fndjsfgosjoijw

7/27/2019 fndjsfgosjoijw

1/23

Rational and Boundedly Rational Behavior in a Binary Choice Sender-Receiver GameAuthor(s): Massimiliano Landi and Domenico ColucciReviewed work(s):Source: The Journal of Conflict Resolution, Vol. 52, No. 5 (Oct., 2008), pp. 665-686Published by: Sage Publications, Inc.Stable URL: http://www.jstor.org/stable/27638633 .

Accessed: 04/12/2012 10:57

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

Sage Publications, Inc. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of

Conflict Resolution.

http://www.jstor.org

This content downloaded by the authorized user from 192.168.82.206 on Tue, 4 Dec 2012 10:57:59 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/action/showPublisher?publisherCode=sagehttp://www.jstor.org/stable/27638633?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/27638633?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=sage


2/23

Journal ofConflictResolutionVolume 52Number 5October 2008 665-686? 2008 Sage PublicationsRational and Boundedly Rational io-h ? !*,?* http://jcr.sagepub.comBehavior in a Binary Choice ^,^,^1Sender-Receiver Game -

Massimiliano LandiSchool ofEconomicsSingapore Management University, SingaporeDomenico ColucciDepartment ofMathematics for DecisionsUniversity ofFlorence, Italy

The authors investigate the strategic rationale behind themessage sent by Osama binLaden on the eve of the 2004 U.S. Presidential elections. They model this situation asa signaling game in which a population of receivers takes a binary choice, the outcome is decided by majority rule, sender and receivers have conflicting interests, andthere is uncertainty about both players' degree of rationality. They characterize thestructure of the sequential equilibria of the game as a function of the parameters governing the uncertainty and find that in all pure strategy equilibria, the outcome mostpreferred by the rational sender is chosen. An explanation of the above-mentionedevents relies crucially on the relative likelihood of rational and naive players: If a sufficient departure from full rationality of the electorate is posited, then our model suggests that bin Laden's pre-electoral message succeeded in tilting the race toward hispreferred outcome.

Keywords: cheap talk; elections; bounded rationality; terrorism

The weekend before the 2004 U.S. presidential elections, the Al Jazeera TVnetwork aired a videotaped address fromOsama bin Laden (OBL, henceforth)targeting the American people. Apparently the message was not about the impendingelections (Al Jazeera 2004):

People of America: This talk of mine is for you and concerns the ideal way to preventanother Manhattan, and deals with thewar and its causes and results.

The timingof themessage and itsvery ending though, leave littledoubts about thefact that itwas indeedmeant to influence theAmerican voters.Quoting again fromthemessage (Al Jazeera 2004):Authors' Note: We thank Vincent Crawford and Domenico Menicucci for discussions and commentson this article.

665

http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


3/23

666 Journal of Conflict Resolution

The wise man doesn't squander his security, wealth, and children for the sake of theliar in theWhite House. In conclusion, I tell you in truth, that your security is not inthe hands of Kerry, nor Bush, nor al-Qaida. No. Your security is in your own hands.

Unfortunately, there are no direct ways to ascertain whether or not the message hada significant impact on the vote. However, several commentators agreed on theconclusion that the message actually helped reelect George W. Bush.1 For example, quoting fromKrauthammer (2004):With the election hanging in the balance, the campaign awaited some improbabledevelopment to tip the scale. Re-enter Osama bin Laden. By reminding us of 9/11 andthe war on terrorism, bin Laden invoked the only thing that could trump Iraq?andsave the President. His chilling reappearance reminded us of our peril, put Iraq in per

spective and played precisely to the President's success and strength?success andstrength that he so squandered in Baghdad. Bin Laden was never one to remotelyunderstand the American mind. He spectacularly misjudged 9/11?and he pulled hisnemesis over the finish line. (p. 50)

A piece of (admittedly indirect) evidence supporting the above view can be foundin Cohen et al. (2005): in late September 2004, the authors conducted an experiment, showing that a shift in the public's attention onto themes such as terrorismand the recollection of the facts of 9/11 would tend to substantially displace votesfromKerry toBush. This is corroborated by pre-electoral polls thatfound Bush tobe clearly favored over Kerry inhandling terrorism (Campbell 2005). These findings suggest thatOBL's sortie in the closing days of the campaign might haveaffected thevote, helping President Bush's reelection.So we take it as a working hypothesis that the tape indeed influenced ballotbehavior. The interpretation f ourmodel also benefits fromassuming that the facevalue ofOBL's words boiled down toopposing George W. Bush and hence endorsing JohnF. Kerry, but this isnot strictlyrequired tobuild themodel.Ultimately, we seek answers to thefollowing questions: Did OBL reallymisunderstand theAmerican voters?Was his implied position onU.S. election (opposingBush and endorsing Kerry) a lie or a truthful evelation of his view? Did theAmerican voters give any credit to the message?

OBL's message is an example of an attempt to influence the result of an electionby an external personality using a costless message. The number of cases in whichsignificantoutside personalities or organizations endorse (more or less explicitly)one candidate is surely large. An example is the Economist's tough stand onMr. Berlusconi (pictured as unfit to lead Italy) before both the2001 and the 2006Italian general elections. Another is subcommander Marcos's endorsement, onnational television inMay 2006, of one of the candidates for the incoming presidential elections inMexico. Nevertheless, OBL's role and his message's timing,content, and echo in themedia make ita rathernovel event in thepolitical scenery



4/23

Landi, Colucci /Binary Choice Sender-Receiver Game 667

and calls, inour opinion, foran explanation thatshould be anchored on empiricallycredible initial responses' behavior.2We consider a game ? laCrawford and Sobel (1982) played between one senderand a population of receivers. The sender has private information about a state variable while the receivers take a binary action after observing a public message fromthe sender.Majority rule determines the choice implemented,which, togetherwiththeprevailing state,determines thepayoffs for all players. In thebenchmark case,sender and receivers have opposite interests.Under this setup, it iswell known thatin any equilibrium, the sender uses only uninformative messages and the receiverschoose the action that,ex ante,maximizes their expected payoff. This being thecase, OBL's message could not have shifted votes in favor of Bush.To account for this possibility, we introduce a structuralmodel of interactionbetween fully rational and behavioral players that resembles Crawford's (2003)analysis of theAllies' successful attempt tomislead theGermans about where theywould land onD-Day in 1944.While Crawford considers the transmission of infor

mation about intended play, we focus on the transmission of private exogenousinformation. The uncertainty for both receivers and sender about the degree ofsophistication of theiropponent induces a game of incomplete informationwhoseequilibria are studied as a function of the parameters governing this uncertainty.The interpretation f the events relies crucially on the relative likelihood ofmortaland sophisticated players. Nonetheless, in all thepure strategies equilibria of thegame, the outcome most preferred by the (rational) sender is selected. These findings extend the results of Crawford to our case. With respect to thementionedcontest between OBL and U.S. voters, ifa sufficientdeparture from full rationalityof the electorate is assumed, then our model supports the idea that OBL's pr??lectoral message succeeded in tilting the race toward the outcome he preferred.The strategic interaction between OBL and theU.S. voterswe have inmind canbe explained as follows: Let the state of the world represent OBL's relativestrength s a terrorist eader and his influence on Islamic communities in theMiddle East and worldwide. This can be seen, from thepoint of view of theAmericanelectorate, as a proxy for the (exogenous) seriousness of OBL's imminent threat toU.S. national security. In a "high" state,OBL is strongand influential and theU.S.voters perceive a higher risk of a terrorist attack. As a result, they see Bush as a better option. Conversely, OBL prefers Kerry and a less aggressive U.S. foreign policy. In a "low" state,OBL is relativelyweak in the sense that his outlook forcontrol on Islamic masses is poor: This makes him prefer aggressive U.S. policiesthat can create conditions that drive the angered toward al-Qaida. Vice versa, theU.S. citizens prefer a foreign policy more inclined toward diplomacy thatwouldprobably cost them less, and see Kerry as a better alternative.Insofar as our motivating story for a two-states scenario is inevitably rather arbitrary, nemight think f resortingto thehistorical events as of 2004 Election Day asa guide to shed lighton things,but this ishardly away to avoid difficulties,because



5/23


itwould require some kind of counterfactual analysis forwhich scientificgroundingishard tofind.At the time of the message, candidates' positions were spelled out clearly, andno time for changes was available. In this respect, we believe that it is a sensible

assumption to focus on the game only between OBL and the U.S. voters.We choose tomodel the type of uncertainty following Crawford (2003), as thisapproach seems quite natural given the idiosyncratic nature of the events. In fact,unlike similar applications of strategic information transmission, such as Sobel(1985) and Benabou and Laroque (1992), thenovelty of the scenario does not leaveroom for learning or reputation considerations. The same lack of repetition rulesout thepossibility for testing theplay ofmixed strategies.3

We therefore assume that each player is either rational or naive. Rational playersplay the incomplete informationgame that is obtained by introducing thepossibilityof naive players and follow the standard equilibrium behavior. Naive playersare built following the logic of level-k thinkingmodels that appear in the behavioral game theory literature see, e.g., Stahl andWilson 1994, 1995). In a nutshell,players form some simplified beliefs about their opponents' play and then bestrespond to that. These players are naive in the sense that they do not use the circular concepts thatare typical of equilibrium reasoning. Different levels of sophistication imply different types of expectations. Thus, at the bottom of the scale, wehave level 0 players, whose behavior is determined by an exogenous motive. In ourcase, the level 0 player is a sender who reveals the true state of the world. One stepabove is the class of level 1 receivers: They believe thepopulation of sender tobeformed by level 0 players only and therefore choose the strategy that is a bestresponse to that. Moving one step above in a recursive way, we can generate aninfinite sequence of behavioral types. Clearly, since we have a finite game, only afinite number of thembehave differently.There is no loss of generality in restricting our analysis only to them, since from thepoint of view of the rational players,it isonly theirbehavior thatmatters.Such a structureof boundedly rational decision rules has been shown to be awell-grounded starting point formodeling initial responses to novel situations.Examples beyond theworks previously cited include first-round ehavior inexperimental guessing games, as in Nagel (1995) and Costa-Gomes and Crawford(2006), and estimated types in a series of two-person normal-form games in CostaGomes, Crawford, and Broseta (2001). The related model of cognitive hierarchydeveloped byCamerer, Ho, and Chong (2004) also shows good fitto empirical datafrom various experimental games.Related research on transmissionof private informationwith possibly boundedlyrational players includesKartik, Ottaviani, and Squintani (2006), and Chen (2006).The first group of authors considers a communication game between a sender and apool of receivers (whomay be partiallynaive) inwhich talk can be inherently ostly.The state of the world is a continuous random variable with an unbounded support.



6/23


Chen is similar but has a state that s uniform in [0, 1] and assumes nonstrategic typesof playerswho are infactoptimizers given theirutilityfunctions (so there is actuallyno bounded rationality in hermodel). Besides the finiteness of the state,message,and action spaces, thepresent article differsfromboth, in the sense thatwe allow fora more complex structure of boundedly rational types for both sender and receivers.

Moreover, inChen, the outcome is not determined by majority rule.Our model also relates to Farrell and Gibbons (1989), where the sender facestwo different receivers. The main differences are that, in our case, the message ispublic and cannot be sent to only one subgroup of receivers, and the outcome isdetermined by majority rule, so that, in particular, the sender's payoff is not additively separable in the actions of differentgroups within the population of receivers. Majority rule per se does not change the outcome with respect to the onereceiver case. The distinctive feature of ourmodel is in the addition of uncertaintyabout players' degree of sophistication.The role of the sophisticated receiversmirrors the equilibrium play of theuninformed independents in Feddersen and Pesendorfer (1996), who mix betweenabstaining and voting for the candidate with less partisan support, to increase theprobability thatthe informed independents be pivotal to the election. Yet, there is asubstantial difference: The behavior of the sophisticated players in ourmodel ispurely driven by their strategic interactionwith theircounterpart.When theprobability of a sophisticated sender is sufficientlyhigh, the sophisticated receiversneed to vote so that the electoral outcome does not depend on the signal observed,which requires them to counterbalance the play of the mortal receivers.Our article ismainly motivated by the interest in the episode of OBL versusvoters at theeve of 2004 elections and thereforeposits a clear clash in the interestsof those actors. Because the analysis might be relevant to interpret other similarevents (some of which are mentioned above) where voters may not be certain aboutwhether the sender's payoff and theirs are negatively or positively correlated, wealso extend ourmodel to allow formore examples from real politics to fit in ourframework. This part is akin to Sobel (1985) and entails that the receivers face anadditional level of uncertainty inchoosing what todo.The article is organized as follows: Section 2 discusses themodel in itsgeneralform. Section 3 extends the model to account for uncertainty about the payoffs correlation between sender and receivers. Section 4 concludes.

The Base ModelSuppose

there are two states of the world, S\ and S2, whose probabilities are,respectively,p and 1 p. Let S = {S\, S2}. There are twogroups of players, a sender(PI), who knows the stateof theworld, and a population (continuum) of receivers(P2), who do not know it.The sender sends a public message m eM = {m\,M2} to



7/23


Figure 1Payoffs Given theActions (Umand Dm) of theMajorityofReceivers and the State ofNature (Two States), with b > 0.pi p^

pn Urn b-b0, 0S^

pn UrnDm 0,0

1,-1

receivers,who in turnchoose an action aeA ? {U,D}. The payoffs forall playersdepend on the stateof nature and on theaction of (themajority of) the receivers andare reported inFigure 1.Notice that, in each state, the sender and the receivers have conflicting interests.A pure strategyfor the sender is a function from the stateof theworld space (S)to themessage space (M). A pure strategyfor the receivers is a function from themessage space (M) to theaction space (A). So both typesof players have fourpurestrategies. Inwhat follows, (mi,m2) denotes the pure strategyby which messagem\ is sent if the state is Si and message m2 if the state is S2. Similarly, (U, D)denotes the pure strategy thatprescribes choosing action U if themessage ism\and action D if themessage ism2.Mixed strategies are probability distributions over the set of pure strategies.Weuse the following notation: for the sender, (oci, ot2, 0C3, 0C4) denotes an element of thethree-dimensional simplexwith respect to the set of pure strategies {(mi, mi), (mi,^2), (^2, ^1), (^2, ^2)}. Similarly, for receivers, (?]? ?2, ?3, ?4) denotes an element of the three-dimensional simplex with respect to the set of pure strategies{(?/, t/),(i/,D),(D,t/),(D,D)}.For example, the sender's mixed strategy (0, 0.5, 0.1, 0.4) means that the sender

assigns probabilities 0, 0.5, 0.1, and 0.4 to the pure strategies (mi, mi), (m}, m2),(m2, mi), and (m2, m2).

Finally let jidenote theposterior probability that the true stateof theworld isS\after observing message mj and r\ theprobability that the state be Si given m2.These probabilities depend on p and on the strategyadopted by the sender, and arecomputed, whenever possible, through Bayes' rule as follows:

p(oLi a2)p(ai +a2) + (l-p)(ai +a3)

_/7(0C3+a4)_/?(a3-r-a4) (l-/?)(oc2 + a4)(1)(2)



8/23


In the remainder of this section,we firstdescribe the equilibria of thegame withonly rational players, then introduce bounded rationality along the lines of Crawford(2003) and look at theequilibria of thegame between the sophisticated players.We shall assume that ph?\?p, which means that, before the message is con

veyed, theexpected payofffromchoosing eitheralternative in theelection is the same.We have inmind a scenario in which, ex ante, voters are indifferent between the candidates,whom are thereforeequally plausible tobe elected.We use this condition asan approximation of the idea of a close race, as itwas thecase in the 2004 presidentialelection. In this respect, ours is amodel of swing voters. One could add partisan voters(i.e., citizens for whom one option is better than the other, irrespective of the realization of the state of nature) into the picture. In this case, a close election could be one

where partisans of both sides are of equal size. As a result, independent voters wouldstill be pivotal to the outcome. Therefore, even in such case, the assumption of

pb?\?p would still be needed to have the electoral outcome ex ante uncertain.4Finally, we assume that receivers choose the action that maximizes their

expected payoff as iftheywere pivotal in theelection. This restriction ismotivatedby the need to reduce the set of equilibria, which is typically large in votinggames.5 The solution concept we work on is that of symmetric sequential equilibrium. The symmetry condition requires all the rational receivers to adopt the same(pure ormixed) strategy.Under full rationality, the following result (which is verymuch in the spiritofCrawford and Sobel 1982) holds:

Proposition 1. In the sequential equilibria of thegame between rational senderandreceivers, the strategies are such that the sender does not reveal information about thetrue state (i.e., a2 = 0C3) and the receivers choose any strategy that induces an electoraloutcome that does not depend on themessage (i.e., ?j + ?2 > 1/2 $2+ ?4> or?,+ ?2< l/2 ?2 ?4,or?!+ ?2 1/2=?2 ?4)

Proof. Consider a mixed strategy for the sender: either oc2^ 0C3 or oc2 0C3.Astrategy for which a2 ^ 0C3cannot be part of an equilibrium. Indeed, supposeoc2> 0C3: the receiver would then optimally choose (U, D). In turn, the sender wouldbest respond playing (rri2, m\). The same logic rules out oc2 1/2.Having ruled out receivers' strategies inducing outcomes that change with themessage, we are leftwith threepossibilities:?,+ ?2> 1/2nd?,+ ?3> 1/2; ,+ ?2< 1/2nd?,+ ?3< 1/2; i+ ?2 l/2?2+ ?4. Finally, observe that?, + ?3 > 1/2ifand only if?2+ ?4 < 1/2. In all these



9/23


cases, the sender is indifferent between sending any message in each state and therefore can mix with a2 = 0C3.

Proposition 1 implies thatwhen players are rational and have conflicting interests, receivers do not change their priors upon receiving the signal, so their choiceat thevoting stage isnot affectedby it.Thus, given the initialuncertainty about thebest possible action, the message sent does not have any impact whatsoever in theway receivers vote. Their choice, with or without any message, remains the same.Therefore, ifwe believe that themessage sentbyOBL had an impacton the electoral outcome, we need to look for alternative frameworks. The one we propose is thefollowing: Players are assumed tobe unsure about thedegree of rationalityof theiropponents,who may be fullyrational (andwe call them "sophisticated") or follow abehavioral norm of conduct (and we call them "mortal"). Mortal players have naivebeliefs about thedistribution of opponents andmaximize theirexpected payoff giventhose beliefs. Iteration of best responses generates the set of behavioral types we consider. We assume that a level 0 sender always chooses (mi, m2). We call this type a"truth teller." The receivers who expect the sender to be a truth teller and thereforeoptimally choose (U, D) are called "believers." A senderwho thinks thepopulationof receivers is made only of believers and thereforeoptimally chooses (m2,m\) iscalled a "liar." Last, the receivers who believe the sender is a liar and therefore optimally choose (D, U) are called "inverters." Notice that therpairs ofbehavioral typescould emerge. For example, a sender who believes that receivers are all inverterswould optimally choose (mi,m2), and receiverswho believe the sender does sowouldoptimally choose (), but theirstrategiesare indistinguishable from the firstpairofmortal types (truthtellers,believers). As a result, there is no loss of generality infocusing our attention on the following pairs of types of sender-receivers:6

truth-teller(rai,ra2) believer (U,D)liar (ni2,m\) inverter (D,U)Let xt, xi, and xs denote the probability a sender is, respectively, a truth teller, aliar, or a sophisticated type. Similarly, y^, yz, and ys denote the population's share ofbelievers, inverters, nd sophisticated types.Sophisticated players are rational in the

standard sense, and their behavior comes from equilibrium considerations, given thatthe structure f thegame (payoffs,strategy, nd uncertaintyabout types) is commonknowledge (among sophisticated players). To understand how sophisticated playerswill behave inequilibrium,we need thefollowing two intermediateresults:Lemma 1. Letting w; be theprobability that the sophisticated sender sendsmessagemi in stateS?, thena sophisticated receiver's best response is given by

(U,D) ifxs(l+wi -w2)>l-2xt,(D, U) ifxs{\+ wi - w2) < 1 2xt.



10/23


Proof. A sophisticated receiver will choose the action maximizing his or herexpected payoff.Upon observing message m\, theprobability of being in state S]and S2 is proportional to, respectively, p(xt + w\xs) and (1?p)(x\ + W2XS).Therefore, since ph?\? p, U is optimal under m\ if

Xt+ W\Xs >X\ +w2xs,that is, if a;v(1 + w\ ? w2) > 1 2xt.

Similarly, upon observing message m2, the probability of being in state S\ and S2is proportional to, respectively, p(xt + (1?

w\)xs) and (1? p)(x\ + (1

?W2)xs).Therefore, U is optimal underm2 if

Xi+(\-W\ )xs > xt+ ( 1 w2)xs,that is, if 1 2xt > (1 + w\ ? w2)xs.

Observe that theonly pure strategybest responses for the sophisticated receiver are(U, D) or (D, U). Therefore, let ? denote theprobability of the sophisticated receiver playing (?/,D) and 1 ? theprobability of playing (D, U).Lemma 2. The sophisticated sender's best response to (0, ?, 1 ?, 0) isgiven by

(mi,m2) if?jy, 2 ~yb(m2,mi) if $ys> - -yh

Proof. The sophisticated sender knows that (U, D) will be played by a share(yb fys) of thepopulation,whereas (D, U) will be played by a share (y? (1? ?)v?).Thus, in state S\, message m\ is best if

>'/, ?>'v


11/23


We are now in the position to characterize the play in the pure strategy sequential equilibria of thegame between sophisticated players. The following propositioncovers all parameters' configurations except the case 1/2 > max{x?, jt/, yb, y,},which is tackled inproposition 3.Proposition 2. In the sequential equilibria of the game between sophisticated

sender and receivers, pure strategies are as follows:

EQ 1 : [(m\,m2), (U,D)} if and only if y?> - and x? < -.

EQ 2 : [(m2,m\), (U,D)\ if and only if y?< - andxr > -.

EQ 3 : [(m\,m2), (D, U)} if and only if yb < - and x? > -.

EQ 4 : [(m2,m\ ), (D, U)\ if and only if yb > - and xt < -.

Furthermore, the outcome preferredby the sophisticated sender is achieved in allthose equilibria.7

Proof. The proof uses Lemma 1 and Lemma 2 above. For example, for [(mi,m2), (?/, D)] to be an equilibrium means (mi, m2) is best given ?=l, whichrequires ys< 1/2- y&, that isy?> 1/2, nd that (?/,D) is best given w\= 1,w2 = 0,implying2x5> \?2xu that isx/ < 1/2. he other cases are done in the sameway.Proposition 2 shows that the (sophisticated) sender can take advantage of theuncertainty, whether it is about the receivers or about himself or herself. Wheneverthepopulation share of one mortal type of receiver is larger than 1/2 (equations 1and 4), the sophisticated sender can just induce themost preferredoutcome by fooling themajority. Although the sophisticated receivers are acting optimally, theycannot affect the outcome because they represent a minority of the voters.

On the other hand, when sophisticated receivers count (equations 2 and 3), andthere is a probability of at least 1/2that the sender is amortal type, the sophisticatedsendermimics the strategy f the least likelymortal and fools the sophisticated receivers.Here, themajority of the receivers are acting optimally, since theprobability offacing a sophisticated sender is smaller than 1/2. et, theoutcome induced is still themost preferredby the sophisticated sender.Last, observe that,fromthepoint of viewof a mortal sender, the outcome favors a liar only inEquations 2 and 4, while itfavors the truth eller inEquations 1 and 3.It remains to consider the case where 1/2 > max{x,, jc/,yb, y,-}. The equilibriumis inmixed strategy, s a quick glance at the best response functions suggests. The

result is as follows:Proposition 3. Assume 1/2 maxj^, x?,y?,yb}. In themixed strategies sequential equilibria, sophisticated senders need tomake sure that no transmission of



12/23


information occurs (i.e., oc2 ^j^1 + 0C3), and the receivers choose any strategythat induces an electoral outcome that does not depend on the message (i.e.,*(?i + ?2) > V2 -yb> ys (?2+ ?4)> or ys (?i + ?2) < 1/2- yb< ys (?2+ ?4), or^(?i + ?2)=V2-^ = v,(?2 + ?4)).

Proof Inspection of the best response functions depicted in Figures 2 and 3shows that,when sophisticated players are pivotal, the equilibrium has to be inmixed strategies. Therefore, theprobabilities about the state of nature have to beleft unchanged by themessage, for otherwise nonequilibrium behavior wouldresult. Note that

_p[x, (ai +0C2K]_p[xt (cLi cl2)xs] (1-p)[xi + ( ? andy/+yJ(?,+?3) > -,< z < zwhich is equivalent to

*(?i + ?2) > l/2-^Observe that in equilibrium, the sophisticated receivers need tomix in a way that (iftheelectoral outcome is tobe U), theymostly add to thebelievers when m 1 so thatys(?i + ?2) + yb> 1/2) and to the inverters hen m2 (so thatys (?2+ ?4) + yb< 1/2).In fact, this equilibrium resembles theone under full rationality (see proposition 1).First,when the shares (probabilities) ofmortal players are equal, the two equilibriacoincide. This is because the twomortal typesperfectlyoffset each other.Second, theequilibriumunder full rationality s the limit f theequilibriumwith mortal and sophisticated players, when the share of the latter goes to 1.In general, theequilibria change continuously with respect to thecomposition ofthe players' population. Thus, our analysis can be carried forward to polar cases.Suppose, for example, that the sender is (known to be) sophisticated (i.e., xs= 1)and plays against a population of mortal and sophisticated receivers. The equilibrium play depends on the share of mortal receivers. When either mortal type constitutes a majority, the sender plays the strategy that fools this typewhile the



13/23


Figure 2Best Responses toPure Strategies for the Sophisticated Sender

y*=i

i2

y.=i i tf=1

sophisticated receiverswill play like theother type (see equations 1 and 4 inproposition 2). If, on the other hand, the sophisticated receivers are in majority, we areleftwith themixed strategyequilibria of proposition 3 inwhich the sender's strategies boil down to oc2 0C3. So assuming a sender who is known to be rational simplifies the strategic interaction.The introduction of behavioral types causes the appearance of pure strategyequilibria thatdo

not existwhen all players considered are rational, in which theoutcome favors the (sophisticated) sender. We can see how our analysis offers aframework to evaluate the happenings of 2004 U.S. elections according to a simplified model of the game thatactually took place between OBL and theAmericanvoters. Depending on one's views on the nature of OBL's rationality, various differentconclusions can be drawn. For example, ifwe think thatOBL was not onlycapable ofmastering the game but also perfectlyknowledgeable about Americanpolitics (in otherwords, ifwe believe OBL was very likely to be fully rational),thenwe can regard the final comment in Krauthammer (2004) thatOBL playedagainst himself throughhismessage as unrealistic. Indeed, even without attachingany specific interpretation to states and actions, the pure-strategy equilibria with a(very likely) sophisticated sender entail thedeception of amajority ofmortal receivers, either believers (inwhich case OBL was lying) or inverters (inwhich caseOBL was telling the truth).



14/23


Figure 3Best Responses toPure Strategies for the Sophisticated Receiver

xf=1

12

\

On the otherhand, ifone believes that, likeKrauthammer (2004) suggests,OBLwas naive about American politics, thenwe should look at the equilibria inwhichthe sender ismost likely a liar or a truth teller: Under those circumstances, thesophisticated American voters would have avoided their mortal counterparts to befooled by the message, whatever the true state of the world.So far,ourmodel depicts the case inwhich the interactionbetween sender andreceivers is clear cut, with a perfect negative payoff correlation. Yet, one can thinkof examples, beside OBL's 2004 message, inwhich thingsaremore nuanced thanour base model permits. As we point out in the introduction, there can be an extralayer of uncertainty in the sense that voters may not know whether their interestsand the sender's are at odds or they are (perhaps partially) the same.8 Less blackand-white scenarios need to be framed in a wider model than the above. Therefore,in thenext section,we extend thebase model to explore thepossibility of uncertainty about the correlation between sender's and receivers' payoffs.

AModel withUncertainty about the Sender's InterestsIn this section,we investigate the case inwhich the sendermay have intereststhat, under some states of the world, coincide with those of the receivers. Assume,



15/23

678 Journal ofConflict Resolution

Figure 4Payoffs Given the Actions (Umand Dm) of theMajorityof Receivers and the State ofNature (Four States)

F2 UnP1

1,-10, 0S,

P11,10,0

P10,01,1

P10, 01,-1

therefore,that the state space is S = {S\, S2, S3, S 4} and let/?/ e theprobability ofstateSi. The message space is stillM = {m\,m2}, and a strategyfor the sender is afunction from S toM.9 Receivers choose an action, aeA = {U,D}, and a strategyfor the receivers is a functionfromM toA. Notations forpure and mixed strategiesfollow the logic of the previous section. For example, (m\, m\, m2, m2) denotes thepure strategy where the sender's message ismi in states S\ and S2 and m2 in statesS3 and S4. In turn, U, D) means that the receiver chooses action U ifmessage ismi, and actionD ifmessage ism2. Last, let (r^) denote theposterior belief aboutstateSi when signalmi (m2) is observed. Remember thattheseprobabilities dependon pi and on the strategyadopted by the sender and are computed by Bayes' rulewhenever possible.

Players' payoffs as a function of the states and of the receivers' actions arereported inFigure 4.

Compared to the previous model, we have added complexity to the payoffsstructure, but we have restricted attention to the case where both alternatives givethe same payoff toboth players. Inwhat follows, we assume that

P\ +P2=P3+P4, (5)meaning there is ex ante indifference between the two candidates, which we adoptas our metric for a close election. We suppose further that either one of the following two inequalities hold

P2+P3 > P\ +P4,P2+P3 p\and p2 > /?4, hereas inequalities 5 and 7 imply thatP3 < p\ and P2< Pa



16/23


This model is similar to Sobel (1985), for there is uncertainty on whether thesender is a friend or an enemy, as well as on which action is best for the receiver.

Given the nature ofM and S, a strategy for the sender can induce posterior probabilities on S thatgenerate one of thefollowing partitions of S:P? = {S}P2 = {{Si,Sh},{Sj,Sk}}.

P? corresponds to totally uninformative messages. Pl corresponds to messageswhere one state, Si, is revealed with probability 1. P2 corresponds tomessages wheretwo states, S? and S/?, are separated from the other two, Sj and S?. There cannot beother partitions because there are four states but only two possible messages. Equilibria can be usefully distinguished in termsof the partitions theygenerate over thestate space. To thisend,we give thefollowing definition:

Definition 1. An equilibrium is said tobe pooling if it induces P? and partiallyrevealing if it induces Pl orP2.To begin with, consider the benchmark case of perfect rationality.Like in thebase model, there are pooling sequential equilibria, inwhich the posterior beliefsare equal to the priors, and receivers choose a strategy that induces an electoral outcome thatdoes not depend on the signal received.

Additionally, therecannot exist partially revealing equilibria that inducePl. Suppose, for example, that there were such an equilibrium where the sender's pure strategy is (mi, ra2, m2, mi). When the receivers observe m\, they know that the true stateisS\.Moreover, when m2 is sent, theyknow that 2 is less likely tobe the true statethanS3 or S4, so theirbest response is (U, D). We now can seewhy the initialproposed strategyfor the sender cannot be an equilibrium strategy:He or shewouldrather send m\ in state S4. In a similar vein, we can see why any other partition of thetype l cannot be part of a pure strategysequential equilibrium.Finally, a partially revealing equilibrium that induces Pl exists only when thestates are partitioned into {Si, S3} and {S2, S4}, and the sender is a friend (i.e., condition 6 holds). In fact, suppose the sender chooses (mi, wi2, m\, m2). The receivers'best response is to choose (D, U) when the sender is a friend,and (U, D) when thesender is an enemy. In the former case, the sender does not want to deviate fromhis or her proposed strategy,whereas in the latter, the sender would ratherpick(jTi2,mi, m2, mi). One can also easily verify that all the other possible partitions ofstates inP2 cannot be supported as an equilibrium: Given thepriors on these states,any strategy inducingP2 implies that it is optimal for the receivers to adopt either([/,D) or (D, U). The only case inwhich the sender does not want todeviate is theone shown above.



17/23


This resultparallels theorem 1 inSobel (1995), which states that if theprobabilityof the sender being a friend is larger than 1/2, then there is an equilibriumin which themost preferred action for the sender is revealed with probability 1.Summing up,we have thefollowing:Proposition 4. Under full rationality, besides the pooling equilibria inwhich

messages are not informative and receivers do not condition their action on the signal, there are partially revealing sequential equilibria, when condition 6 holds, inwhich pure strategies are as follows:

EQ 1 :[(ml,m2,m],m2),(D,U)},EQ 2 :[(m2,ml,m2,m{),(U,D)}.

These results show thatwhen condition 6 holds, a selection issue among multiple(pure strategies) equilibria arise. This is common in communication games, sincelanguage has no intrinsicmeaning and signals are valued only for the informationabout the states of the world they convey.

We now turn to the case in which players have the same type of uncertaintyabout their opponent's rationality as in the previous section, and distinguishbetween the twopolar cases described by inequalities 6 and 7.

The Sender is an EnemyThis case entails P4>pi and p\ > P3. Consider a baseline type of a sender

choosing (mi, m2, mi, m2), which corresponds again to a truth-telling type, in thesense that the strategy separates the states where the sender prefers U from thestates where he or she prefers D. By iterating best responses, we obtain the following pairs of types:

truth-teller (m\,m2,m\,m2) believer {U,D)liar (m2,m\,m2,m\) inverter (D,U)The notation about probabilities of sender's types (xt, x?, and xs) and shares ofreceivers' types (yb, yz, and ys) is as in the previous section.To characterize the behavior of the sophisticated players, letw? denote theprobability that the sophisticated sender sendsmessage m\ in state S? while ? is theprobability a sophisticated receiver chooses (U,D). Then,

Lemma 3. The best response fora sophisticated receiver is given by(U,D) if (xt W]Xs)pi (x? W2Xs)p2 (Xt H'3Xy)p3 (x? W4Xs)p4,(D, U) if (xt WjXs)pi+ (Xj w2xs)p2 {xt W3Xs)pi (Xi W4Xs)p4.



18/23


The best response fora sophisticated sender is given by(mum2,mi,m2) if $ys+yb< ~z,

(m2,mum2,mi) if f>ys+yb> -.

Proof. A sophisticated receiver will choose the action maximizing expectedpayoff.The posteriorprobability of being in stateS, induced bymessage mi isproportional to (xt WiXs)pi if /G {1,3} and to (x? W{Xs)pi if / {2,4}; similarly, theprobability of being in state S, having observed message m2 is proportional toX[ (1? Wi)xs)pi if /G {1,3} and to xt+ (1? Wi)xs)p? if / {2,4}. Using theseprobabilities, it is straightforward overify that (?/,D) is optimal if (xt w\xs)p\ +(x]+ W2XS)p2 > (xt wixs)p3 + (xi w4x5)p4 while (?>, U) is optimal if theopposite inequality holds. Therefore, in equilibrium, the receiver will only play either(?7,D) or (D, ?7).As for the sender, thefractionof those expected toplay (U,D) willfall behind thatof those expected toplay (?>,U) if (yb ?y5) < (yi (1? ?)yj; thatis when ?_y5 y?, < 1/2: In such case, (mj, m2, mi, m2) is the best strategy, whereas itis (m2,mi, m2,mi) when theopposite inequalityholds.

The following proposition characterizes thepure strategy sequential Nash equilibria of thegame:Proposition 5. Suppose conditions 5 and 7 hold. Then in the sequential equili

bria of the game between sophisticated sender and receivers, pure strategies are asfollows:EQ 1: [(m\,m2,m\,m2), (U,D)] if and only if y?> - and x? < -.

EQ 2: l(m],m2,m\,m2), (D, U)] if and only if yh < - and x? > -.

EQ 3: l(m2,m\,m2,m\), (U,D)} if and only if y? < - and xt> -.

EQ 4: [(m2,m\,m2,m.\),(D,U)] if and only if yh > - and xt < - .

Furthermore, the electoral outcome preferred by the sophisticated sender isachieved in all those equilibria, whatever the state.Note thehigh similaritywith the equilibria in the two states case of theprevioussection. There is a difference, though: It is not true anymore that the receivers loseanytime theoutcome preferredby the sender is implemented. In particular, in statesS2 and S3, lettingthe sophisticated senderdetermine theoutcome helps the receiversas well. This is due to the fact that in those states the interests f theplayers are thesame.



19/23


Let us now turn to the equilibria in mixed strategies, which arise whenever1/2 > max{if, X], yb, y,-}. For the sender's message to be uninformative, we need\i? r\i?pi for all i. This condition requires that

xi-xtW\=-h W4XsH;3=W\, W4 = W?,

which corresponds closely to the condition on sender's strategies found in proposition 3.10Observe that the problem for the receivers is the same as before: Theyneed tofind the rightmix so that the electoral outcome is not affectedby thepublicsignal. Therefore, theirequilibrium strategy is the same as inproposition 3.The role of the sophisticated players in themixed strategies equilibrium is analogous to theprevious model: They need toputmore probability on choosing theaction that is consistentwith themortal typewhose probability is smaller. Like inthe base model, if the sender is known to be sophisticated, then the structureofthe equilibria depends on the relative size of themortal receivers. In fact, if eitherof them is large enough to constitute amajority, thepartially revealing pure strategy equilibria of EQ 1 and EQ 4 in proposition 5 apply. Otherwise, ifnone islarge enough so that the sophisticated players are pivotal, thenwe have thepooling equilibria.Therefore, we can conclude that the introduction of uncertainty about correlationin payoffs between sender and receiver does not alter the main result of the basemodel ifthere is at least a 50 percent chance that the sender is indeed an enemy.Wenow turn to the opposite case?that is, when the sender ismore likely to be a friend.

The Sender is a FriendThe sender is a friendwhen condition 6 holds and therefore/?2 P4 and 773p\.

If we assume again that the most naive sender adopts (mi, m2, mi, m2), iteration ofbest responses generates thefollowing pair of players' types:

truth-teller : (m\,m2, m 1,m2) believer (D,U)In thiscase, believers play (D, U) instead of (U, D) as in the restof the article. Thisis because themost likely states are S2 and S3 inwhich choosing, respectively,UandD is best. Also notice thatmortal liars and invertersdo not fit in the scene: Thisis because, maintaining the same naive type as in the rest of the article, the bestresponse to a believer type isnow telling the truth.

Having a simpler structure fmortal types, theconditions describing the sophisticatedplayers' best responses simplifyaccordingly:Lemma 4. Let conditions 5 and 6 hold. The sophisticated receiver's bestresponse is given by



20/23


(U,D) if (xt w{xs)pi + w2xsp2 > {xt w3xs)p3 + wAxspA,(D, U) if (xt w\xs)pi + w2x5/?2 < (xt+ w3xs)p3 + w4x,p4.

The sophisticated sender's best responses are given by(m\,m2,m\,m2) if ?y, < -,

(m2,mi,m2,mi)if ?y5 > -.

The proof is analogous to that of Lemma 3 and is therefore omitted. It is nowstraightforwardto characterize the equilibrium play in thepure strategy sequentialequilibria.

Proposition 6. Let conditions 5 and 6 hold. Then in thesequential equilibria of thegame between the sophisticated sender and receivers, pure strategies are given by

EQ 1 :[(m\,m2,mi,m2), (D, U)] for any parameters' value.EQ 2 :{(m2,m\,m2,m\), {U,D)} if - > max{xt,y?,}.

The electoral outcome most favorable to the sophisticated sender is achieved in allthose equilibria, whatever the state.Remark thatEQ 2 is an equilibrium in which mortal receivers are fooled, butthisdoes not produce any harm since there is amajority of sophisticated receivers.Conversely, with EQ 1, thevote results inunanimity. In both equilibria, however,the outcome is the most preferred by the sender in every state.Similar to the enemy case, when 1/2 max {.*:,,yb}, thereare (mixed strategies)pooling equilibria in which messages are not informative and receivers do not condition their ction on the signal.Unlike theenemy case, note that the setof equilibria does not expand when behavioral types are introduced; rather, itbecomes reduced, provided that the amount ofsophisticated receivers is not too large. As a consequence, and perhaps surprisingly,having less sophistication isgood in the sense that teliminates thepooling equilibriainwhich thepositive correlation in theplayers' payoffs isnot exploited.

ConclusionWe have analyzed amodel with a sender interacting ith a population of receivers.The senderhas informationthat is relevant for thepayoffs of both kinds of players.Receivers take a binary choice, and thefinal outcome is determined bymajority rule.



21/23


In addition, there is uncertainty,similar toCrawford (2003), about the sophisticationof both sender and receivers.We find that thepure strategyequilibria of thegameshare thepropertythat if the sender is sophisticated, theoutcome will be favorable forhim or her, whatever the prevailing state. This result also carries through in the casewhen there are states in which the interests of the sender are not at odds with those ofthe receivers.

Using this framework to analyze OBL's videotape message to the American public on theeve of the2004 U.S. presidential elections can help in tryingto rationalizea seemingly paradoxical event inwhich, according to some views, OBL "pulled hisnemesis over the finish line" (Krauthammer 2004). Although we refrain fromoffering our personal views on thematter, the reader can perhaps usefully fill in the emptyparameters slots regarding the rationality of the various actors to see who won andwho lost according to our model.

The game we investigated models the conflicting interests between U.S. votersandOBL in a simplifiedway alongside a hidden truemotivation for themessage onOBL's part.A possible alternative (and equally interesting)description ofwhat happened can be devised ifone believes thatOBL's ultimate intended audience was not(or not only) theAmerican public but, forexample, (radical) groupswithin the Islamic communities worldwide.11 In thatcase, Farrell and Gibbons's (1989) model ofcheap talkwith two audiences would be the natural startingpoint for the analysis,which can be theobject of further esearchon thetopic.

Notes1. The defeated candidate himself later ascribed a decisive importance to the videotape for theoutcome of thepresidential race. Quoting fromNagourney (2005) "the attacks of Sept. 11were the "central

deciding thing" inhis contest with President Bush and that the release of an Osama bin Laden videotapetheweekend before Election Day had effectively erased any hope he had of victory."2. Another episode that fits well this picture is the strategy adopted by Jos? Maria Aznar (andvoters' response in thepolls) following the2004 Madrid trainbombings thateventually brought him to ahistoric defeat in the general elections 3 days after.Unlike OBL's message, it is not a case of a thirdparty's endorsement. Yet, the events were so close to the elections and novel to call, again, for amodelwith initial responses.3. This approach can be used for sports,where data can be (and in fact are) gathered (see, e.g.,Chiappori, Levitt, and Groseclose 2002; Palacios-Huerta 2003; Walker andWooders 2001).4. On the other hand, it is true thatworking under amore general assumption, such as pb > 1 p,generates a change in the structure of the equilibria. Yet, themain structure of the article remains unaffected, since only the equilibria where the sophisticated receivers mix no longer exist.5. If the population of voters were large but finite, looking at undominated strategies would do.With a continuum of voters the payoff irrelevance of voters' actions can be overcome by the notion ofatom proofness as introduced byMeirowitz (2005).

6. The characterization of behavioral types depends on the definition of the level 0 players. In theworking paper version of Landi and Colucci (2008), we show that themain results hold under a varietyof different but plausible definitions of level 0 types.



22/23


7. We restrict our attention to generic values of the parameters and therefore do not considerequalities.8. The examples ofMarcos in 2006 presidential elections inMexico, theEconomist's frontpageson Berlusconi, and Aznar's message right after theMadrid bombings seem to fit in thismore generalpicture.9.We maintain the assumption of a binary message space because of the kind of application wehave inmind, inwhich themessage clearly signals either one of two possible candidates in an election.10.This condition is presented in terms of behavioral strategies Myerson (1991) rather than standardmixed strategies as in the previous section. Reasons of simplicity motivate this choice because mixedstrategies for the sender are vectors in the fifteen-dimensional simplex. We are reluctant touse the explicitwording about behavioral strategies because of thepossible confusion with the termbehavioral typesthatwe have used extensively in the present article.11. This seems indeed to be the case for various more recent video or audio messages attributed toOsama bin Laden.

ReferencesAl Jazeera. 2004. Full transcript of bin Ladin's speech. http://english.aljazeera.net/English/archive/archive?Archiveld=7403 (accessed October 2006).Benabou, R., and G. Laroque. 1992. Using privileged information tomanipulate markets: Insiders,

gurus, credibility. Quarterly Journal ofEconomics 107 (3): 921-58.Camerer, C, T. Ho, and J.Chong. 2004. A cognitive hierarchy model of games. Quarterly Journal of

Economics 119 (3): 861-98.Campbell, J.E. 2005. Why Bush won thepresidential election of 2004: Incumbency, ideology, terrorism,and turnout.Political Science Quarterly 120 (2): 219-41.Chen, Y. 2006. Perturbed communication games with honest senders and naive receivers. Unpublished

paper, Arizona State University.Chiappori, P. A., S. D. Levitt, and T. Groseclose. 2002. Testing mixed strategyequilibrium when playersare heterogeneous: The case of penalty kicks in soccer. American Economic Review 92 (4): 1138-51.Cohen, F., D. M. Ogilvie, S. Sheldon, J.Greenberg, and T. Pyszczynski. 2005. American roulette: Theeffect of reminders of death on support forGeorge W. Bush in the 2004 Presidential election.

Analyses of Social Issues and Public Policy 5(1): 177-87.Costa-Gomes, M. A., and V. P. Crawford. 2006. Cognition and behavior in two-person guessing games:An experimental study.American Economic Review 96 (5): 1737-68.Costa-Gomes, M. A., V. P. Crawford, and B. Broseta. 2001. Cognition and behavior in normal-form

games: an experimental study.Econometrica 69 (5): 1193-1235.Crawford, V. P. 2003. Lying for strategic advantage: Rational and boundedly rationalmisrepresentationof intentions. American Economic Review 93 (1): 133-49.Crawford, V. P., and J. Sobel. 1982. Strategic information transmission. Econometrica 50 (6): 1431-51.Farrell, J., and R. Gibbons. 1989. Cheap talk with two audiences. American Economic Review 79 (5):1214-23.Feddersen, T. J.,andW. Pesendorfer. 1996. The swing voter's curse. American Economic Review 86 (3):404-24.Kartik, N., M. Ottaviani, and F. Squintani. 2006. Credulity, lies, and costly talk. Journal of Economic

Theory 134 (1): 93-116.Krauthammer, C. 2004. How Bush almost let it slip away. The TimeMagazine, November 15.Landi, M., and D. Colucci. 2008. Rational and boundedly rational behavior in a binary choice senderreceiver game, 2d version (Singapore Management University Economics and Statistics Working

paper series 05-2008). Singapore: Singapore Management University.



23/23

686 Journal ofConflict Resolution

Meirowitz, A. 2005. Informational party primaries and strategic ambiguity. Journal of TheoreticalPolitics 17 (1): 107-36.Myerson, R. 1991. Game theory: Analysis of Conflict. Cambridge, MA: Harvard University Press.Nagel, R. 1995. Unravelling in guessing games: An experimental study. American Economic Review

85(5): 1313-26.Nagourney, A. 2005. Kerry says bin Laden tape gave Bush a lift. The New York Times, January 31.

http://www.nytimes.com/2005/01/31/politics/31kerry.html (accessed October 2006).Palacios-Huerta, I. 2003. Professionals play minimax. Review ofEconomic Studies 70 (2): 395-415.Sobel, J. 1985. A theoryof credibility. Review ofEconomic Studies 52 (4): 557-73.Stahl, D., and P. Wilson. 1994. Experimental evidence on players' models of other players. Journal ofEconomic Behavior and Organization 25 (3): 309-27.-. 1995. On players' models of other players: Theory and experimental evidence. Games andEconomic Behavior 10(1): 218-54.

Walker, M., and J.Wooders. 2001. Minimax play atWimbledon. American Economic Review 91 (5):1521-38.

fndjsfgosjoijw

Documents

Transcript of fndjsfgosjoijw