rules. In: Laffont, J.J. (Ed.), Aggregation and Revelation of Preferences. North-Holland. Sanver, M.R., 2006. Nash implementing non-monotonic social choice rules by awards. Economic Theory 28, 453–460. Sanver, I.O., Sanver, M.R., 2006. Nash implementation via hyperfunctions. Social Choice and Welfare 26, 607–623. Satterthwaite, M., 1975. Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10, 187–217. Vickrey, W., 1961. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance 16, 8–37.
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Implementation via mechanisms with transfers Jianxin Yi School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
article
info
Article history: Received 25 December 2008 Received in revised form 23 July 2010 Accepted 27 October 2010 Available online 2 November 2010 JEL classification: C72 D71 D78
abstract We consider the following problem: suppose that the social choice rule is given, but the planner has the opportunity to redistribute among agents some numeraire commodity (‘‘money’’) by compensatory transfers; can the planner find a transfer such that the social choice rule is Nash implementable by the transfer? In this paper, we establish a necessary and sufficient condition for Nash implementation by transfers. Furthermore, we construct a simplified mechanism, which Nash-implements the desired social choice rule by transfers. © 2010 Elsevier B.V. All rights reserved.
Keywords: Nash implementation Monotonicity Semi-monotonicity Transfers
1. Introduction Implementation theory attempts to answer the following important question. Consider an environment with a set of agents and a set of feasible outcomes, together with a set of possible states of the world. These agents must make a social choice from the set of feasible outcomes according to the states of the world. Then can a mechanism, or game form, be designed for the agents to play such that the equilibrium outcomes coincide with the ones specified by the social choice rule? In general, whether or not a social choice rule is implementable depends on the solution concept. Implementation theory has considered a variety of solution concepts, including equilibrium in dominant strategies, Bayesian equilibrium, and Nash equilibrium. Maskin (1999a) initiated the study of Nash implementation. He showed that a condition on social choice rules called monotonicity is necessary for Nash implementation. Furthermore, when there are at least three agents, any social choice rule which satisfies monotonicity and no veto power is Nash implementable. His work not only provides us with an understanding of what is implementable in Nash equilibrium, but also provides a blueprint for many techniques used in the literature.
E-mail address: [email protected]. 0165-4896/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2010.10.004
Maskin’s monotonicity is a fairly demanding condition, and many well-known social choice rules fail to be Nash implementable (see Examples 1–3). Example 1 (Plurality Rule. From Abreu and Sen, 1991). There are three outcomes, C = {a, b, c }, three agents, I = {1, 2, 3}, and two states, Θ = {θ , ζ }. Preferences are described as follows. u1 (·, θ ) u2 (·, θ ) u3 (·, θ )
a 3, 1, 1,
b 2, 3, 2,
c 1; 2; 3;
a 3, 1, 1,
b 2, 3, 3,
c 1; 2; 2;
u1 (·, ζ ) u2 (·, ζ ) u3 (·, ζ ).
Let F (θ ) = a and F (ζ ) = b. Then, F is not Nash implementable. Example 2 (Condorcet Rule. From Palfrey and Srivastava, 1991). There are three outcomes, C = {a, b, c }, five agents, I = {1, 2, 3, 4, 5}, and two states, Θ = {θ , ζ }. Preferences are described as follows. u1 (·, θ ) u2 (·, θ ) u3 (·, θ ) u4 (·, θ ) u5 (·, θ )
a 3, 3, 1, 1, 1,
b 2, 2, 3, 2, 2,
c 1; 1; 2; 3; 3;
a 3, 3, 2, 1, 1,
b 2, 2, 3, 2, 2,
c 2; 2; 3; 3; 3;
u1 (·, ζ ) u2 (·, ζ ) u3 (·, ζ ) u4 (·, ζ ) u5 (·, ζ ).
Let F (θ ) = b and F (ζ ) = c. Then, F is not Nash implementable.
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J. Yi / Mathematical Social Sciences 61 (2011) 65–70
Example 3 (Borda Rule. From Jackson, 2001). There are three outcomes, C = {a, b, c }, three agents, I = {1, 2, 3}, and two states, Θ = {θ, ζ }. Preferences are described as follows. u1 (·, θ) u2 (·, θ) u3 (·, θ)
a 3, 2, 1,
b 2, 1, 3,
c 1; 3; 2;
a 3, 2, 1,
b 2, 1, 2,
c 1; 3; 3;
u1 (·, ζ ) u2 (·, ζ ) u3 (·, ζ ).
Let F (θ ) = {a, b, c } and F (ζ ) = {c }. Then F is not Nash implementable. On the other hand, the Muller and Satterthwaite (1977) theorem shows the restrictiveness of monotonicity in the voting context. It states that if the outcomes are finite, agents are more than three, the set of admissible preferences includes all strict preferences, then any social choice function that is monotonic is necessarily dictatorial. For these reasons, much of Nash implementation theory has been developed and more positive results are obtained by changing the requirement of implementation. For example, Palfrey and Srivastava (1991) discuss the implementation in undominated Nash equilibrium. Jackson and Palfrey (1992) weakens the solution concept to undominated strategies, and proves that if strict value distinction is satisfied, then any social choice rule is implementable in undominated strategies. Abreu and Sen (1991) consider virtual implementation in Nash equilibrium. Bochet (2007) and Benoit and Ok (2008) consider the problem of Nash implementation with lottery mechanisms. Maskin (1999b), Jackson and Palfrey (2001) and Amorós (2004) examine Nash implementation via renegotiation. Sanver (2006) and Benoit and Ok (2008) discuss Nash implementation by awards. Other interesting works include Sanver and Sanver (2006) and Barlo and Dalkiran (2009). Recall the mechanism design problem under incomplete information. In this setting, the possible states of the world are the profiles of private type. Every agent can observe his type which is not known collectively to the other agents in the society. When agents may hold private information, implementation in dominant strategies (or incentive compatibility) is very important. An advantage of implementation in dominant strategies is that if the mechanism designer is an outsider, he does not need to know the probability density to successfully implement a social choice function. However, Gibbard (1973) and Satterthwaite (1975) show that for a very general class of problems there is no hope of implementing satisfactory social choice functions in dominant strategies. The Gibbard–Satterthwaite theorem is a quite negative result. It implies that transfer functions will be needed for dominant strategy implementation of non-dictatorial social choice functions in some settings. Vickrey (1961), Clarke (1971) and Groves (1973) prove that if the social choice function is efficient, then there exists a transfer function such that the social choice function is incentive compatible by the transfer. On the assumption of unrestricted domain, Roberts (1979) shows that a social choice function is incentive compatible by a transfer if and only if it satisfies PDA. Bikhchandani et al. (2006) establish a difference condition (WMon), which is necessary and sufficient for incentive compatibility of social choice function by transfers on order-based domains. Similarly, to the use of transfers to provide correct incentives mentioned above, we consider the following problems. Suppose that the social choice rule is given and is not Nash implementable, but the planner has the opportunity to redistribute among agents some particular commodity; then, can the planner find a transfer such that the social choice rule with the transfer is Nash implementable? If the answer is negative, what restrictions, if any, must be placed on the SCR? Furthermore, can we require that the transfer is budget balanced?
For convenience, we call a social choice rule T-Nash implementable, if we can find a transfer such that the social choice rule is Nash implementable by the transfer. In this paper, we establish a necessary and sufficient condition which characterize the social choice rules that are T-Nash implementable when there are two or more agents. Especially, a social choice rule must be T-Nash implementable if it is Nash implementable. But the converse is not true. Therefore we successfully implement a wider range of social choice rules. Moreover, for a social choice rule to be T-Nash implemented, we construct a simplified mechanism to implement it. Note that although, in many cases, we can find a transfer function such that the social choice function is incentive compatible by the transfer, but the transfer is not budget balanced. Thus, equilibrium outcomes in certain states will lead to a waste of money and a departure from ‘full’ efficiency. Here, it is shown that, if a social choice rule is T-Nash implementable, it can be T-Nash implementable by a budget-balanced transfer. Furthermore, we can see that T-Nash implementation and individual rationality are compatible. This paper is organized as follows. Section 2 provides definitions. Section 3 discusses the T-Nash implementation of social choice rules. Section 4 establishes two simplified mechanisms. The proofs are in Appendix. 2. Preliminaries Let C be the set of social alternatives or outcomes, I = {1, 2, . . . , n} is the set of agents with n ≥ 2, Θ is the set of possible states of the world. At state θ , each agent i ∈ I is assumed to have a utility function ui (·, θ ) over the set C . We shall refer to the collection (I , C , Θ , {ui }i∈I ) as an environment. A social choice rule (SCR) F is a mapping F : Θ → 2C ∼ {φ}, where F (θ ) is interpreted as the set of alternatives that are deemed socially optimal in state θ . A social choice function (SCF) is a singlevalued SCR, i.e., a function f Θ → C . F is said to be monotonic, if, for any θ , ζ ∈ Θ and any a ∈ F (θ ), we have a ∈ F (ζ ), whenever
{x ∈ C |ui (a, θ ) ≥ ui (x, θ )} ⊆ {x ∈ C |ui (a, ζ ) ≥ ui (x, ζ )} for all i ∈ I. F is said to be semi-monotonic, if, for any θ , ζ ∈ Θ and any a ∈ F (θ ), we have a ∈ F (ζ ), whenever ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . A mechanism in the environment (I , C , Θ , {ui }i∈I ) is a pair (S , g ), where S = S1 × S2 × · · · × Sn is a product of strategy spaces and g: S → C is an outcome function. Let si ∈ Si denote agent i’s strategy. A strategy profile is denoted as s = (s1 , s2 , . . . , sn ). For any s ∈ S, i ∈ I, let s−i = (s1 , · · · , si−1 , si+1 , . . . , sn ). A strategy profile s∗ ∈ S is a Nash equilibrium of mechanism (S , g ) at state θ if ui (g (s∗ ), θ ) ≥ ui (g (s∗−i , si ), θ ) for all i ∈ I and all si ∈ Si . The set of all Nash equilibria at state θ is denoted as NE ((S , g ), θ ). The mechanism (S , g ) Nash-implements F if and only if g (NE ((S , g ), θ )) = F (θ ) for all θ ∈ Θ . F is Nash implementable if there exists a mechanism (S , g ) which Nashimplements F . Given an environment (I , C , Θ , {ui }i∈I ), we consider a new environment (I , C × Rn , Θ , {vi }i∈I ), where vi : C × Rn × Θ → R takes a quasilinear form, i.e.,
vi (x, y1 , y2 , . . . , yn , θ ) = ui (x, θ ) + yi . A transfer is a function p : Θ → Rn . The transfer p is budget balanced if p1 (θ ) + · · · + pn (θ ) = 0 for all θ ∈ Ω .
J. Yi / Mathematical Social Sciences 61 (2011) 65–70
Given a SCR F : Θ → 2C ∼ {φ} and a transfer p: Θ → Rn , we define (F , p) to be a correspondence (F , p)(θ ) = {(a, p1 (θ ), p2 (θ), . . . , pn (θ )) ∈ C × Rn |a ∈ F (θ )} for all θ ∈ Θ . In particular, we rewrite (F , p) as (F , 0) if p(θ ) = (0, 0, . . . , 0) for all θ ∈ Θ . That is, (F , 0)(θ ) = {(a, 0, 0, . . . , 0) ∈ C × Rn |a ∈ F (θ )}. Given a mechanism (S , (g , t )) in the new environment (I , C × Rn , Θ , {vi }i∈I ), where S = S1 × S2 × · · · × Sn is a product of strategy spaces, g: S → C is an outcome function and t : S → Rn a transfer function. We say that a mechanism (S , (g , t )) satisfies the budgetbalanced condition if t1 (s) + · · · + tn (s) = 0 for all s ∈ S. Now, we can describe the primary question precisely for a given SCR F Θ → 2C ∼ {φ} which is not Nash implementable, does there exist a transfer p : Θ → Rn , such that (F , p) is Nash implementable in the environment (I , C × Rn , Θ , {vi }i∈I )? If the answer is affirmative, we call that F is T-Nash implementable and p is feasible with respect to F . 3. T-Nash implementation
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In many cases, it is impossible to implement a social choice rule which satisfies individual rationality constraints. However, it is in the use of transfers, we give a possible result to implement an individually rational SCR. For current purpose, we first present Theorem 2 which characterizes the feasible transfers with respect to a SCR. Theorem 2. Suppose that F is semi-monotonic, then, for any p : Θ → Rn , p is feasible with respect to F if and only if for any θ , ζ ∈ Θ , we have p(θ ) = p(ζ ) whenever either condition (a) or (b) below holds. (a) There is an a ∈ F (θ ), such that ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . (b) There is an a ∈ F (ζ ), such that ui (a, ζ ) − ui (x, ζ ) ≤ ui (a, θ ) − ui (x, θ ) for all i ∈ I and all x ∈ C . Let Mi = infx,θ ui (x, θ ) and ti (θ ) = supη {wi (η)|f (η) = f (θ )}. Moreover, suppose that ti (θ ) < +∞ for any θ ∈ Θ and Mi = infx,θ ui (x, θ ) > −∞.
In this section, we begin to explore the consequences of T-Nash implementation when there are two or more agents. It is known that the two-person problem is different from its many-person counterpart for Nash implementation. Nevertheless, we show that semi-monotonicity is a necessary and sufficient condition for TNash implementation of an n-person social choice rule with n ≥ 2. Then, we characterize completely the class of transfers which are feasible with respect to a given SCR.
Theorem 3. (1) Suppose that the social choice function f is semimonotonic. Let pi (θ ) = ti (θ ) − ui (f (θ ), θ ). Then p is a feasible transfer with respect to f and (f , p) satisfy the individual rationality constraints. (2) Suppose that the social choice rule F is semi-monotonic. Let pi (θ ) = ti (θ ) − Mi . Then, p is a feasible transfer with respect to F and (F , p) satisfy individual rationality constraints.
Theorem 1. Suppose that |I | ≥ 2. Then, there exists a transfer p : Θ → Rn , such that (F , p) is Nash implementable in the environment (I , C × Rn , Θ , {vi }i∈I ), i.e., F is T-Nash implementable if and only if F is semi-monotonic in the environment (I , C , Θ , {ui }i∈I ).
A natural question then is whether optimal transfers exist. In other words, if social choice rule F is semi-monotonic, is there a feasible transfer p, such that F and p satisfy following conditions. (1) (F , p) satisfy the individual ∑n rationality constraints. ∑n (2) p is optimal, that is | i=1 pi (θ )| ≤ | i=1 qi (θ )| for any n θ ∈ Θ and any feasible transfer q : Θ → R which satisfies the individual rationality constraints.
Examples 1–3 above give three social choice rules which are Nash implementable by transfers. In fact, we can check that any transfer p : Θ → Rn is feasible with respect to the Plurality rule F given in Example 1 and the Condorcet rule given in Example 2. Any transfer p : Θ → Rn is feasible with respect to the Borda rule given in Example 3 if p(θ ) = p(ζ ). By definition, we know that monotonicity implies almost monotonicity (Sanver, 2006) and almost monotonicity implies semi-monotonicity, so Nash implementable
⇒ Nash implementable by awards (Sanver, 2006; Benoit and Ok, 2008)
⇒ T-Nash implementable. But the converse is not true. Example 1 provides a counterexample. Up to this point, we have studied whether we can Nashimplement by transfers a social choice rule. But efficiency also requires that none of the numeraire be wasted, that is, the transfer function is budget balanced. If the transfer function is not balanced, there is either waste inside the system of n agents or source of financing from outside it. Fortunately, for a semi-monotonic F , it is possible to find a budget-balanced transfer p, such that (F , p) is Nash implementable. This point can be seen in the proof of Theorem 1. In many applications, agent’s participation in the mechanism is voluntary. As a result, the social choice rule must not only be implemented by a mechanism but also satisfy certain participation (or individual rationality) constraints if it is to be successfully implemented. Formally, Suppose that agent i ∈ I can receive a utility wi (θ ) by withdrawing from the mechanism when the social state is θ . Then, to ensure agent i’s participation, we must satisfy the individual rationality constraints ui (a, θ ) ≥ wi (θ ) for all θ ∈ Θ and all a ∈ F (θ ).
4. Simplified mechanism If (F , p) is monotonic, (F , p) is Nash implementable. Note that, the canonical mechanism which Nash-implements (F , p) uses a huge strategy space. For example, like Maskin’s mechanism (Maskin, 1999a), the agent i’s strategy space Si is of the form Θ × (C × Rn ) × N. According to the Moore and Repullo mechanism (Moore and Repullo, 1990), the agent i’s strategy space Si is more complicated. Now, we shall simplify the canonical mechanism. We propose a simple mechanism in which each agent announces a social state, a feasible social outcome, plus one number in any play. That is, Si = Θ × C × R for each agent i. Such mechanisms bring about a reduction in the communication complexity measured in dimension and has the advantage of informational efficiency of decentralized decision making. Although, semi-monotonicity is necessary and sufficient for the T-Nash implementation of n-person social choice rule with both n = 2 and n ≥ 3. There are also some interesting differences between n = 2 and n ≥ 3. For example, we can find a budgetbalanced mechanism to implement an n-person social choice rule (F , 0) with n ≥ 3. But, we do not know whether there is a budgetbalanced mechanism to implement a 2-person SCR (F , 0). 4.1. Three or more agents Consider the following mechanism (S , (g , t )), where Si = Θ × C × R for each i ∈ I and the outcome functions g : S → C and ti : S → R are defined by the following three rules. For a vector of strategies s = (s1 , s2 , . . . , sn ) ∈ S, R4.1 if si = (θ , c , m) for all i ∈ I and c ∈ F (θ ), then g (s) = c and ti (s) = pi (θ ) for all i ∈ I;
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R4.2 if there exists j ∈ I such that si = (θ , c , m) for all i ̸= j and c ∈ F (θ ), but sj = (θj , cj , mj ) ̸= (θ , c , m), then, (a) if uj (c , θ ) + pj (θ) ≥ uj (cj , θ ) + mj , set g (s) = cj ,
tj (s) = mj
and
ti (s) = −
mj n−1
for each i ̸= j;
ui (a, ζ ) + ti (s∗ ) < ui (g (s∗−i , si ), ζ ) + ti (s∗−i , si ).
(b) if uj (c , θ ) + pj (θ ) < uj (cj , θ ) + mj , set g ( s) = c ,
tj (s) = pj (θ ) and
ti (s) = pi (θ ) for each i ̸= j;
R4.3 in all other cases, let si = (θi , ci , mi ), then g (s) = ck ,
tk (s) = mk
and
ti (s) = −
mk n−1
for each i ̸= k,
where k is the lowest indexed i such that mi ≥ mj for all j ∈ I. Theorem 4. Suppose that |I | ≥ 3. Then (S , (g , t )) Nash-implements F if (F , p) is monotonic. Furthermore, (S , (g , t )) is budget balanced if p is budget balanced. 4.2. Two agents Consider the following mechanism (S , (g , t )), where S1 = S2 = Θ × C × R and the outcome functions g : S → C and ti : S → R are defined by the following rules. Given a s = (s1 , s2 ) ∈ S, where s1 = (θ1 , a1 , λ1 ) and s2 = (θ2 , a2 , λ2 ) (R5.1) if (θ1 , a1 ) = (θ2 , a2 ) = (θ , a) and a ∈ F (θ ), then
(g (s), t1 (s), t2 (s)) = (a, p1 (θ ), p2 (θ )); (R5.2) if (θ1 , a1 ) ̸= (θ2 , a2 ) and λ1 = λ2 = 0, then
(g (s), t1 (s), t2 (s)) = (a2 , p1 (θ2 ) − 1, u2 (a1 , θ1 ) + p2 (θ1 ) − u2 (a2 , θ1 ) − 1); (R5.3) if (θ1 , a1 ) ̸= (θ2 , a2 ), then
(a , λ , λ ) 1 1 1 if u1 (a2 , θ2 ) + p1 (θ2 ) ≥ u1 (a1 , θ2 ) + λ1 (g (s), t1 (s), t2 (s)) = (a2 , p1 (θ1 ), p2 (θ2 )) if u1 (a2 , θ2 ) + p1 (θ2 ) < u1 (a1 , θ2 ) + λ1 when λ1 ̸= 0, λ2 = 0 and
(a , λ , λ ) 2 2 2 if u2 (a1 , θ1 ) + p2 (θ1 ) ≥ u2 (a2 , θ1 ) + λ2 (g (s), t1 (s), t2 (s)) = (a1 , p1 (θ1 ), p2 (θ2 )) if u2 (a1 , θ1 ) + p2 (θ1 ) < u2 (a2 , θ1 ) + λ2 when λ1 = 0, λ2 ̸= 0; (R5.4) in all other cases, let
(g (s), t1 (s), t2 (s)) =
(a1 , λ1 , λ1 ) (a2 , λ2 , λ2 )
Let (S , (g , t )) T-Nash-implements F . Then, there is a vector of strategies s∗ = (s∗1 , s∗2 , . . . , s∗n ) ∈ NE (S , (g , t ), θ ) such that g (s∗ ) = a. It suffices to prove that s∗ ∈ NE (S , (g , t ), ζ ). Suppose that the conclusion does not hold. Then there is an i ∈ I and an si ∈ Si such that
if λ1 ≥ λ2 if λ1 < λ2 .
Theorem 5. Let |I | = 2. If (F , p) is monotonic, (S , (g , t )) Nashimplements F . Acknowledgements I am grateful to two anonymous referees and an associate editor for their helpful comments and suggestions. Appendix Proof of Theorem 1. The necessity part is simple. For any θ , ζ ∈ Θ and any a ∈ F (θ ), suppose that, ui (a, θ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C .
Note that s∗ ∈ NE (S , (g , t ), θ ) implies that ui (a, θ ) + ti (s∗ ) ≥ ui (g (s∗−i , si ), θ ) + ti (s∗−i , si ). Therefore, we have ui (a, θ ) − ui (g (s∗−i , si ), θ ) > ui (a, ζ ) − ui (g (s∗−i , si ), ζ ) which is a contradiction. The sufficiency part will be obtained as a corollary following Lemmas 1 and 2. Lemma 1. If F is semi-monotonic then (F , 0) is monotonic in the environment (I , C × Rn , Θ , {vi }i∈I ). For any θ , ζ ∈ Θ and any (a, 0, . . . , 0) ∈ (F , 0)(θ ), suppose that
{(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, θ ) + 0 ≥ ui (x, θ ) + yi } ⊆ {(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, ζ ) + 0 ≥ ui (x, ζ ) + yi } for all i ∈ I. If ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) is not true for some j ∈ I and some x¯ ∈ C , i.e., uj (a, θ ) − uj (¯x, θ ) > uj (a, ζ ) − uj (¯x, ζ ). Take (z1 , . . . , zn ) ∈ Rn such that uj (a, θ ) − uj (¯x, θ ) > zj − 0 > uj (a, ζ ) − uj (¯x, ζ ). Then
(¯x, z1 , . . . , zn ) ∈ {(x, y1 , . . . , yn ) ∈ C × Rn |uj (a, θ ) + 0 ≥ uj (x, θ ) + yj } whereas
(x0 , z1 , . . . , zn ) ̸∈ {(x, y1 , . . . , yn ) ∈ C × Rn |uj (a, ζ ) + 0 ≥ uj (x, ζ ) + yj }, which is a contradiction. Therefore, a ∈ F (ζ ), which implies (a, 0, . . . , 0) ∈ (F , 0)(ζ ). Lemma 2. Suppose that |I | ≥ 2, (F , p) is Nash implementable iff (F , p) is monotonic in the environment (I , C × Rn , Θ , {vi }i∈I ). Note that, when |I | ≥ 3, the result follows immediately from Maskin’s theorem, since (F , p) automatically satisfies no veto power in the environment (I , C × Rn , Θ , {vi }i∈I ). So we only verify it when |I | = 2. Take B = C × Rn and Ci (a, θ ) = {(x, y1 , y2 ) ∈ C × R2 |ui (a, θ ) + pi (θ ) ≥ ui (x, θ ) + yi }. Moreover, for any a, θ , b, η, let e(a, θ , b, η) = (a, p1 (θ ) − 1, u2 (b, η) − u2 (a, η) + p2 (η) − 1). Then, e(a, θ , b, η) ∈ C1 (a, θ ) ∩ C2 (b, η). Note that there is no ζ , such that
{(x, z1 , z2 ) ∈ C × R2 |u1 (a, θ ) + p1 (θ ) ≥ u1 (x, θ ) + z1 } ⊆ {(x, z1 , z2 ) ∈ C × R2 |u1 (a, ζ ) + y1 ≥ u1 (x, ζ ) + z1 } and
{(x, z1 , z2 ) ∈ C × R2 |u2 (b, η) + p2 (η) ≥ u2 (x, η) + z2 } ⊆ {(x, z1 , z2 ) ∈ C × R2 |u2 (a, ζ ) + y2 ≥ u2 (x, ζ ) + z2 }. It is not difficult to check that B, Ci (a, θ ) and e(a, θ , b, η) satisfy Condition µ2 (Moore and Repullo, 1990). Thus (F , p) is Nash implementable.
J. Yi / Mathematical Social Sciences 61 (2011) 65–70
Proof of Theorem 2. Sufficiency: it suffices to prove that (F , p) is monotonic. Suppose that for any θ , ζ ∈ Θ , and any a ∈ F (θ ),
{(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, θ ) + pi (θ ) ≥ ui (x, θ ) + yi } ⊆ {(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, ζ ) + pi (θ ) ≥ ui (x, ζ ) + yi } for all i ∈ I. It is easy to check that ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . Therefore (F , p)(θ ) = (F , p)(ζ ). Necessity: Suppose that, for any θ , ζ ∈ Θ , there is an a ∈ F (θ ), such that ui (a, θ ) − ui (x, θ ) ≤ ui (a, ζ ) − ui (x, ζ ) for all i ∈ I and all x ∈ C . Then
{(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, θ ) + pi (θ ) ≥ ui (x, θ ) + yi } ⊆ {(x, y1 , . . . , yn ) ∈ C × Rn |ui (a, ζ ) + pi (θ ) ≥ ui (x, ζ ) + yi } Note that (F , p) is monotonic, it follows that a ∈ F (ζ ) and p(θ) = p(ζ ). Therefore, p is feasible with respect to F . Proof of Theorem 3. Theorem 3 will be obtained as a corollary of Theorem 2.
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If s∗ falls into (R4.3), let s∗i = (θi , ci , mi ) for each i ∈ I. Then g (s∗ ) = ck , where k is the lowest indexed i such that mi ≥ mj for all j ∈ I. Note that uk (g (s∗ ), θ ) + tk (s∗ ) = uk (ck , θ ) + mk
< uk (ck , θ ) + l = uk (g (s∗−k , sk ), θ ) + tk (s∗−k , sk ) for any sk = (θk , ck , l) with l > mk . Thus, by announcing a high enough number l > mk , agent k can profit from the deviation from s∗k . This contradicts the assumption that s∗ is a Nash equilibrium. Finally, suppose s∗1 = s∗2 = · · · = s∗n = (ζ , c , m) and c ∈ F (ζ ). For each i ∈ I and each (x, λ1 , λ2 , . . . , λn ) ∈ C × Rn , suppose that ui (c , ζ ) + pi (ζ ) ≥ ui (x, ζ ) + λi . Let si = (η, x, λi ) ̸= (ζ , c , m). By rule (R4.2), we have g (s∗−i , si ) = x, ti (s∗−i , si ) = λi . Note that s∗ ∈ NE (S , (g , t ), θ ) implies ui (g (s∗ ), θ ) + ti (s∗ ) ≥ ui (g (s∗−i , si ), θ ) + ti (s∗−i , si ). Thus
Proof of Theorem 4. First, let us check that
ui (c , θ ) + pi (ζ ) ≥ ui (x, θ ) + λi .
(F (θ), p1 (θ ), . . . , pn (θ )) ⊆ (g , t )(NE (S , (g , t ), θ )) for any θ ∈ Θ .
By hypothesis, it follows that
For any c ∈ F (θ ), let s∗i = (θ , c , 1) for all i ∈ I. Then g (s∗−i , s∗i ) = c and ti (s∗−i , s∗i ) = pi (θ ) for all i ∈ I. For any j ∈ I and any sj = (θj , cj , mj ) ̸= s∗j , if uj (c , θ ) + pj (θ ) ≥ uj (cj , θ ) + mj , then
So, (S , (g , t )) Nash-implements F .
(g , t )(s∗ ) = (c , p1 (ζ ), . . . , pn (ζ )) ∈ (F , p)(θ ).
uj (g (s∗−j , s∗j ), θ ) + tj (s∗−j , s∗j ) = uj (c , θ ) + pj (θ )
≥ uj (cj , θ ) + mj ≥ uj (g (s∗−j , sj ), θ ) + tj (s∗−j , sj ). If uj (c , θ ) + pj (θ ) < uj (cj , θ ) + mj , then uj (g (s∗−j , s∗j ), θ ) + tj (s∗−j , s∗j ) = uj (c , θ ) + pj (θ )
= uj (g (s∗−j , sj ), θ ) + tj (s∗−j , sj ). Thus, (s∗−j , s∗j ) ∈ NE (S , (g , t ), θ ) and (c , p1 (θ ), . . . , pn (θ )) ∈ (g , t ) (NE (S , (g , t ), θ )). Next, let us check that
(g , t )(NE (S , (g , t ), θ )) ⊆ (F (θ ), p1 (θ ), . . . , pn (θ )) for each θ ∈ Θ . Let s∗ = (s∗1 , s∗2 , . . . , s∗n ) ∈ NE (S , (g , t ), θ ). We must show that s∗ falls into (R4.1). If not, then either rule (R4.2) or rule (R4.3) applies to (s∗1 , s∗2 , . . . , s∗n ). If s∗ = (s∗1 , s∗2 , . . . , s∗n ) falls into (R4.2), without loss of generality, we can assume that,
Proof of Theorem 5. We need to show that (F (θ ), p1 (θ ), p2 (θ )) = (g , t )(NE (S , (g , t ), θ )) for any θ ∈ Θ . The proof proceeds in three steps. Step 1: (F (θ ), p1 (θ ), p2 (θ )) ⊆ (g , t )(NE (S , (g , t ), θ )) for any θ ∈ Θ. For any a ∈ F (θ ), let s∗1 = s∗2 = (θ , a, 0), then g (s∗1 , s∗2 ) = a, t1 (s∗1 , s∗2 ) = p1 (θ ) and t2 (s∗1 , s∗2 ) = p2 (θ ). Consider any s1 = (θ1 , a1 , λ1 ). If (θ , a) ̸= (θ1 , a1 ), then when λ1 ̸= 0, u1 (g (s1 , s∗2 ), θ ) + t1 (s1 , s∗2 ) u1 (a1 , θ ) + λ1 u1 (a, θ ) + p1 (θ )
=
if u1 (a, θ ) + p1 (θ ) ≥ u1 (a1 , θ ) + λ1 if u1 (a, θ ) + p1 (θ ) < u1 (a1 , θ ) + λ1
≤ u1 (a, θ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), θ ) + t1 (s∗1 , s∗2 ) when λ1 = 0, u1 (g (s1 , s∗2 ), θ ) + t1 (s1 , s∗2 ) = u1 (a, θ ) + p1 (θ ) − 1
≤ u1 (a, θ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), θ ) + t1 (s∗1 , s∗2 ). Similarly, we can verify that u2 (g (s∗1 , s∗2 ), θ ) + p2 (θ ) ≥ u2 (g (s1 , s2 ), θ ) + t (s∗1 , s2 ) for any s2 = (θ2 , a2 , λ2 ). Therefore, (s∗1 , s∗2 ) ∈ NE (S , (g , t ), θ ) and ∗
s∗1 = s∗2 = · · · = s∗n−1 = (ζ , c , m) ̸= s∗n = (ζn , cn , mn )
and
c ∈ F (ζ ). By rule (R4.2), we have
mn u (c , θ ) − 1 n n−1 if un (c , ζ ) + pn (ζ ) ≥ un (cn , ζ ) + mn u1 (g (s∗ ), θ ) + t1 (s∗ ) = u1 (c , θ ) + p1 (ζ ) if un (c , ζ ) + pn (ζ ) < un (cn , ζ ) + mn . Let s1 = (ζ , c ′ , l) where l > max{m, mn }, then by (R4.3), we have u1 (g (s∗−1 , s1 ), θ ) + t1 (s∗−1 , s1 ) = u1 (c ′ , θ ) + l. Therefore, by announcing a high enough number l, agent 1 profits from the deviation from s∗1 , thereby contradicting the assumption that s∗ is a Nash equilibrium.
(a, p1 (θ ), p2 (θ )) = (g , t )(s∗1 , s∗2 ) ∈ (g , t )(NE (S , (g , t ), θ )). Step 2: For any ζ ∈ Θ , if (s∗1 , s∗2 ) ∈ NE (S , (g , t ), ζ ), where s∗1 = (θ1 , a1 , λ1 ) and s∗2 = (θ2 , a2 , λ2 ), then (θ1 , a1 ) = (θ2 , a2 ), a1 ∈ F (θ1 ) and λ1 = λ2 = 0. To see this, suppose the contrary, that is, following four cases applies to (s∗1 , s∗2 ). Case 1: (θ1 , a1 ) = (θ2 , a2 ) = (θ , a), a ∈ F (θ ) and λi ̸= 0 for some i ∈ {1, 2}. Suppose, without loss of generality, that λ2 ̸= 0. Let s1 = (θ1′ , a′1 , λ′1 ) and (θ1′ , a′1 ) ̸= (θ2 , a2 ), λ′1 ̸= 0, λ′1 > λ2 . Note that, g (s∗1 , s∗2 ) = a, t1 (s∗1 , s∗2 ) = p1 (θ ) and g (s1 , s∗2 ) = a′1 and t1 (s1 , s∗2 ) = λ′1 . Since u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 ) ≥ u1 (g (s1 , s∗2 ), ζ ) + t (s1 , s∗2 ),
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u1 (a, ζ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 )
we have
≥ u1 (g (s1 , s∗2 ), ζ ) + t1 (s1 , s∗2 ) = u1 (x, ζ ) + λ1 .
u1 (a, ζ ) + p1 (θ ) ≥ u1 (a′1 , ζ ) + λ′1 . Note that, this is impossible because u1 (a′1 , ζ ) + λ′1 can be arbitrarily large by choosing λ′1 large enough. Case 2: (R5.2) applies to (s∗1 , s∗2 ), that is (θ1 , a1 ) ̸= (θ2 , a2 ) and λ1 = λ2 = 0. Let s2 = (θ2 , a2 , λ′2 ), where λ′2 ̸= 0 and u2 (a1 , θ1 ) + p2 (θ1 ) ≥ u2 (a2 , θ1 ) + λ′2 > u2 (a1 , θ1 ) + p2 (θ1 ) − 1. Then
If λ1 = 0, let
s1 =
(θ , x, −ε) (ζ , a, −ε)
for any ε > 0. By rule (R5.3), we have g (s1 , s∗2 ) = x and t1 (s1 , s∗2 ) = −ε . By assumption, we have u1 (a, ζ ) + p1 (θ ) = u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 )
≥ u1 (g (s1 , s∗2 ), ζ ) + t1 (s1 , s∗2 ) = u1 (x, ζ ) + λ1 − ε.
u2 (g (s∗1 , s∗2 ), ζ ) + t2 (s∗1 , s∗2 )
= u2 (a2 , ζ ) + u2 (a1 , θ1 ) + p2 (θ1 ) − u2 (a2 , θ1 ) − 1 < u2 (a2 , ζ ) + λ′2 = u2 (g (s1 , s∗2 ), ζ ) + t2 (s1 , s∗2 ). This is a contradiction. Case 3: (R5.3) applies to (s∗1 , s∗2 ). Note that, (θ1 , a1 ) ̸= (θ2 , a2 ) and, suppose, without loss of generality, that λ1 ̸= 0 and λ2 = 0. Let s2 = (θ2 , a2 , λ′2 ) such that λ′2 ̸= 0, λ1 < λ′2 . Then u2 (g (s∗1 , s∗2 ), ζ ) + t2 (s∗1 , s∗2 )
=
u2 (a1 , ζ ) + λ1 u2 (a2 , ζ ) + p2 (θ2 )
Note that the right-hand side of the above expression can be arbitrarily large by choosing λ′2 large enough. But the left-hand side is bounded. This is a contradiction. Case 4: (R5.4) applies to (s∗1 , s∗2 ). Suppose, without loss of generality, that λ1 ≥ λ2 . Then g (s∗1 , s∗2 ) = a1 and t1 (s∗1 , s∗2 ) = λ1 . Choose s1 = (θ1 , a1 , |λ1 | + 1), then , g (s1 , s∗2 ) = a1 and t1 (s1 s∗2 ) = |λ1 | + 1. Note that, u1 (g (s∗1 , s∗2 ), ζ ) + t1 (s∗1 , s∗2 ) ≥ u1 (g (s1 , s∗2 ), ζ ) + t1 (s1 , s∗2 ). This means u1 (a1 , ζ ) + λ1 ≥ u1 (a1 , ζ ) + |λ1 | + 1, which is a contradiction. The combination of (1)–(4) completes the proof of step 2. Step 3: For any ζ ∈ Θ , if (s∗1 , s∗2 ) ∈ NE (S , (g , t ), ζ ), then (g , t ) (s∗1 , s∗2 ) ∈ (F , p)(ζ ). By step 2, we have s∗1 = s∗2 = (θ , a, 0), a ∈ F (θ ), g (s∗1 , s∗2 ) = a, ti (s∗1 , s∗2 ) = pi (θ ) for each i ∈ {1, 2}. For any (x, λ1 , λ2 ) ∈ C × R2 , suppose that u1 (a, θ ) + p1 (θ ) ≥ u1 (x, θ) + λ1 . We need to show that u1 (a, ζ ) + p1 (θ ) ≥ u1 (x, ζ ) + λ1 . If θ = ζ , there is nothing to prove. So consider the case of θ ̸= ζ . If λ1 ̸= 0, let
(θ , x, λ1 ) (ζ , a, λ1 )
It follows immediately that u1 (a, ζ ) + p1 (θ ) ≥ u1 (x, ζ ) + λ1 since ε can be chosen arbitrarily close to zero. Similarly, u2 (a, θ ) + p2 (θ ) ≥ u2 (x, θ ) + λ2 for any (x, λ1 , λ2 ) ∈ C × R2 implies u2 (a, ζ ) + p2 (θ ) ≥ u2 (x, ζ ) + λ2 . By monotonicity of (F , p), (g , t )(s∗1 , s∗2 ) = (a, p1 (θ ), p2 (θ )) ∈ (F , p)(ζ ). References
if u1 (a2 , θ2 ) + p1 (θ2 ) ≥ u1 (a1 , θ2 ) + λ1 if u1 (a2 , θ2 ) + p1 (θ2 ) < u1 (a1 , θ2 ) + λ1
≥ u2 (a2 , ζ ) + λ′2 = u2 (g (s∗1 , s2 ), ζ ) + t2 (s∗1 , s2 ).
s1 =
if x ̸= a if x = a
if x ̸= a if x = a. ,
By rule (R5.3), we know that g (s1 , s∗2 ) = x and t1 (s1 s∗2 ) = λ1 . Therefore, (s∗1 , s∗2 ) ∈ NE (S , (g , t ), ζ ) implies
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