Worst-case mechanism design with undominated strategies

Transcription

1 Worst-case mechanism design with undominated strategies Takuro Yamashita April 26, 2009 Abstract We consider a way to evaluate mechanisms without assuming mutual knowledge of rationality among the agents. More specifically, we assume that each agent can take any undominated strategy under a mechanism, and the mechanism is evaluated by its worst-case scenario. First, we provide some characteristics of a mechanism that implements a strongly monotonic social choice correspondence in an environment with single-crossing preferences. Second, we show that if the mechanism designer is interested in implementation of individually rational outcome mappings in a random valuation model, any mechanism that is not dominant-strategy incentive compatible (DSIC) is outperformed by a DSIC mechanism. Key words: mechanism design, implementation I am grateful to Ilya Segal, Matthew O. Jackson, Paul Milgrom, Jon Levin, Koichi Tadenuma, and seminar participants at Stanford University, Hitotsubashi University, and Yokohama National University. Stanford University. takuroy@stanford.edu 1

2 1 Introduction It is a central problem in the theory of mechanism design and implementation how to construct a mechanism that performs well, given some assumptions for agents behaviors. Among several standard behavioral assumptions, dominant-strategy incentive compatibility is one of the most demanding, and therefore one of the most desirable property for a mechanism to satisfy. However, the dominant-strategy incentive compatibility can be so restrictive in the sense that often times, an outcome that is achievable by a dominant-strategy incentive compatible mechanism (a DSIC mechanism) cannot satisfy other economically desirable properties. For example, in a voting model with three or more alternatives and unrestricted domain of preference for each agent, any DSIC mechanism that is onto is dictatorial (Gibbard (1973) and Satterthwaite (1975)). In a two-sided matching model, there is no DSIC mechanism that always induces a stable matching (Roth (1982)). Given these impossibility theorems, many papers in the literature have evaluated mechanisms using much weaker behavioral assumptions. One of the most standard behavioral assumption is to assume common knowledge of rationality among the agents. Then, depending on a situation, either Nash, Bayesian, or Ex post incentive compatibility becomes a relevant criterion to evaluate a mechanism, rather than the dominant-strategy incentive compatibility. Although the assumption of common knowledge of rationality has been quite useful to enlarge the set of mechanisms under which we can predict what outcomes would be realized, there would be some situations in which the mechanism designer cannot be sure that the agents commonly know that they are rational. Also, under a private valuation setting, even if we assume common knowledge of rationality, if the mechanism designer is not sure as to what kind of beliefs each agent has about the other agents preferences, then the relevant incentive compatibility that a mechanism needs to satisfy (in this case, Ex post incentive compatibility is the relevant one) becomes equivalent to the dominant-strategy incentive compatibility. 2

3 In this paper, we consider a way to evaluate a mechanism without assuming any mutual knowledge of rationality among the agents. More specifically, our behavioral assumption is that each agent can take any undominated strategy. The problem of this approach is that often times, a mechanism has multiple undominated strategies for each agent. In that case, we assume that the mechanism designer would evaluate the mechanism according to its worst-case scenario. The pioneering paper as to this approach is Jackson (1992a). Jackson (1992a) has shown that if we are allowed to use unbounded mechanisms, then basically any social choice correspondence can be implemented, and therefore, we need to focus on bounded mechanisms. A mechanism is said to be bounded if for any agent, that a strategy of the agent is (weakly) dominated implies that there is an undominated strategy that (weakly) dominates the strategy. Specifically, any finite mechanism is bounded. Then in a general implementation problem, Jackson (1992a) has provided a strategyresistance condition as a necessary condition for an implementable social choice correspondence. 1 In this paper, we focus on much restricted environments. First, in an environment with single-crossing preferences, we show that if there is a finite mechanism that implements a strongly monotonic social choice correspondence, then one of such a mechanism is a sort of a revelation mechanism, in the sense that there is an upper bound on the size of the message space for each agent, where the upper bound is a function of the size of the type space. 2 Second, we show that if the mechanism designer is interested in implementation of individually rational outcome mappings in a random valuation model, any mechanism that is not dominant-strategy incentive compatible 1 See also Miyamoto, Watanabe, and Mizuno (1990) and Börgers (1991). 2 In case of a standard revelation mechanism, the size of the message space for an agent is exactly the same as the size of the type space of the agent. The mechanism we analyze in this paper is not a revelation mechanism, but we show that its message space is upper bounded by the size of the type space of an agent. 3

4 (DSIC) is outperformed by a DSIC mechanism. We interpret this second result as a negative result, because even if we adopt our behavioral assumption that is much weaker than dominant-strategy incentive compatibility, we can do no better than what we can do with DSIC mechanisms. The rest of the paper proceeds as follows. In Section 2, we introduce a model in which an agent has a strict preference over all possible alternatives, and in Section 3, we define how the mechanism designer evaluates a mechanism. In Section 4, we show some preliminary results, and in Section 5, we provide some properties of a mechanism that implements a strongly monotonic social choice correspondence in an environment with single-crossing preferences. In Section 6, we show that our main results obtained under the assumption of strict preferences can generalize even if we allow some indifference in agents preferences, but argue that we may need some modification in the model. In Section 7, we show that any non-dsic mechanism is outperformed by some DSIC mechanism in a random-valuation model if the mechanism designer is interested in implementation of individually rational outcome mappings. Section 8 is the conclusion. 2 A model with strict preferences There are N agents, indexed by i = 1,..., N, and there is a finite set of alternatives, X. Each agent has a strict preference ordering, P i, on X, so that xp i y means that agent i prefers x to y for each x, y X. Let xr i y represent either xp i y or x = y. Let P i be the set of the possible preference orderings that can be possessed by agent i, and denote P = i P i. We assume that P i is privately known by agent i, and hence this is a private valuation model. A (finite) mechanism is denoted by (M, g), where M = i M i, each M i is nonempty and finite, and g : M X. M i is called a message set for agent i, and g is called a game form. We define the set of messages that can be chosen by an agent with pref- 4

5 erence P i as follows: B i (P i M, g) = {m i M i m i M i ; m i M i, g(m i, m i )R i g(m i, m i ) m i M i, g(m i, m i)p i g(m i, m i)}. That is, if m i / B i (P i M, g), then there is some message that (weakly) dominates m i in view of P i. Since a finite mechanism is bounded in the sense of Jackson (1992a), if m i / B i (P i M, g), then there is a message in B i (P i M, g) that dominates m i in view of P i. Unless confusing, we may write B i (P i M, g) as B i (P i ). 3 Let σ i : P i M i denote agent i s pure strategy under (M, g), and let Σ UD i be the set of all undominated strategies under (M, g) (i.e., for all σ i Σ UD i and all P i, σ i (P i ) B i (P i )). For each profile of undominated strategies σ = (σ 1,..., σ N ) Σ UD 1..., Σ UD N, an outcome mapping g(σ) : P X is induced, where for all P P, g(σ(p )) = g(σ 1 (P 1 ),..., σ N (P N )) X. The set of all possible outcome mappings attained in this mechanism is given as follows. A(M, g) = {g(σ) σ Σ UD 1..., Σ UD N }. Notice that for any (finite) mechanism (M, g), A(M, g) is nonempty. 3 Worst-case evaluation of a mechanism We assume that the mechanism designer has an evaluation function φ over outcome mappings so that any outcome mapping α is evaluated as φ(α) R. Then, given an evaluation function φ, the mechanism designer evaluates a mechanism (M, g) according to its worst-case outcome mapping, i.e., φ (M, g) min α A(M,g) φ(α). Thus, if there is an optimal mechanism in 3 Notice that in this specification, we do not allow any mixed choice of messages and dominance argument using them, because we only allow ordinal preferences. If we introduce preferences on lotteries according to expected utility specification, we can potentially obtain a smaller set of undominated messages. 5

6 some problem, such a mechanism may be interpreted as a mechanism that gives the best guarantee in terms of φ. Notice that for any φ, if there are two mechanisms (M, g) and (M, g ) that satisfy A(M, g) A(M, g ), then we have min α A(M,g) φ(α) min α A(M,g ) φ(α), and therefore, φ (M, g) φ (M, g ). This specification admits several standard evaluation function. For example, if the mechanism designer has her own utility function w(x P ) and a prior belief µ(p ) for each preference profile (but she is unsure whether it is a common belief among the agent), then we may define φ(α) as her expected utility: φ(α) = w(α(p ) P )µ(p ). P P As another example, if the mechanism designer has a reference point (e.g., the first-best social surplus) and worries about the guaranteed relative performance of α comparing to the reference point, we may define φ(α) = min P P w(α(p ) P ) w(α (P ) P ), where α is the reference outcome mapping. Finally, a (weak) implementation problem is also an incident of this approach. If the mechanism designer has a social choice correspondence F : P 2 X, then we may define { 1 if P ; α(p ) F (P ) φ(α) = 0 otherwise In this case, φ (M, g) = 1 if and only if (M, g) (weakly) implements F, i.e., for all α A(M, g) and all P P, we have α(p ) F (P ). 4 A preliminary result We say that a mechanism (M, g) has a nuisance message if there is i and m i M i that is not in B i (P i M, g) for any P i. A mechanism (M, g) is called 6

7 a mechanism without nuisance messages if there is no nuisance message. The following result shows that in looking for an optimal mechanism, we can focus on a mechanism without nuisance messages. 4 Proposition 1. For an arbitrary mechanism (M, g ), there is a mechanism (M, g) without nuisance messages, and that satisfies A(M, g) A(M, g ). To show this result, we first show the following lemma, which is also useful to prove the other results presented later. Lemma 1. Suppose that in a mechanism (M, g ), m i B i (P i M, g ) dominates m i in view of P i (and so m i / B i (P i M, g )). Let (M, g) be an arbitrary mechanism where M j M j for each j (including i) and m i, m i M i, and g is the restriction of g on M. Then,we have either (i) m i B i (P i M, g) and m i / B i (P i M, g), or (ii) if m i B i (P i M, g), then m i and m i are equivalent, i.e., g(m i, m i ) = g(m i, m i ) for any m i M i. Proof. Since m i dominates m i under (M, g ), in view of P i, we have g(m i, m i )R i g(m i, m i ) for any m i M i, and for some m i, g(m i, m i)p i g(m i, m i). Let M i be the set of all m i such that g(m i, m i)p i g(m i, m i), and hence, for any m i / M i, we have g(m i, m i ) = g(m i, m i ). Suppose that for m i, m i, (i) does not hold under mechanism (M, g). It means that any message m i is excluded from M i. Thus, for any m i M i, we have g(m i, m i ) = g(m i, m i ). Proof. (Proof of Proposition 1) Define M 0 i = M i for each i, and g 0 = g. Then for each k = 1,..., define a mechanism (M k, g k ) so that for each i, Mi k = {m i M k 1 i P i ; m i B i (P i M k 1, g k 1 )}, and that g k is a restriction of g on M k = i M i k, i.e., g k (m) = g (m) for all m M k. Because each M i is finite, there is K < such that Mi K = M K+1 i =... for any i. Define M i = Mi K, and define g as a restriction of g on M = i M i. 4 If we allow indifference in preference, then this property does not necessarily hold. See Dasgupta, Hammond, and Maskin (1979) and Jackson (1992a) for this issue. 7

8 First, under mechanism (M, g), we show that there is no nuisance message. Suppose not. Then there is i and m i M i such that for all P i, m i / B i (P i M, g). Then m i / M K+1 i, and hence M K+1 i is a strict subset of, a contradiction. M K i Second, we show that A(M, g) A(M, g ). Suppose not. Then there is k such that A(M k, g k ) is not a subset of A(M k 1, g k 1 ). It can occur only if there is agent i, P i and m i M k i that satisfies the following: (i) m i / B i (P i M k 1, g k 1 ), (ii) m i B i (P i M k, g k ), and (iii) for any m i B i (P i M k, g k ), there is m i M k i such that g(m i, m i ) g(m i, m i ). However, if such a m i exists, then Lemma 1 induces that there must be m i B i (P i M k 1, g k 1 ) such that under mechanism (M k, g k ), m i is equivalent to m i. Since m i B i (P i M k, g k ), we have to have m i B i (P i M k, g k ), and thus, (iii) is violated, which is a contradiction. In the rest of the paper, a mechanism always refers to a mechanism without nuisance messages. Notice that a DSIC revelation mechanism is a mechanism without nuisance messages such that B i (P i ) is singleton for all i and P i. 5 An optimal mechanism in a restricted environment In this section, we obtain an optimal mechanism in some restricted environment. More specifically, we impose three assumptions on the environment. First, we assume that the mechanism designer is interested in implementation of strongly monotonic social choice correspondence, where a social choice correspondence F is said to be strongly monotonic if for all P i, P i, P i, all x F (P i, P i ) such that L i (x, P i ) L i (x, P i ), and all y F (P i, P i ), we have yr ix (i.e., either x = y or yp i x). Our assumption is that φ (M, g) = 1 if and only if (M, g) implements F. The other two assumptions are imposed on the preference set of each 8

9 agent. Our second assumption is that the preference set of every agent i is single-crossing. We say that P i is single-crossing if there is a linear ordering on P i, say Pi 1,..., P T i i, such that if for some s < t and x, y X, we have xpi s y and ypi t x, then for any s < s and t > t, we have xpi s y and ypi t x. The third assumption is that the preference set for each agent i is connected in the following sense: P i is said to be connected if for any P i, P i P i and any x, y X such that xp i y and yp i x, there is P i P i such that yp i x and [wp i z wp i z]. For example, if X = {a, b, c}, then P i = {abc, bac, bca, cba} satisfies both of the conditions, but P i = {abc, cba} is not connected, and P i = {abc, bac, bca, cba, cab} is not single-crossing. In this environment, we can show a sort of a revelation principle. Theorem 1. For an arbitrary mechanism (M, g ) that implements F, there is a mechanism (M, g) such that (i) A(M, g) A(M, g ) (and so (M, g) implements F ), (ii) B i (P i M, g) 2 for each i and P i, and (iii) B i (P 1 i M, g) = B i (P T i i M, g) = 1. To prove the theorem, we first show some lemmas. In the following, let (M, g ) be a mechanism that implements F, and let B i (P i ) represent B i (P i M, g ) to simplify the notation. Lemma 2. For each i, either (i) there is m i such that m i B i (P 1 i ) and m i B i (P 2 i ), or (ii) there are m i B i (P 1 i ) and m i B i (P 2 i ) such that m i dominates m i in view of P 1 i and m i dominates m i in view of P 2 i. Proof. We show that if both (i) and (ii) do not hold, then we obtain a contradiction. If (i) does not hold, then for any m 1 1 B i (P 1 i ), there is m 1 2 B i (P 2 i ) that dominates m 1 1 in view of P 2 i. If (ii) does not hold either, then m 1 1 cannot dominate m 1 2 in view of P 1 i, and thus, there is m 2 1 B i (P 1 i ) that dominates m 1 2 in view of P 1 i (because m 1 2 / B i (P 1 i ) if (i) does not hold). By the same reasoning, we find two sequences of infinitely many messages m 1 1, m 2 1,..., and m 1 2, m 2 2,..., such that (i) m k 2 dominates m k 1 in view of P 2 i, 9

10 (ii) m k 1 does not dominate m k 2 in view of Pi 1, (iii) m k+1 1 dominates m k 2 in view of Pi 1, and (iv) m k 2 does not dominate m k+1 1 in view of Pi 1. Since (M, g ) is a finite mechanism, we complete the proof by showing m k 1 m l 1 and m k 2 m l 2 for k < l. Since m k 2 dominates m k 1 in view of P 2 i but m k 1 does not dominate m k 2 in view of P 1 i, there is m i such that g (m k 2, m i )P t i g (m k 1, m i ) for both t = 1 and 2. Since m k+1 1 dominates m k 2 in view of P 1 i, g (m k+1 1, m i )P 1 i g (m k 2, m i ), and hence g (m k+1 1, m i )P 1 i g (m k 1, m i ), and therefore, m k+1 1 m k 1. Suppose that for l k + 2, we obtained g (m l 1 1, m i )P 1 i g (m k 1, m i ). Then, since m l 1 2 dominates m l 1 1 in view of P 2 i, g (m l 1 2, m i )R 2 i g (m l 1 1, m i ). Also, since m l 1 dominates m l 1 2 in view of P 1 i, g (m l 1, m i )R 1 i g (m l 1 2, m i ). Thus, if g (m l 1 2, m i )R 1 i g (m l 1 1, m i ), then g (m l 1, m i )R 1 i g (m l 1 1, m i ). On the other hand, if g (m l 1 1, m i )Pi 1 g (m l 1 2, m i ), then g (m l 1 1, m i ) = a and g (m l 1 2, m i ) = b. Thus, g (m l 1, m i )Ri 1 b. That g (m l 1 1, m i ) = a induces that g (m k 1, m i ) is b or worse (so that g (m k 1, m i ) is strictly worse than g (m l 1 1, m i ) = a in view of Pi 1 ). If g (m k 1, m i ) = b, then g (m k 2, m i )P 2 i b, and so g (m k+1 1, m i )P 1 i a, and g (m k+1 2, m i )P 2 i b, and by induction, g (m l 1 1, m i )P i 1a, which is a contradiction. Therefore, bp 1 i g (m k 1, m i ), which implies g (m l 1, m i )P 1 i g (m k 1, m i ). We can show m k 2 m l 2 for k < l by the same reasoning. Lemma 3. Suppose that there is m i B i (Pi t ) and m i B i (P t 1 i ) such that m i dominates m i in view of Pi t, and m i dominates m i in view of P t 1 i. Then either (i) m i B i (P t+1 i ), or (ii) there is m + i B i (P t+1 i ) that dominates m i in view of P t+1 i and m + i B i (Pi t ), or (iii) there are m + i B i (P t+1 i ) and m i B i (Pi t ) such that m + i dominates m i and m i in view of P t+1 i, and m i dominates m + i in view of Pi t. Also, in view of P t 1 i, m i dominates m i in case (iii). Proof. Suppose that neither (i) nor (ii) is true. Then we show that (iii) must be true. Since (i) is not true, m i / B i (P t+1 i ), and thus, there is m + i B i (P t+1 i ) that dominates m i in view of P t+1 i. Since (ii) is not true, there is m i 10

11 such that g (m + i, m i)pi t g (m i, m i ). We also have g (m + i, m i)p t+1 i g (m i, m i ) by definition of m + i. Notice that since F is monotonic, we must have L i (g (m + i, m i), P t+1 i ) L i (g (m + i, m i), Pi t ). Let a, b X be such that api t b and bp t+1 i a. Then g (m + i, m i) = b and bpi t g (m i, m i ). Since m + i / B i (Pi t ), there is m i B i (Pi t ) that dominates m + i, which implies that g (m i, m i )Rig t (m + i, m i). First, g (m i, m i ) can be either a or b, because if g (m i, m i )Pi t a, then L i (g (m i, m i ), Pi t ) = L i (g (m i, m i ), P t+1 i ) implies (by monotonicity) b = g (m + i, m i)r t+1 i g (m i, m i ), which is a contradiction. Second, g (m i, m i ) cannot be a, because if g (m i, m i ) = a, then L i (a, P t 1 i ) L i (a, Pi t ) implies (by monotonicity) g (m i, m i)r t 1 i a, which is a contradiction. Therefore, g (m i, m i ) = b. Let c = g (m i, m i ) and d = g (m i, m i). Then dr t 1 i c and crid. t If c d, then ap t 1 i bp t 1 i dp t 1 i c. Since g (m i, m i ) = b, it contradicts the monotonicity. Thus, we have c = d, and in order to satisfy the monotonicity, we must have ap t 1 i cp t 1 i b. Suppose that m + i does not dominate m i in view of P t+1 i m i such that g (m i, m i)p t+1 i g (m + i, m i)p t. Then there is g (m + i, m i). First, as we have shown above, i g (m i, m i ) induces g (m + i, m i) = g (m i, m i ) = b, and so, we have g (m + i, m i)r t+1 i g (m i, m i). Thus, we can assume g (m i, m i)rig t (m + i, m i). Also, g (m i, m i)rig t (m i, m i) and g (m i, m i)pi t g (m + i, m i). For P t 1 i, we know g (m i, m i)rig t (m i, m i). If g (m i, m i) g (m i, m i), then g (m i, m i) = c and g (m i, m i) = b. Then at P t+1 i, the rank of b = g (m i, m i) must be even higher, and so brig t (m + i, m i) implies br t+1 i g (m + i, m i). Therefore, b = g (m + i, m i) by definition of m + i. Then, we must have g (m i, m i)pi t a and g (m i, m i)p t+1 i a. It is a contradiction for the monotonicity. Therefore, g (m i, m i) = g (m i, m i). By the similar argument, we have g (m i, m i) = g (m i, m i) = g (m + i, m i). Also, g (m i, m i)pi t g (m i, m i ) and g (m i, m i)p t+1 i g (m i, m i ). If g (m i, m i )P t 1 i g (m i, m i), then g (m i, m i ) = c and g (m i, m i) = 11

12 b. It contradicts the monotonicity, because L i (b, P t+1 i ) L i (b, Pi t ). If g (m i, m i )P t 1 i g (m i, m i), then it also contradicts the monotonicity, because we cannot have both L i (g (m i, m i), Pi t ) L i (g (m i, m i), P t 1 i ) and L i (g (m i, m i), Pi t ) L i (g (m i, m i), P t+1 i ). Finally, we show that in view of P t 1 i, m i dominates m i in case (iii). If not, then there is m i such that g (m i, m i)p t 1 i g (m i, m i). By the monotonicity, we must have g (m i, m i) = b. By definition of m i and m i, we have g (m i, m i) = g (m i, m i). By definition of m + i and m i, we have g (m + i, m i) = b, and since m + i dominates m i, bp t+1 i g (m i, m i). By the single-crossing condition, bpi t g (m i, m i). However, by monotonicity, we must have g (m i, m i) = c, which contradicts bp t 1 i g (m i, m i). Proof. (Proof of Theorem 1) We first construct M i from M i as follows. (Step 1) Pick m 1 i B i (P 1 i ) and m 2 i B i (P 2 i ) so that (i) m 1 i = m 2 i if B i (P 1 i ) B i (P 2 i ), or otherwise, so that (ii) m s i dominates m t i in view of = {m 1 i, m 2 i }. Lemma 2 assures the existence Pi s, for s, t = 1, 2. Then let Mi 1 of such messages. (Step 2) Pick m 3 i B i (P 3 i ) so that (i) m 3 i = m 2 i if m 2 i B i (P 3 i ), or otherwise, so that (ii) m 3 i dominates m 2 i in view of P 3 i. In the second case, let ˆm 2 i = m 3 i if m 3 i B i (P 2 i ), or otherwise, let ˆm 2 i B i (P 2 i ) be a message that dominates m 3 i in view of P 2 i and m 3 i dominates ˆm 2 i in view of P 3 i. Then M 2 i = { ˆm 2 i, m 3 i }. (Step k; k = 3,..., T i 1) Pick m k+1 i B i (P k+1 i ) so that (i) m k+1 i = m k i if m k i B i (P k+1 i ), or otherwise, so that (ii) m k+1 i dominates m k i in view of P k+1 i. In the second case, let ˆm k i = m k+1 i if m k+1 i let ˆm k i B i (P k i ) be a message that dominates m k+1 i B i (Pi k ), or otherwise, in view of Pi k and m k+1 i dominates ˆm k i in view of P k+1 i. Then M k i = { ˆm k i, m k+1 i }. Finally, we define M i = T i 1 k=1 M k i, and define g as the restriction of g on M = i M i. Next, we show that B i (Pi 1 M, g) = {m 1 i }, B i (Pi t M, g) = {m t i, ˆm t i}, and B i (P T i i M, g) = {m T i i }. If it is shown, then since mt i, ˆm t i B i (Pi t ), we conclude A(M, g) A(M, g ), and so we complete the proof of the theorem. 12

13 First, the following observation is useful. Lemma 4. Suppose that there is m i B i (Pi s ) and m i B i (Pi t ) such that m i dominates m i in view of Pi s, and m i dominates m i in view of Pi t. Then for any s < s, m i dominates m i in view of Pi s, and for ant t > t, m i dominates m i in view of Pi t. Proof. For any m i, g(m i, m i )R s i g(m i, m i ) and g(m i, m i )R t ig(m i, m i ). By the single-crossing condition, g(m i, m i )Ri s g(m i, m i ), and g(m i, m i )Ri t g(m i, m i ). For m i such that g(m i, m i ) g(m i, m i ), we have g(m i, m i )Pi s g(m i, m i ) and g(m i, m i )Pi t g(m i, m i ) (there is at least one such m i ), and therefore, by the single-crossing condition, g(m i, m i )Pi s g(m i, m i ), and g(m i, m i )Pi t g(m i, m i ). (Proof of Theorem 1, continued) Hence, we have shown ˆm s i / B i (Pi t M, g) and m t i / B i (Pi s M, g) for s < t. Therefore, we need to show ˆm t i / B i (Pi s M, g) and m s i / B i (Pi t M, g) for s < t. We have already shown that ˆm t i / B i (P t 1 i M, g), because ˆm t 1 i dominates ˆm t i in view of P t 1 i. For s t 2, suppose that ˆm t 1 i does not dominate ˆm t i in view of Pi s. Then for some m i, we have g( ˆm t i, m i )Pi s g( ˆm t 1 i, m i ). Since g( ˆm t i, m i )P t 1 i g( ˆm t 1 i, m i ), we have g( ˆm t i, m i )Pi t g( ˆm t 1 i, m i ) and g( ˆm t i, m i )P t+1 i g( ˆm t 1 i, m i ) by the single-crossing condition. For Pi t, we have g(m t i, m i )Rig( t ˆm t 1 i, m i )Pi t g( ˆm t i, m i )Rig(m t t+1 i, m i ). For P t+1 i, we have g(m t+1 i, m i )Pi t g(m t i, m i ). Therefore, we must have g(m t i, m i ) = g( ˆm t 1 i, m i ) and g( ˆm t i, m i ) = g(m t+1 i, m i ), which contradicts g( ˆm t i, m i )P t+1 i g( ˆm t 1 i, m i ). By the similar argument, we can show m s i / B i (Pi t M, g) for s < t. It implies that there is an upper bound on the size of a message space of an optimal mechanism. Corollary 1. M i 2(T i 2) + 2 = 2(T i 1). We interpret this result as a sort of a revelation princile in the followings sense. An original revelation principle says that one of the optimal mechanisms is a revelation mechanism, i.e., a mechanism where the message space 13

14 of agent i is the type space of the agent, P i. It implies, among other things, that there is an upper bound on the size of the message space, which is the size of the type space of the agent. Similarly, the theorem we have shown above states that there is an upper bound on the size of each agent s message space of an optimal mechanism, which is a function of the size of the agent s type space. This result could be useful in searching for an optimal mechanism, in the following sense. First, since there is a finite upper bound on the size of an optimal mechanism, we can search for an optimal mechanism by hand or by computer. The author believes it is a nice property especially if we notice that in many implementation problems, we do not know whether one of the mechanism that implements a desirable social choice correspondence satisfies such an upper-bound property. 5 6 A model with indifference in preference In the previous sections, we have considered a situation in which each agent has a strict preference over X, i.e., all possible alterntives. It would be a reasonable assumption, for example, in a voting situation, but in many economic situations, such as market transaction, auction, matching, and so on, it may be more reasonable sometimes to allow some indifference in preference. For example, in an auction, an individual may care only its own assignment (probability of winning an object and price to be paid), but does not care the other individuals assignments. In this section, we allow some indifference in agents preferences. More specifically, we let X = X 1... X N, and suppose that agent i has a strict preference on X i, 6 but he is indifferent between x = (x 1,..., x N ) and x = (x 1,..., x N ) as long as x i = x i. 5 One of the exceptions is a problem of dominant-strategy implementation, because we have a revelation principle of?. 6 We still keep assuming that agent i s preference does not have any indifference as to X i to simplify the discussion. 14

15 A (finite) mechanism is, as in the previous sections, denoted (M, g), where M = i M i and M i is agent i s message space. One additional detail is that g : M X is decomposed as g(m) = (g 1 (m),..., g N (m)) for each m M, so that g j (m) X j is the assignment for agent j when m = (m 1,..., m N ) are reported by the agents. The objective of this section is to show that we can obtain a similar upper bound proprty in the model with indifference in preference. However, to do so, we need to modify the definition of undominated strategies a little, because without any modification, we cannot show a result that corresponds to Lemma 1. Lemma 1 does not hold in the model of this section because if we obtain m i B i (P i M, g) by cutting some messagies from (M, g ) to obtain (M, g), then, although we can say g i (m i, m i ) = g i (m i, m i ) for any m i M i, we cannot generally say g i (m i, m i ) = g i (m i, m i ), and so m i is not equivalent to m i. That is, by cutting some strategies of others, agent i s set of undominated strategies can essentially be enlarged so that we may no longer have A(M, g) A(M, g ) if we apply the same procedure of cutting messages as in the proof of Theorem 1. In this section, therefore, we first modify the definition of undominated strategies, and then show our result, which is basically the same as Theorem 1. Probably the most straightforward modification is the following: We define P i as a strict preference of i over X so that it is consistent with P i. Then, define B i (P i M, g) as the new set of undominated strategies, which we call the set of undominated strategies of P i for any extended preference, where B i (P i M, g) = {m i M i P i consistent with P i, m i; g(m i, m i ) R i g(m i, m i ) for any m i M i, and g i (m i, m i)p i g i (m i, m i) for some m i M i }. That is, we construct an arbitrary preference over X, not only over X i, and define the set of undominated strategies given the new preferences. This argument could be reasonable in a situation where the mechanism designer cannot be so sure that agent i is really indifferent over X i. If m i domi- 15

16 nates m i in view of any P i that is consistent with P i, then we may say with some confidence that m i dominates m i in view of P i. We then obtain the following lemma. Lemma 5. Bi (P i M, g) = {m i M i m i; g i (m i, m i)p i g i (m i, m i) for some m i M i, and for any m i, either g(m i, m i ) = g(m i, m i ) or g i (m i, m i )P i g i (m i, m i )}. Proof. (IF part) Straightforward. (ONLY IF part) Suppose not. Then there is m i such that g i (m i, m i ) = g i (m i, m i ) and g i (m i, m i ) g i (m i, m i ), and m i dominates m i in view of P i. Let P i be a strict preference of i over X that is consistent with P i, and that satisfies g i (m i, m i ) P i g i (m i, m i ). Then m i no longer dominates m i in view of P i, which is a contradiction. Let Σ i be the set of all σ i : P i σ i (P i ) B i (P i M, g). Then, the set of all possible outcome mappings under (M, g) is defined as follows: Ã(M, g) = {g(σ) σ Σ 1... Σ N }. Then we obtain a lemma that corresponds to Lemma 1. Lemma 6. Suppose that in a mechanism (M, g ), m i B i (P i M, g ) dominates m i in view of P i (and so m i / B i (P i M, g )). Let (M, g) be an arbitrary mechanism where M j M j for each j (including i) and m i, m i M i, and g is the restriction of g on M. Then,we have either (i) m i B i (P i M, g) and m i / B i (P i M, g), or (ii) if m i B i (P i M, g), then m i and m i are equivalent, i.e., g(m i, m i ) = g(m i, m i ) for any m i M i. Proof. Since m i dominates m i under (M, g ), in view of P i, we have g(m i, m i )R i g(m i, m i ) for any m i M i, and for some m i, g(m i, m i)p i g(m i, m i). Let M i be the set of all m i such that g(m i, m i)p i g(m i, m i), and hence, for any m i / M i, we have g(m i, m i ) = g(m i, m i ). Suppose that for m i, m i, (i) does not hold under mechanism (M, g). It means that any message m i is excluded from M i. Thus, for any m i M i, we have g(m i, m i ) = g(m i, m i ). 16

17 First, we show that a mechanism is (weakly) outperformed by a mechanism that does not have a nuisance message. Proposition 2. For an arbitrary mechanism (M, g ), there is a mechanism (M, g) without nuisance messages, and that satisfies Ã(M, g) Ã(M, g ). We omit the proof, because it is basically the same as the proof of Proposition 1. Suppose that for each i, P i satisfies the single-crossing property and connectedness, where those two properties are redefined in terms of X i : We say that P i is single-crossing if there is a linear ordering on P i, say P 1 i,..., P T i i, such that if for some s < t and x, y X i, we have xp s i y and yp t any s < s and t > t, we have xp s i y and yp t i x, then for We say that the pref- i x. erence set for each agent i is connected if for any P i, P i P i and any x, y X i such that xp i y and yp i x, there is P i P i such that yp i x and [wp i z wp i z]. Also, suppose that the mechanism designer is interested in implementation of a strongly monotonic social choice correspondence F, where F is said to be strongly monotonic if for all P i, P i, P i, all x = (x 1,..., x N ) F (P i, P i ) such that L i (x i, P i ) L i (x i, P i ), and all y = (y 1,..., y N ) F (P i, P i ), we have y i R ix i (i.e., either x i = y i or y i P i x i ). Then we obtain the following result. Theorem 2. For an arbitrary mechanism (M, g ) that implements F, there is a mechanism (M, g) such that (i) Ã(M, g) Ã(M, g ) (and so (M, g) implements F ), (ii) B i (P i M, g) 2 for each i and P i, and (iii) B i (P 1 i M, g) = B i (P T i i M, g) = 1. We omit the proof, because it is basically the same as the proof of Theorem 1. Although the modification above would be reasonable in some situations, there may be other situations where another modification would be more reasonable. For example, consider a situation in which an agent may believe what strategies, say σ i, will be taken by the other agents. In that 17

18 case, even if a message m i is dominated by m i in view of P i, agent i may take message m i if g i (m i, m i ) = g i (m i, m i ) for any m i supp(σ i ) (and g i (m i, m i)p i g i (m i, m i) for m i / supp(σ i )). Of course, if we do not restrict σ i that the agent i can believe to be taken, then m i is not dominated by m i unless m i strictly dominates m i (consider σ i whose support is any singleton set), which restricts the scope of analysis very much. Instead, here, we impose a sort of a consistency condition on possible σ i s that the agent i can possess. Namely, σ i itself must be an undominated strategy profile. More precisely, we define the set of undominated strategies of P i robust to any knowledge of the others choices as follows: ˆBi (P i M, g) = {m i M i m i; g i (m i, m i )R i g i (m i, m i ) for any m i, and for any σ i ˆΣ i, there is P i such that g i (m i, σ i (P i ))P i g i (m i, σ i (P i ))}, where ˆΣ i = {σ i : P i σ i (P i ) ˆB i (P i M, g)}. Or equivalently, ˆBi (P i M, g) = {m i M i m i; g i (m i, m i )R i g i (m i, m i ) for any m i, and there is P i such that for any m i ˆB i (P i M, g), g i (m i, m i )P i g i (m i, m i )}, where ˆΣ i = {σ i : P i σ i (P i ) ˆB i (P i M, g)}. This definition is self-referential, and so we have to confirm that (i) it always exists, and (ii) it is unique. The existence is confirmed by giving an algorythm to obtain one: (Step 0) Construct ˆB 0 i (P i M, g) as the standard undominated strategies in view of P i under (M, g), and let ˆΣ 0 i = {σ i : P i σ i (P i ) ˆB 0 i (P i M, g)}. (Step k; k = 1, ldots) Define ˆB i k (P i M, g) as the set of undominated k 1 strategies even if agent i believes that the other agents take any σ i ˆΣ i, and let ˆΣ k i = {σ i : P i σ i (P i ) ˆB i k (P i M, g)}. Lemma 7. ˆBk i (P i M, g) ˆB k 1 i (P i M, g) for each k = 1, 2,.... Proof. Clearly, ˆB0 i (P i ) ˆB i 1 (P i ). Suppose that for l = 1,..., k, ˆBk 1 j (P j ) ˆB j k (P j ) for any j. In the definition of ˆBk+1 i (P i ), we consider any strategy profile σ i ˆΣ k i, which induced by ˆB i(p k i ), while in the definition of ˆB i k (P i ), we consider any strategy profile σ i, which is induced by ˆB k 1 i (P i )( ˆB i(p k i )). Therefore, if m i / m i / ˆB i k (P i ) as well. 18 ˆB k+1 i ˆΣ k 1 i (P i ), then it must satisfy

19 Since ˆB i k (P i ) is a monotonic seqnence, we finally find an integer K such that ˆB i K K+1 (P i M, g) = ˆB i (P i M, g) for any i and P i. Then ˆB i K (P i M, g) satisfies the condition of ˆB i (P i M, g). Lemma 8. If there are two collections of undominated strategy sets robust to knowledge of others choices, say ˆB i (P i M, g) and ˆB i(p i M, g) for each i and P i, then ˆB i (P i M, g) = ˆB i(p i M, g). Proof. If not, then there is m i / ˆB i (P i M, g) such that m i ˆB i(p i M, g). Since m i / ˆB i (P i M, g), there is m i ˆB i (P i M, g) that dominates m i in view of P i. Then there must be P i such that for any m i ˆB i (P i M, g), we have g i (m i, m i )P i g i (m i, m i ), and also, m i / ˆB i(p i M, g). This latter condition implies that for some j i and m j ˆB j (P j M, g), there is m j ˆB j(p j M, g) that dominates m j in view of P j, and g i (m i, m j, m ij ) = g i (m i, m j, m ij ) for some m ij ˆB ij(p ij M, g). Now, since g j (m j, m j )R j g j (m j, m j ) for any m j M j, m j must be in ˆB j (P j M, g). However, it contradicts for any m i ˆB i (P i M, g), we have g i (m i, m i )P i g i (m i, m i ). The set of all possible outcome mappings under (M, g) is defined as follows: Â(M, g) = {g(σ) σ ˆΣ 1... ˆΣ N }. Then we obtain a lemma that corresponds to Lemma 1 (though differently expressed). Lemma 9. From a mechanism (M, g ), suppose we construct a mechanism (M, g) so that M i M i for some i, M j = M j for j i, and g is the restriction of g on M. Suppose that under (M, g), ˆB i (P i M, g) ˆB i (P i M, g ) for each P i. Then for any j i and P j, we have ˆB j (P j M, g) = ˆB j (P j M, g ). We omit the proof because it is clear. Applying the lemma, we first show that a mechanism is (weakly) outperformed by a mechanism that does not have a nuisance message. 19

20 Proposition 3. For an arbitrary mechanism (M, g ), there is a mechanism (M, g) without nuisance messages, and that satisfies Â(M, g) Â(M, g ). We omit the proof, because it is basically the same as the proof of Proposition 1. Suppose that for each i, P i satisfies the single-crossing property and connectedness (as redefined in this section), and suppose that the mechanism designer is interested in implementation of a strongly monotonic social choice correspondence F (also, as redefined in this section). Then we obtain the following result. Theorem 3. For an arbitrary mechanism (M, g ) that implements F, there is a mechanism (M, g) such that (i) Â(M, g) Â(M, g ) (and so (M, g) implements F ), (ii) ˆB i (P i M, g) 2 for each i and P i, and (iii) ˆB i (P 1 i M, g) = ˆB i (P T i i M, g) = 1. We omit the proof, because it is basically the same as the proof of Theorem 1. 7 A random valuation model In this section, we consider another model: Let X i = X + i {o}, where o is called the default assignment. Agent i has the same preference on X + i regardless of the agent s preference type. More precisely, for any P i, P i P i, if x, y X + i and xp i y, then xp i y. Thus, the only difference between P i and P i is the rank of the default assignment. We say that P i is more admissible than P i if xp i o implies xp i o for any x X + i. A cost sharing problem in a public good provision is an incident of this model: Let x = (o,..., o) X denote the alternative in which no public good is provided, let x i X + i denote the payment of i associated with the provision of the public good, and assume that the mechanism designer is unsure about each agent s valuation for the public good. Then it would be reasonable to assume that each agent has the same preference on X + i regardless of the agent s valuation for the public good, and the rank of o 20

21 in agent i s preference list reflects the agent s valuation for the public good (i.e., agent i with more admissible P i has higher valuation). Of course, in this example, any x that has x i = o and x j X + j is impossible. To avoid such an alternative, we would consider an evaluation function φ that induces φ(α) = if α(p ) is such an alternative for some P. A single-unit auction problem is also an incident of this model. Again, the rank of o in i s preference list represents i s valuation for the good, and x i X + i denotes i s payment given that the good is assigned to i. To avoid strange alternatives such as x that has x i X + i and x j X + j for i j, we would consider an evaluation function φ that induces φ(α) = if α(p ) is such an alternative for some P. In this environment, we consider implementation of individually rational social choice correspondences. More specifically, a social choice correspondence F is said to be (ex post) individually rational if for any i, P P, and any x = (x 1,..., x N ) F (P ) we have x i R i o (i.e., x i P i o or x i = o). In this model, we obtain the following result. Theorem 4. For any (M, g) that implements F that is individually rational, there is a DSIC mechanism (M D, g D ) that has A(M D, g D ) A(M, g) (and so (M D, g D ) implements F ). This result may be interpreted as a negative result in our motivation, because we cannot do any better than what we can do with DSIC mechanisms, while in view of dominant-strategy implementation, it may be interpreted as a positive result (by the same reason). Proof. Order agent i s preference as Pi 0,..., P T i i so that Pi 0 be the least admissible preference type of agent i (i.e., o has the highest rank under Pi 0 among all P i s), and Pi t is more admissible P t 1 i for any t = 1,..., T i. Take any m 0 i B i (P 0 i ). If m 0 i B i (P 1 i ), then let m 1 i = m 0 i. Otherwise, let m 1 i B i (P 1 i ) be such that m 1 i dominates m 0 i in view of P 1 i. Then we can show that m 0 i dominates m 1 i in view of P 0 i : Suppose not. Then either (i) m 1 i B i (P 0 i ) or (ii) there is m 0 i B i (P 0 i ) that dominates m 1 i in view of P 0 i. 21

22 Case (i): For any m i such that g i (m 0 i, m i ) o, we have g(m 1 i, m i )R 1 i g(m 0 i, m i ), and so g(m 1 i, m i ) o. Since Pi 0 and Pi 1 have the same preference over X + i, g(m 1 i, m i )Ri 0 g(m 0 i, m i ). For any m i such that g i (m 0 i, m i) = o, either g(m 1 i, m i) = o or g(m 1 i, m i) = a X + i such that api 1 o (there exists such m i; otherwise m 1 i does not dominate m 0 i in view of Pi 1 ). By the individual rationality, we must have api 0 o as well. Therefore, m 1 i dominates m 0 i in view of Pi 0, which contradicts m 0 i B i (Pi 0 ). Case (ii): For any m i such that g i (m 0 i, m i ) o, we have g(m 1 i, m i )Ri 0 g(m 0 i, m i ) as in case (i), and thus, g( m 0 i, m i )R 0 i g(m 1 i, m i )R 0 i g(m 0 i, m i ). For any m i such that g i (m 0 i, m i) = o, either g(m 1 i, m i) = o, g(m 1 i, m i) = b X + i such that opi 0 b and bpi 1 o, or g(m 1 i, m i) = a X + i such that api 0 o (there must exist m i such that g(m 1 i, m i)pi 0 o; otherwise, m 0 i dominates m 1 i in view of Pi 0 ). Since m 0 i dominates m 1 i in view of Pi 0, we have g( m 0 i, m i)r i o if g(m 1 i, m i) = o or if opi 0 g(m 1 i, m i); we have g( m 0 i, m i)r i g(m 1 i, m i) if g(m 1 i, m i)pi 0 o. As a conclusion, m 0 i dominates m 0 i, which contradicts m 0 i B i (Pi 0 ). As a next step, we let m 2 i = m 1 i if m 1 i B i (Pi 2 ), or otherwise, we let m 2 i be a message in B i (Pi 2 ) that dominates m 1 i. In the same way, we obtian m 3 i,..., m T i i. Finally, we define Mi D = {m 1 i,..., m T i i } for each i, and define gd as the restriction of g on M D = i M i D. Then, (M D, g D ) is dominant-strategy incentive compatible, and also A(M D, g D ) A(M, g). 8 Concluding remarks In this paper, we have considered a way to evaluate a mechanism without assuming mutual knowledge of rationality among the agents. More specifically, we have assumed that each agent may take any undominated strategy, and the mechanism designer evaluates a mechanism by its worst-case outcome mapping. In the worst-case mechanism design problem, there are many questions 22

23 that are still open. For example, we need more complete characterization of optimal mechanisms, especially in economically important environments such as auction, two-sided matching, voting, and so on. Also, the author believes that the approach of worst-case mechanism design might be useful in evaluating some mechanisms that are not DSIC, but commonly used in real world. 7 For example, deferred acceptance mechanisms are known to be non-dsic in general, but variants of them are adopted in variety of two-sided matching situations (e.g., student-school matching, hospital-intern matching). Although there are some papers that evaluate those mechanisms assuming common knowledge of rationality among the players, our approach would be useful to know whether they can behave well even without assuming common knowledge of rationality. References Börgers, T. (1991): Undominated strategies and corordination in normalform games, Social Choice and Welfare, 8, Dasgupta, P., P. Hammond, and E. Maskin (1979): The implementation of Social Choice Rules, Review of Economic Studies, 46, Gibbard, A. (1973): Manipulation of voting schemes, Econometrica, 41, Jackson, M. O. (1992a): Implementation in undominated strategies: a look at bounded mechanisms, Review of Economic Studies, 59, (1992b): Economic Letters, 40, Incentive compatibility and competitive allocations, 7 For example, Jackson (1992b) has studied agents behaviors under a large market mechanism, and has shows that each agent s set of undominated strategies (where a strategy is a report of a demand curve) can be arbitrarily small under a large market mechanism. 23

24 Miyamoto, H., T. Watanabe, and S. Mizuno (1990): Implementation in admissible strategies, Discussion Paper 28, Department of Management Science and Engineering, Tokyo Institute of Technology, Technical Report. Roth, A. E. (1982): The economics of matching: stability and incentives, Mathematics of Operations Research, 7, Satterthwaite, M. A. (1975): Strategy-proofness and arrow s conditions: existence and correspondence theorems for voting procedures and social welfare functions, Journal of Economic Theory, 10,