Implementation via Information Design in Binary-Action ... · Daisuke Oyama University of Tokyo...

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Implementation via Information Design in Binary-Action Supermodular Games Stephen Morris Massachusetts Institute of Technology Daisuke Oyama University of Tokyo Satoru Takahashi National University of Singapore Topics in Economic Theory October 19, 2020

Transcript of Implementation via Information Design in Binary-Action ... · Daisuke Oyama University of Tokyo...

  • 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∑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.

    39 / 61

  • 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|.

    40 / 61

  • 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}.

    41 / 61

  • 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).

    43 / 61

  • 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

    44 / 61

  • ▶ 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.

    45 / 61

  • 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.

    46 / 61

  • 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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