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Center for Economic Research

No. 9491

EVOLUTION OF THE MEANINC OF MESSAGES IN SENDER-RECEIVER GAMES: AN EXPERIMENT

By Andreas Blume, Douglas V. DeJong, Yong-Gwan Kim, and Geoffrey B. Sprinkle

November 1994

ISSN 0924-7815

E~'OLt'TIO:~ OF THE :~IEANI~G OF ~IESS.aGES IN SENDER-RECEI~'ER GA~41ES: .a:V EXPERI:~IENT '

Andreas Blume Douglas V. DeJong Yong-Gwan Kim Ceoffrey B. Sprinkle L'niversity of Iowa [owa City, IA 52242

5eptember 6, 1994

'bVe thaak Joyce Berg, Vince Crawford. Robert Forsythe, Joel Horowita, Don Mc- Closlcey, Forrest Neleon, Lazry Samuelson and Joel Sobel for their advice. We benefitted from comments and diacussiona resultíng írom presentstioas at the 1994 summer meet- ings o[ the Econometric Society, the IO~ME~GT Workshop at the University of [owa, the CentER tor Economic Research at the Univeraity otTilburg, aad the Universiey of Bonn. We are gratetul for financial support Gom the College ot Business Adminiatration at the l'niversity of lowa. the Ira B. McGladrey lnstitute and the National Science Fouedation. EVOLUTION OF THE MEANING OF h1ESSACES IN SENDER-RECE[VER GAMES: .4N EXPF.RIMF,NT

Abatract

This paper experimentally investigates the evolution of the meaning of inessages in sender-receiver games of common and divergent interests. Meaning ia viewed as an equilibrium phenomenon and an equilibrium as the reault of a dynamic procesa. Hypotheaea concerning outcomes and dynamica are investigated using evolutionary game theory. Reaulta indicate that efficient communication emerges endogenously; providing population hietory information accelerates this procesa. There is considerable support for the theory in common intereat gamea. Using probit models to investigate the dynamics, population history and past individual atrategies are used by players. Past suc- ceasea are more helpfulin predicting behavior than failures but neither are as important as population history. For games with divergent interesta, there is leas aupport for the theory. Theory is not unanimous in ita predictiona and faila to capture certain salient features of thP data. The size of the message space influences the equilibrium selected: however,the degree of divergence ofinterest between senders and receivera and receivers' prior beliefs about sender types also influence the equilibrium selected. Finally, initial conditiona play an important role by eatabliahing coordination precedents that are deciaive in both game typea. For divergent interest games, initial play impacta the outcome obtained.separating or pooling. I

Two ~avages. who had never beea taught to speak. but had been bred ny r.~motF. from lhr sonelies of inen, would naturally begin to torm ihai Inn~tunqe by which they would endeavour to make their mutual wants mtelligible to euh other. by utterinq certain sounds, whenever they meant to denote certan objects. [Adam Smith]

In Gke manner are languages gradually estabtish'd by human conventions without any promtse. In Gke manner do gold and silver become the common measures ot exchange. ( David Hume]

1 Introduction

We experimentally investigate the endogenous evolution of the meaning of costless messages in a class of simple games of communication called sender- receiver games. [n sender-receiver games a privately informed sender sends a message to a receiver who then acts in response to the message. The sender and receiver obtain payoffs that depend on the private information of the sender and the receiver's action, and not on the message sent. For these games, we derive a series of hypotheses concerning the Gmiting outcomes frortt evolutionary game theory. Inspired by this theory, we formulate hypotheses about the dynamic process itself. We arP interested in communication because of its pervasiveness. [ts importance for mediating economic transactions is apparent; it has been com- pared to the role of money. We choose the game theoretic approach because recent advances in game theory provide both a framework for formal analysis and a source of testable predictions. We choose to study sender-receiver games because communication is a prerequisite for achieving efficiency in these games.t We adopt the evolutionary perspective because it does not assume the existence ot a commonly known language. Given the multitude o(PVOlutionary models o(communication, an experimental investigation can It~ll u~ under what circnntstances these models are approprjate. Further, the empirical approach may provide discipGne and suggest directions for research íu this literature. The evolution o( natural languages, professional languages and transac-

'One could ask why we ue not studying the emergence ot coordination ia strategic torm games: atter all, the games we analyze have a strategic [orm representation. There rre two responses. Fitst, sender-receiver games compriae a rich class of games with a weH- drvrloped theory. Second, when we report the history ot play to the players, we report actions snd not strategies. This diatinction seems to be important in prutice and would disappear if we went directly to strategic totm games. z

tion specific languages are of interest to economists. Pidgins and creoles ara examples of natural languages which evolved when European languages came into contact with local languages during the European mercantile ex- pansion. Professional languages, e.g. the language used by a manager who must provide a financial report on his firm for its shareholders, continue to evolve. So do languages relating to specific transactions such as job in- terviews. Whinnom [1971] and Coulmas [199Y] stress the analogy betwPen biological and Gnguistic processes and the need to adopt "... an evolution- ist view of language which assumes that speech communities adapt their language to the communication needs at hand. ..." Coulmas (1992]. Communication plays a prominent role in the general game theoretic Gterature. It has been cited, Aumann [ 1989], as a mechanism for equilibnum attainment in a game. Second, adding communication to a game may strictly enlatge the set of equilibrium payoffs, equilibrYUm enlargement, Aumann [1974], Myerson [1989] and Forges (1990a,1990b]. Thirdly, communication may play a role in selecting among multiple equilibria, equilibrium selection, Schelling [1960]. Sender-receiver games, introduced by Crawford and Sobel [1982] and Green and Stokey [1980], comprise a rich class of games extensively studied in theory and widely used in applied situations to study issues of information disclosure in accounting, economics, finance and political science, e.g. Dar- rough and Stoughton [1990], Stein [1989], Farrell and Gibbons [1990] and Austen-Smith (1990]. Communication is essential in sender-receiver games because it allows the receiver to condition his actions on the sender's message; unlike in complete information coordination games, communication cannot stop once coordination is achieved. Without communication the receiver would simply best respond to his prior beliefs. In common interest games, for example, where the incentives of the sender and the receiver are perfectly aligned, communication significantly enlarges the set of equilibria relative to the trivial game in which the receiver can only rely on his prior beliefs. However, while communication enlarges t11e set of equilibria, there is no guarantee that it will be effective in the sense that messages have distinct meanings. All the original uninformative equiGbrium outcomes remain; the receiver can simply ignore the sender's message and therefore the sender does not care about which message to send and "babbling" is a best response. Standard refinement techniques are of little help. Babbling equilibria are perfect and even proper, Blume (1993). Even strategic stability does not rule out uninformative equilibria in general. One response to this difficulty 3

i: to admit messages whose meaning is given outside oC the game, Farrell f 199aJ. ~latthews. Okuno-Fujiwara and PostlPwaite [1991]. Alternatively, ecolutionarc game theory lets meaning evolve endogenously. This requires one to conceptualize a dynamic process of how meaning becomes attached to messages. The advantage of this approach is that one deals with a closed model which holds the promise of providing a more solid foundation for making predictions in sender-receiver games.z Dynamic stories, e.g. Can- ning [1992] and Ndldeke and Samuelson [1992), are attractive because they offer the promise of a theory of both equiGbrium attainment and equiGbrium ;election. Static solution concepts, e.g. W~rneryd [1993] and Blume, him and 5obel ( 1993], which formaGze the key intuition of the dynamic stories are attractive because they do not require faith in a specific dynamic rule and because they are easier to manipulate. We study two classes of games: games of common interest, with a unique Pfficient payoff point, and games of divergent interest, where the sender's and the receiver's interests over equilibtia are opposed. Among the common interest games we focus on two types, one in which the receiver has a pooGng action distinct from his separating actions and one where he does not. We ask the following questions: How does meaning get attached to messages? How well do the dynamic theories and static solution concepts perform? :~re there "styGzed facts" about how players accompGsh meaning-message association? Does providing players with more information help them in their task? Do more messages help or does a larger message space add to confu- sion? What is the role of endogenously generated focal points? How well do evolutionary stories per(orm off their domain, e.g. with human subjects who process history information and engage in some form of deductive reasoning. These questions cannot be answered by a priori reasoning alone. An experimental study can provide useful clues as to what features a dynamic story should attempt to model and what characteristics might be missing in rhP rurrently available stoties. bíoreover, there are a number of predictions ~~~miwm tu all the presently available dynamic and implicitly dynamic stories. ;`t the cen~ least we wish to know how these predictions hold up under experimental scrutiny. We find considerable support for the theory in common interest games;

7The applicstions of sender-receiver games predate the evolutionary literatute and instead rely on ad hoc rules for equifibrium selectioa. Besides being ad hoc, ehese rules tend to conflict with equally ad hoc rules tor refining equitibria in seadeo-receiver gamea based on literal meanings. The evolutionsry approach holds the poteatial [or a more solid [oundation for making predictions in theae games. 4

meaningful and therefote efficient communication emerges endogenously and providing population history information accelerates this process. K'e use probit models to investigate the dynamic process. Population histon~ and past individual strategies help predict players' behavior. Past succPSSPS in individual matchPS are more helpíul in predicting behavior than past fallufes but neither are as important as information about population history. for games with divergent interests there is less support for theory. The theory is not unanimous in its predictions and fails to capture certain saGent features of the data. C'onsistent with theory, the size of the message space influences the equiGbrium selected, separating or pooGng. Contrary to theory, for the pooGng outcomes, the way in which senders conceal their types varies by message space size. The degree of divergence of interest between senders and receivers and receivers' prior beliefs about sender types also influence the equiGbrium ,elected. Finally, initial conditions play an important role in both types of games by estabGshing decisive coordination precedents.

2 Theory and Hypotheses

ln the games we analyze, the sender's private information is described bv the sender's type. A type t is drawn from a finite support T according to a distribution a(.); in our discussion types are equally likely. The sender sends a message m E M to the receiver, who takes an action a E A in response to the sender's message; M and A are assumed. to be finite sets. Payoffs do not depend explicitly on the messages sent. a strategy pair for the sender and the receiver is a Nash equilibrium if the strategies respond optimally to one another. If an equilibrium is such that each type of the sender is identified through his message, an equiGbrium is called separating.3 at az tt 0,0 ï 00. ï00

t2 700,700 0,0

Figure 1

~Levna [I969) introduced a special clasa of sender-receiver qames in which the receiver haa as many actiona as the sender haa types and payoffa for both players are maximized i( and only if the receiver's ution "matchea" the sender's type. Lewis refers to a separating equilibrium as a signaling convention if it is used regularly in tepeated sender-receiver gamea. According to him such conventiona can come about either through expGcit :gree- ment, salieace or precedent. 5

Figure I gives the oayoffs of a simple common interest game ( Came I) as a funrtion of the sender's type, t„ and the receiver's action, a~. If, for example, the true type is tt, then the receiver's optimal action is a.z. .an example of a separating equilibrium in this game is one where the sender sends mt E Af if he is of type tt and m2 otherwise. and where the recF~iver takes action a2 after message m~, and at otherwise. All of the P xplicit and implicit dynamic theories that we consider predict separation in Game l. Canning ( 1992] and NSldeke and Samuelson [1992] look at expGcit rules for adjusting behavior in a setting where players are repeatedly matched to play a given sender-receiver game.4 Common to these works is the assumption of random mutations which results in an ergodic process with a unique stationary distribution.s Predictions are in terms of the Gmiting stationary distribution, as the mutation rate goes to zero. Both papers show that in a subset of these games ( which is itself a subset of the class of common interest games Gke Game 1) the Gmiting distributions of their dynamic procPSSes support only efficient outcomes.s The evolutionary stable strategy ( ESS) [:vlaynard Smith, 1982] is the first static solution concept that was used in evolutionary game theory, and has inspired numerous others..' An ESS is characterized by stabiGty against invasion as long as the proportion of invaders is sufficiently small. To illustrate the role of ESS in Game 1 suppose that the message space is given by .N -{mt, mZ}. Then this game has two separating equilibria and infinitely many uninformative equilibria. The separating equilibria are ~trict and therefore satisfy every conceivable refinement; there are uninformative equilibria which are strategically stable and consequently proper and perfect.a Because the separating equilibria are strict, they cannot bP invaded; any other equilibrium can be invaded because it will involve multiple

'They build on work by Foster and Young [1993], Young [1993] and Kandori, Mailath and Rob [1993]. ' l,inguists have recognized thst '... evolutionary processes in biology and Gnguistics arr rcplicable and deuribable in terms ot minimal mutations ..., whèther intrusions from thr outaide or generated within the system, whose spread is controlled by tutors of 'advantage' or lack o( disadvantage. which are mesaurable only relatively. by reterence to the envirooment." [Whinnom. 1971, p.91] sStrictly speaking, Canning usumes that there is a nominal sigealing cost for every message. 'Hoth ESJ and NSS (discusaed below) are defined tor a single-population model but ran br rxtended W a two-population model in which agenta play an aaymmetric game. "F'arrell's refinement u not appropriate both because ot the limited size o[ the mesaage space and becaux messages have no Gteral meaning; with a large message space and meaningful messagea Farrell would predict separation. 6

best repGes.

One problem of ESS is that if wP Pnlarge thP mPSSage sparP in this gamP no E'.SSexist, b~~-ause all equilibria will invulve mull.iple best replies. :1 pure ,trateqy separating ayuilibrium can bP invaded by a strategy profile that rc- sponds differently to unused messages. One might argue that this instabilitv is spurious, since this invasion does not change the players' payoffs. One way of dealing with this problem is to simply declare invasions of this kind as benign. More formally, a neutrally stable strategy (NSSJ is a symmetric Nash equilibrium with the additional property that the incum- bent strategy is at least as good against a potential invader as the invader is against itself. Wárneryd [1993] shows that in games like Game 1. with an arbitrary finite message space, for a pure strategy equilibrium to be an ,1~SS, it must be efficient.

Blume, h:im and Sobel [1993] partially solve thr nonexistence problem of by ESS using a set-valued solution concept, following and approach developed by h:im and Sobel [1992]. Set valuedness allows for drift in an equilibrium component and guarantees the existence result for a class of sender-receiver games with a large message space. The drift also implies a greater degree of instability for an inefficient equilibrium and they derive a stronger efficiency result than Wtirneryd. Somewhat more formally, the solution concept is Swinkels' [1992] equilibrium evolutionarily stable (EES) set. A set of equilibria is an EES set if it contains all profiles (potential entrants) which are best replies to the post-entry population; here the post- entry population is composed of primarily a population using a profile in the candidate stable set and a small population of potential entrants. In common intenest games, i.e. games with a unique Pareto efficient point, every EES strategy profile must be an efficient equilibrium, and conversely that the set of efficient equilibria is equilibrium evolutionarily stable. The above discussion about static solution concepts and explicit dynam- ical systems in evolutionary game theory yields our first hypothesis: IJplay- ers ane repeatedly and rnndomly matched to play Came I with a message space of size two. ~(M) - 2, observed play will converye to a sepamting equilibrium. 9

'While many of the solution concepts were developed to overcome the restrictions mesaage spue of ESS, we start our design with the minimum message space neceaaary solutioe for all concepta to predict the aame equilíbrium outcome in Game 1. 7

n2 ti 0.0 ï00.700 100,.100

tZ ï00.ï00 0,0 400.100

Figure 2

~~~árnen~d's efftciency result does not extend to games where the cardi- nalitv of the action space exceeds the cardinality of the type space and~or where we admit mixed strategies.tu Game 2, with payoffs as in Figure 2, for example also has a neutrally stable strategy which supports the pooGng outcome in which the sender mixes over all messages and the receiver takes action a3 after even. message. In Blume. Iíim and Sobel, the issue is not the stability of a single equilibrium but the stability of an entire component as was discussed above. Thus, in our Game 2, a pooGng .YSS would not be part of an EES set because it belongs to an unstable component. ESS predicts separating equilibria since they are the only strict Nash equilibria in Game 2. The works of Canning and Vdldeke and Samuelson also ptedict separating equilibria. From this we derive our second hypothesis: !J playera are repeatedly nnd rnndornly matched to play Came 2 wíth ~(M) - 2, obaerved play urill conrerge to a separating equilibrium.

al a2 a3

tt 0.0 200.700 400.400

tZ 200.100 0,0 400.~t00

Figure 3

~~~hen we move from games of common interest to games in which the .rudrr'. and th~~ recPiver's interesta over equilibria are opposed, the predic- tiuns uf the wlutiun concepts are not as clear cut. Both ESS and EES sets are sensitive to the size o( the message space. Let the payoffs in Game 3 ht~ a., in Figure 3. With only two messages, thPre are two ESS's and EES set; rorresponding to the two separating equiGbria. With more than two messages no ESS or EES set exists in this game. Blume, Kim and Sobel show that if there is a nominal signaling cost which varies across messages

~o[Inlike Wàrneryd, we use neutrally stable serategies in mixed strategies in our discussion of theory. 8

only the pooling outcome is supported by an EES set in Game 3 with three messages.'t t1 [t can be also shown that both pooGng and separating equilibria are VSS's no matter what the size of the message space. tiSldeke and Samuelson's theory predict separation with two and pooGng with three messages, provided one limits drift through nominal signaling costs or similar means. Here, for a message space of a given size, we find that different solution concepts predict different equilibrium outcomes and these predictions change for certain solution concepts depending on the size of the message space. Essentially, there is more theoretical support for the separating outcome with two than with three messages because the separating outcome is strict with two but not with three messages. We formulate our third and fourth hypotheses to evaluate these solution concepts: !f players are repeatedly and mndomly matched to play Came .y with ~(~L~) - 2, observed play will converge to a sepamting equilibrium. Ij players are repeatedly and rondomly matched to play Game ,I unth ~(:N) - 3, observed play will converge to a pooling equilibrium. The final hypothesis is inspired by the explicit dynamic theories and the intuition underlying the static theories. The explicit theories combine the current state and a fixed behavioral rule to determine the probability distribution over future states. They vary in what to include in the current state, e.g. information from the last period only or from other prior periods, information on population behavior or individual behavior, etc.. To motivate the investigation of these issues, we propose the following working hypothesis: IJ players am repeatedly and randomly matched to play a sender-receiver game, then (i) in games oj common interest, the speed of convergence to a separoting equilibrium depends upon whether information on the past population dtistributtion oj stmtegies is available to the players, and (iiJ in general, an individual's observed play in any given period will

t~An ESS does not exiat even íf there ia a nominal signalíng cost. '~Blume (1994] ahowa for a different aolution concept, perturhed persutenca that the aame elfect can be achieved with costleas measagea under an ejjeehve lanquage assumption- Peraiatence, introduced by Kalai and Samet (1984], is a another static solution concept based on intuition about dynamics. Peraistence predicta xparation in Gamea 1 and 2, and in Came 3 with 2 mesaagea. Depending on whether one subxnbea to peroiatent equilibria or perautent sets, it allowa both pooling and aeparating equilibria, or makea no prediction in Game 3 with three meesagea. The aame predidions are obtained from a closely related set-va(ued solution concept, curó (cfoaed under rot~onol behaviar) sets. Curb sets were introduced by Baau and Weibull [I991]. Hurkena [1994] ptopoeed an explicit dynamic which convergea almost aurely to a curb set. 9

dFpsnd npon last period's population distribution of strotegies. lasl perind's persona! strategy choice and its success or Jailure.

3 Design

i'he origin of evolutionary solution concepts is in biology although others, e.g. Crawford (1991], ~Iailath (1992], have given these solution concepts a learning interpretation. In either interpretation there is a tendency for successful strategies to replace unsuccessful ones. There are no knowledge requirements in the biological interpretation of these models. The learning interpretation however requires some knowledge on the part of the agents. For example, players need to be able to compare their actions with the per- ceived consequences of their actions. This also implies certain informational requirements; players need to know the current distribution of strategies in order to judge the performance of their own strategies. Knowledge requirements also differ across the evolutionary solution concepts. For example, in Blume, Kim and Sobel's approach they are considerably more stringent than in W'~rneryd's theory. Potential entrants anticipate the consequences of their entry in Blume, Kim and Sobel. For this to occur, players need to know the structure of the game including the payoffs of both senders and receivers. .as the above discussion suggests, there is no uniformly accepted view of the knowledge and information requirements across the evolutionary theories. ~Teither has there been any experimental work addressing the evolution of ineaning in communication games. However, there is a considerable Gt- erature on communication in standard game theory, e.g. Brandts and Holt (1992]. Cooper et al. [1989], Dickhaut et al. [1992]. We build on the structure of the Cooper et al. design which has addressed issues such as common knowledge, pairings of players and communication structures. In so doing, we construct a design that satisfies the knowledge requirements of the evolutionary theories to be evaluated. In the expetiment, players are asked to participate in a sender-receiver game such as the one in Figure 1. Each player's payoffs contingent upon the sender's type are common knowledge and the numerical payoffs represent a player's utility. We induce payoffs in terms of utility using the Roth-Malouf procedure. Roth and l~talouf [1979] and Berg et al. [1986].t3 With this

~~Binmore and Samuelson [1993j discuss the relationship between the notion of repro- ductive success in biological game theory and atility. io

procedure, each player's payoff is given in poínts and these points determine rhe probability of winning a monetary prize. At the end of each period of each game, a lottety is conducted where winning players receive ~1 and losing players receive zero.t4 The experiment is conducted using a series of poputation cohorts. .all players are recruited Gom undergraduate (sophomore and above) and grad- uate classes at the University of [owa- Upon their arrival, players are seated at separate computer terminals and each is given a copy of the instructions for the experiment. Instructions for the treatments are found in appendix A. Since these instructions are also read aloud, we assume that the information contained in them is common knowledge.ts A cohort consists of twelve players, six senders and six receivers. Each player is randomly designated a sender or receiver at the beginning of a game and remains so for all periods of the game. In each period, senders and receivers are paired using a random matching procedure. Sender types are equally likely and independently and identically drawn in every period for each sender. Senders are privately informed of their type at the beginning of each period. All random matchings and drawings (except for the lottery) are done by the computer; players also eommunicate and rhoose actions through the computer. Hence, no player knows the identity of the players with whom he is paired, in either past, current or future pairings. The above information is included in the instructions. Each cohort participates in two separate sessions.'g In Session I, all players participate in 20 repetitions (periods) of the two stage game whose payoffs are shown in Figure 1. Prior to the first stage, senders are informed about their respective types. In the first stage, senders send a message of either A or B, M- {A, B}, to the receiver they are paired with. In the second stage, receivers take action at or a2 after receiving the message ,.1 or B from the player they are paired with. Each sender and receiver pair then learns their respective type, message sent, action taken and payoffs received. All players next receive certain aggregate information about the population; in particular, all the sender types that are drawn and all the messages sent

~~Eac6 player waa given an oppor[unity to draw the lottery ticket the opportunity was rotated among the playen. "To provide additional aasurance regarding common knowledge, players are aaked to anawer a set ot queations related to the inatructions, see Appendix A. These quesGons and related anawen are reviewed betore a game ia conducted. 18 Each cohort completed the two sessions in about one and one-hal[ to two and one-halt houra. Paymenta to participanta ranged trom S9 to 532. 11

bv the respecrive sender types. This information is displayed for the current and all prvciou; periods of the session. In Session [I, all players participated in an additional 20 repetitions (40 repetitions for Game 3 treatments) of a two stage game which differs from the game played in Session I. An essential part of the Session I[ design is the message space and its abiGty to ensure that at the beginning of the session there is no common understanding of inessages among senders and receivers, senders alone and receivers alone. This is guaranteed in the design by en- dowing each player with his own representation of the message space and order of inessage space representation.t~ The representation and ordering is stable over the entire session and players are informed about the essential properties of the representation in the instructions. The mapping used to generate the player specific representations is best understood in terms of an underlying, unobservable message space, M-{A, B}. Using the structure of this underlying message space, fifty percent of the players see A as ~ and the other fifty percent see it as s. Similarly fifty percent of the players see B as ~ and the other fifty percent see it as s. Regarding the order of the message presentation on the screen, it is structured to have the effect that for half of the players their representation of A appears first and for the other half their representation of B appears first. For example, consider a sender-receiver pair in which A maps into ~ for the sender and s for the receiver; and the order of the message presentation is ~,s for the sender and ~, s for the receiver. When the sender sends message ~ because it is the message on the left (s because it is the message on the right), it will be received as s(~) by the receiver and appear as the right (left) message. This procedure destroys all conceivable focal points that players might use for a priori coordination.18 As in Session I, all players receive the same aggregate

"Schelling was the first to emphasize the role ot labeling as a[ocal poiat and Crawtord and Haller(1940] were the firat to syatematically atudy games in which players did not have common knowledge ot labels aad positiooa. '" Aa indicsted in the inatructions, terminology waa desigaed [or its neutrality. Sender typea ue designated as column players. C1 and C2; receivers ue deaigaated aa row players who take utioas Rl, R2 and (in Game 2 and Game 3) R3. The efficieat separating equilibrium also appeua on the oR disgonal elements o[ the matrices. The observable messaga ~ aad . appear on the top aet of computer keys ia that ordet, leít and rigàt thus the design destroys whatever [ocal poinu ue created by the keyboard prexatation. Technically, with a cohort of size 1'? and two messages the mapping aad ordering are as follows tor the senders, detailed by player numbet, mapping ( A,B) and ordering ( left,right): 21..íkl lik..). 3 1~.~1 (~.~).6l~,lk) 1i4t,~), e (~k..) (t~,~), l0 (~,ÁI) ('.i't) and 12 l;'k. ~1 l;rk, ~). For receiven, t (.. fk) (;fk. ~), 3 (;rk. ~). (~. ~k), 5 (~,;'k) (~k, ~), 7 ( ~, ~) (At, s), 9(., jk) (.,~) and 11 (~,.) (;bé,.). This mappiag and ordering destroys all tocal points. 12

information about the population; all the sender types that are drawn and all the messages sent by the respective sender types. This information is displayed in terms of each player's own representation of the message space and the order o( the message presentation. The information is provided for the current and all previous periods of the session. We choose the structure of Session I to provide players with experience with experimental procedures, to ensura thP subject, understand the structure of a sPnder-receiver game, message space and population history and finally to faciGtate coordination.19 As detailed in the results section, there are interesting coordination issues even in Session I. For Session II, we constructed the representation of the message space and the order of inessage presentation so that all conceivable focal points that players might use for a ptiori coordination ( e.g. Session [) are destroyed. Thus, Session II fo- cuses on the rohorts' abiGty to develop a new language as a function of the ~ame being play~~d. We only report population historirs ahuut seuders to all players (rPCeivPr information is not reported) bt~cause in a set of failed pilots we reported complete histories. It is our conjecture that it is harder to process the additional information. [f the full history is displayed, there is an additional coordination problem; the two histories may suggest two ways to coordinate. By only displaying sender history we implicitly suggest which history to follow. The design is structured such that treatments consist of three repGca- tions. Each repGcation involves two sessions, as described above. Session [ is common to all treatments and repGcations. Session [I varies by treatment. The treatments are: 1) Game 1 with two messages and population history, 2) Game 1 with two messages but without population history, 3) Game 2 with population history and two messages, 4) Game 3 with population history and two messages, 5) Game 3 with population history but with three messages.

4 Results and Analysis

We present results for the treatments in two parts. First, we examine the equilibria selected. Second, we use the data to describe the dynamics from

Por example, if type C1 always choases the lett mesaage and type C2 always chooaea the right message, there will be a 50-50 apGt on the ordering that appears on the sender and receiver screens. 19fnitial pilot studies with a common interest game showed ehat players could not coordinate without Session [. 13

the :Pnder-receit.Pr games. The data are organized by Session [. which is common to all treatments, Game 1 with and without population history, Game 2. and Game 3 with two and three messages. Figures ~l to 10 graph on a period by period basis the (raction of senders and receivers who coordinate on the equilibrium selected in the respective games (type-message and message-action association) . We group all fifteen Session I's together and the three replications for Game I with and without population history and Game 2. We performed Gkelihood ratio tests and could not reject the hypothesis that the data from the repfications came from a common process.~o For visual presentation purposes for Game 3 with two messages, we graph each repGcation separately; with three messages, we graph the first replication separately from the other two.

4.1 Equilibrium Selection

When examining the equilibria sefected, we focus on the last five periods for the Session I game, Game t and Game 2, and the last ten periods for Game 3.21 For Session I(Figure 4), 93~ of the play is the separating equilibrium.32 In seven repGcations of Session [, type tt sent message A and in the other eight repGcations type tt sent message B. Thus, players make no use of a"natural" focal point in Session I(e.g. message A might mean type tt and message B might mean type t~ based on the natural orderings of type and message space) and instead rely ón population history to achieve coordination on the separating equilibrium selected. For Game 1 with population history (Figure 5), 98010 of the play cor- responds to the separating equilibrium. By repGcation, it wbs 95qo, 98oïo

'o As detailed in the dyaamica section, we used probit analyaia to inveatigate the dynamic structure o( the treatmenta and replicationa. We pool by treatmeot since the likelihood ratio teats írom ehe probit analysea could not reject the hypothesis thst individual replications were generued ttom a common proceaa. . ~ " We uae Fuher'a exut teat to determine whethet or not period dependence exiaq for thr coordiaation reaulte. For the Seaaion f game. Came 1 and Game 2, we could reject the hypothesis of independence compuing the firot ten and laat ten periods tor all three treatmenta. Except [or Came 2, thu hypothesis u rejected when comparing periods 11- IS and 16-10. However. we failed to reject indepeadence for the laat Rve periods ot the treatmenta. llaing a aimilar procedure tor Gune 3, we failed to reject indepeadence [or the last ten perioda tor all rephutions ot the tteatmenta. The level ot significance uaed fot the Fisher exsce teat and all other testa included in the paper is p-.05, unlesa otherwiu indicated. ~~ Fot ehe fi[teen replicationa, twelve equaled or exceeded 909ó, two replicationa exceeded 80"Jo and one exceeded 70q. 14

and IOOq. ~I'hus, from the Session [ game and Game 1, we have substantial support for our first hypothesis; observed play in Game 1 with two messages converges to a separating equilibrium. When comparing Game 1 with population history to the game without it ( Figure 5), we find support for the first part of hypothesis five: the speed of convergence to a separating equilibrium will depend on whether information on past population distributions of strategies is provided. We chose Game 1 as the treatment here because players have a strong incentive to coordinate in this game. In the treatment without population history, ~2q of play is separating equilibrium play while it was 9801o in the history treatment. Comparing the frequency of the separating equilibrium, itself, it is 96q with population histor,y and 66oJ'o with just individual history.23 We also have an empirical measure of the contributions o( the different levels of history in the development of a language for coordination. Given the expected frequency of the separating equilibrium with no individual or population history is 30q, individual history increases the frequency to 66Q1o and population history increases it to 9R~o. tn terms of payoffs to the players, the contributions of the different levels of history are 350. d95 and 672 points respectively. In Game 2( Figure 6) there are two equilibrium outcomes, the separating one and the pooling one. Our second hypothesis states the observed play in Game 2 with two messages will converge to the separating equilibrium. 97010 of play follows the separating equilibrium. By repGcation, it is 92q, 100010 and 100q. While there was some initial play of a3, it declined quickly and play of a3 in the last five periods was all due to one player. Game 3 is a game in which the interests of sender and receiver over equilibria are opposed; a separating equilibrium that is preferred over the pooling equilibrium by the receiver and vice versa for the sender. Hypothe- ses three and four state that with two messages, play will converge to the separating equilibrium and with three messages, play will converge to the pooGng equilibrium. Figures 7 and 8 present the results for two messages. Replications 1 and 2 reach a separating outcome and replication 3 reaches a pooling outcome.2~ By repfication equilibrium play is 99010, 78~ and 84q.

~'Comparing the treatments by repfication, the frequency of separating eyuilibnum play and the separating equilibrium itself were always higher in the p„pulation hinturv treatment. " ~'Replication 1 was stable and reuhed the separating equilibrium outcome in the lut 10 periode. When conducting repGcation 2, it became dear that the game was not stable in the firot 20 periods. It waa then decided to conduct the rep6cation and all remaimng Game 3 repócations for 40 periods. 15

For the three message space treatment (Figures 9 and 10), replication 1 concPrqes to a separating outcome while the pooóng outcome occurs in the other two repGcations. EquiGbrium play is ï3q. 77q and 95q by replication. Thus, the separating outcome occurs in two thirds of the replications in the two message treatment and the pooGng outcome occurs in two thirds of the repGcations in the three message treatment. Similarly, comparing the frequency of equilibrium separating play by treatment, it is ó401o for the two message treatment and 34q for the three message treatment. We conclude that there is some support for hypotheses three and four; separation occurs more often in the two message treatment, pooGng tuore often with three messages. The separating equilibrium is strict in the two message treatment but not in the three message treatment. Once the players are in a strict separating equilibrium, their best replies are unique. It is impossible to leave such an equiGbrium via a sequence of best replies. With three messages, there are multiple best repGes. Theory does not account for all aspects of the data. The pooling outcomes differ across treatments; both messages are used in the two message treatment while only one message is used in the three message treatment. As detailed in the next sections, the initial play of receivers is also related to whether a pooGng or separating outcome is obtained in the treatments.

4.2 Dynamics

In addition to the equilibria selected, we examine the processes which lead to those equilibria. Figures 4-10 show a process which starts from coordination by chance and progresses over time to almost complete coordination. This is interesting because there areintuitively plausible rules for playing the game, similar to those investigated by Crawford and Haller [1990) and Kramarz [1992), which would in most cases lead to full coordination in period 2. One such rule is to calculate the equilibrium on which most senders are coordinated in the first period and to play according to that equilibrium from then on. While we find no evidence that players follow this rule, the initial coordination precedent predicts the equilibria selected in 24 out of 27 sessioas, and in 10 out of L'2 Session II's, excluding the no history treatment. The (act that convergence is not instantaneous suggests that players adjust their strategies over time. Evolutionary game theory investigates a number of dynamics, including the replicator dynamic and Nëldeke and Samuelson's dynamics. These dynamics are highly styGzed and, purposely, not designed to quantify the 16

contribution of a broad range of driving forces. For example, the repGca- tnr dvnamic is bnilt on the law of genetic inheritance, and it is difficult to provide a learning interpretation for it. For that reason, it says little about van.ing information conditions. Which factors matter in these adjustments is primarify an ampirical question. We use hypothesis five as a guide to investigate these factors [f players are repeatedly and randomly matched to play a sender-receiver game, then (i) for common interest games, the speed of convergence to a separating equilibrium will depend on whether information is provided on the past population distribution of strategies, and (ii) in general, an individual's observed play in any given period will depend upon last period's population distribution of strategies, last period's personal strategy choice and its success or failure. We use a series of ptobit regressions to examine the relationships outGned in hypothesis five.js We ask what factors determine the probability that message mt (action ar) is sent (taken). [n the probit model this probability depends on an intensity index mi (ai) which itself is a linear function of a number of explanatory variables a; and an error with a standard normal distribution; e.g., for senders

m~ - ao f~ a;z; ~ e. ~-t The intensity index m~ itself is unobserved. It exerts its influence through determining m,. [f we abuse notation and let mt - 1 denote "message m~ is chosen," and ml - 0"message mt is not chosen," then

J 1 ifm; ~0 m;-l 0 otherwise . We estimate a similar relationship for the receivers by treatment, with a replacing m and d; replacing a;. 26 ~sCrawford (1993j uses ordercd probits to analyze game dynamics usinq experimental data. Our approach is simílar in spirit; however, Crawford's specific techniques cannot be dírectly appfied to our xtting because his data have an additional linear that his structure techniques could exploit. Roth and Erev [1993] compare experimental simulated data with data on game dynamics. While appropriate [or their settings, their does not dynamic readily accommodate the information condítiona of our design. ~tExttpt for the Game 3 treatments, we pool by treatment since Gkelíhood ratio tescs could not reject the hypothesu ehat individual repGcations were generated by a common procesa. For Came 3, we can pool the xparating rep6cations in the two and message treatment the pooling repfications ia the three message treatment. The specific intensity index we use for the sender is: m~ -

Qo constant

} tY t [!t (~(mt.tt)s~(m7.t71)-I } ft7(~(m7.ttlstk(mt.t:l)-t~

} ns [fh (lÍm,.f1l } !(~n7,t7) )-1 } !t7(llmt.t7) } !(m7,tt ) )-t ~

} á3 [(!m, )-1~

} pq [!h (lÍmt.tl)!a7 } !(mLt7)!a, )-I

}!t7(f(mt,t7)!a, }!(,nz,ttlla7)-i~

} os [Ih(llm7,ttllat }!(m,.t7)!a7)-t

}it7(lÍmt,tt)fat } llm7,t7)!a7)-I~ } E

This model is used for the Session I game and Game 1. For Games 2 and 3 the following variable is added: as[(lm,fa,)-t~. The independent variables are population, separation inertia, mes.sage inertia. success and Jailure (and the pooling action in Game 2 and 3). Population measures the fraction of the population of players who use mt to indicate their type. !t, is an indicator variable that equals 1 if the (~(m`'t')ttc(„`7't7) )-t sender is type t; this period and 0 othetwise; 6 and (~fm7 t`)s~(m''t7))-1 measure the population frequency of revealing behavior in the previous period. The separation inertia variable is constructed analugously with (l~,,,~,t~) } I(,,,7,t7))-t and (!(m~,t7) }!(„y tt))-t measuring the ittdividual's type-message association from the previous period. ï'he messagc incrtia variable is an indicator variable that equals 1 if the sender sent mt last period and 0 otherwise. The success variable (essentially) measures whether last period a player used mt successfully to indicate his present period type. The foilure variable (essentially) measures whether last period a player failed to induce his preferred separating action with the alternative to mt. In Games 2 and 3 we use a sixth explanatory variable, pooling induction, whicó measures whether the sender in question was successful in inducing a3 with mt last period. is

ThP specific intensity index we use for the receiver is: ~i -

~3o constant

(!m ( fklmt.tz)s~lmz.tt) )-~ f ~Im2.t:)tAtlmi.ti) !m2( 6 )- 1~

t i3z ~l,n,(!(mi.nt) f !(mz.az))-t i- lezll(.ni.,~zl f !(mz.n,))-t~

f i3J [(!at)-t~

f ;3q ~l,n,(!(mt.a,)!t~ fl~,nz.az)!tt)-t

~lmx(!(mz.a,)ftz f llm~~a])!t, )-1~

f l~s ~l,n,(!(mt,a~)!ts f-I(mz~ai)!tt)-t

flmz(!(mi.az)!tz f !(mt.a,)!t, )-t~

For the pooling repGcations of Game 3 the intensity variable for at is replaced with the one for a3 and the variable ( !,t )-t is replaced with Gn, (~;` )-t t l,,,z (~ 6' )-t t 1,,,~ (~ s' )-t. This jrequency variable measures the frequency of the currently received message in the last period. The underlying notion is that in order for a message to be sent with very high frequency, it must essentially be sent independent of type and should therefore induce action 3. The other independent variables are analogous to those for the sender; population, sepamtion inertia, actíon inertia, success and jailune. The indicator variables for the current period are in terms of the message received and the lagged indicator variables record information about last period's messages, actions and message-action associations. Table 1 and 2 present the probit coefficients for senders and receivers.27 Interpretation of the coefficients is best understood through an example. Looking at Table 1 for the Session I game, suppose we wanted to determine "We used a runs test to evaluate tóe serial correlation of the error terms and failed to reject the hypothesia of seria! iadependence. fieteroekedasticity of the errors across players u also not a problem in the probite. 19

rhe marginal effect that an increase in population last period of .33 has on the probabilit~ of inessage mr being sent this period. Also, let us suppose that la;t period population was .S and the individual did not usè mt to indicate his present t}~pe, sent message l and failed. Csing the results of the 5ession 1 game probit, the marginal effect is

0(-1.~6f 1.63(.5)f.16t.69)(L63)(.33)-.21 where o( ~) is the ordinate value of the standard normal density. Thus, a.33 increase in pupulation last period increases the Gkelihood of an individual ;ending n:t this period by .2L C'sing Table 2, one could go through an analogous calculation for a receiver. .-1 quick interpretation of the magnitude of the coefficients can be obtained as Collows. Take as a reference point the case where all values of the independent variables except the constant equal zero. In that case the probability of inessage mt being sent equals d'(ao), where ~(.) denotes the CDF of the standard normal distribution. This amounts to a small probability if o is negative and large in magnitude. For example, the probability o( message mt being sent when all the other independent variables take on values of zero and ao --1.76 is ~(-1.76) which equals .04. If we could change population all by itself from 0 to 1, it would increase the probabifity of mt to ~( - L76 f 1.63) -.~t0. Thus, the population variable almost off- sets the negative constant all by itself. Since all independent variables vary between 0 and l and most assume only these two values, this exercise can be conducted to show that most of the statistically significant coefficients are also economically significant. For common interest games, the impact of providing population history to players is evaluated by comparing the two treatments of Game 1, with and without population history. Consistent with hypothesis five, separation inertia, suecea and failure but not population are significant in explaining r~~urdinati~m without history for hoth senders aud receivers. .alternatively, with history, population enters with a significant coefficient. Separation inertia is still significant; however, failure is not and success is only significant for the receivers. [t appeats that when provided, population history dominates individual success and failure in determining whether message mt will be sent. ~iessage and action inertia play no role in the process. For the common interest games (Session I, Game 1 with history and Game 2), we next evaluate the impact of population, separation inertia, success and failure on the probability of inessage mt being sent and action zo

at being taken. Population and separation inertia have large and statistirally significant coefficients for senders and receivers. Further, the message (action) inertia variable is not significant for five of the six probits. The impact of previoas success and failure is not as clear cut. For the Session I game, previous success and failure are significant for both senders and receivers. For Game 'l, previous success and failure are significant for the senders but for receivers only success is significant. For Game 1, only success is significant and only for receivers. We know that population (senders) history dominates success and failure from the no history treatment. Interpreting the results in light of this observation, say, a player encounters a failure while conforming with the emerging equilibrium. Responding to this failure implies disregarding the population norm. Since receivers can respond directly to this information, receivers are less likely to be swayed by failure than senders. On the other hand, senders do not directly respond to history but best respond to what they expect receivers' responses to be. Success is reinforcing for both senders and receivers. Thus, we conclude that previous population and individual strategies are important in understanding dynamics in the common interest games. Previous success and failure matters for senders but for receivers success is more important. Finally, message (action) inertia has no explanatory power. Evaluating the game in which sender and receiver have opposing interests over equilibria, we divide our analysis between the separating and pooGng outcomes. The dynamics of the separating outcomes share many of the characteristics of the common interest games. Population is significant for both senders and receivers. Separation inertia is significant in three of the (our probits for senders and receivers. However, failure is not significant and success is only significant for receivers in the two message treatment. In the three message treatment, the importance of inessage inertia and pooGng induction is due to the portion of the sender population attempting to pool rather than separate. Players develop different strategies in the pooling replications. In the two message treatment of Game 3, senders develop a successful strategy of uniformly distributing messages across types. This is accomplished by having a given sender use a separating type-message association but having the association be different for different senders.28 This is reflected in the '"In the nezt xction, we describe a variation of Game 3 with two messages in which the pooling outcome occurs in all three repGcstions. The xnden in all repficationa exhibit same the randomization o[ strategiea at the population leve! as uaed in the repGcation being zl

~Pnder probit wherP rhe roe!ficient on separatiun inertia is positivP and ;ig- nilirant. l-h~~ ruPtfirirnt on population is signifirant, nPga[ivP and largr~ in Ittagltllqr)P- IIIIIIrating sPndPri 5~'sLPlnaLlfally p1lrsltPd a stratPgy of tfl~ing to uffsPt rendencies to separatP in the population. Previous success is ;ig- nificant and negative for receivers. The intensity index for receivers is the pooGng action whirh Pxplains why the coefficient for previous success (at separating) is negative. For the three message treatment, senders develop a strategy of always ;ending the same message independent of type. The choice of which message to wnd is Pstablished by thP mPSSagP most (requently used in the initial period,. l his is Pxartly t.hP typP of pooling Pquilibrinm which is predirtPd by BIumP, him and Sobel and by VSldeke and Samuelson. This behavior manifests itself in significant message inertia and pooGng induction variables. For receivers. the frequency variable is significant and success (at separating) is negatively associated with the choice of the pooGng action. The negative .ign for the failure variable is caused by the variable's construction; e.g. it is zero whenecer the pooGng action is taken in the previous period.29

4.3 Game 3 Revisited

In Game :3, we find that initial actions by receivers predict the equilibrinm. In thP initial periods, thP typP-message mappings do not statistirally diffPr across the repGcations. However, the frequencies o( action :3 differ substantially across the pooGng and separating replications. In the first ten periods, the number of pooling actions were 4 and 16 out of 60 for the two separating repGcations of the two message space treatment. However, 39 of 60 actions were action 3 in the pooling repGcation. Similarly, for the three message discussed. -1~'e rlso ~nnsider higher ordet lags o( the independent variables. Aa a first pass the ~~rubn. wrrr rr-run by adding all the appropriately defined lag two independent variables. h:~t)mating both models over the same periods, the average pseudo r-squared increbaed trom .a0 to 4ti. ~~'ith the number o[ independent variables increaaed by 100q and the ~ ollinearity problems encountered by their introduction, particularly population history, the mcrease m t-squated was deemed immaterial. .~s an attempt to mitigate the collinearity problems with our only (stepwise) continuous r~ariable. population history. we difterenced the second and third lag to determine their margtnal corttribution beyond the first lag. Estimating the models over the same periods, the average r-squared is .41, .42 and .43 (or lag one, Iag one plua the marginal (or lag two, and lag one plue the marginal (or lag two and three, respectively. Assessing the economic significance o( lag two and lag three by r-squared, they aze at best marginal. T'he important (first order) effects are captuted by the lag one variables. 22

treatment, the separating repGcation had 12 of 60 actions being action 3 while the pooling replications had 25 and '23 actions being action 3. We conjecture that initial play is related to the degree of divergence of interests between the sender and receiver and receivers' prior beliefs about sender types rather than exclusively a function of inessage space size as sug- gested by theory. In the spirit of Nóldeke and Samuelson's best response dynamics, receivers begin with uniform priors and as history accumulates they revise their beliefs, which become self-confirming in the :Vash equilibrium chosen later. For Game 3, if the receiver's belief p that the sender's type is tt falls in the range 3~7 G p G~~7, then his unique best reply is the pooling action a3. Outside of this pooling range, one of the separating actions is optimal. The observed outcomes in the replications are consistent with these priors as measured by initial play in the replications of Game 3. Suppose we construct a new game, Game 4, from Game 3 by substituting the sender payoff of 500 for ï00 in the separating equilibrium, leaving all else the same. This change enlarges the pooling range to l~5 ~ p ~~1~5. This substantially increases the Gkelihood of a pooGng action initially and as a consequence the pooling outcome. With a message space of size two, ~(M) - 2, we predict more pooling in Came 4 than Game 3. The theorv of section 2 still predicts separation. - We conducted another treatment, Game 4 with population history and two messages, using the same procedures as previous treatments. Again, the observed outcomes in the replications are consistent with the priors as measured by initial play. In equilibrium, 8ó010 of play followed the pooling outcome; by replication, it was 93q, ï8q and 88e1o.~ In the initial ten periods by replication, the pooling action was taken 30, 32, and 22 of 60 times respectively. As in Game 3, senders developed a successful strategy of distributing messages across types. Thus, we conclude that while message space size influences the equilibrium outcomes so does the degree of divergence between senders and receivers and rec~ivers' prior beliefs about sender types.

5 Conclusions

We investigated evolution of the meaning of inessages in two classes ot sender-receiver games, games of common and divergent interests. Evolu- 'oUaias the same ptocedures as in the other treatmenta, we failed to reject independenc.. [or the laat ten perioda for all replications. 23

tiunarv ~ame theory was used to develop the predictions for the treatments conducted. ~~'e found considerable support for the theory in games of common interest. In the absence of other means to coordinate, the fact that we obsPrve convergence to an informative equilibrium itself suggests players best respond to population history, albeit with inertia. Using probit models to study the dynamic process, we find population history significant if and only if players ate informed about it. Players continue to use strategies they have used in the past, separation inertia. Past success in individual matches are more helpful in predicting behavior than past failures and more so for senders than receivers. However, neither are as important as information about population history. For the games with divergent interests, there is less support for theory. This is due in part to the fact that theory is not unanimous in its predictions in these games. Consistent with theory, we find support fot the prediction that message space size influences the equi6brium outcome. How- ever, the degree of divergence of interest between the sender and receiver and receivers' prior beliefs about sender types significantly also influence the equilibrium selected. We observe both types of equilibrium outcomes, separation and pooling, in both the two and the three message treatment. The nature of the poo6ng outcomes differs across treatments. In the three message treatment, senders adopt the strategy of all choosing the same message, which is consistent with theory. For two messages, senders successfully distribute messages across tvpes to prevent identification. Theory does not predict pooling here and makes no prediction about the message strategies used. For the separating outcomes, the dynamics are similar to those of the common interest games. Population history and separation inertia are important; however, previous failure is not and success is not as important. [nitial conditions play an important role in both types of games. We tind it surptising that in the 5ession [ game, the "focal equilibrium" whete tt ;ends A was observed in only about 50q of the replications. This makes tiession 1 interesting in its own right. In games with separating outcomes, the initial coordination precedent is decisive in 23 out of 24 replications, excluding the no-history treatment. For the pooling outcomes in the three message treatment, the precedent for the commonly used message was es- tablished in the initial periods. Finally, for the divergent intetest games, the initial frequency of the pooling action impacts the outcome, pooGng or ,eparation. 24

The analysis and results suggest additional issues to address. titanv games of applied and theoretical interest are games of partia! common interest; the sender wants to reveal some but not all information. Thus, these games combine aspects of common and divergent interest games. The theories considered here make predictions for such games. [n view of our results, it is an open question whether these predictions hold up under experimental scrutiny. We would also Gke to investigate the dynamics in games of partial interest, given the new type of equilibrium outcome in these games and our observation that dynamics appear to be outcome dependent. Currently, sender (but not receiver) history is displayed to all players. What happens if we inform some or all of the players about the partial or complete history of play? Varying this design feature should provide insigát about how human learning differs from biological evolution. Finally, we plan to examine different matching rules, e.g. one where individual sender- receiver pairs remain matched for the entire series of repetitions, Crawford and Haller [1990], Kramarz [1992J. zs

References

(1] .aumann. R.J. [19ï~1], "Subjectivity and Correlation in Randomized Strategies." Journal of 1(athematical Economics, 1, 67-96.

[2] .aumann, R.J. [1989]. Lectures on Game Theory, Boulder: Westview Press.

[3] .austen-Smith, D. (1990], "Information Transmission in Debate," .4mer- ican Journal of Politica! Science. 34, 124-152.

[4] Basu. IC. and J.W. Weibull [1991], "Strategy Subsets Closed under Rational Behavior." Economics Letters, 38, 141-146.

[.i] Berg, J. E.. L. Daley, J. Dickhaut and J. 0'Brien (1986], "Controlling Preferences for Lotteries on Units of Experimental Exchange," Quar- terl}- Journal of Economics, 101, 281-306.

[6] Binmore, E~.G. and L. Samuelson [1993], ":Vluddling Through: Noisy Equilibrium Selection," University College London and University of Wisconsin.

[ï] Blume, A. [1993], "Equilibrium Refinements in Sender-Receiver Games," forthcoming in Journal of Economic Theory.

(8] Blume, A. [1994], "Neighborhood Stability in Sender-Receiver Games," forthcoming in Games and Economic Behavior.

(9] Blume, A., Y.-G. Kim and J. Sobel [1993], '`Evolutionary Stability in Games of Communication." Games and Economic Behavior, 5, 547-575.

[10] Brandts, J. and C.A. Holt [1992], "An Experimental Test of Equilib- rium Dominance in Signaling Games," American Economic Review, 82, 13~i0-1'36~.

[ I I ~ Canning, D. [ 1992], "Learning Language ('onventions in Common In- tPrest SignaGng Games," ('olumbia University.

[1'?] Cooper, R., D.V. DeJong, R. Forsythe and T.W. Ross [1989], "Commu- nication in the Battle of the Sexes Game," Rand Journal of Economics, 20. 568-í8ï.

[13] Coulmas, F. (1992], Language and Economy, Oxford: Blackwell. 26

[I 1] Crawford. V. [1991], "An 'Evolutionary' Interpretation o( Van Huvck. BattaGo, and Beil's Experimental Results on Coordination," Games and Economic Behavior. 3, 25-59.

[15] Crawford, V. (1993] "Adaptive Dynamics in Coordination Games." forthcoming in Econometrica.

[16] Crawford, V. and H. Haller [1990], "LearninR how to Cooperate: Op- timal Play in Repeated Coordination Games," Eronometrica, 58, :i81- 596.

[17] Ctawford, V. and J. Sobel [1982], "Strategic [nformation Transmis- sion," Econometrica, 50. 1431-1452.

[18] Darrough, M. and N. Stoughton [1990], "Financial Disclosure PoGcy in an Entry Game," Journal of Accounting and Economics, 12, 219-243.

[l9] Dickhaut, J., K. McCabe and A. Mukherji [1992), ".An ExperimPntal Study of Strategic [nformation Transmission," University of Minnesota.

[20] Farrell, J. [1993], "Meaning and Credibility in Cheap-Talk Games," Games and Ecoaomic Behavior, 5, 514-531.

[2l] Farrell, J. and R. Gibbons [1990], "Union Voice," Cornell University Working Paper.

[22] Forges, F. [1990a], "Universal Mechanisms," Econometrica. 58, 1341- 1364.

[23] Forges, F. [1990b], " Equilibria with Communication in a Job Market Example," Quarterly Journal of Economics, 105, 375-398.

[24] Foster, D. and H. P. Young [1990]. "Stochastic Evolutionary Game Dynamics," Theoretical Population Biology, 38. 219-232.

(25] Green, J. and N. Stokey [1980], ",1 Two-Person Game of [nformation Transmission," Harvard University Discussion Paper, ~larch.

(26] Hume, D. [1964], "On the Origin of Justice and Property," in A Treatise olHuman Nature, Book 3, 258-273 (reprint of second edition). Aalen: Scientia Verlag. 27

['?ïJ Hurkens. S. [1994]. " Learning by Forgetful Players: From Primitive Formations to Persistent Retracts," CentER for Economic Research Di,cus;ion Paper. Tilburg [?niversity

[18] Kalai. E. and U. Samet [1984], "Persistent Equilibria in 5trategic Games,'~ International Journal of Came Theory, 13, l`29-1~44.

[29] Kandori. ~L. G. blailath and R. Rob [1993], "Learning, Mutation, and Long-Run Equilibria in Games," Econometrica, 81, Y9-56.

[30] Kim. Y.-G. and J. Sobel [1992], "An Evolutionary Approach to Pre- Play Communication," University of Iowa Working Paper.

[:31] Kramarz. F. [1992], "When Repeated Cheap Talk Generates a Common Language," INSEE.

[32] Lewis, D.K. [1969], Convention: A Philosophical Study, Cambridge, ~IA: Harvard Cniversitv Press.

[33] Mailath, G. [1991], "lntroduction: Symposium on Evolutionary Game Theory," Journal o( Economic Theory, 57, 259-277.

[34] titatthews, S., M. Okuno-Fujiwara and A. Postlewaite [1991], "Refining Cheap-Talk Equilibria," Journal of Economic Theory, 55, 24ï-273.

[35] ylaynard Smith. J. [1982J, Evolution and the Theory of Cames, Cam- bridge: Cambridge University Press.

[36] ~fyerson, R. [1989], "Credible Negotiation Statements and Coherent Plans," Journal of Economic Theory, 48, 264-291.

[3i] '.~lbldeke, G. and L. Samuelson [1992], "The Evolutionary Foundations of Backward and Forward Induction," [Jniversitv of Wisconsin.

[S8] Roth. A.E. and I. Erev [1993], "Learning in Extensive Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term," CTniversity of Pittsburgh Working Paper.

[:39] Roth. A. and bí. W. K. 3ialout [1979], "Game-Theoretic Models and the Role of Infortnation in Bargaining," Psychological Review, 88, 57~t-594.

(-10] Schelling, T.C. [1960], The Strategy of Conflict, Cambridge: Harvard L niversity Press. 28

[41] Smith. A. [1983], "Considerations Concerning the First Formation of Languages," in J.C. Bryce (ed.), Lectures on Rhetoric and Belles Let- tres, Oxford: Oxford University Press.

(-12] Stein, J. [1989], "Cheap Talk and the Fed: A Theory of Imprecise PoGcy Announcements," American Economic Review, 79, 32-4'2.

(43] Swinkels, J.M. [1992], "Evolutionary Stability with Equilibrium En- trants," Journal of Economic Theory, 57, 306-332.

[44] W~tneryd, K. (1993], "Cheap Talk, Coordination, and Evolutionary Stability,~ Cames and Economic Behavior, 5, 532-546.

[45] Whinnom, K. [1971]. "Linguistic Hybridization and the 'Special Case' of Pidgins and Creoles," in Pidginization and Creolization of Languages, (D. Hymes, ed.), Cambridge: Cambridge University Press.

[46] Young, H. P. [1993], "The Evolution of Conventions," Econometrica, 81, 57-84. 29

Fiqur~ ~

Coordination By Period~ 8~ssion i

~: Coordination is defined for both senders and receivers for each period in terms of the final ( last period) equilibrium outcome for the 15 replications. 30

aiqur~ S

Coordination By Period~ 3aa~ i with and Mithout Yopulation History

~: Coordination is defined for both senders and receivers for period in terms of the final each (last period) equilibrium outcome for the 3 replications of each treatment. 31

Biqur~ 6

Coordination By Period~ (iam~ 2

. - o - - senders - a - reCelvers o -~-e '-~-e-e-e-e-e-~-e .o .1 ` ~o ~ J~. A ~~--r-ti ,o-a ~ .e -~ -e d

- a- receivers choosing action 3

2 4 6 8 !0 12 14 !6 18 20 Period

~: Coordination is defined for both senders and receivers for each period in terms of the final (last period) equilibrium outcome for the 3 replications. 32

aiqur~ 7

Coordination By Period~ aas~ 3 Rith 2 l[~ssaq~s -- Roplications i and 2

~: Coordination is defined for both senders and receivers for each period in terms of the final (last period) equilibrium outcome for each replication. 33

Fiqur~ 8

Coordination By Period~ Qams 3 witb 2![asaaqss -- Roplication 3

~: The frequencies in the top qraph represent the percentaqe of receivers choosinq actions (1 and 2) or 3. In the bottom qraph, sender coordination is defined in terms of an arbitrary type- messaqe mapping, (t~,m~). Had we used the other mapping, (t~,mZ), we would obtain the mirror image of the bottom qraph. 34

liqur~ 9

Coordination By Period~ ciaaa 3 with 3 I[assaqa' -- Rapliaatioa 1

- . e - - senoers - . - receivers ~~ ~ ~e , , ,, , , - o - receivers choosing ; ', ; ; ; ; action 3 ; ~ , ~ r ~, , , ~ e .á ~ ; ~ . .~.1~ ~' I 1 11 .~ ~ i.~ e~

.6 ~

.4~

.2 ~

o- T- 4 8 12 16 20 24 28 32 36 40 Period

~: Coordination is defined for both senders and receivers for each period in terms of the final (last period) equilibrium outcome. 35

Fiqure 10

Coordination Sy Period~ Game 3 Nith 3 xessaqes -- Replicatioas 2 i 3

~: The frequencies represent the percentaqe of senders choosinq a single messaqe and the percentage of receivers choosinq actions (1 and 2) or action 3. 36

N h O

4~. ~? -M ~1 .T M ~ ~

á ó -. ê ti ~NO, ~ R M Cp V io a ~- M N N m N ~. ~ `..~ M v eT v Q v V L ~ N i, a N ' Q~ ~ d ~ V y ~ ~ ~ C V ~}' K; N ~D N c0 .-N-~ M uJ f~ O T `r :~l Y ~y„ w ~3 ~ v M ~ ~ ~ w W `d C~ ~ G O~iO ~ G ' ~ ,~ `n N ~ 00 ~ G! QJ N~ N ~ ~D p ~ O ~ ~ Q ap Eo u ó E .- CO ~ Q M N ~ ~ , .-, o ~ ? ry.

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~ J y f- ~ N .V ~D ~

Oi ~ f- ~ :V M `7 N x .j .r N ~ ~ , ,. I -. I "

~ r c..7 ~ ~ O t- 00 ~ .~-, I r`! O N ~ ~ ~ ~

~ ,y ~ .'~V ~` ~ u7 T ~ M --~ ry ~O ~ c7 ~j ~ - ~ - r `- .-. -... ? .-. .ti .-- . .-. ry ,.-. o. T t- ~ M ~ N O O c0 ,") ,v ~Y .y C ~ M

VI .V :C ~ a -~ ti r- -- a . ? . pf ~- :~l "-' :V -- p r :V -`~ N N N ~ T N N :d I `. I .- I -,. I ~.. ~ `.. P .-...... -.. c ro ~ u G. 6. O O i ii cWi~ á a~. 00

z - i, ï ~ z ~ :J .. ,i U U C7 :7 C7 38

APPENDIX A: INSTRUCTIONS FOR TREATMENTS

INSTRUCTIONS FOR SESSION I (common to all treatmants) Ceneral You are about to participate in an experiment in the economics of decision making. If you follow these ínstructions carefully and make good decisions, you might earn a considerable amount of money that will be paid to.you in cash at the end of the experiment. The experíment will consíst of a series of separate decísion making periods. Each períod consists of two phases. In Phase I you w111 be paired wíth another person and, based upon the choíces you make, you will be able to earn points. In Phase II, you will have the opportunity to earn dollars based upon the points you earn in Phase I. We begin by describing Phase II so that you understand how the points you earn affect the number of dollazs you earn. Then, we describe Phase I ín detail so that you understand hw to earn points. Phase II Instructiona At the end of Phase I, you will have earned between 0 and 1000 points accordíng to the rules we will discuss below. The number of dollars you earn in Phase II will depend partly on the number of points you earned 1n Phase I and partly on chance. Specifically, we have a box which contains lottery tickets numbered 1 to 1000. In Phase II, a ticket will be randomly drawn from the box. If the number on the ticket IS LESS THAN OR EQUAL TO the number of points you hava earned in Phase I, you WIN 51.00. If the number on this tícket IS GREATER THAN the number of points you have earned in Phase I, you WIN 50.00. For example, if you have 600 points, you will have a 601 chance of winning 51.00. Notíce that the more points you have, the larger will be your chance of winning the ~1.00 prize.

Phase I Instructiona You will either be designated a Column player or a Row player at the start of the experiment. Once you are designated a Column player, you will remain a Column player from one peziod to the next. Similarly, once you are designated a Row player, you will remain a Row player. In your folder you will find a record and profít sheet. On this sheet you wíll indicate, based on the message received on your terminal, whether you are a Row or Column player. You will be participating in a series of separate sessions during today's experiment. The current session vill consist of 20 periods. In each decision making petiod, you will be paired wíth another person. Each period, the computer randomly determines which player you are paired with. If you are a Column player you will be randomly paired with a Row player. If you are a Row player, you will be randomly paired with a Column player. For example, if there are 12 people participating ín the sessíon, 6 will be Column players and 6 wlll be Row players. In any period, there is a 17i chance a particular column player will be palred with a particular row player. There is a 31 chance these two players will be paired together a second period, a.5s chance they wiil be paired for three periods and so forth. You will not know the identification of the person you are playing agaínst in any period. Similarly, nobody in your decision making pair will know your identification in any period. Further, you will not be told who these people are 39

either during or after the experiment. Each period, the game wi11 proceed as follows: Flrst, the Column player will Learn which one of two payoff tables is being used this period. Next, the Column player can send a message to the Row player. Fínally, the Row player selects an action which determines the points earned for both the Column player and the Row player for the period. More specifically, at the beginning of each period, the payoff table is randomly chosen by the computer. Each table is equally likely and as such has a 1 in 2(508) chance of occurring each period. Also, the payoff table for each pair of players is determined by a separate, independent drawing. Since each table is ídentified by a label-- CL or C2, the computer wíll inform the Column player of [he table drawn for the period. The camputer will not inform the Row player of the table drawn. Next, the Column player will be given the opportunity to send a message to the Row player about the table that was drawn. The Column player may choose to send a message of A or B. After the Column player sends a message, the Row player will choose an ac[ion. The Row player's action and the payoff table determined by the table drawn at the beginning of the period determines the number of points earned by the Row player and the Column player. In the next part of the instructions we will be referring to specific numbers of points. These numbers are the same as you will be usíng !n the first session of today's experiment.

Specific Instructíons to the Columi Player As a Column player, you will first be told the table drawn for the períod. If for i nstance the C1 table was drawn, you would receive the following message:

The true table is ~-

Record this information on your record and profit sheet. The Row player will not be ínformed of the payoff table drawn. Nessage Stage After the computer determines the true table, you will be gíven an opportunity to send a message. You will be asked: Vhat vould you like to send9 A B

If you choose A, hit the A key. If you choose B, hit the B key. You may not choose both A and B. After deciding which message to send, you will again be nsked co verífy your response. Actton Stage After sending your message to the Row player, you vill enter the action stage of Phase I. In this stage, the Row player will choose an action which will determine the poínts you earn and the points the Row player earns for the period. The Row player wíll choose action R1 or action R2. The Row player's action choice will be dísplayed on your terminal by highlíghting the appropriate row on the true table. The number of points you may earn is given by the folloving tables for each action the Row player might select and each possible payoff table. (Note that these tables appear with the appropriate label on your screen.) 40

Numbar of Points aarnad by tha Column Player

501 C1 S08 C2

Rov player's R1 0 R1 700 action R2 700 R2 0

To read these tables, notice that the likelíhood of a given table occurring appears in the upper left hand corner of each table. Since each table is equally likely, each has a 1 ln 2 chance (508) of occurring. Next, suppose the payoff table drawn was C1 and the Row player chose actíon R2. You would then earn 700 points. Similarly, suppose the payoff table drawn was C2 and the Row player chose action R2. You would then earn 0 points. In a líke manner, you can use these tables to determíne the number of points you would earn for all other The Row combínations of the Row player's actions and payoff tables drawn. players also earn poínts dependíng upon the payoff table drawn and the action they select. Once the Row player has selected his or her action and entered it into the computer, the computer will determine the number of points earned by you based on the table that was drawn. The result is then sent to you via your terminal. The message will look like the one below:

You raceiwd - points.

At the end of the period, you are to record your point earnings for Phase I on your record and profit sheet. The computer will also inform you about the action the taken by the Row player and the payoff table drawn at the beginning of period. You should record the payoff table drawn, the message sent to the Aow player and the action choice of the Row player on your record and pzofit sheet. Make aure you check your earnings in points against the computer's calculations.

Spscific Instructions for tha Rov Player As a Row player, you will begin by receiving a message from the Column player. The Column player can send a meesage of either A or B but not both. For inatance, if the Column playar chose to send B, you would receive the message: The columm playar sent: B

Aftar receiving this message, you will choose an action. You may choose action R1 or action R2. The number of points you earn is given by the following tables for each action you might select and each possible payoff table. (Note that these tablea appear with the appropriate label on your screen.)

Numbar of Points aarnad by the Row Player SOt C1 508 C2

Rov playar'a R1 0 A1 700 action R2 700 R2 0 41

To read these tables, notice that the likelihood of a gíven table occurríng appears in the upper left hand corner of each table. Since each table is equally likely, each has a 1 in 2 chance (508) of occurring. Next, suppose the payoff table drawn was C1 and you chose action R2. You would then earn 700 points. Símilarly, suppose the payoff table drawn was C2 and you chose action R2. You would Chen earn 0 points. In a like manner, you can use these tables to determine the number of points you would earn for a11 other combinations of your actions and payoff tables drawn. The Column players also earn points dependíng upon the payoff table drawn and the action you select. When it is time for you to select an action, you will receive a message stating: You may choose 1-2. Your decision9

Enter the number of the row (1 or 2) you wish to choose. A light bar wíll highlight your choice on the screen. You will then be given an opportunity to verify your response. Once you have selected an action and entered it into the computer, the computer will determine the number of points earned by you based on the table that was drawn. The result is then sent to you via your terminal. The message wíll Look like the one below:

You recaived - points.

At the end of the period, you are to record your point earnings foz Phase I on your record and profit sheet. The computer will also inform you about the payoff table drawn at the beginníng of the period. You should record the table drawn, the message sent by the Column player and your action choice on your record and profit sheet. Make sure you check your earnings i n points against the computer's calculations.

History Ac the end of each period, a history ínformation screen will display the payoff tables drawn and messages sent by all the Column players. A history sheet wíll be provided-- on this sheet you are to record the frequencies for each period. For instance, suppose that for period 1 in the session, 3 of the Column players drew table C1 and 3 drew table C2. AL1 6 of the Column players sent B and consequently, all 6 of the Row players received B. A Column player's hístoty screen will show

Table C1 C2 Hessage A B A B Period 1 3 3

A Row player's history screen will show

Table C1 C2 Message A B A B 42

Period 1 3 3

Phase II Recording Rulas After completing your record and profit sheet for Phase I of a given decision making period, you are to use the last two columns to record the dollars you earn in Phase II. One person will be asked to draw a lottery ticket from the box. Before he~she returns the ticket to the box, the number on the ticket will be announced. You should record the number of the ticket in the second to the last column of your record and profit sheet. If the number drawn IS LESS THAN OR EQUAL TO the number of points earned in Phase I, círcle 51.00 ín the last column; otherwise circle 50.00 in that column. Pay careful attention to what you circle. Any erasure will invalidate your earnings for the period. If you do make a mistake and circle the wrong number, call it to the experimenter's attention. To help you with your record keeping, the computer will summazize your point earnings and lottery outcome at the end of each period. Use thís to verify the information you recorded on your record and profit sheet. At the end of the session, add up your total profit in dollars. All dollars on hand at the end of the session in excess of S 0.00 are yours to keep. This is [he amount of dollars you have earned in this session. In summary, your earnings in tha experíment will be the total of the amounts you win in all Phase II lotteries. The amount of money you earn will depend partly upon luck and partly upon whether you have made good decisions in Phase I. Notice that the more points you earn in Phase I, the more likely you wíll win in Phase II. Are there any questions?

Questions

1. In each period, how does the computer pair a Row player with a Column player?

2. What is the chance that you will be paired with a particular player for

a. one period?

b. two periods7

c. three periods7

3. In each period, does the computer determine the payoff table for each column player separately or do all column players have the same payoff table? 4. In each pariod, how does the computer determine the payoff table for each column player?

5. For each column player, what is the chance in each period thet the payoff table is

a. C1? b. C2? 43

6. 4Then the computer determines the payoff table for a column player,

a. does the column player know which table was chosen? b. does the row player know which table was chosen?

7. When the column player sends the message A, what message will be received by the row player?

8. When the column player sends the message B, what message will be received by che row player?

9. Suppose for period 1 in the session, 3 of the column players drew C1 and 3 of the column players drew C2. All 6 of the column player send A.

a. what does the column player see on his`her history i nformation screen

Table C1 C2 Message A B A B Period

b. what does the row player see on hís`her information screen

Table C1 C2 Message A B A B Períod

1 44

INSTRUCTIONS FOR SESSION II Came 1

[The alternative order of inessage space presentatíon is in brackets]

This session of the experiment will again consist of a series of separate decision making periods. The current sessíon wíll consist of 20 periods. Each period will again consist of two phasea. In Phase II you will be able to earn dollars based upon the poínts you earned i n Phase I in exactly the same way you díd in the first session. In Phase I, you will again be paired wíth another person and, based upon the choices you make, you will be able to earn points. However, Phase I in this session differs from Phase I in the previous session in the following way:

Specific Instructions for tha Columa Player Nessage Stage After the computer determinas the true table, you will be given opportunity to a an send message, e or ~ ( ~ or aJ. When you send a massage, a particular row player will always receive that message i n the same way. To select the message to send to the Row player, you will be asked: fihat vould you lika to sandT e N (,~ ,~~ If you choose e, hold down the shift kéy and then hít the a key. If you choose u, hold down the shift key and then h1t the p key. You may not choose both ~ and o. After deciding which message to send, you will again be asked to response. verify your Phase I than proceeds i n exactly the same way as in the previous session. Specific Instructions for the Rov Playez As a Row player, you will begin by receiving a message from the Column player. A partícular message from a Column player will always be received by you ín the same vay. You will receive the message: The colvmit player sent: e oz The column player snnt: K After receiving this message, you wíll choose an action. Phase I then proceeds i n exactly the same way ae in the previoua session. Histozy At the end of each period, a history i nformation screen will display the payoff tables dravn and messagea sent by ell the Column players. The history screen can be usad ín the same way as the previoua aeasion's history acreen. The hístory of the messages sent by the Column players are presented to you in your own representation of the messages. Thus, hístory inforos you of the frequency with which Column playera who drew payoff tables C1 and C2 send messages ~ and a. A hlstory sheet wíll be provided-- on this sheet you are to record the frequencies for each period. For instance, suppose for period 1 in the session, 3 of the Column players drew table C1 and 3 drew table C2. The messages sent by the Column players terms of your own representation in of the messages show that all 6 of the players sent X. Column 45

If you are a Column player, your history screen will show

Table CL C2 Message ~(~sJ x(~] ~[u] ~[~] Period

1 3 3

If you are a Row player, your history screen will show

Table C1 C2 Message ~[a] ~(~1 ~(a) ~(~l Period

1 3 3

All dollars on hand at the end of the session i n excess of S 5.00 are youzs to keep.

Questions

1. When the column player sends a particular message which i s represented as a ~ to the row player, what message will be received by the row playert

2. When the column player sends a partícular message which is represented as a n to the row player, what message wíll be received by the row player?

3. Suppose for period 1 in the session, 3 of the column players drew C1 and 3 of the column players drew C2. The messages sent by the column players in terms of a player's representation of the messages show that all 6 of the column players send ~.

a. if the player í s a column player, what does he`she see on his`her histozy informacion screen

Table C1 C2 Message ~[u] t~[~] ~(~1 e[~1 Period 1

b. íf the player i s a row player, what does he`she see on his`her ínformatíon screen

Table C1 CZ Message ~(u] s[~1 e(~1 s[~1 Period i 46

INSTRUCTIONS FOR SESSION II Game 2 and Game 3 Tfessage Space (e ~v)

[The alternative order of inessage space presentation i s in brackets] (For Game 3 replace Game 2 payoffs with those for Game 3]

This sessíon of the experíment will agaín consist of a series of separate decisíon making periods. The current seasion will consist of 20 periods. Each period will again consist of two phases. In Phase II you will be able to earn dollars based upon the points you earned in Phase I in exactly the same way you did in the first session. In Phase I, you will again be paired with another person and, based upon the choices you make, you will be able to earn points. However, Phase I in this session differs from Phase I i n the previous session in the following way:

Specifíc Instructions for the Column Player Iiessage Stage After the computer determines the true table, you will be given an opportunity to send a message, u or e. When you send a particular message, a row player will always receive that message in the same way. To select the message to send to the Row player, you will be asked:

What vould you like to send? ~ e ~,s ,w] If you choose ~, hold down the shift key and then hit the ~ key. If you choose t~, hold down the shift key and then hit the t~ key. You may not choose both ~ and a. After decidíng which message to send, you will again be asked to verify your response. Acrion Srage After sending your message to the Row player, you will enter the action stage of Phase I. In this stage, the Row player will choose an action which wíll determine the points you earn and the points the Row player earns for the period. The Row player will have three actions to choose from, R1, R2 or R3. The number of points you may earn ís gíven by the following tables for each action the Row player might select and each possible payoff table. (Note that these tables appear with the appropriate label on your screen.)

Numbar of Points earnad by the Column Player sot cl soa cs

Rov player's R1 0 R1 700 actioa R2 700 R2 0

R3 400 R3 400

These tables are constructed in the same manner as the tables in Sesaion I. Notice ehat the likelihood of a given table occurring appears in the upper left hand corner of each table. Sínce each table is equally likely, each has a 1 in 47

2 chance (508) of occurring. Next, suppose the payoff table drawn was C1 and the Row player chose action R2. You would then earn 700 points. Similarly, suppose the payoff table drawn was C2 and the Row player chose action R2. You would then earn 0 points. In a like manner, you can use these tables to determine the number of points you would earn for all other combinations of the Row player's actions and payoff tables drawn. The Row players also earn points depending upon the payoff table drawn and the action they select. Phase I then proceeds ín exactly the same way as in the previous sessíon.

Specific Instructions for the Row Player As a Row player, you will begin by receiving a message from the Column player. A parcicular message from a Column player will always be receíved by you in the same way. You will receive the message: The column player sent: ~ or The column player sent: e After receiving thís message, you will choose an action. You may choose action Rl, R2 or action R3. The number of points you earn is given by the following tables for each action you might select and each possible payoff table. (Note that these tables appear with the appropriate label on your screen.)

Number of Points earned by tha Rov Playez

SOi C1 SOá C2

Row player's R1 0 R1 700 action R2 700 R2 0

R3 400 R3 400

These tables are constructed in the same manner as the tables in Session I. Notice that the Likelihood of a gíven table occurring appears in the upper left hand corner of each table. Sínce each table is equally likely, each has a 1 in 2 chance (508) of occurríng. Next, suppose the payoff table drawn was C1 and you chose action R2. You would then earn 700 points. Similarly, suppose the payoff table drawn was C2 and you chose action R2. You would then earn 0 points. In a like manner, you can use these tables to determine the number of poínts you would earn for all other combinations of your actíons and payoff tables drawn. The Column players also earn poínts depending upon the payoff table drawn and the action you select. Phase I then proceeds in exactly the same way as in the previous session.

Nistory At the end of each period, a hístory information screen wíll display the payoff tables drawn and messages sent by all the Column players. The history screen can be used in the same way as the previous session's history screen. The history of the messages sent by the Column players are presented to you in your own representation of the messages. Thus, history informs you of the frequency with whích Column players who drew payoff tables C1 and C2 send messages ~ and x. A history sheet will be províded-- on this sheet you are to 48

record the frequencies for each period. For instance, suppose that for period 1 in the session, 3 of the Column players drew table C1 and 3 drew table C2. The messages sent by the column players i n terms of your own representation of the messages show that all 6 of the Column players sent n.

If you are a Column player, your history screen will show

Table C1 C2 Message ~[a] n[~] ~t(~] u(~] Period 1 3 3

If you are a Row player, your history screen will show

Table C1 C2 Message ~[s~l ~[~1 ~[~1 ~[~l Period 1 3 3

All dollars on hand at the end of the session i n excess of S 5.00 aze yours to keep.

Questiona

Same as Session II Game 1 Questions 49

INSTRUCTIONS FOR SESSION II Came 3 Nessage Space (t M 8)

[The alternative orders of inessage space presentation are in brackets]

This session of the experiment will again consist of a series of separate decísíon makíng periods. The current session will consist of 20 periods. Each period will again consist of two phases. In Phase II you will be able to earn dollars based upon the points you earned in Phase I in exactly the same way you did in the first session. In Phase I, you will again be paired with another person and, based upon the choíces you make, you will be able to earn points. Nowever, Phase I ín this sessian differs from Phase I in the previous session in the following way:

Specific Instructions for the Column Player Nessage Stage After the computer determines the true table, you will be given an opportunity to send a message, a, 8 or ~. When you send a particular message, a row player will always receive that message in the same way. To select the message to send to the Row player, you will be asked:

Vhat would you like to sendT ~r k t [K 8 ~] [8 e ~1 If you choose ~, hold down the shíft key and then hit the a key. If you choose x, hold down the shift key and then hit the a key. If you choose 8, hold down the shift key and then hit the 8 key. You may not símultaneously choose any two you will again be or all three messages. Af[er deciding which message to send, asked to verify your response. Actton Stage After sending your message to the Row player, you will enter the action stage of Phase I. In this stage, the Row player will choose an actíon whlch will period. determine the points you earn and the points the Row player earns for the The Row player wíll have three actions to choose from, R1, R2 or R3. The number action the Row of points you may earn is gíven by the following tables for each player might select and each possible payoff table. (Note that these tables appear with the appropriate label on your screen.)

Number of Points earned by the Column Player

508 C1 508 C2

Rov player'a R1 0 R1 200 action R2 200 R2 0

R3 400 R3 400 50

These tables are constructed in the same manner as the tables in Session I. Notice that the Likelihood of a given table occurring appears in the upper left hand corner of each table. Since each table is equally likely, each has a 1 ín 2 chance (508) of occurring. Next, suppose the payoff table drawn was C1 and the Rov player chose action R2. You would then earn 200 points. Similarly, suppose the payoff table drawn was C2 and the Row player chose action R2. You would then earn 0 points. In a like manner, you can use these tables to determine the number of points you would earn for all other combínations of the Row player's actions and payoff tables drawn. The Row players also earn points depending upon the payoff table drawn and the action they select. Phase I then proceeds in exactly the same way as in the previous session.

Specific Instructions for the Rov Player As a Row player, you will begin by receiving a message from the Column player. A particular message from a Column player will always be received by you in the same way. You will receive the message: Ths colvmn player sent: e or The column playnr sent: e or The eolumn player sent: t After receiving this message, you will choose an action. You may choose actíon R1, R2 or action R3. The number of points you earn is given by the following tables for each action you might select and each possible payoff table. (Note that these tables appear with the appropriate label on your sczeen.)

Number of Points earnsd by the Rov Player

S08 C1 S0~ C2

Row player's R1 0 R1 700 action R2 700 R2 0

R3 400 R3 400

These tables are constructed in the same manner as the tables in Session I. Notice that the likelihood of a given table occurring appears ín the upper left hand corner of each table. Since each table is equally líkely, each has a 1 ín 2 chance (508) of occurring. Next, suppose the payoff table drawn vas C1 and you chose actíon R2. You would then earn 700 points. Similarly, suppose the payoff table drawn was C2 and you chose action R2. You would then earn 0 points. In a like manner, you can use these tables to determine the number of points you would earn for all other combinations of your actions and payoff tables drawn. The Colwen players also earn points depending upon the payoff table drawn and the action you select. Phase I then proceeds in exactly the same way as in the previous session.

Nistory At the end of each period, a history information screen will display the payoff tables drawn and messages sent by all the Column players. The history 51

screen can be used in the same way as the previous session's hístory screen. The hístory of the messages sent by the Column players are presented to you in your own representation of the messages. Thus, hístory informs you of the frequency with which Column players who drew payoff tables C1 and C2 send messages ~, ~ and 8. A history sheet will be provided-- on this sheet you are to record the frequencíes for each perïod. For instance, suppose that for period 1 in the session, 3 of the Column players drew cable C1 and 3 drew table C2. The messages sent by the column players in terms of your own representation of the messages show that all 6 of the Column players sent ~.

If you are a Column player, your history screen will show [a 8~] (8 ~~]

Table C1 CZ Message t ~ 8 ~ ~ 8 Period 1 3 3

show (a 8~] [8 If you are a Row player, your hístory screen will

Table C1 Cz t Message ~ ~ 8 ~ 8 Period 1 3 3

All dollars on hand at the end of the session in excess of S 0.00 are yours to keep.

Questions

message which is represented as ~ 1. When the column player sends a particular by the row player? to the row player, what message will be received which is represented as ~ 2. When the column player sends a particular message row playerl to the row player, what message will be received by the is represented as 8 3. When the column player sends a particular message which the row player? to the row player, vhat message will be received by the column players drew C1 and 3 4. Suppose for period 1 in the session, 3 of sent by the column players in of the column players drew C2. The messages messages show that all 6 of the terms of a player's representation of the colwnn players send ~. what does he`she see on his`her a. if the player is a column player, history ínformation screen [~ 8 ~J(8 ~ ~J 52

Table C1 C2 Hessage ~ ~ g ~ 8 Period 1

b. if the player is a row player, what does he`she see on his`her information screen [r t~J(s e e]

Table C1 C2 Message ~ ~ S ~ ~ B Períod 1

5. Suppose for period 1 in the aession, 3 of the column players drew C1 and 3 of the column players drew C2. The messages sent by the column players in terms of a player's representation of the messages show that all 6 of the column players send 8.

a. if the player i s a column player, what does he`she see on his`her history information screen [a 8~J[8 ~ rJ

Table C1 C2 Message ~ a E ~ e e Period 1

b. if the playar is a rov playsr, what doea he`she see on hís`her information screen [a ~~][t e wJ

Table C1 C2 Message ~ a e ~ 8 Period 1 Discussion Paper Series, CentF,R, Tilburg University, The Netherlands:

(For previous papers please consult previous discussion papers.)

No. Author(s) Title

9409 P. Smit Arnoldi TyDe Methods for Eigenvalue Calculation: Theory and Experíments

9410 J. Eichberger and Non-additive Beliefs and Game Theory D. Kelsey

9411 N. Dagan, R. Serrano A Non-cooperative View of Consistent Bankruptcy Rules and O. Volij

9412 H. Bester and Coupons and Oligopolistic Price Discrimination E. Petrakis

9413 Q Koop, J. Osiewalski Bayesian Efficiency Analysis with a Flexible Form: The AIM and M.F.J. Steel Cost Function

9414 C. Kilby World Bank-Borrower Relations and Project Supervision

9415 H. Bester A Bargaining Model of Financial Intermediation

9416 J.J.G. Lemmen and The Price Approach to Financial Integration: Decomposing S.C.W. I?ijllingcr Europcan Money Market Interesi Rate Differentials

9417 J. de la I lurra and Sensitivity to Prior Independence via Farlie-Gumbel C Fernander. -Morgenstern Model

9418 D. Talman and Z. Yang A Simplicial Algorithm for Computing Proper Nash Equilibria of Finite Games

9419 H.J. Bierens Nonparametric Cointegration Tests

9420 G. van dcr Laan, Intersection Theorems on Polytopes D. 'Falman and Z. Yang

9421 R. van den Brink and Ranking the Nodes in Directed and Weighted Directed Graphs R.I'. tiillcs

94" A.~~an tiuca Youlh Minimum Wage Rates: The Dutch Experience

94?) N. U;i~:m and ll. Vulij I;ilatcr:d Cumparisuns and Cunsislent Fair Division Rules in the Contcxt of Bankruptcy Problems

9424 R. van dcn Brink and Digraph Competitions and Cooperative Games P. Rurm

9425 P.ILM. Ruys and The Interdependence between Production and Allocation R.P. Gilles

9426 T. Callan and Family Labour Supply and Taxes in Ireland A. van Soest No. Author(s) Title

9427 R.M.W.J. Beetsma Macroeconomic Stabilisation and Intervention Policy under an and F. van der Ploeg Exchange Rate Band

9428 J.P.C. Kleijnen and Two-stage versus Sequential Sample-size Determination in W. van Grcenendaal Regression Analysis of Simulation Experiments

9429 M. Pradhan and Household Labour Supply in Urban Areas of a Developing A. van Scest Country

9430 P.J.J. Herings Endogenously Determined Price Rigidities

9431 H.A. Keuzenkamp and On Tests and Significance in Econometrics ].R. Magnus

9432 C. Dang, D. Talman and A Homotopy Approach to the Computation of Economic Z. Wang Equilibria on the Unit Simplex

9433 R. van den Brink An Axiomatization of the Disjunctive Perrnission Value for Games with a Permission Structure

9434 C. Veld Warrant Pricing: A Review of Empirical Research

9435 V. Feltkamp, S. Tijs and Bird's Tree Allocations Revisited S. Muto

9436 G.-J. Otten, P. Borm, The MC-value for Monotonic NTU-Games B. Peleg and S. Tijs

9437 S. Hurkens Leaming by Forgetful Playets: From Primitive Formations to Persistent Retracts

9438 1.-J. Herings, D. Talman, The Computation of a Continuum of Constrained Equilibria and Z. Yang

9439 E. Schaling and D. Smyth The Effects of Inflation on Growth and Fluctuations in Dynamic Macroeconomic Models

9440 J. Arin and V. Feltkamp The Nucleolus and Kernel of Veto-rich Tnjnsferable Utility Games

9441 P.-J. Jost On the Role of Commitment in a Class of Signalling Problems

9442 J. Bendor, D. Mookherjee, Aspirations, Adaptive Learning and Cooperation in Repeated and D. Ray Games

9443 G. van der Laan, Modelling Cooperative Games in Pennutational Structure D. Talman and Z. Yang

9444 G.J. Almekinders and Accounting for Daily Bundesbank and Federal Reserve S.C. W. Eijffinger Intervention: A Friction Model with a GARCH Application

9445 A. De Waegenaere Equilibria in Incomplete Financial Markets with Portfolio Constraints and Transaction Costs No. Aulhor(s) Titlc

944G G Schaling and D. Smylh fhc El7ccls uf Inllatiun on Gruwlh and I luctualiuns in Dynamic Macroeconomic Models

9447 G. Koop, J. Osiewalski Hospital Efficiency Analysis Through Individual Effects: A and M.F.J. Steel Bayesian Approach

9448 H. Hamers, J. Suijs, The Split Core for Sequencing Games S. Tijs and P. Bonn

9449 G.-J. Otten, H. Peters, Two Characterizations of the Uniforrn Rule for Division and O. Volij Problems with Single-Peaked Preferences

9450 A.L. Bovcnbcrg and 'I~ransilional Impacts uf Environmental Policy in an Endogenous S.A. Smulders Growth Model

9451 F. Verboven Intemational Price Discrimination in the European Car Market: An Econometric Model of Oligopoly Behavior with Product Differentiation

9452 P.J.-J. Herings A Globally and Universally Stable Price Adjustment Process

9453 D. Diamantaras, A Note on the Decentralization of Pareto Op[ima in Economies R.P. Gilles and with Public Projects and Nonessential Private Goods S. Scotchmer

9454 1~. de Jong, I. Nijman Price Eftècts of Trading and Components of the Bid-ask Spread and A. R6ell on the Paris Bourse

9455 F. Vella and M. Verbeek Two-Step Estimation of Simultaneous Equation Panel Data Models with Censured Endogenous Variables

9456 H.A. Keuzenkamp and Simplicity, Scientific Inference and Econometric Modelling M. McAleer

9457 K. Chatterjee and Rubinstein Auctions: On Competition for Bargaining Partners B. Dut[a

9458 A. van den Nouwcland, Axiomatic Characterizations of the Walras Correspondence for R. Prlcp, and S. 'I ijs Gencralized Ecunumies

94i9 ' I. Icn Raa and I..N. W~~IIT Outsourcing uf Services and Productivity Growth in Goods Industries

9460 G.J. Almekindcrs A Pusitive Theory of Central Bank Intervention

9461 J.P. Choi Standardization and Experimentation: Ex Ante Versus Ex Post Standardization

946? .LP. Choi Ilerd Behavior, the "Penguin Effect", and the Suppression of Informational Diffusion: An Analysis of Informational Externalities and Payoff Interdependency

9463 R.I L Gurdun and Why is Capital so Immobile lntemationally?: Possible A.i.. Bovcnberg E~planations and Implications for Capital lncome Taxation No. Author(s) Title

9464 E. van Damme and Games with Imperfectly Observable Commitment S. Hurkens

9465 W. Guth and E. van Damme Information, Strategic Behavior and Fairness in Ultimatum Bargaining - An Experimental Study -

9466 S.C.W. Eijffinger and The Catching Up of European Money Markets: The Degree J.J.G. Lemmen Versus the Speed of Integration

9467 W.B. van den Hout and The Power-Series Algorithm for Markovian Queueing Networks J.P.C. Blanc

9468 H. Webers The Location Model with Two Periods of Price Competition

9469 P.W.J. De Bijl Delegation of Responsibility in Organizations

9470 T. van de Klundert and North-South Knowledge Spillovers and Competition. S. Smulders Convergence Versus Divergence

9471 A. Mountford Trade Dynamics and Endogenous Growth - An Overlapping Generations Model

9472 A. Mountford Growth, History and International Capital Flows

9473 L. Meijdam and Comparative Dynamics in Perfect-Foresight Models M. Verhoeven

9474 L. Meijdam and Constraints in Perfect-Foresight Models: The Case of Old-Age M. Verhceven Savings and Public Pension

9475 Z. Yang A Simplicial Algorithm for Testing the Integral Property of a Polytope

9476 H. Hamers, P. Bortn, The Chinese Postman and Delivery Games R. van de Leensel and S. Tijs

9477 R.M.W.J. Beetsma Servicing the Public Debt: Comment

9478 R.M.W.J. Beetsma InFlation Versus Taxation: Representative Democracy and Party Nominations

9479 J.-J. Herings and Intersection Theorems with a Continuum of Intersection Points D. Talman

9480 K. Aardal Capacitated Facility Location: Separation Algorithms and Computational Experience

9481 G.W.P. Charlier A Smoothed Maximum Score Estimator for the Binary Choice Panel Data Model with Individual Fixed Effects and Application to Labour Force Participation

9482 J. Bouckaert and Phonebanking H. Degryse No. Author(s) Title

9483 B. Allen, R. Deneckere, Capacity Precommitment as a Barrier to Entry: A Bertrand T. Faith and D. Kovenock -Edgeworth Approach

9484 J.-J. Herings, Equilibrium Adjustment of Disequilibrium Prices G. van der Laan. D. 7"ahnan, and R. Venniker

9485 V. Bhaskar Informational Constraints and the Overlapping Generations Model: Folk and Anti-Folk Theorems

9486 K. Aardal, M. Labbé, On the Two-level Uncapacitated Facility Location Problem J. Leung, and M. Queyranne

9487 W.B. van den Iloul and The Power-Series Algorithm for a Wide Class of Markov J.P.C. Blanc Processes

9488 F.C. Drost, Adaptive Estimation in Time-Series Models C.A.J. Klaassen and B.J.M. Werker

9489 Z. Yang A Simplicial Algorithm for Testing the Integral Property of Polytopes: A Revision

9490 H. Huizinga Real Exchange Rate Misalignment and Redistribution

9491 A. Blume, D.V. DeJong, Evolution of the Meaning of Messages in Sender-Receiver Y.-G. Kim, and Games: An Experiment G.B. Sprinkle P(~ R(~X ~n1~~ ,~Cl(1 I F TII Rl 1RC~ THF NFTHFRLAND~ Bibliotheek K. U. Brabant II~I~~NNIYIIIIYII~~II~III!II