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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
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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
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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
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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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