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On the Equivalence of Bayesian and Dominant Strategy Implementation

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NES 20th Anniversary Conference, Dec 13-16, 2012 …

NES 20th Anniversary Conference, Dec 13-16, 2012
On the Equivalence of Bayesian and Dominant Strategy Implementation (based on the article presented by Alexey Kushnir at the NES 20th Anniversary Conference).
Authors: Alex Gershkov, Jacob K. Goeree, Alexey Kushnir,
Benny Moldovanu, Xianwen Shi
Authors: Alex Gershkov, Jacob K. Goeree, Alexey Kushnir, Benny Moldovanu, Xianwen Shi

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  • 1. On the Equivalence of Bayesian andDominant Strategy Implementation Alex Gershkov, Jacob K. Goeree, Alexey Kushnir, Benny Moldovanu, Xianwen Shi NES 20th Anniversary Conference, December 2012
  • 2. This paper subsumes Gershkov, Moldovanu, and Shi “Bayesian and Dominant Strategy Implementation Revisited” (2011) Goeree and Kushnir “On the Equivalence of Bayesian and Domi- nant Strategy Implementation in a General Class of Social Choice Problems” (2011)
  • 3. This paper subsumes Gershkov, Moldovanu, and Shi “Bayesian and Dominant Strategy Implementation Revisited” (2011) Goeree and Kushnir “On the Equivalence of Bayesian and Domi- nant Strategy Implementation in a General Class of Social Choice Problems” (2011) ...forthcoming in Econometrica
  • 4. Motivation Bayesian Implementation VS Dominant Strategy Implementation
  • 5. MotivationWilson (1987)’s critique of Bayesian approach: ”... it (game theory) isdeficient to the extent it assumes other features to be common knowledge,such as one agent’s probability assessment about another’s preferences orinformation”.
  • 6. Main result For any Bayesian IC and interim IR mechanism there exists a dominant strategy IC and ex post IR mechanism that delivers a) the same interim expected utilities to agents, b) the same ex ante expected social surplus. Environment: social choice, linear utilities, and independent, one-dimensional, private values
  • 7. Related literature Mookherjee and Reichelstein (1992) single crossing utility functions any monotone BIC social choice rule is DIC implementable
  • 8. Related literature Mookherjee and Reichelstein (1992) single crossing utility functions any monotone BIC social choice rule is DIC implementable Vincent and Manelli (2010) 1-unit auctions, IPV, symmetric and asymmetric for any BIC mechanism there is DIC mechanism that yields the same interim expected allocation probabilities the same interim expected utilities of agents
  • 9. Related literature Mookherjee and Reichelstein (1992) single crossing utility functions any monotone BIC social choice rule is DIC implementable Vincent and Manelli (2010) 1-unit auctions, IPV, symmetric and asymmetric for any BIC mechanism there is DIC mechanism that yields the same interim expected allocation probabilities the same interim expected utilities of agents Goeree and Kushnir (2012) a geometric approach to mechanism design characterize the set of interim expected agent values implementable with an IC and IR mechanisms
  • 10. Model
  • 11. Model Set of agents I = {1, ..., I }
  • 12. Model Set of agents I = {1, ..., I } Set of social alternatives K = {1, ..., K }
  • 13. Model Set of agents I = {1, ..., I } Set of social alternatives K = {1, ..., K } uik (xi , ti ) = aik xi + ti aik ≥ 0 xi distributed according to λi with support Xi = [xi , xi ] ¯ ti agent’s transfer ¯
  • 14. Model Set of agents I = {1, ..., I } Set of social alternatives K = {1, ..., K } uik (xi , ti ) = aik xi + ti aik ≥ 0 xi distributed according to λi with support Xi = [xi , xi ] ¯ ti agent’s transfer ¯ x ∈ X = Πi ∈I Xi , vectors are bold-faced
  • 15. Model Direct mechanism (q, t): q k : X → [0, 1], k ∈ K, ∑k ∈K q k (x) = 1 ti : X → R, i ∈ I
  • 16. Model Direct mechanism (q, t): q k : X → [0, 1], k ∈ K, ∑k ∈K q k (x) = 1 ti : X → R, i ∈ I Qik (xi ) = Ex−i (q k (xi , x−i )) interim probability alternative k is chosen
  • 17. Model Direct mechanism (q, t): q k : X → [0, 1], k ∈ K, ∑k ∈K q k (x) = 1 ti : X → R, i ∈ I Qik (xi ) = Ex−i (q k (xi , x−i )) interim probability alternative k is chosen Ti (xi ) = Ex−i (ti (xi , x−i )) the expected transfer to agent i
  • 18. Model Direct mechanism (q, t): q k : X → [0, 1], k ∈ K, ∑k ∈K q k (x) = 1 ti : X → R, i ∈ I Qik (xi ) = Ex−i (q k (xi , x−i )) interim probability alternative k is chosen Ti (xi ) = Ex−i (ti (xi , x−i )) the expected transfer to agent i Agent i’s interim expected utility from truthful reporting: ui (xi ) = Ex−i (∑k ∈K aik q k (x)xi + ti (x)) = ∑k ∈K aik Qik (xi )xi + Ti (xi )
  • 19. Model To simplify notation: vi ( x ) = ∑k ∈K aik qk (x) Vi (xi ) = ∑k ∈K aik Qik (xi ) and v = (v1 , ..., vI ), V = (V1 , ..., VI ). Agent i’s interim expected utility from truthful reporting: ui (xi ) = Vi (xi )xi + Ti (xi )
  • 20. ModelDefinition (Equivalent mechanisms)Two mechanisms (q, t) and (q, ˜) are equivalent if they deliver ˜ t
  • 21. ModelDefinition (Equivalent mechanisms)Two mechanisms (q, t) and (q, ˜) are equivalent if they deliver ˜ t the same interim expected utilities to all agents,
  • 22. ModelDefinition (Equivalent mechanisms)Two mechanisms (q, t) and (q, ˜) are equivalent if they deliver ˜ t the same interim expected utilities to all agents, the same ex ante expected social surplus.
  • 23. Main theorem
  • 24. Preliminary stepsFact 1. (e.g. Myerson, 1981) A mechanism is BIC iff (i) for all i ∈ I and xi ∈ Xi , Vi (xi ) is non-decreasing in xi and (ii) agents interim expected utilities satisfy xi ui (xi ) = ui (x i ) + Vi (t )dt xiFact 2. (e.g. Laffont and Maskin, 1980) A mechanism is DIC iff (i) for all i ∈ I and xi ∈ Xi , vi (xi , x−i ) is non-decreasing in xi and (ii) agents utilities satisfy xi ui ( x i , x − i ) = ui ( x i , x − i ) + vi (t, x−i )dt xi
  • 25. BIC-DIC equivalenceTheoremLet (q, ˜) be a BIC and interim IR mechanism. An equivalent DIC and ex ˜ tpost IR mechanism is given by (q, t), where q solves min Ex ||v(x)||2 {q k }k ∈K  k  q (x) ≥ 0 ∀k, x   s.t. ∑ q k (x) = 1 ∀x (Program)  k  Vi (xi ) = Vi (xi ) ∀i, xi 
  • 26. BIC-DIC equivalenceTheoremLet (q, ˜) be a BIC and interim IR mechanism. An equivalent DIC and ex ˜ tpost IR mechanism is given by (q, t), where q solves min Ex ||v(x)||2 {q k }k ∈K  k  q (x) ≥ 0 ∀k, x   s.t. ∑ q k (x) = 1 ∀x (Program)  k  Vi (xi ) = Vi (xi ) ∀i, xi and transfers t follow from xi ti (xi , x−i ) = ti (x i , x−i ) + vi (xi , x−i )x i − vi (xi , x−i )xi + vi (t, x−i )dt. ¯ xiwhere ti (x i , x−i ) = (vi (x i , x−i )/Vi (x i ))Ti (x i ).
  • 27. BIC-DIC equivalence. Proof. Prove vi (xi , x−i ) = ∑k ∈K aik q k (xi , x−i ) is non-decreasing in xi for q – solution to (Program), Steps of the proof:
  • 28. BIC-DIC equivalence. Proof. Prove vi (xi , x−i ) = ∑k ∈K aik q k (xi , x−i ) is non-decreasing in xi for q – solution to (Program), Steps of the proof:1. Discrete and uniformly distributed types (Lemma 1) an extension of the theorem due to Gutmann et al. (1991)
  • 29. BIC-DIC equivalence. Proof. Prove vi (xi , x−i ) = ∑k ∈K aik q k (xi , x−i ) is non-decreasing in xi for q – solution to (Program), Steps of the proof:1. Discrete and uniformly distributed types (Lemma 1) an extension of the theorem due to Gutmann et al. (1991)2. Continuous and uniformly distributed types (Lemma 2)
  • 30. BIC-DIC equivalence. Proof. Prove vi (xi , x−i ) = ∑k ∈K aik q k (xi , x−i ) is non-decreasing in xi for q – solution to (Program), Steps of the proof:1. Discrete and uniformly distributed types (Lemma 1) an extension of the theorem due to Gutmann et al. (1991)2. Continuous and uniformly distributed types (Lemma 2)3. Arbitrary type distributions (Lemma 3)
  • 31. BIC-DIC equivalence. Proof of Lemma 1. Lemma 1. Xi is discrete, λi is uniform. Let q be a solution of (Program) then vi (xi , x−i ) is non-decreasing in xi for all i ∈ I , x ∈ X .Proof outline.
  • 32. BIC-DIC equivalence. Proof of Lemma 1. Lemma 1. Xi is discrete, λi is uniform. Let q be a solution of (Program) then vi (xi , x−i ) is non-decreasing in xi for all i ∈ I , x ∈ X .Proof outline. 1. The set of solutions to (Program) is not empty.
  • 33. BIC-DIC equivalence. Proof of Lemma 1. Lemma 1. Xi is discrete, λi is uniform. Let q be a solution of (Program) then vi (xi , x−i ) is non-decreasing in xi for all i ∈ I , x ∈ X .Proof outline. 1. The set of solutions to (Program) is not empty. 2. Assume vi (xi , x−i ) is not non-decreasing.
  • 34. BIC-DIC equivalence. Proof of Lemma 1. Lemma 1. Xi is discrete, λi is uniform. Let q be a solution of (Program) then vi (xi , x−i ) is non-decreasing in xi for all i ∈ I , x ∈ X .Proof outline. 1. The set of solutions to (Program) is not empty. 2. Assume vi (xi , x−i ) is not non-decreasing. 3. Construct feasible q that delivers lower value to Ex ||v(x)||2 .
  • 35. BIC-DIC equivalence. Proof of Lemma 1. Lemma 1. Xi is discrete, λi is uniform. Let q be a solution of (Program) then vi (xi , x−i ) is non-decreasing in xi for all i ∈ I , x ∈ X .Proof outline. 1. The set of solutions to (Program) is not empty. 2. Assume vi (xi , x−i ) is not non-decreasing. 3. Construct feasible q that delivers lower value to Ex ||v(x)||2 . 4. Contradiction.
  • 36. BIC-DIC equivalence. Proof of Lemma 1.Proof. 1. The set of solutions of (Program) is not empty. min Ex ||v(x)||2 (Program) {q k }k ∈K  k  q (x) ≥ 0 ∀k, x s.t. ∑ q k (x) = 1 ∀x  k Vi (xi ) = Vi (xi ) ∀i, xi
  • 37. BIC-DIC equivalence. Proof of Lemma 1.Proof. 1. The set of solutions of (Program) is not empty. min Ex ||v(x)||2 (Program) {q k }k ∈K  k  q (x) ≥ 0 ∀k, x s.t. ∑ q k (x) = 1 ∀x  k Vi (xi ) = Vi (xi ) ∀i, xi 2. Suppose vj (xj , x−j ) > vj (xj , x−j ) for some j, xj > xj , and some x−j .
  • 38. BIC-DIC equivalence. Proof of Lemma 1.Proof. 1. The set of solutions of (Program) is not empty. min Ex ||v(x)||2 (Program) {q k }k ∈K  k  q (x) ≥ 0 ∀k, x s.t. ∑ q k (x) = 1 ∀x  k Vi (xi ) = Vi (xi ) ∀i, xi 2. Suppose vj (xj , x−j ) > vj (xj , x−j ) for some j, xj > xj , and some x−j . 3. {q k }k ∈K being BIC means Vj (xj ) = Vj (xj ) = Ex−j v (xj , x−j ) is ˜ non-decreasing.
  • 39. BIC-DIC equivalence. Proof of Lemma 1.Proof. 1. The set of solutions of (Program) is not empty. min Ex ||v(x)||2 (Program) {q k }k ∈K  k  q (x) ≥ 0 ∀k, x s.t. ∑ q k (x) = 1 ∀x  k Vi (xi ) = Vi (xi ) ∀i, xi 2. Suppose vj (xj , x−j ) > vj (xj , x−j ) for some j, xj > xj , and some x−j . 3. {q k }k ∈K being BIC means Vj (xj ) = Vj (xj ) = Ex−j v (xj , x−j ) is ˜ non-decreasing. 4. ∃ x−j such that vj (xj , x−j ) < vj (xj , x−j ). Consider allocations at 4 type-points.
  • 40. BIC-DIC equivalence. Proof of Lemma 1. Allocation Space q(xj , x−j ) q(xj , x −j ) q(xj , x−j ) q(xj , x −j )
  • 41. BIC-DIC equivalence. Proof of Lemma 1. V-Space v(xj , x−j ) Does q satisfy:  k v(xj , x −j )  q (x) ≥ 0 ∀k, x ∑ q k (x) = 1 ∀x ?  k ˜ Vi (xi ) = Vi (xi ) ∀i, xi v(xj , x−j ) v(xj , x −j )
  • 42. BIC-DIC equivalence. Proof of Lemma 1. The above argument shows q is feasible. Direct calculations also show Ex (||v (x)||2 − ||v(x)||2 ) < 0 This contradicts to {q k }k ∈K being a solution to (Program). QED
  • 43. The limits of BIC-DIC equivalence
  • 44. The limits of BIC-DIC equivalence Stronger notion of the equivalence counterexample to equivalence based on Qik (xi ) = Qik (xi ) Correlated values (see Cremer and McLean, 1988) Multidimensional types Non-linear utilities Interdependent values
  • 45. Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir A discrete version of Maskin (1992)’s example
  • 46. Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir A discrete version of Maskin (1992)’s example Single-unit auction, 2 bidders, 2 types {x, x }
  • 47. Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir A discrete version of Maskin (1992)’s example Single-unit auction, 2 bidders, 2 types {x, x } K = 3 states: assign to bidder 1, bidder 2, or not at all
  • 48. Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir A discrete version of Maskin (1992)’s example Single-unit auction, 2 bidders, 2 types {x, x } K = 3 states: assign to bidder 1, bidder 2, or not at all Bidder i’s value vi (xi , xj ) = xi + αxj
  • 49. Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir A discrete version of Maskin (1992)’s example Single-unit auction, 2 bidders, 2 types {x, x } K = 3 states: assign to bidder 1, bidder 2, or not at all Bidder i’s value vi (xi , xj ) = xi + αxj Compare BIC with EPIC (ex post incentive compatibility)
  • 50. Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir A discrete version of Maskin (1992)’s example Single-unit auction, 2 bidders, 2 types {x, x } K = 3 states: assign to bidder 1, bidder 2, or not at all Bidder i’s value vi (xi , xj ) = xi + αxj Compare BIC with EPIC (ex post incentive compatibility) v v v 1 α=0 α= 2 α=2 EPIC EPIC EPIC BIC BIC BIC v v vFeasible (yellow shaded), BIC (light gray), and EPIC (dark gray)
  • 51. Conclusion
  • 52. Conclusion BIC-DIC equivalence for social choice problems in settings with independent private values, linear util- ities, one-dimensional types, and for general type distributions (including discrete type-space).
  • 53. Conclusion BIC-DIC equivalence for social choice problems in settings with independent private values, linear util- ities, one-dimensional types, and for general type distributions (including discrete type-space). The proof is short and constructive.
  • 54. Conclusion BIC-DIC equivalence for social choice problems in settings with independent private values, linear util- ities, one-dimensional types, and for general type distributions (including discrete type-space). The proof is short and constructive. Identify limits of BIC-DIC equivalence: stronger no- tion of equivalence, interdependent values, correlated values, multi-dimensional types, non-linear utilities.

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