Implementation in undominated strategies by bounded mechanisms: The Pareto Correspondence

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1 Implementation in undominated strategies by bounded mechanisms: The Pareto Correspondence Saptarshi Mukherjee Eve Ramaekers Arunava Sen September 5, 2016 Preliminary and Incomplete Abstract We show that the Pareto correspondence can be implemented in weakly undominated strategies by bounded mechanisms provided there are at least three agents. 1 Introduction In this paper, we show that the Pareto correspondence can be implemented in weakly undominated strategies by bounded mechanisms provided that there are at least three agents. This addresses an issue raised in Börgers (1991). 2 Implementation There is a finite set of agents N and a finite set of alternatives A. To avoid triviality, we assume A 2. Each agent i N has preferences over A, represented by a complete, transitive, and anti-symmetric binary relation P 1 For each a, b A, if a P i b or a = b, we write a R i b. For each a, b A, if a P i b, we say that a is better than b and if a R i b, we Indian Institute of Technology, New Delhi, India. Université catholique de Louvain, CORE, Voie du Roman Pays, 34, B-1348 Louvain-la-Neuve, Belgium. Indian Statistical Institute, 7 SJS Sansanwal Marg, New Delhi , India. 1 Binary relation P i is complete if for each a, b A, either a P i b, b P i a, or both, transitive if for each a, b, c A, if a P i b and b P i c, then a P i c, and anti-symmetric if for each a, b A with a b, either a P i b or b P i a. 1

2 say that a is at least as good as b. Let P be the set of all strict orderings on A. We let P N and P i, i N denote the N -fold and N 1 -fold Cartesian product of P respectively. A preference profile P (P i ) i N is an element of P N. A social choice correspondence SCC F associates a non-empty subset of alternatives with every preference profile P, i.e. F : P N 2 A \. A mechanism consists of a message space M i N M i and an outcome function g : M A. Let P D N and i N. Message m i M i weakly dominates message m i M i at P i in mechanism (M, g) if for each m i M i, we have g(m i, m i ) R i g(m i, m i ) and for at least some m i M i, we have g(m i, m i ) P i g(m i, m i ). 2 Further, m i is undominated at P i in (M, g) if no m i M i weakly dominates m i at P i in (M, g). Let U(P i ) be the set of all undominated messages at P i in (M, g). Let m (m i ) i N M be a typical message profile. Let U(P ) i N U(P i ) and by abuse of notation, g(u(p )) {a A there is m U(P ) such that g(m) = a}. Definition 1 A SCC F is implementable in undominated strategies if there is a mechanism (M, g) such that for each P P N, we have F (P ) = g(u(p )). Remark: Jackson (1992) proved the following surprising result - it is possible to implement any SCC in undominated strategies. However the canonical mechanism used to prove this result involves infinite strings of messages, each of which dominates the earlier without an undominated message at the end. Thus, in such a mechanism, an agent s best-response according to the criteria specified, does not exist. This is unsatisfactory. To avoid such situations (following Jackson (1992)), we require that for each agent s dominated message, there is an undominated message that weakly dominates it. Definition 2 A mechanism (M, g) is bounded if for each i N, each P i P and each m i / U(P i ), there exists m i U(P i ) that weakly dominates m i at P i in (M, g). If a SCC is implementable in undominated strategies by a bounded mechanism we say that it is USBM implementable. Some necessity conditions for USBM implementability exist. Jackson (1992) shows that strategy-resistance is a necessary condition and Yamashita (2012, 2015) generalises this condition to chain dominance. Börgers (1991) shows that there is no selection of the Paretocorrespondence that satisfies a compromise property which is USBM implementable if there are either two agents or three alternatives. Certain SCCs are known to be USBM implementable. For instance, strategy-proof social choice functions (SCFs) such as dictatorial SCFs are trivially implementable. The union 2 Let M i l N\{i} M l be the message sets of all the agents but i and m i (m l ) l N\{i} be the messages of all the agents but i. 2

3 of best-ranked alternatives at every profile (a strict subset of the Pareto Correspondence) is USBM implementable (Börgers (1991)). However, no general sufficiency result exists. Mukherjee, Muto and Ramaekers (2016) provide a general solution to the problem where an additional behavioural assumption (partial honesty) is imposed on agents. We show below that the entire Pareto correspondence can be USBM implemented. 3 Main Result The Pareto Correspondence selects all alternatives a at profile P with the property that there does not exist another alternative b such that bp i a for all i N. Theorem 1 The Pareto correspondence is USBM implementable for N 3. Proof : We describe a mechanism that implements the Pareto correspondence. In order to do so, some preliminaries are required. 1. Let > be an ordering on N. 2. For each i N, let i be an ordering on all distinct pairs of admissible orderings of i. 3. For each i N and each P i, P i P, let x i (P i, P i ), y i (P i, P i ) A be such that x i (P i, P i ) P i y i (P i, P i ) and y i (P i, P i ) P i x i (P i, P i ). We now describe the mechanism. Each agent i N announces two orderings P i, P i P with P i P i, a color C i {G, B, R} together with a shade s i {1, 2,...}, and an integer λ i {1, 2,..., max P i F (P i, P i ) }. For each i N, let m i = (P i, P i, C i, s i, λ i ) M i be a typical announcement. The outcome is determined as follows. For each l N and each m l = (P l,., C l, s l,.) M l, we say that m = (m l ) constitutes a trigger message profile for i N if: (i) there are x, y A and Pi, Pi P such that: P i is unanimous on (y, x), i.e. for each j N\{i}, we have y = τ 1 (P j ) and x = τ 2 (P j ), (x, y) is associated with (Pi, Pi ), i.e. x = x i (Pi, Pi ) and y = y i (Pi, Pi ), (ii) for each j, k N\{i}, we have C j = C k = R and there is t {1, 2,...} such that t = s j = s k. 3

4 Case I There is only one i N for whom announced m = (m l ) l N constitutes a trigger message profile. Thus, there are x, y A and Pi, Pi P such that Condition (i) holds. Break ties as follows. If (x, y) is associated with (P i, P i ), assume P i = Pi and P i = Pi. If (x, y) is associated with (P i, P i ), assume P i = Pi and P i = Pi. Otherwise, break ties by means of ordering Also, there is t {1, 2,...} such that for each l N\{i}, we have t = s l. The outcome is x, y or τ 1 (P i ), depending on C i, on how announced (P i, P i ) relates to (Pi, Pi ), and if C i = B, on how s i relates to t. Distinguish five cases. I.a P i = P i = P If i announces G, R, or B with s i < t, then the outcome is x. If i announces B with s i = t, then the outcome is y. If i announces B with s i > t, then the outcome is τ 1 (P i ). I.b P i = P i = P If i announces G, R, or B with s i t, then the outcome is y. If i announces B with s i > t, then the outcome is τ 1 (P i ). I.c P i = P i P If i announces G, R, or B with s i t, then the outcome is x. If i announces B with s i > t, then the outcome is τ 1 (P i ). I.d P i = P i P If i announces G, R, or B with s i t, then the outcome is y. If i announces B with s i > t, then the outcome is τ 1 (P i ). I.e P i P i and P i P If i announces G, R, or B with s i t, then the outcome is τ 1 (P i, {x, y}). If i announces B with s i > t, then the outcome is τ 1 (P i ). Case II There are i, j N with i j for whom announced m = (m l ) l N constitutes a trigger message profile. Since N 3, there is y A such that for each l N, we have y = τ 1 (P l ), and the outcome is y. Case III There is j N announcing B, and Case I does not hold. Then, the outcome is τ 1 (P h ), where h announces B and is the winner, among all announcing B, of the integer game for announced shades (s l ) l N, i.e. s h = max {l N Cl =B} s l. Break ties by means of ordering >. Case IV Case I, II nor III holds. Arbitrarily order F (P ). W.l.o.g. let a 1 F (P ) be the alternative ranked first, a 2 F (P ) be the alternative ranked second, and so on. Then, 4

5 the outcome is a k, where k is the winner of the modulo game for announced (λ l ) l N, i.e. k l N λ l mod F (P ) + 1. We show that the mechanism described above implements the Pareto correspondence. We proceed in several steps. [Step 1:] True ordering, any secondary ordering, blue, any shade, and any integer announcement weakly dominates this secondary ordering, any other ordering as secondary ordering, any color, any smaller shade, and any integer announcement at true ordering. Let i N, P i P, m 1 i = (P i, P i, B, s i, λ i ) M i, and m 2 i = (P i, P i, C i, s i, λ i) M i such that s i < s In what follows, we prove that for each m i M i, we have g(m 1 i, m i ) R i g(m 2 i, m i ). In Step 2, we prove that there is m i M i such that g(m 1 i, m i ) P i g(m 2 i, m i ). Thus, m 1 i weakly dominates m 2 i at P For each l N\{i}, let m l = (P l, P l, C l, s l, λ l ) M l. Note that (m 1 i, m i ) constitutes a trigger message profile for i if and only if (m 2 i, m i ) constitutes a trigger message profile for i. Also, (m 1 i, m i ) constitutes a trigger message profile for no j N\{i}. Thus, distinguish the following six cases. Case A: Assume that (m 1 i, m i ) and (m 2 i, m i ) constitute trigger message profiles for i only (Case I, Case I). Let x, y A, Pi, Pi P, and t {1, 2,...} be such that Conditions (i) and (ii) hold (cf. page 3). Assume s i > t. Then g(m 1 i, m i ) = τ 1 (P i ). Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). Otherwise, assume s i t. Then, g(m 1 i, m i ) {x, y, τ 1 (P i )}, depending on how announced (P i, P i ) relates to (Pi, Pi ), and on how s i relates to t. Also, g(m 2 i, m i ) {x, y, τ 1 (P i )}, depending on C i, on how announced (P i, P i ) relates to (Pi, Pi ), and if C i = B, on how s i relates to t. Distinguish five cases. A.1 P i = P i = P = Pi, Case i ). As s i t, hence s i < t, we have g(m 2 i, m i ) = y. By If s i < t, then g(m 1 i, m i ) = x. If s i = t, then g(m 1 i, m i ) = y. As P i I.b or I.d holds for (P i, P assumption, (x, y) is associated with (Pi, Pi ) and P i = P g(m 1 i, m i ) R i g(m 2 i, m i ). A.2 P i = P i = P i, hence x P i y. Thus, As s i t, we have g(m 1 i, m i ) = y. As P i = Pi, Case I.a or I.c holds for (P i, P i ). As s i t, hence s i < t, we have g(m 2 i, m i ) = x. By assumption, (x, y) is associated with (Pi, Pi ) and P i = Pi, hence y P i x. Thus, g(m 1 i, m i ) P i g(m 2 i, m i ). A.3 P i = P i P As s i t, we have g(m 1 i, m i ) = x. As P i = Pi, hence P i Pi, and P i Pi, Case I.e holds for (P i, P i ). As s i t, hence s i < t, we have g(m 2 i, m i ) = τ 1 (P i, {x, y}). 5

6 By assumption, (x, y) is associated with (Pi, Pi ) and P i = Pi, hence x P i y. Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). A.4 P i = P i P As s i t, we have g(m 1 i, m i ) = y. As P i = Pi, hence P i Pi, and P i Pi, Case I.e holds for (P i, P i ). As s i t, hence s i < t, we have g(m 2 i, m i ) = τ 1 (P i, {x, y}). By assumption, (x, y) is associated with (Pi, Pi ) and P i = Pi, hence y P i x. Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). A.5 P i P i, P As s i t, we have g(m 1 i, m i ) = τ 1 (P i, {x, y}). Either case holds for (P i, P i ). As s i t, hence s i < t, in Case I.a or I.c, g(m 2 i, m i ) = x, in Case I.b or I.d, g(m 2 i, m i ) = y, and in Case I.e, g(m 2 i, m i ) = τ 1 (P i, {x, y}). Altogether, g(m 2 i, m i ) {x, y}. Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). Case B: Assume that (m 1 i, m i ) constitutes a trigger message profile for i only and (m 2 i, m i ) constitutes a trigger message profile for i and some j N\{i} (Case I, Case II). Let x, y A, Pi, Pi P, and t {1, 2,...} be such that Conditions (i) and (ii) hold (cf. page 3). In Case I.a, g(m 1 i, m i ) {x, y, τ 1 (P i )}. In Case b or d, g(m 1 i, m i ) {y, τ 1 (P i )}. In Case I.c, g(m 1 i, m i ) {x, τ 1 (P i )}. In Case I.e, g(m 1 i, m i ) {τ 1 (P i, {x, y}), τ 1 (P i )}. Altogether, g(m 1 i, m i ) {x, y, τ 1 (P i )}. Note that g(m 1 i, m i ) = x only if x P i y. Also, τ 1 (P i ) = y and for each l N\{i}, we have τ 1 (P l ) = y, hence g(m 2 i, m i ) = y. Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). Case C: Assume that (m 1 i, m i ) constitutes a trigger message profile for no j N, and (m 2 i, m i ) constitutes a trigger message profile for some j N\{i} (Case III, Case I). Let h N announce B and be the winner, among all announcing B in (m 1 i, m i ), of the integer game for announced shades (s l ) l N, i.e. s h = max {l N Cl =B} s l. Break ties by means of ordering >. Then, g(m 1 i, m i ) = τ 1 (P h ). Note that h {i, j}, and h = j only if C j = B. Let x, y A, Pi, Pi P, and t {1, 2,...} be such that Conditions (i) and (ii) hold. Then, g(m 2 i, m i ) {x, y, τ 1 (P j )}, 3 depending on C j, on how announced (P j, P j) relates to (Pj, Pj ), and if C j = B, on how s j relates to t. Distinguish two cases. If C j {G, R}, or C j = B and s j < s i or s j = s i and i > j, then g(m 1 i, m i ) = τ 1 (P i ). If C j = B and s j > s i or s j = s i and j > i, hence s j > s i = t, then g(m 1 i, m i ) = g(m 2 i, m i ) = τ 1 (P j ). Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). 3 In Case I.a, g(m 2 i, m i) {x, y, τ 1 (P j )}. In Case b or d, g(m 2 i, m i) {y, τ 1 (P j )}. In Case I.c, g(m 2 i, m i) {x, τ 1 (P j )}. In Case I.e, g(m 2 i, m i) {τ 1 (P j, {x, y}), τ 1 (P j )}. 6

7 Case D: Assume that (m 1 i, m i ) constitutes a trigger message profile for no j N, and (m 2 i, m i ) constitutes a trigger message profile for some j, k N\{i} with j k (Case III, Case II). For each l N\{i}, we have C l = R, hence g(m 1 i, m i ) = τ 1 (P i ). Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). Case E: Assume that neither (m 1 i, m i ) nor (m 2 i, m i ) constitutes a trigger message profile for some j N, and there is some j N announcing B in (m 2 i, m i ) (Case III, Case III). Let h, k N announce B and be the winner, among all announcing B in (m 1 i, m i ) and (m 2 i, m i ) resp., of the integer game for announced shades (s l ) l N and (s i, s i ) resp., i.e. s h = max {l N Cl =B} s l, and s k s i and s k = max {l N\{i} Cl =B} s l. Break ties by means of ordering >. Then, g(m 1 i, m i ) = τ 1 (P h ) and g(m 2 i, m i ) = τ 1 (P k ). Distinguish two cases. If h = i, then g(m 1 i, m i ) = τ 1 (P i ). If h i, then h = k, hence g(m 1 i, m i ) = g(m 2 i, m i ) = τ 1 (P h ). Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). Case F: Assume that (m 1 i, m i ) nor (m 2 i, m i ) constitutes a trigger message profile for some j N, and no j N announcing B in (m 2 i, m i ) (Case III, Case IV). For each l N\{i}, we have C l {G, R}, hence g(m 1 i, m i ) = τ 1 (P i ). Thus, g(m 1 i, m i ) R i g(m 2 i, m i ). [Step 2:] True ordering, any secondary ordering, any color, any shade, and any integer announcement is undominated at the true ordering. Let i N, P i P, and m 1 i = (P i, P i, C i, s i, λ i ) M i and m 2 i = (P i, P i, C i, s i, λ i) M i be such that m 1 i m 2 In what follows, we prove that either there is m i M i such that g(m 1 i, m i ) P i g(m 2 i, m i ) or for each m i M i, we have g(m 1 i, m i ) = g(m 2 i, m i ). Thus, m 2 i does not weakly dominate m 1 i at P We distinguish four cases. Case A: C i {G, R} and C i = B. Let x, y A and m i = (P l, P l, R, t, λ l) l N\{i} M i be such that P i is unanimous on (y, x) and (x, y) is associated with (P i, P i ). Then, (m 1 i, m i ) and (m 2 i, m i ) constitute trigger messages profile for i. If y P i x, assume t > s i. As y P i x, hence P i P i, Case I.b, I.d, or I.e holds for (P i, P i ), or Case II holds. In Case I.b or d, g(m 1 i, m i ) = y. In Case I.e, g(m 1 i, m i ) = τ 1 (P i, {x, y}), hence g(m 1 i, m i ) = y. In Case II, g(m 1 i, m i ) = y. Altogether, g(m 1 i, m i ) = y. Case I.a holds for (P i, P i ) and t > s i, hence g(m 2 i, m i ) = x. Thus, g(m 1 i, m i ) P i g(m 2 i, m i ). If x P i y, assume t = s i. As x P i y, hence P i P i, Case I.a, I.c, or I.e holds for (P i, P i ). In Case I.a or I.c, g(m 1 i, m i ) = x. In Case I.e, g(m 1 i, m i ) = τ 1 (P i, {x, y}), hence g(m 1 i, m i ) = x. Altogether, g(m 1 i, m i ) = x. Case I.a holds for (P i, P i ) and s i = t, hence g(m 2 i, m i ) = y. Thus, g(m 1 i, m i ) P i g(m 2 i, m i ). Case B: C i = B and C i = B. 7

8 First, assume P i = P i and P i = P i, and s i = s i. Then, for each m i M i, we have g(m 1 i, m i ) = g(m 2 i, m i ). Second, assume P i P i and P i P i, and s i = s i. Assume that for each P i D i, each x, y X, and each Pi, Pi D i such that P i is unanimous on (y, x) and (x, y) is associated with (Pi, Pi ), we have x P i y if and only if x P i y. Also, assume τ 1 (P i ) = τ 1 (P i ). Then, for each m i M i, we have g(m 1 i, m i ) = g(m 2 i, m i ). Otherwise, either w.l.o.g. x P i y and y P i x or τ 1 (P i ) τ 1 (P i ) (or both). If the former holds, let m i = (P l, P l, R, t, λ l) l N\{i} M i, x, y X, and Pi, Pi D i be such that P i is unanimous on (y, x) and (x, y) is associated with (Pi, Pi ). Then, (m 1 i, m i ) and (m 2 i, m i ) constitute trigger messages profile for i only. Also, assume t > s i and t > s i. As x P i y, hence P i Pi, Case I.a, I.c, or I.e holds for (P i, P i ). As t > s i, in Case I.a or c, g(m 1 i, m i ) = x, and in Case I.e, g(m 1 i, m i ) = τ 1 (P i, {x, y}), hence g(m 1 i, m i ) = x. As y P i x, hence P i Pi, Case I.b, I.d, or I.e holds for (P i, P i ). As t > s i, in Case I.b or d, g(m 2 i, m i ) = y, and in Case I.e, g(m 2 i, m i ) = τ 1 (P i, {x, y}), hence g(m 2 i, m i ) = y. Thus, g(m 1 i, m i ) = x and g(m 2 i, m i ) = y, hence g(m 1 i, m i ) P i g(m 2 i, m i ). If the latter holds, let m i = (P l, P l, G, s l, λ l ) l N\{i} M Case III holds for (m 1 i, m i ) and (m 2 i, m i ), hence g(m 1 i, m i ) = τ 1 (P i ) and g(m 2 i, m i ) = τ 1 (P i ). Thus, g(m 1 i, m i ) P i g(m 2 i, m i ). Case C: C i {G, R} and C i {G, R}. Let m i = (P l, P l, G, s l, λ l ) l N\{i} M i, x, y A, and P i P be such that x P i y and y P i x, P i is unanimous on (x, y), and for each l N\{i}, if τ 1 (P i ) x, then τ 3 (P l ) = τ 1 (P i ). Note that F (P i, P i ) = {x, τ 1 (P i )} and {x, y} F (P i, P i ). Then, further assume that if l N λ l mod F (P i, P i ) + 1 = k, then a k = τ 1 (P i ), and if λ i + l N\{i} λ l mod F (P i, P i ) + 1 = k, then a k τ 1 (P i ). Case IV holds for (m 1 i, m i ) and (m 2 i, m i ), hence g(m 1 i, m i ) = τ 1 (P i ) and g(m 2 i, m i ) τ 1 (P i ). Thus, g(m 1 i, m i ) P i g(m 2 i, m i ). Case D: C i = B and C i {G, R}. Let m i = (P l, P l, G, s l, λ l ) l N\{i} M i, x, y A, and P i P i be such that x P i y and y F (P i, P i ), 4 and if λ i + l N\{i} λ l mod F (P i, P i ) + 1 = k, then a k = y. Case III holds for (m 1 i, m i ), hence g(m 1 i, m i ) = τ 1 (P i ). Case IV holds for (m 2 i, m i ), hence g(m 2 i, m i ) = y. As x P i y, hence τ 1 (P i ) y, we have g(m 1 i, m i ) P i g(m 2 i, m i ). [Step 3:] Any alternative is the outcome of an undominated message profile at any ordering profile if and only if it belongs to the Pareto correspondence for this profile. Let P P N and P ar(p ) be the set of alternatives selected by the Pareto correspondence for P. In what follows, we prove g(u(p )) = P ar(p ). Let x P ar(p ). Let m = (P l, P l, G, s l, λ l ) l N M be such that if l N λ l mod F (P ) + 1 = k, then a k = x. By Step 2, for each l N, we have m l U(P l ). 4 E.g. let P i be unanimous on (y, x). 8

9 For each l N, let m l = (P l, P l, C l, s l, λ l ) U(P l ). By Steps 1 and 2, P l = P l. If there is i N for whom m = (m l ) l N constitutes a trigger case, then there are x, y A such that P i is unanimous on (y, x) and g(m) {x, y, τ 1 (P i )}. 5 As P i is unanimous on (y, x), and g(m 1 i, m i ) = x only if x P i y, if g(m 1 i, m i ) = x, then x P ar(p ). Also, y, τ 1 (P i ) P ar(p ). If there is no i N for whom m = (m l ) l N constitutes a trigger case and for each l N, we have C l {G, R}, then there is a k P ar(p ) such that k is the winner of the modulo game for announced (λ l ) l N, i.e. k l N λ l mod F (P ) + 1, and g(m) = a k. Otherwise, there is i N such that C i = B and g(m) = τ 1 (P i ). Thus, τ 1 (P i ) P ar(p ). Altogether, g(m) P ar(p ). [Step 4:] The mechanism is bounded. Let i N, P i P, and m i = (P i, P i, C i, s i, λ i ) U(P i ). By Step 2, P i P Let m i = (P i, P i, C i, s i, λ i) M i be such that C i = B and s i > s By Step 2, m i U(P i ). By Step 1, m i weakly dominates m i at P 4 Discussion A useful starting point for a discussion of the mechanism is the pseudo-random dictatorship mechanism constructed in Börgers (1991). Agents play the well-known modulo game and the winner gets to choose an alternative; equivalently they announce an ordering and the winner gets the best outcome according to the announced ordering. Announcing a false preference ordering and an arbitrary integer is weakly dominated by announcing her true ordering and the same integer. Moreover announcing her true ordering and an arbitrary integer is an undominated strategy. This mechanism implements a sub-correspondence of the Pareto correspondence, the correspondence consisting of the best elements of all agents at every profile. The drawback of the mechanism as pointed out in Börgers (1991) is that always excludes compromises. Our result shows that we can successfully implement correspondences that include all compromises. The mechanism is a suitable generalisation of the pseudo-random dictatorship mechanism. Consider the following first attempt to include compromises by modifying the pseudorandom dictatorship mechanism. Agents again play the modulo game and the winner gets to choose and alternative from the Pareto set at that profile. This attempt clearly fails because the strategy proposing a compromise will be weakly dominated by the strategy proposing the agent s best alternative at her true preference ordering. The basic idea behind our mechanism is ensure that this does not happen, i.e. the strategy proposing a compromise and the strategy 5 In Case I.a, g(m) {x, y, τ 1 (P i )}. In Case b or d, g(m) {y, τ 1 (P i )}. In Case I.c, g(m) {x, τ 1 (P i )}. In Case I.e, g(m) {τ 1 (P i, {x, y}), τ 1 (P i )}. 9

10 proposing the best alternative at a preference ordering are both undominated. Our mechanism works as follows. All agents announce two orderings, one primary and one secondary, one of three colours, Green (G), Blue (B) and Red (R) and two positive integers (one called a shade ). If all agents announce G, the outcome is an element of the Pareto set at the announced profile using the primary preference announcements. The actual alternative chosen is determined by one of the announced integers in a modulo-game manner and all alternatives can be chosen by varying announcements. If some agents announce B, the outcome is the maximal alternative according to the primary preference ordering announced by the agent announcing the highest shade. 6 It is important to ensure that announcing B and the true ordering does not weakly dominate announcing G and the true ordering. this is achieved by the R messages. If all agents other than some i announce a primary ordering where alternative y is first-ranked and x is second ranked, the colour R and the same shade t, the message profile is referred to as a trigger profile for i for a pair of orderings (Pi, Pi ) where xpi y and ypi x. The outcome at such a profile depends on the primary and secondary orderings of i, the colour and the shade i announces. For example, suppose i announces (Pi, Pi ). If her colour announcement is either G or R, the outcome is x. If she announces B, the outcome is x if her shade is less than t, y if her shade is equal to t and the maximal element according to Pi if her shade is greater than t. A key feature of this construction is that if Pi is i s true ordering, by announcing G and an arbitrary shade, i gets x while by announcing B and the same shade she gets y at the trigger profile for i where all the others are announcing this shade. Since xpi y, announcing G and the true ordering is not dominated by announcing B and the true ordering. Suppose i s true ordering is Pi. Consider an arbitrary message by i where her primary announcement is P i Pi. Then this message is weakly dominated by another message where i s primary announcement is Pi, her secondary announcement is P i, her colour is B and her shade announcement is higher. Also announcing her true ordering as her primary ordering, an arbitrary secondary ordering, colour, shade and integer is undominated. This suffices to show that the mechanism USBM implements the Pareto correspondence. The mechanism exploits several key properties of the Pareto correspondence. The first is tops inclusivity, i.e. τ 1 (P i ) F (P i, P i ) for all (P i, P i ). The second is the following. Let P be a profile, i be an agent and P i be an ordering. Suppose (i) xp i y and yp i x, y = τ 1 (P i ) P i i, P i ) for all and (ii) x = τ 2 (P i ). Then x, y F (P ), y F (P i, P i ) and max P i {x, y} F (P P. In other words, if all agents other than i have y and x as their first and secondranked alternatives respectively, then both x and y belong to the Pareto correspondence when i ranks x above y. Also y belongs to the Pareto correspondence when y is above x. In general the better alternative between x and y in any ordering of i belongs to the Pareto correspondence. Finally, the correspondence satisfies the strict form of unanimity: if all 6 Ties are broken according to a fixed ordering over agents. 10

11 agents rank y first, then the only y is picked by the correspondence. In fact, the mechanism can be used to implement any SCC defined over an arbitrary domain that satisfies these properties. References [1] Börgers, T. (1991): Undominated strategies and coordination in normal-form games, Social Choice and Welfare, 8, [2] Carroll, G. (2014): A complexity result for undominated-strategy implementation, working paper, Stanford University. [3] Jackson, M. O. (1992): Implementation in undominated strategies: A look at bounded mechanisms, Review of Economic Studies, 59, [4] Mukherjee, S, Muto, N and E. Ramaekers (2016): Implementation in undominated strategies with partially honest agents, mimeo. [5] Ohseto, S. (1994): Implementation of the plurality correspondence in undominated strategies by a bounded mechanism, The Economic Studies Quarterly, 54, [6] Yamashita, T. (2012): A necessary condition for implementation in undominated strategies, with applications to robustly optimal trading mechanisms, Mimeo, Toulouse School of Economics. [7] Yamashita, T. (2015): Implementation in weakly undominated strategies: Optimality of second-price auction and posted-price mechanism, Review of Economic Studies, 82,