Informationally Simple Incentivesweb.stanford.edu/~agathep/Gleyze_Pernoud_ISI.pdfInformationally Simple Incentives Simon GLEYZEy Agathe PERNOUDz Abstract We consider a mechanism design

Informationally Simple Incentives∗

Simon GLEYZE† Agathe PERNOUD‡

Abstract

We consider a mechanism design setting in which agents can acquire costlyinformation on their preferences as well as others’. The choice of the mecha-nism generates informational incentives as it affects what information is acquiredbefore play begins. A mechanism is informationally simple if agents have no in-centive to learn about others’ preferences. This property is of interest for tworeasons: First, the extended game has an equilibrium in dominant strategiesonly if it is informationally simple. Second, this endogenizes an “independentprivate value” property of the interim information structure. Our main result isthat a mechanism is informationally simple only if it is de facto dictatorial. Thisholds for generic environments and any smooth cost function which satisfiesan Inada condition. Hence even interim strategy-proof mechanisms incentivizeagents to learn about others, and do not admit dominant strategies at the ex-antestage. Moreover, the Independent Private Value assumption is unlikely to ariseendogenously, though we show that full surplus extraction is infeasible.

Keywords: Mechanism Design, Information Acquisition, Strategy-Proofness, Sim-plicity, Independent Private Values

JEL classification codes: C72, D01, D82

∗We are grateful to Piotr Dworczak, Fuhito Kojima, Shengwu Li, Elliot Lipnowski, MikeOstrovsky, Eduardo Perez-Richet, Doron Ravid, Al Roth, Ilya Segal, Olivier Tercieux and es-pecially Matt Jackson, Philippe Jehiel and Paul Milgrom as well as seminar participants atPSE, Stanford, Akbarpour–Milgrom discussion group, YES2020 for helpful conversations andcomments.†Paris School of Economics, Paris 1 Pantheon Sorbonne. Email: [email protected]‡Department of Economics, Stanford University. Email: [email protected]

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mailto:[email protected]:[email protected]

1 INTRODUCTION

Scholars have long understood that institutions can have a strong impact onthe formation of preferences. Surprisingly little attention, however, has beenpaid to how institutions shape the ways in which we constitute our knowl-edge and acquire information, that is, about how agents’ informational incen-tives are affected by institutional rules. This can sometimes be of primary im-portance because heterogeneous informational incentives can lead to unequalopportunities in voting, labor market outcomes, education choices, invest-ment decisions, etc. Conversely, little is known about how these informationalincentives constrain the type of institutions that are actually implementable.In this paper, we make progress toward addressing these questions.

We investigate what kind of mechanisms lead to simple informational in-centives, and why simple informational incentives matter for the design ofinstitutions. Informational simplicity is defined as acquiring signals on one’sown preferences only, as opposed to signals on the preferences of the entirepopulation. We show that whenever the mechanism is de facto non-dictatorialand for a rich set of technologies of costly information acquisition, players al-ways have an incentive to learn about others’ preferences even though theyare not directly payoff-relevant. In particular, even strategy-proof mechanismsincentivize players to acquire information on others’. Moreover, we showthat such informational incentives make strategy-proof mechanisms no longerdominant solvable at the ex-ante stage, when players decide what informa-tion to acquire. These results hold whenever the cost of information satisfiesan Inada condition—which makes it never optimal to become fully informedabout any state—and a smoothness condition—which guarantees that agentscan fine-tune the informativeness of signals without discontinuously chang-ing their cost.

The intuition behind our main result is the following: The set of outcomesthat a player can bring about, call this her choice set, depends on other play-ers’ reports to the designer, and hence on the entire vector of fundamentalsdetermining the preferences of the population. Since the value of information

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on her own preferences depends on her choice set, it indirectly depends onother players’ preferences as well. If gathering information about any agent’spreferences (including her own) is relatively costly, then it is generically opti-mal for the agent to devote resources to acquiring information about others’preferences first. It helps them predict the report of others, inducing betterinformation acquisition on one’s own preferences.

Finally, we explore the implications of our result for mechanism design.First, it appears that strategic simplicity, as captured by strategy-proofness,is more limited than previously thought. Much of the literature focused onstrategic simplicity at the interim stage, that is once players have acquiredtheir private information. Our result shows that interim strategy-proofnessdoes not guarantee strategic simplicity of the ex ante information acquisi-tion stage. Indeed, only dictatorial mechanisms admit equilibria in dominantstrategies in the extended game.

Second, a direct corollary of our main result is that the standard Indepen-dent Private value assumption is unlikely to arise endogenously. Of course,this assumption is usually understood as a technically convenient approxima-tion of reality—nothing more. Nevertheless, our result makes precise why thisis unlikely to hold in practice, and why departures from IPV in the standardframework lead to discontinuities such as Crémer and McLean (1988)’s fullsurplus extraction result. Instead, our approach restores a form of continuity:side bets at the interim stage reduce the amount of information acquisition atthe ex-ante stage—in particular, side bets prevent the efficient amount of in-formation acquisition. Hence constrained surplus extraction is feasible, but fullsurplus extraction is not because players internalize the informational incen-tives generated by side bets.

1.1 Related Literature

The literature on information acquisition in mechanism design can be broadlydivided into two categories: information acquisition with fixed mechanisms,and optimal mechanism design under information acquisition. We omit the

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large literature on optimal information disclosure and information design be-cause, in our model, the designer is uninformed of the underlying state of theworld, and information acquisition is decentralized.1

Information Acquisition with Fixed Mechanisms Matthews (1977) showsthat the incentives to acquire information are identical between the first priceand second price auctions. This finding is corroborated by Hausch and Li(1993) who prove a similar result for all independent private value auctions.Stegeman (1996) shows that the second price auction induces efficient infor-mation acquisition in the independent private value setting. However, Persico(2000) makes clear that it relies on restrictive parametric assumptions on thecost function by proving a representation theorem of the demand for infor-mation. Compte and Jehiel (2007) show that dynamic auctions induce morerevenue than static auctions. Finally, Bergemann et al. (2009) show that withinterdependent values the equilibrium level of information acquisition is inef-ficient under VCG. More recently, Bobkova (2019) investigates the incentivesto learn about private versus common value components in auctions. Kimand Koh (2017) show that when learning cost is small, signal correlation islow, or value interdependence is weak, the first price auction generates higherrevenue than the second price auction.

The closest paper to ours is Larson and Sandholm (2001) in the computerscience literature. They introduce a model in which agents can devote compu-tational resources to discover their own as well as others’ valuation. Playerschoose deterministic or stochastic algorithms to compute valuations at a costwhich is increasing in the complexity of the algorithm. For several auction for-mats, they show that players compute the valuation of others in equilibrium.

Optimal Mechanism Design with Information Acquisition Most papersconsider the implementation of the efficient allocation in quasi-linear envi-ronments. Bergemann and Välimäki (2002) show that in a standard allocationproblem with monetary transfers, private values, and one-dimensional infor-

1The reader is referred to Bergemann and Morris (2019) and the references therein.

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mation acquisition (players can only acquire information on their own prefer-ences), the ex-ante welfare-maximizing allocation is implemented by the VCGmechanism. Hatfield et al. (2018) further strengthen this result by showingthat strategy-proofness and ex-post efficiency—which characterize VCG bythe Green-Laffont-Holmström theorem—are not only sufficient but also nec-essary for ex-post efficient mechanisms to induce ex-ante efficient informationacquisition. They only investigate information acquisition on one’s own pref-erences. Instead, a corollary of our result is that VCG induces ex-ante inefficientinformation acquisition because information acquisition endogenously leadsto interdepent values at the interim stage.

In the case of a unique non-divisible object and under restrictive assump-tions on the cost function, some attempts have been made to characterize rev-enue maximizing auctions with information acquisition. Shi (2012) derivesthe optimal auction in a private value setting with one-dimensional informa-tion acquisition. Lu and Ye (2013) derive an efficient and revenue maximizingtwo-stage auction, the first stage being an entry right allocation mechanism.Li (2019) studies the design of ex-ante efficient mechanisms when agents cancovertly acquire information at some cost before participating in a mechanism.

In school choice settings, Immorlica et al. (2018) look for mechanisms thatare stable and induce students to acquire information efficiently. They showthat information deadlocks can arise, i.e. situations in which no student wantsto acquire additional information before others’ do, preventing the mecha-nism to move forward.

Rappoport and Somma (2017) study a principal-agent problem with moralhazard in which realized signals are contractible and show that the first-bestcan be achieved under mild conditions. Roesler and Szentes (2017) and Ravidet al. (2019) consider monopoly pricing when buyers can flexibly acquire in-formation. In their papers, the seller chooses the mechanism after the buyerchooses her information strategy, whereas in our paper the designer ex-antecommits to a mechanism. Hence in their model the buyer must internal-ize the seller’s strategy, whereas in our paper the designer must internalizethe agents’ future decisions (what we refer to as “informational incentives”).

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Mensch (2019) considers a screening problem with informational incentivesand provides a Myersonian characterization of the optimal mechanism.

2 MOTIVATING EXAMPLE

Two cities must decide on the provision of a public good, such as a high-schoolor a hospital. City representatives (the players) are uncertain about the needsof their citizens. In city i, the demand for a new school might be either highωi ∈ Ωi or low ωi ∈ Ωi, and similarly in city j. Cities have linear preferencesin their own state, and we denote the state space by Ω = Ωi × Ωj . Moreover,players have a common prior belief on the state space µ0 ∈ ∆Ω.

Cities use a VCG mechanism to decide on the provision of the public good.Contrary to the standard approach, cities can engage in costly information ac-quisition about ω = (ωi, ωj) ex ante. Importantly, they can privately acquireinformation on any fundamental. At the interim stage, cities must report theirexpected valuation m̂i = Eµi [ωi] where µi is the posterior belief of city i result-ing from the information acquisition stage.

The VCG mechanism selects the efficient allocation: x(m̂i, m̂j) = 1 if m̂i +m̂j ≥ 0, and x(m̂i, m̂j) = 0 otherwise. We assume that ωi + ωj > 0, ωi + ωj > 0and ωi + ωj < 0. Therefore, the efficient allocation if cities were informed ofthe realized state is:

x∗ ωi ωi

ωj 1 1

ωj 1 0

The VCG payments which guarantee truth-telling at the interim stage are:

ti(m̂i, m̂j) = −m̂jx(m̂i, m̂j) tj(m̂i, m̂j) = −m̂ix(m̂i, m̂j).

Do city representatives have an incentive to acquire information about thedemand of the other city? We show that the answer is generically yes, even

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though the mechanism is strategy-proof at the interim stage. Indeed, if playeri learns that demand is high in city j, then there is really no point in asking thepreferences of her own citizens: the representative of city j will most likelylearn that he has a high demand, he will report a high valuation, and thepublic good will be provided irrespective of i’s report. Conversely, if demandis low in city j, then city i has an incentive to learn about the preferences ofher own citizens, as her report will have an impact on the outcome. Hence,in equilibrium, each city learns about the demand of the other city, because ittells her how much she should invest in learning about the preferences of herown citizens. Our main result says that this is generically true as soon as themechanism is de facto non-dictatorial.2

3 SETUP

Environment. We consider good allocation problems with transferable util-ities. Let N = {1, . . . , N} be the set of agents, and K the finite set of goods tobe allocated.3 There is a finite set of possible states of the world, or fundamen-tals, that has a product structure Ω =×i∈N Ωi. Each player’s preferences overgoods depend on his own fundamental only: ui : K × Ωi −→ R. The priorprobability distribution µ0 ∈ ∆(Ω) is common knowledge among the play-ers and the designer, and satisfies independence: µ0(ω) =

∏i∈N µ

i0(ωi). The

superscript µi corresponds to the marginal on dimension ωi. Should players’preferences depend on the entire vector of fundamentals, or be correlated, wecould not make the distinction between player i acquiring information on herpreferences or on others’. Our assumption ensures that statements such as“player i acquires information on player j” have a proper meaning. Hence inwhat follows, an agent’s payoff will depend on others’ fundamentals ω−i onlyindirectly through the mechanism.

2That is, non-dictatorial over the set of messages that are actually sent in equilibrium.3For ease of exposition, we suppose there is only one unit of each good, but the analysis

easily extends to multi-unit settings. As soon as the supply is restrictive, such that changingthe allocation of an agent affects that of another, then our results go through.

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Though we state our results for good allocation problems, they easily ex-tend to public good problems, and to more general settings in which there isan abstract set of outcomes and agents have arbitrary preferences over theseoutcomes. The results also extend to environments without transfers (e.g.,matching). We focus on good allocation problems and quasi-linear utilitiesfor ease of exposition, and to make our impossibility result stronger as trans-fers give more leeway to the designer.

Mechanism. A designer ex-ante commits to a mechanism. A mechanism Γconsists of a finite set of messages for each player Mi, as well as allocationfunctions and transfer functions:

xi : M1 × · · · ×MN −→ [0, 1]K

ti : M1 × · · · ×MN −→ R

In our baseline model, we consider non-wasteful mechanisms in which allgoods are allocated:

∑i∈N xik = 1 for all k.

The ex-post utility of agent i in state ω under message profile m is quasi-linear in the transfer: ∑

k∈K

xik(m)ui(k, ωi)− ti(m).

Strategies. At the ex-ante stage, players can acquire costly information aboutany state. Information acquisition is equivalently represented by the choice ofan experiment (a signal) s̃i, or of a distribution over posterior beliefs πi ∈ ∆∆Ω(Kamenica and Gentzkow, 2011).4 At the interim stage, players send a mes-sage to the designer conditional on their posterior belief. Let σi : ∆Ω −→ Midenote player i’s reporting strategy. See Figure 1 for the timing of the game.

4Formally, an experiment is a Ω×S-valued random variable, where S is a set of signal real-izations of cardinality at least |Ω| to guarantee full flexibility. Any distribution over posteriorbeliefs πi satisfying the Bayesian requirement that the expected posterior is equal to the priorcan be achieved by choosing an appropriate signal.

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State ω isdrawn from µ0

Designer announces Γ

Informationstrategy πi

Update µi(ω)

Reportingstrategy σi(µi)

Allocation xi(µi, µ−i)Transfers ti(µi, µ−i)

Figure 1: Timing of the game

Player i’s ex-ante utility is:

Eπi

[∑ω∈Ω

µi(ω)∑k∈K

Eπ−i|ω[xik(σi(µi), σ−i(µ−i))

]ui(k, ωi)− Eπ−i|ω

[ti(σi(µi), σ−i(µ−i))

]]− c(πi)

where c(·) is the cost of information acquisition. This formulation makes clearthat information acquisition on ω−i helps player i predict the report of others.

Assumptions on the Cost Function. We consider cost functions which admita posterior-separable representation:

c(πi) = H(µ0)− Eπi [H(µi)]

where H : ∆Ω −→ R is a strictly concave functional which can be interpretedas a measure of uncertainty.5 The class of posterior separable cost functionsis very rich and particularly appealing because we can use the first-order-condition approach to find players’ equilibrium strategies (Caplin and Dean,2013). Mensch (2018) provides an axiomatic characterization of such func-tions, and De Oliveira et al. (2017) a behavioral characterization. Zhong (2020)shows that posterior separable cost functions can be understood as reducedform representations of a process of sequential information cost minimization.

We make two extra assumptions which are crucial for the derivation of our

5The measure of uncertainty H is allowed to also depend on the prior µ0, as is the case inExample 2 below. However, since this is not relevant for the rest of the analysis, we do notmake this explicit in the notation.

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results. First we assume that the cost function is smooth—excluding kinks andjumps—which allows players to fine-tune the informativeness of signals. Thisin particular rules out the possibility that learning about others’ preferences isdiscontinuously costlier than learning about oneself.6 Second we assume thatthe marginal cost of becoming fully informed about any fundamental goes toinfinity (though the actual cost can be bounded). This Inada condition guaran-tees that players’ best response is interior (conditional on acquiring informa-tion).

Assumption 1 (Regularity Conditions).

(i) H is twice continuously differentiable on the interior of ∆Ω.

(ii) For all ω and for all µ̂ such that µ̂(ω) = 0, limµi→µ̂ ∂H(µi)/∂µi(ω) =∞.7

Example 1 (Entropic Cost). Sims (2003) proposes a cost function based on Shan-non’s entropy which measures the informativeness of a signal in terms of the expectedreduction in entropy. The functional H is Shannon’s entropy:8

H(µi) = −∑ω∈Ω

µi(ω) log µi(ω).

Entropy-based cost functions are appropriate to capture information processing orinformation attention, namely a situation in which all the information is in principleavailable but the agent exerts costly effort to pay attention to it.

Example 2 (Log-Likelihood Ratio Cost). Pomatto et al. (2019) show that infor-mation production can be naturally expressed in terms of KL-divergence. If the costfunction satisfies an extra additivity requirement then there exist non-negative coeffi-cients βωω′ such that

H(µi) = −∑

ω,ω′ ∈Ω

βωω′µi(ω)

µ0(ω)log

µi(ω)

µi(ω′).

6We discuss and relax this assumption in Section 4.27The limit is defined (for instance) with respect to the total variation distance.8See DeDeo (2016) for an illuminating introduction to information theory.

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Log-likelihood ratio cost functions are appropriate to capture information production,namely situations where the difficulty to distinguish states intrinsically limits theinformativeness of signals.

Solution Concept. We conclude by defining a Nash equilibrium in this game.A Nash Equilibrium is a strategy profile (π∗, σ∗) such that, for all i ∈ N ,

(σ∗i , π∗i ) ∈ arg max

σi∈Σi,πi∈∆∆ΩEπi

[∑ω∈Ω

µi(ω)∑k∈K

Eπ∗−i|ω[xik(σi(µi), σ

∗−i(µ−i))

]ui(k, ωi)

− Eπ∗−i|ω[ti(σi(µi), σ

∗−i(µ−i))

]]− c(πi).

Existence of Nash equilibria follows from standard arguments that we omit.

4 THE GENERIC COMPLEXITY OF

INFORMATIONAL INCENTIVES

In this section we address the following question: Which mechanisms provideplayers with simple informational incentives, i.e. incentives to only acquireinformation on their own preferences? We show that players have informa-tionally simple incentives only if the mechanism is de facto dictatorial.

Informational simplicity is captured by the following refinement of theNash equilibrium: an equilibrium is informationally simple if agents only ac-quire information on their own payoff-relevant fundamental ωi.

Definition 1 (Informationally Simple Equilibrium). An Informationally Sim-ple Equilibrium (ISE) is a Nash equilibrium (σ∗, π∗) such that, for all i, the signals̃∗i associated with π∗i satisfies

s̃∗i ⊥ ω̃−i.

This refinement is of interest for several reasons. Informational simplic-ity captures a notion of strategic simplicity which is a priori distinct from

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strategy-proofness. As it turns out, we will show that informational simplicityis a necessary condition for ex-ante strategy-proofness of the extended gamethat includes an information acquisition stage. Second, players’ interim infor-mation structure satisfies the independence private value assumption if andonly if the equilibrium is informationally simple. Therefore, if such equilib-rium does not generically exist then players’ private information will gener-ally be correlated.

Fix an arbitrary indirect mechanism Γ. Without loss of generality, we di-rectly work with probability distributions over messages conditional on states:Pi : Ω −→ ∆Mi. This object is obtained from (σi, πi) using Bayes’ rule:

Pi(mi | ω) =∑

µi∈supp πi

1{σi(µi) = mi}µi(ω)

µ0(ω)πi(µi).

In words, the information strategy and reporting strategy (σi, πi) lead player ito send message mi in state ω with probability Pi(mi | ω).

At the ex-ante stage, player i chooses a stochastic choice rule Pi given P ∗−iso as to solve the following program:

maxPi:Ω→∆Mi

∑ω∈Ω

µ0(ω)EPi(·|ω),P ∗−i(·|ω)[ui(xi(mi,m−i), ωi)− ti(mi,m−i)

]−

∑mi∈ suppPi

H̄[Pi(mi | ·) ◦ µ0(·)]

subject to∑

mi∈ suppPi

Pi(mi | ω) = 1 ∀ω

Pi(mi | ω) ≥ 0 ∀ω,mi

where9

H̄[Pi(mi | ·) ◦ µ0(·)] ≡ −H[

Pi(mi|·)µ0(·)∑ω∈Ω Pi(mi|ω)µ0(ω)

]∑ω∈Ω

Pi(mi|ω)µ0(ω).

9See Denti (2020) (Lemma 4) for how to rewrite H as a function of the stochastic choicerule.

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Conditional on acquiring some information, the first-order conditions with re-spect to Pi(mi|ω)µ0(ω) yield necessary restrictions on player i’s best response:for all mi in the support of P ∗i and for all ω ∈ Ω,

EP−i(·|ω)[ui(xi(mi,m−i), ωi)− ti(mi,m−i)

]+γi(ω)

µ0(ω)=∂H̄[P ∗i (mi|·) ◦ µ0(·)]∂P ∗i (mi|ω)µ0(ω)

(?)

where γi is the Lagrange multiplier associated with the constraint that thestochastic choice rule must sum to one. The left-hand side captures the marginalgain from sending messagemi in state ω, whereas the right-hand side capturesthe marginal cost.

We can rearrange and substitute out the Lagrange multiplier to obtain aninterpretation of the FOC in terms of value of information. Take the FOC withrespect to Pi(m′i | ω)µ0(ω) and substract from the previous FOC. This givesthe marginal gain from reporting mi relative to m′i in state ω, net of marginalcosts:

EP−i(·|ω)[ui(xi(mi,m−i), ωi)− ti(mi,m−i)− [ui(xi(m′i,m−i), ωi)− ti(m′i,m−i)]

]=∂H̄[P ∗i (mi | ·) ◦ µ0(·)]∂P ∗i (mi | ω)µ0(ω)

− ∂H̄[P∗i (m

′i | ·) ◦ µ0(·)]

∂P ∗i (m′i | ω)µ0(ω)

.

In general the value of information for i seems to depend on other agents’ re-ports, as m−i impacts the chosen outcome. In an informationally simple equi-librium, however, player i’s strategy must be independent of ω−i: P ∗i (·|ωi, ω−i) =P ∗i (·|ωi, ω′−i) for all ωi, ω−i and ω′−i. This requires the value of information forplayer i to be independent of other players’ realized state. We now show thatthis independence cannot generically be satisfied.

We consider two definitions of genericity, one coming from measure the-ory and the other from topology.10 A set U ⊂ Rn is of measure zero if it hasLebesgue measure zero. Conversely, a set is of full measure if its Lebesgue

10In general the two concepts cannot be compared—in fact, they can lead to opposite con-clusions. This is illustrated by the following striking result: The real line can be decomposed intotwo complementary sets A and B such that A is a meager set and B is of measure zero. For furtherdiscussion of the connections between topological and measure spaces, see Oxtoby (1980).

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measure is equal to the dimension of the ambient space n. A set U ⊂ Rn

is meager if it can be represented as a countable union of nowhere densesets. Conversely, a set is comeager if it is the complement of a meager set.Comeager sets are dense (by Baire category theorem), but not all dense setsare comeager.11 A set U ⊂ Rn is generic if it has full measure and is comeagre.A set U ⊂ Rn is degenerate if it has zero measure and is meagre.

We show that Informationally Simple equilibrium imply that the mecha-nism is, de facto, dictatorial.

Definition 2. A mechanism is dictatorial on the support of P ∗, or de facto dic-tatorial, if there exists at most one agent i for whom there exist mi,m′i ∈ suppP ∗isuch that xik(mi,m−i) 6= xik(m′i,m−i) for some m−i ∈ suppP ∗−i, for some k ∈ K.

Theorem 1. For a generic set of utility functions in the quasilinear domain U ⊆R

∑i |K×Ωi|, if P ∗ is an Informationally Simple equilibrium of Γ then Γ is dictatorial

on the support of P ∗.

Proof Sketch: The proof strategy consists of showing that the number of en-dogenous variables is smaller than the number of necessary conditions for theexistence of an ISE. Denote Φ the functional which maps endogenous vari-ables (i.e. stochastic choice rules, transfers and Lagrange multipliers) andpreferences to the FOCs (?). Namely, the system of FOCs (?) is written asΦ(P, γ, t;u) = 0. By the Inada conditions Φ(P, γ, t;u) = 0 is a necessary con-dition for the existence of an ISE. We want to compute the dimension of thesolution set Φ−1(0;u), i.e., the dimension of the set of solutions to the FOCs.We need to show that the functional Φ is invertible at 0, i.e. that the FOCs arelocally linearly independent. Because the FOCs are highly non-linear, we needto use a generalization of the rank-nullity theorem. It can be shown that the Ja-cobian of Φ has full rank, i.e. rankDΦ = |×i∈N Ωi|×∑i∈N | suppP ∗i |. Hence bythe Parametric Transversality theorem Φ is invertible at 0 for a comeager andfull measure set of preferences. Finally, a generalization of the Implicit Func-tion theorem is used to compute the dimension of the manifold Φ−1(0;u). The

11For instance Q is dense in R but Q it is not comeager in R. Hence comeagerness is astronger requirement than density.

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dimension of the solution manifold corresponds to the number of endogenousvariables minus the number of (locally) linearly independent equations of Φ,i.e.∑

i∈N

[| suppP ∗i |× (|Ωi|+ |Ω−i|)

]−|×i∈N Ωi|×∑i∈N | suppP ∗i | < 0. Q.E.D.

The economic intuition is that players have uncertainty on their “choiceset,” i.e. which outcomes they can bring about. Their choice set dependson others’ preferences, which makes it optimal to first acquire informationon others’—as the marginal cost is zero at the prior due to the smoothnesscondition—, and then on their own preferences if it turns out to be valuable.12

Therefore, an information strategy on one’s own state which condition on oth-ers’ preferences dominates an information strategy restricted to informationacquisition on one’s own preferences. This holds even when the mechanismis interim strategy-proof as the Inada condition makes it never optimal to be-come fully informed on one’s own preferences.

Importantly, the theorem is not a statement about the primitives: if P ∗ isan Informationally Simple equilibrium of Γ then the mechanism need not bedictatorial, but under P ∗ the mechanism acts as if it were dictatorial. For in-stance, when the marginal gain of information acquisition is very small for allbut one player, it is very possible that even if all agents can impact the cho-sen outcome in the mechanism, only one decides to acquire information inequilibrium. We can restate the theorem as follows: if there exists an Infor-mationally Simple equilibrium in Γ then we can find an outcome-equivalentmechanism Γ′ which is dictatorial.

That being said, the theorem is sufficient for mechanism design purposes.The point is that informational simplicity is impossible to achieve unless thedesigner’s objective is trivial and only requires the elicitation of a single agent’sinformation. Whether or not the mechanism is truly dictatorial or only dicta-torial de facto is immaterial.

12Note that agents do not care about others’ preferences ω−i per se, but only because ithelps them predict others’ report to the designer. Hence if agents were allowed to acquireinformation on what others know—i.e. what signal realization s−i they received—as in Denti(2018), then they would do so instead of learning about their underlying preferences ω−i.

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Remark 1. Without additional assumptions on the cost of information, dic-tatorial mechanisms need not admit Informationally Simple equilibria (i.e. theconverse of Theorem 1 can be false). This is the case when the marginal costof information acquisition on one’s own preferences depends on others’ fun-damentals: it is then optimal to condition information acquisition on theseas well. For instance, the optimal choice rule with log-likelihood ratio cost(Pomatto et al., 2019) solves:

µ0(ω)[Vi(mi, ω)− Vi(m′i, ω)

]=−

[∑ω′∈Ω

βωω′ logP ∗i (mi|ω′)P ∗i (mi|ω)

+ βω′ωP ∗i (mi|ω′)P ∗i (mi|ω)

]

+

[∑ω′∈Ω

βωω′ logP ∗i (m

′i|ω′)

P ∗i (m′i|ω)

+ βω′ωP ∗i (m

′i|ω′)

P ∗i (m′i|ω)

]

with Vi(mi, ω) = EP−i(·|ω)[ui(x(mi,m−i), ωi) − ti(mi,m−i)

], for every state ω

and every messages mi,m′i ∈ suppP ∗i . Clearly, P ∗i depends on ω−i throughthe prior µ0(ω) even if the ex-ante utility Vi is independent of ω−i. We believe,however, that this form of prior dependence to payoff-irrelevant states is eco-nomically unlikely. A cost function is independent of irrelevant states if, forall ω,

Vi(mi, (ωi, ω−i)) = Vi(mi, (ωi, ω′−i)) ∀mi =⇒ P ∗i (· | ωi, ω−i) = P ∗i (· | ωi, ω′−i)

In words, the cost function is independent of irrelevant states if the inducedoptimal choice rule does not depend on payoff-irrelevant states.13 The mostnotable example satisfying this condition is the entropic cost function. Indeed,this function is “invariant under compression” (Caplin et al. (2017)), which is astronger requirement than our independence of irrelevant states.14 The latter

13Our notion of independence to irrelevant states is conceptually distinct from prior inde-pendence of the cost function. Prior independence is too strong for our purposes, and Zhong(2020) shows that it is incompatible with posterior separability when there are more than twostates.

14Invariance under compression requires that splitting a state into two payoff-equivalentstates should not change how costly it is to learn about it. In particular, splitting a state ωiinto | Ω−i | payoff-equivalent states should not change agent i’s optimal strategy, which is

16

indeed only requires the cost function to be invariant under compression ofsome dimensions of the state space (i.e. only of Ω−i), which is reminiscent ofa similar condition in Hébert and La’O (2020).

If the cost function is independent of irrelevant states, then the converseof Theorem 1 is also true: a mechanism Γ admits an informationally simpleequilibrium P ∗ if and only if it is dictatorial on the support of P ∗.

4.1 Two Illustrative Examples

We first go back to our motivating example and show that the equilibrium un-der VCG is not informationally simple. We then go through a knife-edge casefor which our main result does not hold. Indeed, since the latter in a generic-ity result, there exist degenerate utility functions under which an equilibriumcan be both informationally simple and non-dictatorial.

Back to Our Motivating Example. To illustrate Theorem 1, we show thatcity representatives acquire information on the demand of the other city inany non-dictatorial equilibrium. We consider the entropic cost function:

H(µi) = −λ∑ω∈Ω

µi(ω) log µi(ω),

where λ is a parameter that scales the cost of information. At the interimstage, it is a dominant strategy for city representatives to report their trueexpected valuation to the designer. We look for an equilibrium in which bothagents acquire information.15 It can be shown that each agent only has twoposterior beliefs in the support of his equilibrium strategy: one that puts moreweight on state ω̄i, and the other more weight on state ωi. Denote them byµi and µi, respectively. Let mi = Eµi [ωi] be agent i’s expected valuation atbelief µi, and define mi similarly. For agents to have an incentive to acquiresome information, it has to be that reporting these valuations does not always

our above condition.15Note that if an agent does not acquire information, he always sends the same message to

the designer and hence never impacts the outcome—the equilibrium is dictatorial.

17

yield the same outcome, that is x∗(mi,mj) = x∗(mi,mj) = x∗(mi,mj) = 1 butx∗(mi,mj) = 0.

Following Matějka and McKay (2015), we know that equilibrium stochasticchoice rules follow a logit rule. Simple algebra yields that the stochastic choicerule verifies:

Pi(mi|ωi, ωj) =Pi(mi)

Pi(mi) + Pi(mi) exp[− 1

λ

(1− Pj(mj|ωi, ωj)

) (ωi +mj

) ](1)The expression

(1 − Pj(mj|ωi, ωj)

) (ωi +mj

)captures the value of reporting

a high valuation m̄i in state ω relative to reporting mi. If the other agent islikely to report a high valuation himself— Pj(mj|ωi, ω−i) is close to 1—thisvalue is zero as agent i’s report very rarely impacts the chosen outcome. Thelogit rule makes transparent why the equilibrium cannot be InformationallySimple. Indeed, Pi(mi|ωi, ωj) is independent of ωj only if the RHS of (1) is aswell, which is the case only when agent j’s strategy does not depend on ωj .Hence agent i does not learn about j only when j does not learn about himself.

For the sake of tractability, we characterize what the equilibrium convergesto as the relative cost of information λ goes to zero.16

(ωi, ωj) (ωi, ωj) (ωi, ωj) (ωi, ωj)

Pi(mi|ωi, ωj) 1 1 23 0

Pj(mj|ωi, ωj) 1212

1 0

Each agent receives the correct signal whenever the other’s valuation is low, asthis is when information makes a difference: Pi(mi|ω̄i, ωj) −→ 1 and Pi(mi|ωi, ωj)−→ 0, and reciprocally for agent j. The equilibrium strategy is more sub-tle in state (ωi, ωj). As (1) suggests, Pi(mi|ωi, ωj) is a decreasing function ofPj(mj|ωi, ωj): as one agent learns more about his valuation and becomes more

16This simplifies the analysis as it ensures both agents do in fact acquire information inequilibrium. For non-vanishing λ, we would have to make sure that agents do have thesetwo posteriors in the support of their equilibrium strategy, which involves checking that a setof inequalities is satisfied (Caplin et al. (2018)).

18

likely to send a high report to the designer, the other has less of an incentiveto learn about himself. Such informational free-riding leads to an asymmet-ric equilibrium in which one agent—agent i in the table above—becomes in-formed about his high valuation, ensuring that the school is built with proba-bility close to 1 in this state, and the other remains uninformed.

A Knife-Edge Example. There are three goods {A,B,C} to be allocated tothree agents {i, j, k}. For simplicity, there are no transfers. The mechanismused is a simultaneous version of the serial dictatorship: Agents report theirpreferences, agent i gets her favorite good, then agent j her favorite goodamong the remaining ones, and k is allocated whatever remains.

Agent i’s most preferred good is either A or B: ui = (uiA, uiB, uiC) ∈{(4, 2, 1), (2, 4, 1)}. Agent j always values good A and B equivalently: uj ∈{(2, 2, 0), (0, 0, 2)}. Since agent k has no impact on the outcome, his prefer-ences are irrelevant. Agent i and j each have two possible messages they cansend to the designer: Mi = {mA,mB} and Mj = {mAB,mC}. Intuitively, youcan think of a message ml as indicating to the designer that the agent wantsgood l. The allocation function is then:

xi(mi,mj) mAB mC

mA (1, 0, 0) (1, 0, 0)

mB (0, 1, 0) (0, 1, 0)

xj(mi,mj) mAB mC

mA (0, 1, 0) (0, 0, 1)

mB (1, 0, 0) (0, 0, 1)

As in the previous example, consider the entropic cost function. Sinceagent i’s allocation does not depend on agent j’s report, she only learns abouther own preferences in equilibrium. More surprisingly, agent j also onlylearns about his own preferences, despite the fact that his allocation is im-pacted by i’s report. That is because his utility function has some symmetrythat makes the value of information on ωj independent of mi. Indeed, if agenti picks good A, agent j is left choosing between B and C, and the utility valueof making the correct choice is 2. Similarly, if agent i picks good B, agent jis left choosing between A and C, and the utility value of making the correct

19

choice is, again, 2. The equilibrium is informationally simple, even thoughtwo agents acquire information and impact the outcome in equilibrium.

4.2 Discussion of the Assumptions on the Cost Function

The proof of the impossibility result leverages the assumption that the costof information is smooth. This seems to be a relevant approximation in somesettings. For instance, consider a school choice problem in which studentsmust send a rank-order list of schools to a central authority, and can before-hand acquire information on the different schools. They can learn about theirown preferences over schools—e.g., by looking at the set of courses offeredand whether they look interesting to them—but also about others’—e.g., byasking about the popularity of the school, and admission cutoffs. Arguably,acquiring some information about a school’s popularity is not very costly, andonly requires a quick Google search.

A natural concern is that in other settings, the smoothness assumptionmay be missing relevant factors, and in particular overlooks the possibilitythat learning about others may be discontinuously harder than learning aboutoneself. This would mechanically make Informational Simplicity easier toachieve, and we investigate the robustness of Theorem 1 to such discontinuity.

Consider the same smooth cost of information as in our main setup, butsuppose that as soon as an agent decides to learn a bit about others, it has topay an additional fixed cost κ. Let UIS(κ) ⊆ R

∑i |K×Ωi| be the set of utility

functions for which (i) there exists an Informationally Simple equilibrium P ∗

of Γ, and (ii) Γ is not dictatorial on the support of P ∗.

Proposition 1. Let ρ(κ) be the Lebesgue measure of UIS(κ). The function ρ is in-creasing, and continuous in κ, with ρ(0) = 0.

Theorem 1 corresponds to the corner case in which κ = 0: it is gener-ically impossible to have an equilibrium that is informationally simple andnon-dictatorial. As κ increases, the set of preferences for which informationalsimplicity can be achieved by non-dictatorial mechanisms grows. Interest-ingly, it grows continuously, so our benchmark with κ = 0 is not a knife-edge

20

case: Adding a small cost to learning about others does make InformationalSimplicity easier to achieve, but only in very few settings. It is only as κ tendsto infinity that Informational Simplicity becomes generically feasible.

We close this section with a brief comment on the assumption that themarginal cost of information is unbounded as agents become fully informedabout some states of the world. If this assumption is not satisfied, then agentswill choose to become fully informed about their preferences for a set of util-ity functions that is non-degenerate, but non-generic either. In those cases,they will have no interest in learning about others as long as the mechanismis interim strategy-proof.

5 IMPLICATIONS FOR MECHANISM DESIGN

5.1 The Limits of Strategy-Proofness

Strategic simplicity is valued in mechanism design for robustness and for lev-eling the playing field across players. A lot of attention has been given to in-terim strategy-proof mechanisms, i.e., mechanisms for which revealing one’sprivate information is a dominant strategy. Little is known, however, aboutthe strategic complexity of the acquisition of agents’ private information atthe ex-ante stage. This is important as many inequalities may arise due tosuboptimal information acquisition and strategic mistakes.

We show that informational simplicity is a necessary condition for ex-antestrategy-proofness.

Proposition 2. Assume that the cost of information is independent of irrelevantstates.17 If P ∗ is an equilibrium in dominant strategy of Γ, then P ∗ is informationallysimple.

The standard notion of strategy-proofness ensures that agents have a dom-inant strategy once they have acquired information, but the stronger require-ment of Informational Simplicity is needed to guarantee agents also have a

17Namely, the induced optimal choice rule does not depend on payoff irrelevant states. SeeRemark 1 above for a formal definition.

21

dominant strategy when choosing what information to acquire. The intuitionis that information about others is valuable only insofar as it helps predicttheir reports at the interim stage. Hence if an agent learns about another, herequilibrium information strategy has to depend on the strategy of the otherplayer: the mechanism is not ex-ante strategy proof.

Together, Proposition 2 and Theorem 1 yield that, generically, P ∗ is an equi-librium in dominant strategy of Γ only if Γ is dictatorial on the support of P ∗.Therefore, in the extended game that includes information acquisition, agentsvirtually never have a dominant strategy under non-dictatorial mechanisms.

5.2 Independent Private Values

A direct corollary of Theorem 1 is that the standard Independent Private Valueassumption is unlikely to arise endogenously.

Corollary 1. For a generic set of utility functions U ⊆ R∑

i |K×Ωi|, the equilibriumposterior beliefs (µ̃i, µ̃−i) are (unconditionally) independent across players only if Γis dictatorial on the support of P ∗.

Therefore the interim information structure is endogenously correlated,which creates interdependent values across players. Put differently, the IPVassumption does not arise endogenously whenever the mechanism is non-dictatorial and the technology of information satisfies our conditions. This isin contrast with the case of privately known preferences with costly informa-tion acquisition only on others’ preferences. In this case, IPV endogenouslyarises whenever the mechanism is strategy-proof. Therefore it is essential forour result that players do not know their own preferences, and that becomingfully informed on their own preferences is never optimal by the Inada condi-tion.

This result, however, raises an intriguing question: is full surplus extrac-tion à la Crémer-Mclean always feasible (in Nash equilibrium) when privateinformation is endogenous?

Crémer and McLean (1988) show that as soon as there is some correlationin agents’ ex-ante private information, the designer can extract all surplus by

22

constructing appropriate side bets. This result highlights that the independenceof private information across players is necessary for them to earn an infor-mation rent as in Myerson (1981). The limits of that result to risk aversion,limited liability, collusion among the agents, etc. have been explored exten-sively. However what has been explored less is how such results rely on theexogenous nature of private information: if agents anticipate the designer willexploit the correlation structure in their information, why would they acquiresuch information in the first place? Indeed we show that full surplus extrac-tion is generically impossible to achieve when taking into account informa-tional incentives.

First, we need to properly define what full surplus extraction means ina setting where private information is endogenous. We say full-surplus ex-traction is feasible if there exists a mechanism that can extract the maximalsurplus that can be generated in the economy. Namely, given an environmentand an information technology, there exists a maximal total surplus that canbe generated, which balances total gains from the outcome and total informa-tion costs. Full surplus extraction requires that we reach an equilibrium thatgenerates this surplus, and then extract it entirely using transfers.

As in Crémer and McLean (1988), there is one good to be allocated.18 Letthe ex-post efficient allocation at belief profile µ = (µi)i∈N be the allocationthat maximizes total expected welfare:

x∗(µ) = arg maxx∈∆N

∑i∈N

∑ω∈Ω

µi(ω)ui(ωi)xi.

The maximum total surplus that can be generated in the economy equals:

Max. Total Surplus = maxπ∈×i ∆∆Ω

∑i∈N

Eπ

[∑ω∈Ω

µi(ω)x∗i (µ)ui(ωi)

]− c(πi).

Let π† be the information strategy profile that maximizes total surplus. A(direct revelation) mechanism extracts the full surplus if it induces an equi-

18Our result easily extends to multiple goods.

23

librium π∗ such that:∑i∈N

Eπ∗ [ti(µi, µ−i)] = Max. Total Surplus,

while satisfying incentive and individual rationality constraints. Incentiveconstraints are of two sorts here: agents should be incentivized to reveal theirprivate information to the designer at the interim stage, and should find it op-timal to acquire the socially efficient level of information at the ex-ante stage.The former is the standard IC constraint in mechanism design, and from nowon, suppose it holds. The latter, which is the one limiting the possibility of fullsurplus extraction in this setting, writes:

Eπ†i ,π†−i

[∑ω∈Ω

µi(ω)x∗i (µ)ui(ωi)− ti(µi, µ−i)

]− c(π†i )

≥ Eπi,π†−i

[∑ω∈Ω


]− c(πi) for all πi, i.

Finally, the mechanism should satisfy the following ex-ante and interim indi-vidual rationality constraints:

Eπ†i ,π†−i

[∑ω∈Ω


]− c(π†i ) ≥ 0 for all i.

Eπ†−i

[∑ω∈Ω


∣∣∣µi] ≥ 0 for all i.Hence to extract the full surplus, the mechanism must (i) induce agents toacquire the socially efficient level of information, (ii) pick the ex-post efficientallocation given reported posterior beliefs, and (iii) have transfers that extractall surplus net of information acquisition costs. The last two requirements arefamiliar from Crémer and McLean (1988), whereas the first one is new butnecessary to make sense of ex-post efficiency.

Observe that in some extreme cases full surplus extraction is possible. For

24

instance, consider a setting in which the efficient allocation is the same in ev-ery state ω. This means that x∗(µ) = x∗ is independent of agents’ posteriorbeliefs and that the efficient information strategy is to acquire no informationat all. The mechanism that always selects outcome x∗ irrespective of agents’reports, and has transfers ti =

∑ω µ0(ω)x

∗iui(ωi) is individually rational, in-

centive compatible, and extracts full surplus.However informational incentives generally limit the possibility of full

surplus extraction. To prove this, we show that the three requirements ex-posed above translate into necessary conditions that are generically mutuallyincompatible. We first focus on requirements (i) and (ii) of full surplus ex-traction, namely that the mechanism induces socially efficient information ac-quisition and implements the ex-post efficient allocation. Conditional on π∗−i,agent i’s optimal information strategy solves:

maxπi∈∆∆Ω

Eπi,π∗−i

[∑ω∈Ω


]− c(πi).

The standard approach to incentivize efficient information acquisition is touse Clarke pivot rule:

ti(µ) = −∑j 6=i

∑ω∈Ω

µj(ω)x∗j(µ)uj(ωj).

We show that such transfers are actually the only one inducing agents to ac-quire the socially efficient level of information. This result is reminiscentof Hatfield et al. (2018) who extend the Green–Laffont–Holmström theoremby showing that VCG mechanisms with ex-ante costly investments are theunique efficient and strategy-proof mechanisms. To obtain a Crémer-McLeanmechanism and enforce the third requirement of full surplus extraction, weadd side-bets bi :×j 6=i ∆Ωj −→ R. Therefore the transfers write:

tCMi (µ) = −∑j 6=i

∑ω∈Ω

µj(ω)x∗j(µ)uj(ωj) + bi(µ−i).

25

However such side bets generically distort informational incentives, that is in-centives to acquire the efficient level of information. This reduces the totalsurplus generated by the mechanism, preventing full surplus extraction.

Theorem 2. The set of preferences in the quasilinear domain U ⊆ R∑

i |K×Ωi| forwhich full surplus extraction is feasible is degenerate.

The impossibility of full surplus extraction in our setting contrasts with thepossibility result in Bikhchandani (2010): he shows that when multiple playerscan acquire information on the preferences of others, full surplus extraction isfeasible. Our result suggests that this is a consequence of the rigidity of infor-mation acquisition in Bikhchandani (2010): players only have access to a finiteand pre-determined set of signals, whereas here players acquire informationflexibly.

In this section, we took an ex ante perspective to full surplus extraction,requiring that the mechanism extracts the maximal surplus that can be gener-ated in the economy. Another approach would be to ask whether ex post fullsurplus extraction is possible: Does there exist a mechanism such that, in equi-librium, the ex post efficient allocation is implemented and all the associatedsurplus is extracted from agents? Here the answer is always yes: the constantmechanism that always picks the efficient allocation at the prior x∗(µ0) andhas transfers always equal to ti =

∑ω µ0(ω)x

∗i (µ0)ui(ωi) induces no informa-

tion acquisition, and does extract all ex post surplus in equilibrium. Even ifwe restrict attention to equilibria in which agents acquire some information,it seems to be always possible to find a mechanism extracting all ex post sur-plus in equilibrium. This, however, is no guarantee on the magnitude of thesurplus that is extracted by the seller: it can very well be that the generatedsurplus in equilibrium is very small.

6 CONCLUSION

In this paper, we investigate players’ informational incentives in mechanism de-sign, namely how the choice of the mechanism impacts information acquisi-

26

tion in equilibrium. A mechanism is informationally simple if agents have noincentives to acquire information on others’ preferences. Our main result isthat, for any smooth technology of information acquisition satisfying an In-ada condition, a mechanism has an Informationally Simple equilibrium onlyif the mechanism is de facto dictatorial. This result holds generically in both ameasure-theoretic sense and a topological sense. The intuition is that the out-comes players can bring about depend on others’ preferences, which makesit optimal to acquire information on them before investing in information ac-quisition on one’s own preferences.

This result has two implications for mechanism design. First, we show thata mechanism is ex-ante strategy-proof only if it has an informationally simpleequilibrium, hence only if the mechanism is de facto dictatorial. This points toa limitation of strategy-proofness as a concept of strategic simplicity. Indeed,even interim strategy-proof mechanisms incentivize players to acquire infor-mation about others’ and to best respond to beliefs about opponents’ playat the ex-ante stage. Second, our result suggests that the independent pri-vate value assumption is unlikely to arise endogenously. This however doesnot mean full surplus extraction is possible using side bets as in Crémer andMcLean (1988), as these would distort players’ incentives when acquiring in-formation

There are several avenues for future research. One source of informationacquisition that we ignored so far is communication among players. On theone hand, our result suggests that some players would benefit from informa-tion aggregation in a communication stage after the information acquisitionstage due to the correlation of the information structures. On the other hand,adding such a communication stage modifies informational incentives and free-riding may arise in the information acquisition stage. This raises an interestingquestion: Under what conditions does communication facilitate implementa-tion and would arise endogenously from a coalition of players?

These considerations suggest that informational incentives may have im-portant and concrete implications for the design of institutions—which re-main largely unexplored to this day.

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APPENDIX A PROOFS

Proof of Theorem 1. By contradiction, suppose that there exists an IS equilib-rium P ∗ of Γ but Γ is not dictatorial over the set of equilibrium message pro-files M∗ =×i∈N M∗i with M∗i = suppP ∗i . Then there exist (at least) two agentsi and j such that |M∗i | ≥ 2 and |M∗j | ≥ 2, and

xik(mi,m−i) 6= xik(m′i,m−i) for some mi,m′i ∈M∗i ,m−i ∈M∗−i, k ∈ K;

xik(mj,m−j) 6= xik(m′j,m−j) for some mj,m′j ∈M∗j ,m−j ∈M∗−j, k ∈ K.

Denote N∗ the set of agents who send at least two messages in equilibrium(without loss of generality, this is the set of agents who acquire informationin equilibrium). For i ∈ N∗, let Ω∗i be the coarsest partition of Ωi such thatPi(· | ωi) is constant across all ωi in the same element of the partition. That is Ω∗ibundles together all states ωi at which agent i plays the exact same stochasticchoice rule.19 In what follows we slightly abuse notations, and use ωi to denotea representative element of the partition to which ωi belongs.

We restrict attention to the FOCs of messages in M∗, hence we ignore thenon-negativity constraint on equilibrium probabilities. By the Inada conditionwe know that the equilibrium stochastic choice rule P ∗i for all i ∈ N∗ must beinterior. Define Φ : P × U −→ Rm the function which maps stochastic choicerules with support M∗ and Lagrange multipliers (P, γ, t) ∈ P together withpreferences u ∈ U to the following vector:

EP−i(·|ω)[ui(xi(mi,m−i), ωi)− ti(mi,m−i)

]+γi(ω)

µ0(ω)− ∂H̄[P

∗i (mi|·) ◦ µ0(·)]

∂P ∗i (mi|ω)µ0(ω)

for all mi ∈ M∗i , ω ∈ Ω∗. By definition the vector of transfers is an elementof R

∑i∈N∗ |M∗|. By Blackwell principle of irrelevant information, it is easy to

see that |M∗i | ≤ |Ω∗i | as information is valuable only insofar as it changes the19Note that for this to be the case in equilibrium, agent i’s preferences over outcomes she

can bring about must be the same in all states that are bundled together.

28

optimal action. Therefore we have

dim P̃ =∑i∈N∗

[(|M∗i | − 1)× |Ω∗i |+ |Ω∗i |+ |M∗i × Ω∗−i|

]=∑i∈N∗

[|M∗i | × (|Ω∗i |+ |Ω∗−i|)

]The FOCs can be written as Φ(P ∗, γ, t;u) = 0. Hence, the set of IS stochas-

tic choice rules (together with Lagrange multipliers and transfers) which solvethe agents’ FOCs is Φ−1(0;u). We show that this set is a manifold of negativedimension, and hence is empty, for almost all u ∈ U. This is done by suc-cessively applying the Transversality theorem (to show that the non-linearequations in this system are locally linearly independent at 0 for almost all u),and the Regular Value theorem (to show that the solution set is a manifold ofnegative dimension).20

In order to apply the Transversality theorem we need to show that 0 is aregular value of Φ, i.e. the number of locally linearly independent equations ofthe system evaluated at 0 is maximal: Φ(P, γ, t;u) = 0 =⇒ rankDΦ(P, γ, t;u) =|×i∈N∗ Ω∗i | ×∑i∈N∗ |M∗i |. where D is the Jacobian. Denote by Du the restric-tion of the Jacobian corresponding to the derivative with respect to the param-eter u only. Note that DuΦ has |×i∈N∗ Ωi| ×∑i∈N∗ |M∗i | rows—one for eachmi ∈ M∗i and ω ∈ Ω∗—and |K| ×

∑i∈N∗ |Ω∗i | columns—each corresponding to

the derivative of Φ with respect to ui(k, ωi).Since ui(k, ωi) only enters the FOC of agent i in state ωi ∈ Ω∗i , the Jacobian

is block diagonal with each block corresponding to one state ωi for agent i:

DuΦ =

Bi(ωi) 0 . . .

0. . .

... Bj(ωj) 0

0. . .

In a block Bi(ωi), each row corresponds to a possible message mi ∈ M∗i for

20Mas-Colell (1989) Chapter 1 (section H) and especially Chapter 8 provide an introductionto differential topology. A formal statement of the results we use here can be found on page320.

29

player i ∈ N∗ together with a state for other players ω−i ∈ Ω∗−i; and eachcolumn corresponds to a possible outcome k ∈ K:

Bi(ωi) =

∑m−i∈M∗−i

P−i(m−i | ω−i)xik(mi,m−i)∑

m−i∈M∗−i

P−i(m−i | ω−i)xik′(mi,m−i) . . .

∑m−i∈M∗−i

P−i(m−i | ω′−i)xik(mi,m−i)∑

m−i∈M∗−i

P−i(m−i | ω′−i)xik′(mi,m−i) . . .

......

. . .∑m−i∈M∗−i

P−i(m−i | ω−i)xik(m′i,m−i)∑

m−i∈M∗−i

P−i(m−i | ω−i)xik′(m′i,m−i) . . .

∑m−i∈M∗−i

P−i(m−i | ω′−i)xik(m′i,m−i)∑

m−i∈M∗−i

P−i(m−i | ω′−i)xik′(m′i,m−i) . . .

......

. . .

To show that DuΦ has full rank, it is enough to show that each block Bi(ωi)

has full rank. Observe that a row gives the equilibrium distribution over cho-sen outcomes conditional on a report mi by agent i, and other agents’ stateω−i. Recall that we here only look at agents who have the ability to changethe outcome: for each i, there exists mi, m′i, and m−i such that xik(mi,m−i) 6=xik(m

′i,m−i), and these messages must be sent in equilibrium. Hence agent

i’s report must impact the distribution over chosen outcome, and the rowscorresponding to different reports of i are linearly independent.

It remains to show that all rows where mi ∈ M∗i is fixed are also linearlyindependent. Note that from the FOC, for all j 6= i, Pj(mj | ωj) = Pj(mj | ω′j)only if∑m−j

P−j(m−j | ω−j)uj(xj(mj,m−j), ωj) =∑m−j

P−j(m−j | ω−j)uj(xj(mj,m−j), ω′j)

which is satisfied only for a degenerate set of uj by definition of Ω∗j .Thus, if Γ is non-dictatorial over the support of P , rank DΦ(P, γ, t;u) =

|×i∈N∗ Ω∗i | × ∑i∈N∗ |M∗i | hence 0 is a regular value of Φ. The ParametricTransversality theorem states that, except for a nullset and meagre set of u ∈ U,

30

0 is a regular value of Φ(·;u). Then by the Regular Value theorem, Φ−1(0;u) isa smooth manifold of dimension

dim Φ̃−1(0;u) =∑i∈N∗

[|M∗i | × (|Ω∗i |+ |Ω∗−i|)

]− |×

i∈N∗Ω∗i | ×

∑i∈N∗|M∗i | < 0.

Therefore we conclude that for a comeagre and full measure set of preferencesu ∈ U, the set of stochastic choice rules (together with Lagrange multipliersand transfers) solving the FOCs is empty.

Proof of Proposition 1. Let U(κ) ⊆ R∑

i |K×Ωi| be the set of utility functions forwhich (i) there exists an IS equilibrium P ∗ of Γ, and (ii) Γ is not dictatorial onthe support of P ∗.

The case with κ = 0 is the baseline case considered in this paper, for whichTheorem 1 applies: U(0) has Lebesgue measure zero and ρ(0) = 0.

To show that ρ is increasing, we prove that for any κ, κ′ with κ′ ≥ κ,U(κ) ⊆ U(κ′). Take any u ∈ U(κ). By definition, we know that there existsa non-dictatorial IS equilibrium P ∗. We need to show that P ∗ remains an equi-librium if we increase the fixed cost from κ to κ′. Now that we have introduceda discontinuity in the objective function of agents, the FOCs (4) are no longersufficient to characterize an equilibrium. There are two possible types of equi-librium strategies for an agent: either she learns about others or not. If shedoes, then her strategy must satisfy (4). If she does not, then her IS strategymust solve:

Eµ0(EP ∗−i(·|ω−i)

[ui(xi(mi,m−i), ωi)− ti(mi,m−i)

]+γi(ω)

µ0(ω)− ∂H̄[P

∗i (mi|·) ◦ µ0(·)]

∂P ∗i (mi|ω)µ0(ω)

)= 0

for all ωi and all mi ∈ suppP ∗i . These two sets of FOCs define two possibleequilibrium strategies for agent i, yielding two different expected payoffs. Inequilibrium, agent i learns about others only if the gap between these twoexpected payoffs ∆(u) more than compensate the fixed cost κ. Since P ∗ is

31

informationally simple by assumption, we know that this gap is lower thanκ. it is hence also lower than κ′, and P ∗ remains a equilibrium under κ′: u ∈U(κ′), for all u ∈ U(κ).

Finally, we show that ρ is continuous in κ. By contradiction, suppose it isnot: there exists κ∗ and δ > 0 such that, for all ε > 0, either ρ(κ∗) − ρ(κ∗ −ε) > δ or ρ(κ∗ + ε) − ρ(κ∗) > δ. Consider the latter case21 — the function ρdiscontinuously jumps up at κ∗ — and pick any ε > 0. By assumption thereis a difference of at least δ between the Lebesgue measure of U(κ∗ + ε) andthat of U(κ∗). Consider any u ∈ U(κ∗ + ε) \ U(κ∗). For these utility functions,there exists a non-dictatorial IS equilibrium P ∗ under κ∗ + ε but not under κ∗.Hence κ∗ < ∆(u) < κ∗ + ε: it is worth learning about others given that theyplay P ∗−i if the associated fixed cost is κ∗ but not if it is κ∗ + ε. As ε tends tozero, this means that any u ∈ U(κ∗ + ε) \ U(κ∗) must satisfy ∆(u) = κ∗. Thisequality defines a line in the domain of preferences, and hence the Lebesguemeasure of the set of utility functions satisfying it is zero. This contradicts theassumption that the measure of U(κ∗ + ε) \ U(κ∗) must be above δ even forvanishing ε.

Proof of Proposition 2. Let P ∗ be an equilibrium in dominant strategy of Γ. Foreach agent, there are two cases: either P ∗i is a corner strategy in which theagent does not acquire any information, in which case it is trivially IS, or itis en interior strategy and must solve equations (4). Since it is a dominantstrategy, it must solve (4) for all possible strategies P−i of others. This impliesthat for any P−i, P ′−i,

EP−i(·|ω)[ui(x(mi,m−i), ωi)− ti(mi,m−i)

]= EP ′−i(·|ω)

[ui(x(mi,m−i), ωi)− ti(mi,m−i)

].

Hence the LHS of (4) must be independent of P−i if P ∗i is a dominant strategyfor agent i. It must then also be independent of ω−i, since others’ state entersagent i’s FOCs only through their strategies. This entails P ∗i is independent ofω−i as well.

21The proof is similar for the other case.

32

Proof of Theorem 2. We find necessary conditions for full surplus extraction thatare non generic in the space of quasi-linear utilities. The ex-ante efficientamount of information acquisition solves

maxPi:Ω→∆∆Ω ∀i

∑ω∈Ω

µ0(ω)∑

µ∈suppP

[Pi(µi|ω)P−i(µ−i|ω)

∑j∈N

x∗j(µ)uj(ωj)]

−∑j∈N

∑µj∈suppPj

H̄[Pj(µj | ·) ◦ µ0(·)]

s.t.∑

µi∈suppPi

Pi(µi|ω) = 1 ∀ω, i

Pi(µi|ω) ≥ 0 ∀µi, ω

The FOC with respect to Pi(µi|ω)µ0(ω) writes

EP−i(µ−i|ω)

[∑j∈N

x∗j(µ)uj(ωj)

]− ∂H̄(Pi(µi|·) ◦ µ0(·))

∂Pi(µi|ω)µ0(ω)+ζi(ω)

µ0(ω)= 0,(2)

for all µi ∈ suppPi and for all ω ∈ Ω, where ζi is the Lagrange multiplierassociated with the constraint that Pi(·|ω) sums to one. Hence the efficientstochastic choice rule (P †i )i—or equivalently information strategy (π

†i )i with

π†i (·) ≡∑

ω µ0(ω)P†i (·|ω)—must satisfy the above equation for all i, µi, and ω.

Conversely, the FOCs from the individual decision problem write

EP−i(µ−i|ω)[x∗i (µ)ui(ωi)− ti(µi, µ−i)

]− λ∂H̄(Pi(µi|·) ◦ µ0(·))

∂Pi(µi|ω)µ0(ω)+γi(ω)

µ0(ω)= 0.(3)

Surplus extraction requires that the information strategy chosen by the agentcoincides with P †i . Hence P

†i must solve both (2) and (3). Subtracting (3) from

(2) yields:

∑µ−i

P †−i(µ−i|ω)ti(µi, µ−i) = −∑µ−i

P †−i(µ−i|ω)∑j 6=i

x∗j(µ)uj(ωj)−ζi(ω)− γi(ω)

µ0(ω)

33

which implies:

Eµ0,P †(·|ω) [ti(µi, µ−i)] = −Eµ0,P †(·|ω)

[∑j 6=i

x∗j(µ)uj(ωj)

]−∑ω∈Ω

(ζi(ω)− γi(ω)

).

(4)

Hence efficient information acquisition requires this particular VCG mecha-nism. To extract full surplus, each agent’s expected transfer must sum to hernet utility:

Eµ0,P †(·|ω)[ti(µi, µ−i)] = Eµ0,P †i (·|ω) [x∗i (µ)ui(ωi)]−

(H(µ0)− Eπ†i [H(µi)]

)(5)

Transfers must extract the agent’s expected utility given her type while com-pensating her for the ex-ante investment in information acquisition. Not com-pensating for these costs would violate the ex-ante IR constraint. Combining(4) and (5), (P †i )i must solve:

Eµ0,P †(·|ω)

[∑j∈N

x∗j(µ)uj(ωj)

]= H(µ0) +

∑µi∈suppP †i

H̄[P †i (µi | ·) ◦ µ0(·)]−∑ω∈Ω

(ζi(ω)− γi(ω)

)Finally taking expectations over µi and ω in equation (2) yields:

Eµ0,P †(·|ω)

[∑j∈N

x∗j(µ)uj(ωj)

]= Eµ0,P †(·|ω)

[∂H̄(P †i (µi|·) ◦ µ0(·))∂P †i (µi|ω)µ0(ω)

]−∑ω∈Ω

ζi(ω) ∀i.

Combining the above two equations entails that, for all i ∈ N , P †i must solve:

Eµ0,P †(·|ω)

[∂H̄(P †i (µi|·) ◦ µ0(·))∂P †i (µi|ω)µ0(ω)

]+

∑µi∈suppP †i

H̄[P †i (µi | ·) ◦ µ0(·)]−H(µ0)−∑ω∈Ω

γi(ω) = 0

(6)

That is, they together require that the total cost of information equals the ex-pected marginal cost at the efficient solution. We show that this condition,however, is non-generic.

34

Define Φ̂ the functional which maps a profile of stochastic choice rules, La-grange multipliers and preferences to the LHS of the system of equations (2)and exactly one equation in the LHS of the system of equations (6).22 Hencethe necessary conditions (2) and one necessary condition in (6) are jointly writ-ten Φ̂(P, γ;u) = 0.

In order to apply the Transversality theorem we need to show that 0 isa regular value of Φ̂, i.e. the number of infinitesimally linearly independentequations of the system evaluated at an equilibrium point is maximal:

Φ̂(P, γ;u) = 0 =⇒ rankDΦ̂(P, γ;u) = 1 + |Ω| ×∑i∈N

| suppP †i |.

It is sufficient to show that DuΦ̂ has full rank. DuΦ̂ has∑

i∈N |Ω|×K columns,since u specifies a utility value from every object k in each state ω for eachagent i. For any column corresponding to some agent i, object k, and state ω,the row entries in this column equal the derivative of the system (2) and oneequation from (6) w.r.t. uik(ω). The last row corresponds to the derivative ofone equation of system (6) w.r.t. uik(ω), and hence only has zeros. All the otherrows, corresponding to the derivative of the system (2), is block diagonal:

(2) fixing ω

(2) fixing ω′

...

(2) fixing ω′′

(6)

B̂(ω) 0 · · · 0

0 B̂(ω′) · · · 0

......

. . ....

...... B̂(ω′′)

0 0 · · · 0

In each block B̂(ω) a row corresponds to a report of a particular agent µi and

22If it is generically impossible to find a solution to the system of equations where we onlyconsider one equation from (6), then it is also generically impossible to find a solution to thesystem with all equations from (6). We only include one equation because when computingDuΦ̂ the LHS of the system of equations (6) would all be zero hence not linearly independent.

35

a column to the derivative w.r.t. the utility of each agent:

B̂(ω) =

∑µ−i

P−i(µ−i|ω)x∗i (µi, µ−i)∑µ−j

P−j(µ−j |ω)x∗j (µj , µ−j) · · ·∑µ−i

P−i(µ−i|ω)x∗i (µ′i, µ−i)∑µ−j

P−j(µ−j |ω)x∗j (µ′j , µ−j) · · ·

......

. . .

HenceDuΦ̂ has full rank if and only if each of the diagonal block has full rank.It follows directly from Blackwell’s principle of irrelevant information that therows are linearly independent. Thus 0 is a regular value of Φ̂.

The Transversality theorem states that the set of u ∈ U such that 0 is aregular value of Φ̂(·;u) has full Lebesgue measure and is comeager. Then bythe Regular Value theorem, Φ̂−1(0;u) is a smooth manifold of dimension

dim Φ̂−1(0;u) = |Ω| ×∑i∈N

| suppP †i | −(

1 + |Ω| ×∑i∈N

| suppP †i |)< 0.

Therefore we conclude that for a comeager and full measure set of prefer-ences u ∈ U, the set stochastic choice rules solving (2) and (6) is empty.

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IntroductionRelated Literature

Motivating ExampleSetupThe Generic Complexity of Informational IncentivesTwo Illustrative ExamplesDiscussion of the Assumptions on the Cost Function

Implications for Mechanism DesignThe Limits of Strategy-ProofnessIndependent Private Values

ConclusionProofs

Documents

Informationally Simple Incentivesweb.stanford.edu/~agathep/Gleyze_Pernoud_ISI.pdfInformationally Simple Incentives Simon GLEYZEy Agathe PERNOUDz Abstract We consider a mechanism design