on the equivalence of bayesian and dominant strategy implementation

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

This paper subsumes

Gershkov, Moldovanu, and Shi “Bayesian and Dominant StrategyImplementation Revisited” (2011)

Goeree and Kushnir “On the Equivalence of Bayesian and Domi-nant Strategy Implementation in a General Class of Social ChoiceProblems” (2011)

...forthcoming in Econometrica

This paper subsumes

Gershkov, Moldovanu, and Shi “Bayesian and Dominant StrategyImplementation Revisited” (2011)

Goeree and Kushnir “On the Equivalence of Bayesian and Domi-nant Strategy Implementation in a General Class of Social ChoiceProblems” (2011)

...forthcoming in Econometrica

Motivation

Bayesian Implementation

VS

Dominant Strategy Implementation

Motivation

Wilson (1987)’s critique of Bayesian approach: ”... it (game theory) is

deficient to the extent it assumes other features to be common knowledge,such as one agent’s probability assessment about another’s preferences orinformation”.

Main result

For any Bayesian IC and interim IR mechanismthere exists a dominant strategy IC and ex post IRmechanism 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

Related literature

I Mookherjee and Reichelstein (1992)

I single crossing utility functionsI any monotone BIC social choice rule is DIC implementable

I Vincent and Manelli (2010)I 1-unit auctions, IPV, symmetric and asymmetricI for any BIC mechanism there is DIC mechanism that yields

I the same interim expected allocation probabilitiesI the same interim expected utilities of agents

I Goeree and Kushnir (2012)I a geometric approach to mechanism designI characterize the set of interim expected agent values implementable with

an IC and IR mechanisms

Related literature

I Mookherjee and Reichelstein (1992)

I single crossing utility functionsI any monotone BIC social choice rule is DIC implementable

I Vincent and Manelli (2010)I 1-unit auctions, IPV, symmetric and asymmetricI for any BIC mechanism there is DIC mechanism that yields

I the same interim expected allocation probabilitiesI the same interim expected utilities of agents

I Goeree and Kushnir (2012)I a geometric approach to mechanism designI characterize the set of interim expected agent values implementable with

an IC and IR mechanisms

Related literature

I Mookherjee and Reichelstein (1992)

I single crossing utility functionsI any monotone BIC social choice rule is DIC implementable

I Vincent and Manelli (2010)I 1-unit auctions, IPV, symmetric and asymmetricI for any BIC mechanism there is DIC mechanism that yields

I the same interim expected allocation probabilitiesI the same interim expected utilities of agents

I Goeree and Kushnir (2012)I a geometric approach to mechanism designI characterize the set of interim expected agent values implementable with

an IC and IR mechanisms

Model

Model

I Set of agents I = {1, ..., I}

I Set of social alternatives K = {1, ..., K}

I uki (xi , ti ) = aki xi + ti

I aki ≥ 0I xi distributed according to λi with support Xi = [x

¯i , xi ]

I ti agent’s transfer

I x ∈ X = Πi∈IXi , vectors are bold-faced

Model

I Set of agents I = {1, ..., I}

I Set of social alternatives K = {1, ..., K}

I uki (xi , ti ) = aki xi + ti

I aki ≥ 0I xi distributed according to λi with support Xi = [x

¯i , xi ]

I ti agent’s transfer

I x ∈ X = Πi∈IXi , vectors are bold-faced

Model

I Set of agents I = {1, ..., I}

I Set of social alternatives K = {1, ..., K}

I uki (xi , ti ) = aki xi + ti

I aki ≥ 0I xi distributed according to λi with support Xi = [x

¯i , xi ]

I ti agent’s transfer

I x ∈ X = Πi∈IXi , vectors are bold-faced

Model

I Set of agents I = {1, ..., I}

I Set of social alternatives K = {1, ..., K}

I uki (xi , ti ) = aki xi + ti

I aki ≥ 0I xi distributed according to λi with support Xi = [x

¯i , xi ]

I ti agent’s transfer

I x ∈ X = Πi∈IXi , vectors are bold-faced

Model

I Direct mechanism (q, t):I qk : X → [0, 1], k ∈ K, ∑k∈K qk (x) = 1I ti : X → R, i ∈ I

I Qki (x

′i ) = Ex−i (q

k(x ′i , x−i )) interim probability alternative k is chosen

I Ti (x ′i ) = Ex−i (ti (x′i , x−i )) the expected transfer to agent i

I Agent i ’s interim expected utility from truthful reporting:

ui (xi ) = Ex−i (∑k∈K aki qk(x)xi + ti (x))

= ∑k∈K aki Qki (xi )xi + Ti (xi )

Model

I Direct mechanism (q, t):I qk : X → [0, 1], k ∈ K, ∑k∈K qk (x) = 1I ti : X → R, i ∈ I

I Qki (x

′i ) = Ex−i (q

k(x ′i , x−i )) interim probability alternative k is chosen

I Ti (x ′i ) = Ex−i (ti (x′i , x−i )) the expected transfer to agent i

I Agent i ’s interim expected utility from truthful reporting:

ui (xi ) = Ex−i (∑k∈K aki qk(x)xi + ti (x))

= ∑k∈K aki Qki (xi )xi + Ti (xi )

Model

I Direct mechanism (q, t):I qk : X → [0, 1], k ∈ K, ∑k∈K qk (x) = 1I ti : X → R, i ∈ I

I Qki (x

′i ) = Ex−i (q

k(x ′i , x−i )) interim probability alternative k is chosen

I Ti (x ′i ) = Ex−i (ti (x′i , x−i )) the expected transfer to agent i

I Agent i ’s interim expected utility from truthful reporting:

ui (xi ) = Ex−i (∑k∈K aki qk(x)xi + ti (x))

= ∑k∈K aki Qki (xi )xi + Ti (xi )

Model

I Direct mechanism (q, t):I qk : X → [0, 1], k ∈ K, ∑k∈K qk (x) = 1I ti : X → R, i ∈ I

I Qki (x

′i ) = Ex−i (q

k(x ′i , x−i )) interim probability alternative k is chosen

I Ti (x ′i ) = Ex−i (ti (x′i , x−i )) the expected transfer to agent i

I Agent i ’s interim expected utility from truthful reporting:

ui (xi ) = Ex−i (∑k∈K aki qk(x)xi + ti (x))

= ∑k∈K aki Qki (xi )xi + Ti (xi )

Model

I To simplify notation:

vi (x) = ∑k∈K aki qk(x)

Vi (xi ) = ∑k∈K aki Qki (xi )

and v = (v1, ..., vI ), V = (V1, ..., VI ).

I Agent i ’s interim expected utility from truthful reporting:

ui (xi ) = Vi (xi )xi + Ti (xi )

Model

Definition (Equivalent mechanisms)

Two mechanisms (q, t) and (q, t) are equivalent if they deliver

I the same interim expected utilities to all agents,

I the same ex ante expected social surplus.

Model

Definition (Equivalent mechanisms)

Two mechanisms (q, t) and (q, t) are equivalent if they deliver

I the same interim expected utilities to all agents,

I the same ex ante expected social surplus.

Model

Definition (Equivalent mechanisms)

Two mechanisms (q, t) and (q, t) are equivalent if they deliver

I the same interim expected utilities to all agents,

I the same ex ante expected social surplus.

Main theorem

Preliminary steps

Fact 1. (e.g. Myerson, 1981) A mechanism is BIC iff (i) for all i ∈ I andxi ∈ Xi , Vi (xi ) is non-decreasing in xi and (ii) agents interimexpected utilities satisfy

ui (xi ) = ui (x i ) +∫ xi

x i

Vi (t)dt

Fact 2. (e.g. Laffont and Maskin, 1980) A mechanism is DIC iff (i) for alli ∈ I and xi ∈ Xi , vi (xi , x−i ) is non-decreasing in xi and (ii)agents utilities satisfy

ui (xi , x−i ) = ui (x i , x−i ) +∫ xi

x i

vi (t, x−i )dt

BIC-DIC equivalence

Theorem

Let (q, t) be a BIC and interim IR mechanism. An equivalent DIC and expost IR mechanism is given by (q, t), where q solves

min{qk}k∈K

Ex||v(x)||2

s.t.

qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀x

Vi (xi ) = Vi (xi ) ∀i , xi

(Program)

and transfers t follow from

ti (xi , x−i ) = ti (x i , x−i ) + vi (x¯ i , x−i )x i − vi (xi , x−i )xi +

∫ xi

x i

vi (t, x−i )dt.

where ti (x i , x−i ) = (vi (x i , x−i )/Vi (x i ))Ti (x i ).

BIC-DIC equivalence

Theorem

Let (q, t) be a BIC and interim IR mechanism. An equivalent DIC and expost IR mechanism is given by (q, t), where q solves

min{qk}k∈K

Ex||v(x)||2

s.t.

qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀x

Vi (xi ) = Vi (xi ) ∀i , xi

(Program)

and transfers t follow from

ti (xi , x−i ) = ti (x i , x−i ) + vi (x¯ i , x−i )x i − vi (xi , x−i )xi +

∫ xi

x i

vi (t, x−i )dt.

where ti (x i , x−i ) = (vi (x i , x−i )/Vi (x i ))Ti (x i ).

BIC-DIC equivalence. Proof.

I Prove vi (xi , x−i ) = ∑k∈K aki qk(xi , x−i ) is non-decreasing in xifor q – solution to (Program),

I Steps of the proof:

1. Discrete and uniformly distributed types (Lemma 1)I 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)

BIC-DIC equivalence. Proof.

I Prove vi (xi , x−i ) = ∑k∈K aki qk(xi , x−i ) is non-decreasing in xifor q – solution to (Program),

I Steps of the proof:

1. Discrete and uniformly distributed types (Lemma 1)I 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)

BIC-DIC equivalence. Proof.

I Prove vi (xi , x−i ) = ∑k∈K aki qk(xi , x−i ) is non-decreasing in xifor q – solution to (Program),

I Steps of the proof:

1. Discrete and uniformly distributed types (Lemma 1)I 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)

BIC-DIC equivalence. Proof.

I Prove vi (xi , x−i ) = ∑k∈K aki qk(xi , x−i ) is non-decreasing in xifor q – solution to (Program),

I Steps of the proof:

1. Discrete and uniformly distributed types (Lemma 1)I 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)

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.

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.

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.

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.

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.

BIC-DIC equivalence. Proof of Lemma 1.

Proof.

1. The set of solutions of (Program) is not empty.

min{qk}k∈K

Ex||v(x)||2 (Program)

s.t.

qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀xVi (xi ) = Vi (xi ) ∀i , xi

2. Suppose vj (xj , x−j ) > vj (x ′j , x−j ) for some j , x ′j > xj , and some x−j .

3. {qk}k∈K being BIC means Vj (xj ) = Vj (xj ) = Ex−j v(xj , x−j ) isnon-decreasing.

4. ∃ x′−j such that vj (xj , x′−j ) < vj (x ′j , x′−j ). Consider allocations at 4

type-points.

BIC-DIC equivalence. Proof of Lemma 1.

Proof.

1. The set of solutions of (Program) is not empty.

min{qk}k∈K

Ex||v(x)||2 (Program)

s.t.

qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀xVi (xi ) = Vi (xi ) ∀i , xi

2. Suppose vj (xj , x−j ) > vj (x ′j , x−j ) for some j , x ′j > xj , and some x−j .

3. {qk}k∈K being BIC means Vj (xj ) = Vj (xj ) = Ex−j v(xj , x−j ) isnon-decreasing.

4. ∃ x′−j such that vj (xj , x′−j ) < vj (x ′j , x′−j ). Consider allocations at 4

type-points.

BIC-DIC equivalence. Proof of Lemma 1.

Proof.

1. The set of solutions of (Program) is not empty.

min{qk}k∈K

Ex||v(x)||2 (Program)

s.t.

qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀xVi (xi ) = Vi (xi ) ∀i , xi

2. Suppose vj (xj , x−j ) > vj (x ′j , x−j ) for some j , x ′j > xj , and some x−j .

3. {qk}k∈K being BIC means Vj (xj ) = Vj (xj ) = Ex−j v(xj , x−j ) isnon-decreasing.

4. ∃ x′−j such that vj (xj , x′−j ) < vj (x ′j , x′−j ). Consider allocations at 4

type-points.

BIC-DIC equivalence. Proof of Lemma 1.

Proof.

1. The set of solutions of (Program) is not empty.

min{qk}k∈K

Ex||v(x)||2 (Program)

s.t.

qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀xVi (xi ) = Vi (xi ) ∀i , xi

2. Suppose vj (xj , x−j ) > vj (x ′j , x−j ) for some j , x ′j > xj , and some x−j .

3. {qk}k∈K being BIC means Vj (xj ) = Vj (xj ) = Ex−j v(xj , x−j ) isnon-decreasing.

4. ∃ x′−j such that vj (xj , x′−j ) < vj (x ′j , x′−j ). Consider allocations at 4

type-points.

BIC-DIC equivalence. Proof of Lemma 1.

BIC-DIC equivalence. Proof of Lemma 1.

Does q′ satisfy:qk(x) ≥ 0 ∀k, x

∑k qk(x) = 1 ∀xVi (xi ) = Vi (xi ) ∀i , xi

?

BIC-DIC equivalence. Proof of Lemma 1.

I The above argument shows q′ is feasible.

I Direct calculations also show

Ex (||v′(x)||2 − ||v(x)||2) < 0

I This contradicts to {qk}k∈K being a solution to (Program). QED

The limits of BIC-DIC equivalence

The limits of BIC-DIC equivalence

I Stronger notion of the equivalenceI counterexample to equivalence based on Qk

i (xi ) = Qki (xi )

I Correlated values (see Cremer and McLean, 1988)

I Multidimensional types

I Non-linear utilities

I Interdependent values

Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir

I A discrete version of Maskin (1992)’s example

I Single-unit auction, 2 bidders, 2 types {x , x}

I K = 3 states: assign to bidder 1, bidder 2, or not at all

I Bidder i ’s value vi (xi , xj ) = xi + αxj

I Compare BIC with EPIC (ex post incentive compatibility)

v

vα = 0

EPIC

BIC

v

vα = 1

2

EPIC

BIC

v

vα = 2

EPIC

BIC

Feasible (yellow shaded), BIC (light gray), and EPIC (dark gray)

Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir

I A discrete version of Maskin (1992)’s example

I Single-unit auction, 2 bidders, 2 types {x , x}

I K = 3 states: assign to bidder 1, bidder 2, or not at all

I Bidder i ’s value vi (xi , xj ) = xi + αxj

I Compare BIC with EPIC (ex post incentive compatibility)

v

vα = 0

EPIC

BIC

v

vα = 1

2

EPIC

BIC

v

vα = 2

EPIC

BIC

Feasible (yellow shaded), BIC (light gray), and EPIC (dark gray)

Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir

I A discrete version of Maskin (1992)’s example

I Single-unit auction, 2 bidders, 2 types {x , x}

I K = 3 states: assign to bidder 1, bidder 2, or not at all

I Bidder i ’s value vi (xi , xj ) = xi + αxj

I Compare BIC with EPIC (ex post incentive compatibility)

v

vα = 0

EPIC

BIC

v

vα = 1

2

EPIC

BIC

v

vα = 2

EPIC

BIC

Feasible (yellow shaded), BIC (light gray), and EPIC (dark gray)

Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir

I A discrete version of Maskin (1992)’s example

I Single-unit auction, 2 bidders, 2 types {x , x}

I K = 3 states: assign to bidder 1, bidder 2, or not at all

I Bidder i ’s value vi (xi , xj ) = xi + αxj

I Compare BIC with EPIC (ex post incentive compatibility)

v

vα = 0

EPIC

BIC

v

vα = 1

2

EPIC

BIC

v

vα = 2

EPIC

BIC

Feasible (yellow shaded), BIC (light gray), and EPIC (dark gray)

Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir

I A discrete version of Maskin (1992)’s example

I Single-unit auction, 2 bidders, 2 types {x , x}

I K = 3 states: assign to bidder 1, bidder 2, or not at all

I Bidder i ’s value vi (xi , xj ) = xi + αxj

I Compare BIC with EPIC (ex post incentive compatibility)

v

vα = 0

EPIC

BIC

v

vα = 1

2

EPIC

BIC

v

vα = 2

EPIC

BIC

Feasible (yellow shaded), BIC (light gray), and EPIC (dark gray)

Interdependent Valuesbased on ”A Geometric Approach to Mechanism Design” (2012) Goeree and Kushnir

I A discrete version of Maskin (1992)’s example

I Single-unit auction, 2 bidders, 2 types {x , x}

I K = 3 states: assign to bidder 1, bidder 2, or not at all

I Bidder i ’s value vi (xi , xj ) = xi + αxj

I Compare BIC with EPIC (ex post incentive compatibility)

v

vα = 0

EPIC

BIC

v

vα = 1

2

EPIC

BIC

v

vα = 2

EPIC

BIC

Feasible (yellow shaded), BIC (light gray), and EPIC (dark gray)

Conclusion

Conclusion

BIC-DIC equivalence for social choice problems insettings with independent private values, linear util-ities, one-dimensional types, and for general typedistributions (including discrete type-space).

The proof is short and constructive.

Identify limits of BIC-DIC equivalence: stronger no-tion of equivalence, interdependent values, correlatedvalues, multi-dimensional types, non-linear utilities.

Conclusion

BIC-DIC equivalence for social choice problems insettings with independent private values, linear util-ities, one-dimensional types, and for general typedistributions (including discrete type-space).

The proof is short and constructive.

Identify limits of BIC-DIC equivalence: stronger no-tion of equivalence, interdependent values, correlatedvalues, multi-dimensional types, non-linear utilities.

Conclusion

BIC-DIC equivalence for social choice problems insettings with independent private values, linear util-ities, one-dimensional types, and for general typedistributions (including discrete type-space).

The proof is short and constructive.

Identify limits of BIC-DIC equivalence: stronger no-tion of equivalence, interdependent values, correlatedvalues, multi-dimensional types, non-linear utilities.