Implementation via Information Design inBinary-Action Supermodular Games

Stephen MorrisMassachusetts Institute of Technology

Daisuke OyamaUniversity of Tokyo

Satoru TakahashiNational University of Singapore

Topics in Economic Theory

October 19, 2020

Implementation via Information Design

▶ Fix payoff functions ui(a, θ), where a ∈ A and θ ∈ Θ.

▶ What outcomes (i.e., joint distributions over A×Θ) can beimplemented by choosing an information structure?

▶ Partial implementation:An outcome is partially implementable if it is induced by someequilibrium of some information structure.

▶ Well known (Bergemann and Morris 2016):An outcome is partially implementable if and only if it satisfiesan “obedience” constraint,

or it is a Bayes correlated equilibrium (BCE).

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Full and Smallest Equilibrium Implementation

This paper characterizes full implementation and smallestequilibrium implementation in binary-action supermodular (BAS)games.

▶ An outcome is fully implementable if it is induced byall equilibria of some information structure.

▶ An outcome is smallest equilibrium implementable if it isinduced by the smallest equilibrium of some informationstructure.

▶ Well defined in supermodular games.

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Main Results

Under a dominance state assumption,

an outcome is smallest equilibrium implementable if and only ifit satisfies not only obedience but also sequential obedience.

▶ “Sequential obedience”:

▶ Designer recommends players to switch to action 1 fromaction 0 according to a randomly chosen sequence;

▶ each player has a strict incentive to switch when told to do soeven if he only expects players before him to have switched.

▶ Full implementation requires “reverse sequential obedience” inaddition.

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Applications

0. Simpler condition for potential games:

In potential games, sequential obedience is equivalent toan even simpler coalitional obedience condition.

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Applications

1. Information design with adversarial equilibrium selection:

supT

minBNE

E[V (a, θ)]

where designer’s objective V (a, θ) is increasing in a.

(Hoshino 2018; Bergemann and Morris 2019; Mathevet, Perego,

and Taneva 2020; Inostroza and Pavan 2020)

▶ Worst equilibrium = Smallest equilibrium▶ Optimization problem rewritten as max

ν∈SIEν [V (a, θ)]

▶ Identify conditions on the designer’s and players’ payoffs underwhich solution satisfies the perfect coordination property:

either all players choose action 1 or they all choose action 0.

▶ Characterize the optimal solution for this case.

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Applications

2. Joint design of information and transfers:

In a context of team production, solve for the minimum bonusto players to always play action 1.

▶ inf(total bonus) subject to (“always play 1”) ∈ SI

(Winter 2004; Moriya and Yamashita 2020; Halac, Lipnowski, and

Rappoport 2020)

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Setting

▶ I = {1, . . . , |I|}: Set of players

▶ Θ: Finite set of states

▶ µ ∈ ∆(Θ): Common prior

▶ Without loss of generality, assume µ(θ) > 0 for all θ.

▶ Ai = {0, 1}: Binary action set for player i (A = {0, 1}I)

▶ ui : A×Θ → R: player i’s payoff, supermodular:

di(a−i, θ) = ui(1, a−i, θ)− ui(0, a−i, θ)

increasing in a−i.

▶ Dominance state:There exists θ ∈ Θ such that di(0−i, θ̄) > 0 for all i.

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Information Structures▶ Ti: Countable set of types for player i (T =

∏i∈I Ti)

▶ π ∈ ∆(T ×Θ): Common prior▶ We require π to be consistent with µ ∈ ∆(Θ):∑

t∈T π(t, θ) = µ(θ) for all θ ∈ Θ.

▶ With I,Θ, µ, A, (ui)i∈I fixed, an information structureT = ((Ti)i∈I , π) defines a Bayesian game:▶ σi : Ti → ∆(Ai): Strategy of player i▶ Bayesian Nash equilibrium (BNE) is defined as usual.▶ E (T ): Set of BNEs.▶ σ = (σi)i∈I : Smallest (pure-strategy) BNE

▶ The outcome ν ∈ ∆(A×Θ) induced by information structureT and strategy profile σ:

ν(a, θ) =∑t

π(t, θ)∏i∈I

σi(ti)(ai).

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Partial implementation

▶ ν is partially implementable if there exist an informationstructure T and an equilibrium σ that induce ν.

▶ ν satisfies consistency if∑

a ν(a, θ) = µ(θ) for all θ ∈ Θ.

▶ ν satisfies obedience if∑a−i,θ

ν(ai, a−i, θ)(ui(ai, a−i, θ)− ui(a′i, a−i, θ)) ≥ 0

for all i ∈ I and all ai, a′i ∈ Ai.

Proposition 1 (Bergemann and Morris (2016))

ν is partially implementable if and only if it satisfies consistencyand obedience.

▶ Write BCE for the set of implementable outcomes.

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Smallest Equilibrium Implementation

▶ ν is smallest equilibrium implementable (S-implementable) ifthere exists an information structure T such that (T , σ)induces ν.

▶ Write SI for the set of S-implementable outcomes.

▶ Clearly, SI ⊂ BCE .

▶ Our paper characterizes SI and its closure SI .

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Two-Player Two-State Example (Symmetric Payoffs)From Bergemann and Morris (2019)

▶ I = {1, 2}

▶ A1 = A2 = {NI , I }

▶ Θ = {B,G}, µ(B) = µ(G) = 12▶ Payoffs:

B NI I

NI 0 0

I −1 −1 + ε

G NI I

NI 0 0

I x x+ ε

0 < x < 1, 0 < ε < 12(1− x)

▶ Designer’s objective: maximize the expected number ofplayers who invest.

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Optimal BCE

B NI I

NI 1−x−2ε2(1−ε) 0

I 0 x+ε2(1−ε)

G NI I

NI 0 0

I 0 12

▶ In the direct mechanism:▶ “Always obey the recommendation” is an equilibrium.▶ “Always play NI ” is also an equilibrium (smallest equilibrium).

▶ In fact, this outcome is not S-implementable.

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Sequences of Players

▶ Our characterization of SI involves “hidden variables” νΓ.We care about who eventually plays action 1 in the smallestBNE, but the characterization is based on the order in whichplayers change actions along the iteration procedure.

▶ Let Γ be the set of all finite sequences of distinct players;for example, if I = {1, 2, 3}, then

Γ = {∅, 1, 2, 3, 12, 13, 21, 23, 31, 32, 123, 132, 213, 231, 312, 321}.

▶ An “ordered outcome” is a distribution over sequences andstates νΓ ∈ ∆(Γ×Θ).

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▶ For γ ∈ Γ, ā(γ) denotes the action profile where player i playsaction 1 iff player i appears in γ.

▶ Each “ordered outcome” νΓ ∈ ∆(Γ×Θ) induces outcomeν ∈ ∆(A×Θ) by forgetting the ordering, i.e.,

ν(a, θ) =∑

γ∈Γ:ā(γ)=a

νΓ(γ, θ).

▶ Let Γi = {γ ∈ Γ | player i appears in γ}.

▶ For γ ∈ Γi, a−i(γ) denotes the action profile of player i’sopponents where player j plays action 1 iff player j appears inγ before player i.

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Sequential Obedience

Definition 1

▶ Ordered outcome νΓ ∈ ∆(Γ×Θ) satisfies sequentialobedience if∑

γ∈Γi,θ∈ΘνΓ(γ, θ)di(a−i(γ), θ) > 0

for all i such that νΓ(Γi ×Θ) > 0.

▶ Outcome ν ∈ ∆(A×Θ) satisfies sequential obedience ifthere exists ordered outcome νΓ ∈ ∆(Γ×Θ) that induces νand satisfies sequential obedience.

▶ Weak sequential obedience: “≥” in place of “>”.

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Sequential Obedience vs. Obedience

▶ Sequential obedience captures the iterative procedure at theoutcome level.

▶ Sequential obedience is a strengthening of “upper obedience”:∑a−i,θ

ν(1, a−i, θ)(ui(1, a−i, θ)− ui(0, a−i, θ))

=∑γ,θ

νΓ(γ, θ)(ui(1, ā−i(γ), θ)− ui(0, ā−i(γ), θ))

≥∑γ,θ

νΓ(γ, θ)(ui(1, a−i(γ), θ)− ui(0, a−i(γ), θ))

> 0,

where ā−i(γ) is the action profile of player i’s opponentswhere player j plays action 1 iff player j appears in γ(regardless of his relative position to player i).

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Main Results

Theorem 1

1. If ν ∈ SI , then it satisfies consistency, obedience, andsequential obedience.

2. If ν with ν(1, θ) > 0 satisfies consistency, obedience, andsequential obedience, then ν ∈ SI .

(SI = (Set of smallest equilibrium implementable outcomes))

Corollary 1

ν ∈ SI if and only if it is satisfies consistency, obedience, and weaksequential obedience.

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Necessity of Sequential Obedience▶ Suppose that ν is smallest equilibrium implementable.▶ Let T = ((Ti)i∈I , π) be an information structure whose

smallest equilibrium induces ν.

▶ Starting from the constant 0 strategy profile, apply sequentialbest response in the order 1, 2, . . . , |I|.

▶ For each type ti ∈ Ti, let▶ ni(ti) = n if ti changes from action 0 to action 1 at n-th step;▶ ni(ti) = ∞ if ti never changes.

▶ Let T (γ) = {t ∈ T | (ni(ti))i∈I is ordered according to γ},and define

νΓ(γ, θ) =∑

t∈T (γ)

π(t, θ).

▶ Because this process converges to the smallest equilibrium,we know that νΓ induces ν.

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▶ To show sequential obedience, note that for each ti ∈ Ti withni(ti) < ∞, we have∑

t−i,θ

π((ti, t−i) , θ)di(a−i(ti, t−i), θ) > 0,

where a−i(ti, t−i) is the action profile of player i’s opponentsin the sequential best response process when i switches; soplayer j plays action 1 iff nj(tj) < ni(ti).

▶ By adding up the inequality over all such ti, we have

0 <∑

ti : ni(ti) 0.19 / 61

Sufficiency of Sequential Obedience

▶ Let νΓ ∈ ∆(Γ×Θ) satisfy sequential obedience.

▶ We construct an information structure as follows.

▶ Ti = {1, 2, . . .} ∪ {∞}▶ By the assumption ν(1, θ) > 0,

νΓ(γ̄, θ) > 0 for some sequence γ̄ of all players.

Take ε > 0 such that ε < νΓ(γ̄, θ).

▶ m drawn from Z+ according to the distribution η(1− η)m,where 0 < η ≪ ε.

▶ γ drawn from Γ according to νΓ.▶ Player i receives signal ti given by

ti =

{m+ (ranking of i in γ) if γ ∈ Γi∞ otherwise.

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▶ To initiate contagion, re-arrange probabilities:▶ Replace νΓ(γ̄, θ) with νΓ(γ̄, θ)− ε.▶ Allocate ε|I|−1 to (t, θ) such that 1 ≤ t1 = · · · = t|I| ≤ |I| − 1.▶ Since η ≪ ε, types ti ∈ {1, . . . , |I| − 1} will assign high

probability to θ.

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▶ Show by induction that action 1 is the unique action survivingiterated deletion of dominated strategies for all types ti < ∞.

▶ Initialization step:If ti ∈ {1, . . . , |I| − 1}, the player assigns high probability toθ = θ, and by Dominance State, action 1 is a dominant action.

▶ Induction step:For τ ≥ |I|, Suppose all types ti ≤ τ − 1 play action 1.

Then type ti = τ knows that all players before him in therealized sequence play action 1, so his payoff to 1 is at least∑

γ∈Γi,θ∈ΘνΓ(γ, θ)di(a−i(γ), θ)× (constant) > 0 as η ≈ 0,

where the inequality is by Sequential Obedience.

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Two-Player Two-State Example (Symmetric Payoffs)

B NI I

NI 0 0

I −1 −1 + ε

G NI I

NI 0 0

I x x+ ε

▶ S-implemetable outcome:

B NI I

NI 1−x−ε2−ε + δ 0

I 0 2x+ε2(2−ε) − δ

G NI I

NI 0 0

I 0 12

▶ The limit as δ → 0 attains the supremum when the objectiveis to maximize the expected number of players who invest.

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B NI I

NI 0 0

I −1 −1 + ε

G NI I

NI 0 0

I x x+ ε

▶ By symmetry, consider the symmetric ordered outcome:

B G

∅ 12 − 2p 0

12 p 14

21 p 14

▶ Derive p such that weak sequential obedience is satisfied withequality.

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Two-Player Two-State Example (Asymmetric Payoffs)

b Not Invest

Not 0, 0 0,−8

Invest −7, 0 −4,−5

g Not Invest

Not 0, 0 0, 1

Invest 2, 0 5, 4

▶ µ(b) = µ(g) = 12▶ Supermodular (payoff gain of 3 to investing if the other player

invests)

▶ Both players have dominant action to invest in good state andnot invest in bad state

▶ Asymmetric: Row player 1 gets higher payoff (+1) frominvesting relative to column player 2

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Optimal BCE

b Not Invest

Not 0, 0 0,−8

Invest −7, 0 −4,−5

g Not Invest

Not 0, 0 0, 1

Invest 2, 0 5, 4

▶ Optimal BCE when the objective is to maximize the expectednumber of players who invest:

b Not Invest

Not 0 0

Invest 11025

g Not Invest

Not 0 0

Invest 0 12

(Asymmetric)

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Smallest Equilibrium Implementation

b Not Invest

Not 0, 0 0,−8

Invest −7, 0 −4,−5

g Not Invest

Not 0, 0 0, 1

Invest 2, 0 5, 4

▶ The following outcome ν is S-implementable for any δ > 0:

b Not Invest

Not 14 + δ 0

Invest 0 14 − δ

g Not Invest

Not 0 0

Invest 0 12

(“Perfect coordination outcome”)

▶ The limit as δ → 0 attains the supremum when the objectiveis to maximize the expected number of players who invest.

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Risk-Dominance

Complete information game conditional on both being told toinvest (and δ = 0):

Not Invest

Not 0, 0 0,−2

Invest −1, 0 2, 1

▶ (Invest, Invest) is (just) risk-dominant, which can be fullyimplemented by an Email-game information structure.

▶ Higher probability of investment cannot be fully implemented(Kajii and Morris 1997).

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Sequential Obedience

b Not Invest

Not 0, 0 0,−8

Invest −7, 0 −4,−5

g Not Invest

Not 0, 0 0, 1

Invest 2, 0 5, 4

▶ The following ordered outcome νΓ (which induces ν) satisfiessequential obedience:

b g

∅ 14 + δ 0

12 16 − δ13

21 11216

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Construction of Information Structureb

t1\t2 1 2 3 4 · · · ∞

1 η(

16

− δ)

2 η 112

η(1 − η)(

16

− δ)

3 η(1 − η) 112

η(1 − η)2(

16

− δ)

4 η(1 − η)2 112

. . .

.

.

.. . .

∞ 14

+ δ

g

t1\t2 1 2 3 4 · · · ∞

1 ε η(

13

− ε)

2 η 16

η(1 − η)(

13

− ε)

3 η(1 − η) 16

η(1 − η)2(

13

− ε)

4 η(1 − η)2 16

. . .

.

.

.. . .

∞

▶ η ≪ ε30 / 61

Dual Characterization of Sequential Obedience▶ Recall:

ν ∈ ∆(A×Θ) satisfies sequential obedience if there existsνΓ ∈ ∆(Γ×Θ) that induces ν and satisfies∑

γ∈Γi,θ∈ΘνΓ(γ, θ)di(a−i(γ), θ) > 0

for all i such that νΓ(Γi ×Θ) > 0. (♯♯)

Proposition 2

ν satisfies sequential obedience if and only if∑a∈A,θ∈Θ

ν(a, θ) maxγ:ā(γ)=a

∑i∈S(γ)

λidi(a−i(γ), θ) > 0

for all (λi)i∈I ≥ 0, (λi)i∈I(ν) ̸= 0. (♯)

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Proof

▶ Fix ν ∈ ∆(A×Θ).

▶ Let NΓ(ν) = {νΓ ∈ ∆(Γ×Θ) |∑

γ:ā(γ)=a νΓ(γ, θ) = ν(a, θ)}and Λ(ν) = {λ ∈ ∆(I) |

∑i∈I(ν) λi = 1}.

(Both are convex and compact.)

▶ For νΓ ∈ NΓ(ν) and λ ∈ Λ(ν), let

D(νΓ, λ) =∑i∈I

λi∑

γ∈Γi,θ∈ΘνΓ(γ, θ)di(a−i(γ), θ)

=∑

γ∈Γ,θ∈ΘνΓ(γ, θ)

∑i∈S(γ)

λidi(a−i(γ), θ)

=∑

a∈A,θ∈Θ

∑γ:ā(γ)=a

νΓ(γ, θ)∑

i∈S(a)

λidi(a−i(γ), θ).

(Linear in each of νΓ and λ.)

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▶ By the Minimax Theorem, D has a value D∗:

minλ∈Λ(ν)

maxνΓ∈NΓ(ν)

D(νΓ, λ) = D∗ = max

νΓ∈NΓ(ν)min

λ∈Λ(ν)D(νΓ, λ).

▶ ν satisfies sequential obedience⇐⇒ ∃ νΓ ∈ NΓ(ν) ∀λ ∈ Λ(ν): D(νΓ, λ) > 0⇐⇒ D∗ = maxνΓ∈NΓ(ν)minλ∈Λ(ν)D(νΓ, λ) > 0

▶ (LHS of (♯)) = maxνΓ∈NΓ(ν)D(νΓ, λ) for each λ ∈ Λ(ν)Hence,

(♯) holds ⇐⇒ D∗ = minλ∈Λ(ν)maxνΓ∈NΓ(ν)D(νΓ, λ) > 0

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Application 0: Simplifying Sequential Obedience inPotential Games

▶ In potential games,the dual condition (♯) (hence sequential obedience) isequivalent to a simpler coalitional obedience condition.

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Potential Games

Definition 2The game is a potential game if there exists Φ: A×Θ → R suchthat

di(a−i, θ) = Φ(1, a−i, θ)− Φ(0, a−i, θ).

▶ For each ν ∈ ∆(A×Θ), we define a potential for thatoutcome:

Φν(a) =∑a′,θ

ν(a′, θ)Φ(a ∧ a′, θ)

where b = a ∧ a′ is the action profile such that bi = 1 if andonly if ai = a

′i = 1.

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Potential Games

▶ For simplicity, we focus on outcomes ν such thatν({1} ×Θ) > 0.

Definition 3Outcome ν satisfies coalitional obedience if

Φν(1) > Φν(a)

for all a ̸= 1.

Proposition 3

In a potential game, an outcome satisfies sequential obedienceif and only if it satisfies coalitional obedience.

▶ Show that coalitional obedience is equivalent to the dualcondition (♯) of sequential obedience.

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Convex Potential

▶ Normalize: Φ(0, θ) = 0 for all θ.

▶ Denote n(a) = |{i ∈ I | ai = 1}|.

Definition 4The potential Φ satisfies convexity if

Φ(a, θ) ≤ n(a)|I|

Φ(1, θ)

(=

(1− n(a)

|I|

)Φ(0, θ) +

n(a)

|I|Φ(1, θ)

)for all θ.

▶ Because of supermodularity, this is automatically satisfied if Φis symmetric.

▶ The potential is convex if and only if the game is not tooasymmetric.

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Investment Game

▶ Θ = {1, . . . , |Θ|}▶ di(a−i, θ) = R(θ) + hn(a−i)+1 − ci

▶ hk: increasing in k▶ R(θ): strictly increasing in θ▶ R(|Θ|) + h1 > ci for all i ∈ I

Dominant state is satisfied with θ = |Θ|▶ c1 ≤ c2 ≤ · · · ≤ c|I|

▶ This game has a potential:

Φ(a, θ) = R(θ)n(a) +

n(a)∑k=1

hk −∑

i∈S(a)

ci.

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▶ Φ satisfies convexity if and only if

1

ℓ

ℓ∑k=1

(hk − ck) ≤1

|I|

|I|∑k=1

(hk − ck)

for any ℓ = 1, . . . , |I| − 1.

▶ In particular, a sufficient condition for convexity is:

hk − ck ≤ hk+1 − ck+1

for any k = 1, . . . , |I| − 1.

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Regime Change Game

▶ Θ = {1, . . . , |Θ|}

▶ di(a−i, θ) ={1− ci if n(a−i) + 1 > |I| − k(θ),−ci if n(a−i) + 1 ≤ |I| − k(θ)

▶ 0 < ci < 1▶ k : Θ → N: strictly increasing, k(1) ≥ 1▶ k(|Θ|) = |I|

Dominant state is satisfied with θ = |Θ|

▶ This game has a potential:

Φ(a, θ) =

{n(a)− (|I| − k(θ))−

∑i∈S(a) ci if n(a) > |I| − k(θ),

−∑

i∈S(a) ci if n(a) ≤ |I| − k(θ).

▶ Φ satisfies convexity if and only if c1 = · · · = c|I|.

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Grand Coalitional Obedience and Perfect Coordination

Definition 5Outcome ν satisfies grand coalitional obedience if

Φν(1) > Φν(0) = 0,

or equivalently,∑a∈A,θ∈Θ

ν(a, θ)Φ(a, θ) > 0.

Definition 6Outcome ν satisfies perfect coordination if ν(a, θ) > 0 only fora ∈ {0,1}.

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Proposition 4

Suppose that the potential satisfies convexity.A perfectly coordinated outcome satisfies sequential obedienceif and only if it satisfies grand coalitional obedience.

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Application 1: Information Design with AdversarialEquilibrium Selection

▶ Information designer’s objective function: V : A×Θ → R▶ V (a, θ): increasing in a▶ Normalization: V (0, θ) = 0 for all θ▶ Optimal information design problem with adversarial

equilibrium selection:

supT

minσ∈E(T )

∑t∈T,θ∈Θ

π(t, θ)V (σ(t), θ)

= supT

∑t∈T,θ∈Θ

π(t, θ)V (σ(t), θ).

▶ This is equivalent tosupν∈SI

∑a∈A,θ∈Θ

ν(a, θ)V (a, θ) = maxν∈SI

∑a∈A,θ∈Θ

ν(a, θ)V (a, θ).

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Restricted ConvexityDefinition 7Designer’s objective V satisfies restricted convexity with respect topotential Φ if

V (a, θ) ≤ n(a)|I|

V (1, θ)

whenever Φ(a, θ) > Φ(1, θ).

Special cases of interest

▶ Linear preferencesV (a, θ) = n(a)

▶ Full coordination preferences

V (a, θ) =

{1 if a = 1

0 otherwise

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▶ Regime change preferences:▶ Potential

Φ(a, θ) =

{n(a)− (|I| − k(θ))−

∑i∈S(a) ci if n(a) > |I| − k(θ)

−∑

i∈S(a) ci if n(a) ≤ |I| − k(θ)

▶ Φ(a, θ) > Φ(1, θ) holds only when n(a) ≤ |I| − k(θ).▶ The objective

V (a, θ) =

{1 if n(a) > |I| − k(θ)0 if n(a) ≤ |I| − k(θ)

satisfies restricted convexity with respect to Φ.

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Perfect Coordination Solution

Theorem 2Suppose that Φ satisfies convexity and V satisfies restrictedconvexity with respect to Φ.Then there exists an optimal outcome of the adversarialinformation design problem that satisfies perfect coordination.

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Proof

▶ Consider the problem

max(ν(1,θ))θ∈Θ

∑θ∈Θ

ν(1, θ)V (1, θ)

with respect to perfect coordination outcomes,

subject to

▶ consistency, and▶ grand coalitional obedience.

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▶ Easy to characterize the solution to this problem:▶ Relabel the states as Θ = {1, . . . , |Θ|} in such a way that

Φ(1,θ)V (1,θ) is increasing in θ.

▶ Ignoring integer issues,find θ∗ that solves

∑θ∗≥θ

µ(θ)Φ(1, θ) = 0

= ∑θ∗≥θ

µ(θ)Φ(0, θ)

.▶ Let

ν∗(a, θ) =

µ(θ) if a = 1 and θ ≥ θ∗,µ(θ) if a = 0 and θ < θ∗,

0 otherwise.

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▶ We want to show that ν∗ is an optimal outcome ofthe adversarial information design problem.

▶ Take any ν ∈ SI .▶ Show that there exists a perfect coordination outcome ν ′

satisfying consistency such that

▶ grand coalitional obedience is satisfied (by convexity of Φ), and▶ ∑

a,θ ν′(a, θ)V (a, θ) ≥

∑a,θ ν(a, θ)V (a, θ)

(by restricted convexity of V ).

If ν(a, θ) > 0 for a ̸= 0,1, split ν(a, θ) to (0, θ) and (1, θ)appropriately.

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Application 2: Adding Transfers

Bonus contracts for team production

(Winter (2004, AER), Moriya and Yamashita (2020, JEMS))

▶ Team project by |I| agents, effort level ai ∈ {0, 1}

▶ c: (Common) cost of effort

▶ Θ: Set of states, µ ∈ ∆(Θ)

▶ p(n, θ): Probability of success when n agents make effort

▶ p(n, θ): nondecreasing in n▶ p(|I|, θ) > p(0, θ) for some θ

▶ Denote ∆p(n, θ) = p(n, θ)− p(n− 1, θ).

▶ Assume strategic complementarities: ∆p(n, θ) ≤ ∆p(n+ 1, θ).

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▶ b = (b1, . . . , b|I|): bonus scheme (to be chosen by the principle)▶ Agent i’s payoff:

▶ p(n(a−i) + 1, θ)bi − ci for ai = 1▶ p(n(a−i), θ)bi for ai = 0

▶ With normalization, define

di(a−i, θ; bi) = ∆p(n(a−i) + 1, θ)−cibi.

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▶ ν̄: Full-effort outcome (i.e., ν̄(1, θ) = µ(θ) for all θ)

▶ Principal’s objective:Design a bonus scheme b and an information structure Tthat minimize the total payment

while implementing ν̄ in the smallest (hence unique)equilibrium:

infb:ν̄∈SI (b)

∑i∈I

bi.

(Moriya and Yamashita 2020, with |I| = 2, |Θ| = 2, and symmetricbonuses)

▶ A bonus scheme b∗ = (b∗i )i∈I is optimal if∑

i∈I b∗i is equal to

this infimum and ν̄ ∈ SI (b∗ + ε) for every ε > 0

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▶ Dominance state counterpart:Let θ̄ ∈ Θ be a state such that ∆p(1, θ̄) ≥ ∆p(1, θ) for allθ ∈ Θ.

Assume

∆p(1, θ̄) ≥∑θ∈Θ

µ(θ)p(|I|, θ)− p(0, θ)

|I|.

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Relaxed Problem

▶ The base game (di(a−i, θ; bi))i∈I given b = (bi)i∈I has apotential

Φ(a, θ; b) = p(n(a), θ)− p(0, θ)−∑

i∈S(a)

c

bi.

▶ Consider the relaxed minimization problem subject toweak grand coalitional obedience∑

θ∈Θ µ(θ)Φ(1, θ; b) ≥∑

θ∈Θ µ(θ)Φ(0, θ; b) = 0:

minb

∑i∈I

bi

subject to∑i∈I

c

bi≤

∑θ∈Θ

µ(θ)(p(|I|, θ)− p(0, θ)).

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▶ By the strict convexity of x 7→ 1x ,an optimal solution to this relaxed problem is unique.

▶ It is given by b∗ = (β∗, . . . , β∗) with

β∗ =|I|c∑

θ∈Θ µ(θ)(p(|I|, θ)− p(0, θ)).

Proposition 5

The unique optimal bonus scheme is given by b∗ = (β∗, . . . , β∗).

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Proof

We want to verify that ν̄ ∈ SI (b∗ + ε) for any ε > 0.

▶ Potential Φ(·; b∗ + ε) satisfies convexity.⇒ Grand coalitional obedience is equivalent to sequentialobedience.

▶ Dominant State is satisfied under b∗ + ε.

▶ Therefore, it follows from Theorem 1 that ν̄ ∈ SI (b∗ + ε).

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Literature

▶ Winter (2004)

▶ Moriya and Yamashita (2020)

▶ Halac, Lipnowski, and Rappoport (2020)

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Full Implementation

▶ Outcome ν is fully implementable if there existsan information structure T such that (T , σ) induces ν forall σ ∈ E (T ).

▶ Under supermodularity, full implementation in fact requiresE(T ) to be a singleton.

▶ Reverse sequential obedience:Reverse version of sequential obedience, where actions 1 and0 are exchanged.

▶ Add a symmetric dominance state assumption:there exists θ ∈ Θ such that di(1−i, θ) < 0 for all i ∈ I.

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Theorem 3

1. If ν ∈ FI , then it satisfies consistency, sequential obedience,and reverse sequential obedience.

2. If ν with ν(1, θ) > 0 and ν(0, θ) > 0 satisfies consistency,sequential obedience, and reverse sequential obedience, thenν ∈ FI .

(FI = (Set of fully implementable outcomes))

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S-Implementation and Full Implementation

▶ By definition, FI ⊂ SI .

▶ In general, FI ⫋ SI .

Proposition 6

For any ν ∈ SI , there exists ν̂ ∈ FI that first-order stochasticallydominates ν.

(I.e.,∑

a′≥a ν′(a′, θ) ≥

∑a′≥a ν(a

′, θ) for all a and θ.)

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Optimal Information Design under Full Implementation

▶ By Proposition 6, if V (a, θ) is increasing in a, then

maxν∈FI

∑a∈A,θ∈Θ

ν(a, θ)V (a, θ) = maxν∈SI

∑a∈A,θ∈Θ

ν(a, θ)V (a, θ).

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