persuasion games with higher-order uncertainty

Journal of Economic Theory 110 (2003) 393–399

Persuasion games with higher-order uncertainty

Frederic Koessler�

THEMA (CNRS, UMR 7536), Universite de Cergy-Pontoise, 33 Boulevard du Port,

F-95011 Cergy-Pontoise Cedex, France

Received 20 June 2001; final version received 5 March 2002

Abstract

Shin (J. Econom. Theory 64 (1994) 253–264) showed that a perfectly revealing equilibrium

fails to exist in persuasion games when the decision maker is uncertain about the interested

party’s payoff-relevant information. By explicitly integrating higher-order uncertainty into the

information structure, this note shows that a perfectly revealing equilibrium does exist when

disclosures are not restrained to intervals of the payoff-relevant state space.

r 2003 Elsevier Science (USA). All rights reserved.

JEL classification: C72; D82

Keywords: Strategic information revelation; Persuasion games; Higher-order uncertainty; Provability

Persuasion games are sender–receiver games in which message sending is cheap,non-binding, and in which some information can be proved (or certified) byparticular messages.1 In such games, interested parties (such as salesmen, regulatedfirms, plaintiffs or defendants) try to influence a decision maker (such as a consumer,a regulator, a government, or an arbitrator) by strategically providing or concealinginformation relevant to the decision. The pioneering contributions of Grossman [1],Grossman and Hart [2] and Milgrom [6], show that efforts to manipulate a decisionmaker’s choice by concealing or distorting information do not always succeed.Indeed, by looking at situations where a buyer is to purchase a product having anunknown quality, these former papers prove that in every sequential equilibrium theseller reveals all relevant information to the buyer. The intuition of this result is thatthe buyer will believe the worst information reported by the seller and the seller will

�Fax: +33-1-3425-6233.

E-mail address: [email protected], one can assume that information can be verified or that penalties against lying agents do

exist.

0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0022-0531(03)00036-X

only report favorable information. Therefore, voluntary communication results infull revelation insofar as the buyer can make rational inferences which take intoaccount the seller’s incentives to withhold unfavorable information. The argumentbehind this fully revealing equilibrium is known as the unravelling argument.Full revelation relies on the implicit assumption that it is common knowledge that

the interested party is perfectly informed. Actually, the fragility of the unravellingargument has been emphasized by Shin [8,9] who considered persuasion games inwhich the decision maker is uncertain about the interested party’s fundamental(payoff-relevant) information. By assuming that only fundamental events can beproved, he showed that a perfectly revealing equilibrium does not exist. Indeed, theunravelling argument fails in such a setting because the decision maker has no way todistinguish an interested party who possesses information and remains silent from aninterested party who does not possess the information in the first place. Nonetheless,Shin [9] noticed that ‘the only substantial assumption involved in my model is thatannouncements are of intervals of the set (of payoff-relevant states). This is arestriction, but the corresponding gain in terms of the simplicity of the model wouldseem to justify its use. Future work may examine relaxing this assumption’ [9, p. 62].In this note, we scrutinize the failure of the unravelling argument by suggesting away to explicitly integrate higher-order uncertainty in the information structure andin communication possibilities. We establish that if we relax the assumption thatannouncements are of intervals of the payoff-relevant spate space, then theunravelling argument works again, i.e., a perfectly revealing equilibrium alwaysexists. This result does not depend on the information structure as long as theinterested party is more informed about the fundamentals than the decision maker. Inparticular, the perfectly revealing equilibrium is robust to the precision of theinterested party’s information and to the depth of knowledge in the informationstructure.2

We consider two players: an interested party (player 1) and a decision maker(player 2).3 The decision maker is interested in predicting the value of a payoff-relevant state or state of Nature tAR by choosing an action VAR as close as possibleto t: The interested party tries to persuade the decision maker that the true state ofNature is high by revealing him some information. Let O be a finite set of states of

the world and p a full support probability distribution on O: Player i’s partition overO is denoted by Hi; and his information set at oAO is denoted by hiðoÞ: Let t :O-R

be the function which assigns a state of Nature tðoÞAR to each state of the worldoAO: Given a state of the world oAO and a decision VAR; the utility derived byplayer iAf1; 2g is uiðo;VÞ: The interested party’s utility function u1 is strictly

increasing in V : The decision maker’s utility is given by u2ðo;VÞ ¼ �ðV � tðoÞÞ2:

2For more details concerning higher-order uncertainty and the depth of knowledge in an information

structure, see [3,7].3Shin [8] considered two interested parties (a defendant and a plaintiff) whereas Shin [9] considered only

one interested party (a firm) and two decision makers (two shareholders). Almost the same logic applies

with one interested party and one decision maker.

F. Koessler / Journal of Economic Theory 110 (2003) 393–399394

After observing his private signal, h1ðoÞ; the interested party chooses what todisclose to the decision maker. Let X be the s-algebra generated by the interestedparty’s partition H1: Since revealed information is proved by the interested party, hechooses to reveal a message (event) xAX such that the true state o belongs to x: Bydenoting X ðoÞ ¼ fxAX :oAxg the set of messages containing o; the interestedparty’s disclosure strategy is given by a H1 measurable function c :O-X satisfyingcðoÞAXðoÞ for each oAO: After observing the announcement x; the decisionmaker’s belief at o about the state of the world o0 is denoted by mðo0 j x; h2ðoÞÞ: Thestrategy of the decision maker is a function s :X O-R such that sðx; �Þ is H2

measurable for all x: Call a disclosure strategy c perfectly revealing if for all o; o0AOwe have mðo0 j cðoÞ; h2ðoÞÞ ¼ pðo0 j JðoÞÞ; where JðoÞ is the element of the Join,J ¼ H13H2; of players’ partitions.As usual, a sequential equilibrium of the persuasion game is a tuple ðs; c; mÞ such

that ðs; c; mÞ is consistent,4 and both players are sequentially rational, i.e., for alloAO and xAXðoÞ;

sðx;oÞ ¼Xo0AO

tðo0Þmðo0 j x; h2ðoÞÞ; ð1Þ

and Xo0Ah1ðoÞ

pðo0 j h1ðoÞÞsðcðoÞ;o0ÞXX

o0Ah1ðoÞpðo0 j h1ðoÞÞsðx;o0Þ: ð2Þ

For any event yDO; let Epðt j yÞ �P

oAO tðoÞpðo j yÞ: We assume throughout thatthe interested party is more informed about the fundamentals than the decision maker inthe sense that Epðt j h1ðoÞÞ ¼ Epðt j JðoÞÞ for all oAO: In other words, if the interestedparty were able to combine his information with the decision maker’s information, hisexpectation about the fundamentals would never change.5 The information structure ofShin [8,9] always satisfies this property since he assumed that the decision maker has noinformation at all about the state of the world, i.e., H2 ¼ fOg:

Theorem 1. If the interested party is more informed about the fundamentals than the

decision maker then there always exists a sequential equilibrium ðs; c; mÞ of the

persuasion game in which the disclosure strategy c is perfectly revealing.

Proof. The proof is made in two steps. First, we characterize a belief function m suchthat the interested party has no incentive to deviate from the perfectly revealingdisclosure strategy c satisfying cðoÞ ¼ h1ðoÞ for all oAO: Then, we showthat ðs; c; mÞ satisfies the consistency condition of Kreps and Wilson [5].6

4For a general definition, see [5].5Our assumption is implied by the assumption that the interested party is more informed (about the

state of the world) than the decision maker (i.e., H1 is finer than H2) but is not equivalent because we allow

the interested party to ignore how the decision maker is informed.6 It is worse noticing that in the persuasion games considered here consistency is not necessarily satisfied

if we apply Bayes’ rule whenever possible. The equivalence is only satisfied if H1 ¼ ffo1g; fo2g; fo3g;ygand H2 ¼ fOg; as it is the case in sender–receiver games in which the sender knows his type and thereceiver has no information.

F. Koessler / Journal of Economic Theory 110 (2003) 393–399 395

For all oAO and xAXðoÞ; let Iðx;oÞ � fo0Ah2ðoÞ-x: Epðt jh1ðo0ÞÞpEpðt j h1ðo00ÞÞ; 8o00Ah2ðoÞ-xg: The set Iðx;oÞ is the set of states of the worldthe decision maker conceives as possible at o when the interested party discloses x

(i.e., states in h2ðoÞ-x) such that the interested party’s expectation of t is the worstas possible. Since the decision maker’s expectation of t determines the interestedparty’s payoff, Iðx;oÞ is the set of states consistent with the decision maker’sinformation at o; the disclosure x and the interested party’s partition that generatethe worst payoff for the interested party. Now, consider the belief m satisfying

mðo0 j x; h2ðoÞÞ ¼ pðo0 j Iðx;oÞÞ; ð3Þ

for all o; o0AO and xAXðoÞ: Since IðcðoÞ;oÞ ¼ JðoÞ; the interested party isrational if for all oAO and xAX ðoÞ we have

Po0Ah1ðoÞ pðo0 j h1ðoÞÞEpðt j Jðo0ÞÞXP

o0Ah1ðoÞ pðo0 j h1ðoÞÞEpðt j Iðx;o0ÞÞ: To verify this property it is sufficient to showthat Epðt j JðoÞÞXEpðt j Iðx;oÞÞ for all oAO and xAXðoÞ: From the definition of

the set Iðx;oÞ and the fact that the interested party is more informed about thefundamentals than the decision maker, we have Epðt j Jðo0ÞÞ ¼ Epðt j Jðo00ÞÞ for allo0; o00AIðx;oÞ: Since Iðx;oÞ can be written as the union of elements of the Join, J;we get Epðt j Iðx;oÞÞ ¼ Epðt j Jðo0ÞÞ for all o0AIðx;oÞ; i.e., Epðt j Iðx;oÞÞ ¼Epðt j h1ðo0ÞÞ for all o0AIðx;oÞ: But now, from the definition of the set Iðx;oÞ wehave Epðt j h1ðo0ÞÞpEpðt j h1ðoÞÞ since oAh2ðoÞ-x; so Epðt j Iðx;oÞÞpEpðt j h1ðoÞÞ ¼ Epðt j JðoÞÞ: In consequence, the interested party has no incentive to deviatefrom his disclosure strategy c:To show that ðs; c; mÞ is consistent, we define a sequence of perturbed persuasion

games in which the interested party reveals every possible message with positiveprobability. We then show that the decision maker’s belief obtained via Bayes’ ruletends to the belief m given by (3). To allow perturbed games we have to define mixeddisclosure strategies. A mixed disclosure strategy of the interested party is a H1

measurable function p :O-DðXÞ such that suppðpðoÞÞDXðoÞ for all oAO; wheresuppðpðoÞÞ is the support of the probability distribution pðoÞ: Let P be the set of

mixed disclosure strategies and let P0 be the set of all strictly positive mixed

disclosure strategies, i.e., P0 � fpAP : pðxjoÞ40 for all oAO and xAXðoÞg: IfptAP0; then we can associate a belief mt with pt via Bayes’ rule. More precisely, if

ptAP0; then for all oAO and xAXðoÞ we have

mtðo0 j x; h2ðoÞÞ ¼0 if o0eh2ðoÞ-x;

ptðxjo0Þpðo0ÞPo00Ah2ðoÞ

pðo00Þptðxjo00Þ otherwise:

8<: ð4Þ

Let pAP be the mixed disclosure strategy associated with the perfectly revealingdisclosure strategy c; i.e., pðcðoÞjoÞ ¼ 1 for all oAO: Let m be the belief defined by(3). We have to show that there exists a sequence fðmt; ptÞgt such that p

tAP0 for all t;

mt is associated with pt via Bayes’ rule, and ðm; pÞ ¼ limt-Nðmt; ptÞ: If o0eh2ðoÞ-x;then o0eIðx;oÞ and thus pðo0 j Iðx;oÞÞ ¼ 0 ¼ limt-N mtðo0jx; h2ðoÞÞ by Eq. (4).

Therefore, we must find a sequence fptgtDP0 such that for all oAO; xAXðoÞ; and


o0Ah2ðoÞ-x we have limt-N ptðcðoÞ joÞ ¼ 1 and

limt-N

ptðx jo0Þpðo0ÞPo00Ah2ðoÞ pðo00Þptðxjo00Þ ¼ pðo0 j Iðx;oÞÞ; ð5Þ

i.e.,

limt-N

ptðx jo0ÞPo00Ah2ðoÞ pðo00Þptðx jo00Þ ¼

0 if o0eIðx;oÞ;1=ð

Po00AIðx;oÞ pðo00ÞÞ if o0AIðx;oÞ:

(ð6Þ

This last equality is satisfied if for all o00AIðx;oÞ we have

o0AIðx;oÞ ) limt-N

ptðx jo0Þptðx jo00Þ ¼ 1; ð7Þ

o0eIðx;oÞ ) limt-N

ptðx jo0Þptðx jo00Þ ¼ 0: ð8Þ

Note that the fractions given just above are well defined since o00AIðx;oÞ ) o00Ax:

Hence, xAX ðo00Þ; which implies that ptðxjo00Þa0 because ptAP0: Let fetgt

be a sequence such that limt-N et ¼ 0: To simplify the notations, we drop the

subscript t in et: Let I1 ¼ fhAH1 : EpðtjhÞpEpðtjh0Þ8h0AH1g; I2 ¼ fhAH1\I1 :

EpðtjhÞpEpðtjh0Þ 8h0AH1\I1g; I3 ¼ fhAH1\ðI2,I2Þ : EpðtjhÞpEpðtjh0Þ 8h0AH1\

ðI2,I2Þg; and so on.7 For all oAO; let lðoÞ be the integer l satisfying oAIl : Weconsider the following ‘trembling’ disclosure strategy: For all oAO;

ptðxjoÞ ¼elðoÞ if xacðoÞ; xAX ðoÞ;1� ðjX ðoÞj � 1ÞelðoÞ if x ¼ cðoÞ;0 if xeXðoÞ:

8><>: ð9Þ

Hence,P

xAX ðoÞ ptðxjoÞ ¼ 1 and limt-N ptðcðoÞ joÞ ¼ 1 for all oAO: To show that

conditions (7) and (8) are satisfied, let o00AIðx;oÞ; o0Ah2ðoÞ-x; and xAXðoÞ\cðoÞ(the result is obvious if x ¼ cðoÞ). If o0AIðx;oÞ; then lðo0Þ ¼ lðo00Þ; which impliesthat limt-N

ptðxjo0Þptðxjo00Þ ¼ limt-N

elðo0 Þ

elðo00 Þ ¼ 1: If o0eIðx;oÞ; then lðo0Þ4lðo00Þ; which

implies that limt-N

ptðxjo0Þptðxjo00Þ ¼ 0: This completes the proof. &

In the following examples, we illustrate the theorem with first-, second-, and third-order uncertainty.

Example 1 (First-order uncertainty). Let H1 ¼ ffo1g; fo2gg; H2 ¼ fOg andtðo1Þotðo2Þ: This is the standard information structure in which it is commonknowledge that the interested party is perfectly informed. Beliefs associated with the

7For example, if H1 ¼ fh1;y; h5g and Epðtjh1ÞoEpðtjh2Þ ¼ Epðtjh3Þ ¼ Epðtjh4ÞoEpðtjh5Þ; then I1 ¼fh1g; I2 ¼ fh2,h3,h4g; and I3 ¼ fh5g:


perfectly revealing disclosure strategy are

mðo0 j x; h2ðoÞÞ ¼pðo0 j fo2gÞ ifx ¼ fo2g;pðo0 j fo1gÞ otherwise:

(

Example 2 (Second-order uncertainty). Let H1 ¼ ffo1g; fo2g; fo3; o4gg; H2 ¼fOg and tðo1Þ ¼ tðo3Þotðo2Þ ¼ tðo4Þ: This is the kind of information structureconsidered by Shin [8,9] where the decision maker does not know whether theinterested party is informed about the fundamentals. Beliefs associated with theperfectly revealing disclosure strategy are

mðo0 j x; h2ðoÞÞ ¼pðo0 j fo2gÞ if x ¼ fo2g;pðo0 j fo1gÞ if o1Ax;

pðo0 j fo3;o4gÞ otherwise:

8><>:

It is to be noticed that if the event fo3;o4g cannot be revealed by the interestedparty, i.e, if we restrict the set of available messages to payoff-relevant events, thenwe get on the contrary the partially revealing equilibrium obtained by Shin [8,9].

Example 3 (Third-order uncertainty). Let H1 ¼ ffo1;o2;o3;o4g; fo5;o6g;fo7;o8gg; H2 ¼ ffo1;o3;o5;o7g; fo2;o4g; fo6;o8gg and tðo1Þ ¼ tðo2Þ ¼ tðo5Þ¼ tðo6Þotðo3Þ ¼ tðo4Þ ¼ tðo7Þ ¼ tðo8Þ: With this information structure, thedecision maker may or may not know whether the interested party is informed,and the interested party does not know whether the decision maker knows that theinterested party is informed. Beliefs associated with the perfectly revealing disclosurestrategy are

mðo0 j x; fo1;o3;o5;o7gÞ ¼pðo0 j fo7gÞ if x ¼ fo7;o8g;pðo0 j fo5gÞ if o5;o6Ax;

pðo0 j fo1;o3gÞ otherwise;

8><>:

mðo0 j x; fo2;o4gÞ ¼ pðo0 j fo2;o4gÞ for all xAX ðo2Þ ¼ Xðo4Þ;

mðo0 j x; fo6;o8gÞ ¼pðo0 j fo8gÞ if o5;o6ex;

pðo0 j fo6gÞ otherwise:

(

As shown in Koessler [4], if we assume on the contrary that only payoff-relevantinformation can be revealed, then we get in this example a partially revealingequilibrium similar to the one obtained by Shin [8,9].

In this note, we have shown that full revelation of information occurs and isrobust in persuasion games as long as fundamental and non-fundamental events canbe disclosed. As a consequence, even without specifying the information structureand higher-order uncertainty into details, predictions are stable: we only need toknow the configuration of provability. An advantage is that this latter detail is often


available in economic or legal situations we consider, contrary to details concerningthe precise information that players possess when they are called to make a choice.

Acknowledgments

I am grateful to an associate editor and an anonymous referee for very helpfulsuggestions. I have also benefited from comments from Fran-cois Laisney, PatrickRoger, Gisele Umbhauer and Anthony Ziegelmeyer.

References

[1] S.J. Grossman, The informational role of warranties and private disclosure about product quality,

J. Law Econom. 24 (1981) 461–483.

[2] S.J. Grossman, O.D. Hart, Disclosure laws and takeover bids, J. Finance 35 (1980) 323–334.

[3] S. Hart, A. Heifetz, D. Samet, ‘Knowing whether’, ‘knowing that’, and the cardinality of state spaces,

J. Econom. Theory 70 (1996) 249–256.

[4] F. Koessler, Persuasion games with higher order uncertainty, Working Paper 2001-14, BETA,

Strasbourg, 2001.

[5] D.M. Kreps, R. Wilson, Sequential equilibria, Econometrica 50 (1982) 863–894.

[6] P. Milgrom, Good news and bad news: representation theorems and applications, Bell J. Econom. 12

(1981) 380–391.

[7] S. Morris, A. Postlewaite, H.S. Shin, Depth of knowledge and the effect of higher order uncertainty,

Econom. Theory 6 (1995) 453–467.

[8] H.S. Shin, The burden of proof in a game of persuasion, J. Econom. Theory 64 (1994) 253–264.

[9] H.S. Shin, News management and the value of firms, Rand J. Econom. 25 (1994) 58–71.


persuasion games with higher-order uncertainty

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