Economics Letters 70 (2001) 357–362www.elsevier.com/ locate /econbase

Repetition and signalling: experimental evidence from gameswith efficient equilibria

a b ,*Kenneth Clark , Martin Seftona

School of Economic Studies, University of Manchester, Manchester, UKbDepartment of Economics, University of Newcastle, Newcastle-upon-Tyne, NE1 7RU, UK

Received 21 July 1999; accepted 16 December 1999

Abstract

We report experiments where subjects generally fail to attain the efficient equilibrium of a one-shot game, butattain the efficient equilibrium of the repeated version. The results suggest that in the repeated game actions areused to signal future intentions. 2001 Elsevier Science B.V. All rights reserved.

Keywords: Experimental games; Coordination; Repetition; Signaling

JEL classification: C92

1. Introduction

Many economic problems involve agents attempting to coordinate on an efficient equilibrium (see,e.g., Cooper, 1999). We report an experiment using two symmetric two-player games, each with anefficient equilibrium. Previous experiments suggest that players fail to coordinate on this equilibriumwhen the games are played as one-shot games. We replicate this finding, but demonstrate that whensubjects play repeatedly against the same opponent efficient play is more likely than in a sequence ofone-shot games.

Such a result might be explained by the differences in learning opportunities given by the twoprotocols. However, this explanation is incomplete. We also find that the increase in efficient play isapparent even in the first stage of the repeated game, and so a complete explanation must refer tosome amount of forward-looking behavior on the part of subjects. One such explanation is thatsubjects recognize the desirability of the efficient outcome and use the early stages of a repeated gameto signal to their opponent that they intend to play the associated strategy.

*Corresponding author. Tel.: 1 44-11-5846-6130; fax: 1 44-11-5951-4159.E-mail address: martin.sefton@nottingham.ac.uk (M. Sefton).

0165-1765/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S0165-1765( 00 )00381-5

358 K. Clark, M. Sefton / Economics Letters 70 (2001) 357 –362

Fig. 1. Payoff matrix for coordination game.

2. The stage games

2.1. The Coordination Game

Fig. 1 illustrates a coordination game in which coordinating on ‘Risky’ forms an efficientequilibrium. However, coordinating on ‘Safe’ forms another equilibrium, and there is also a mixedstrategy equilibrium.

In experiments where subjects play this game repeatedly against changing opponents Cooper et al.(1992) found that subjects converge on the inefficient equilibrium, a result that has been reproducedby others (for example Clark et al., in press). An explanation for this can be given in terms ofstrategic uncertainty. Players are uncertain about which strategy will be played and favor the safestrategy. Harsanyi and Selten (1988, p. 89) suggest that the efficient equilibrium may be more likelyto be attained with pre-play communication, and indeed Cooper et al. confirm this in experiments.

2.2. The Mechanism Game

The second game studied here is the ‘Mechanism Game’. This applies a mechanism (Abreu andMatsushima, 1992) to the Coordination Game in order to induce the efficient outcome as a uniquerationalizable outcome. The mechanism implements the efficient outcome by modifying the game intwo ways. First, players play the game in pieces. Instead of simultaneously making a single choice,each player simultaneously chooses a sequence of T choices. These sequences are then matched, andfor each pair of elements payoffs are as given in Fig. 1, divided by T. Second, a player who includes achoice of ‘Safe’ in his or her sequence has to pay a small fine, F, unless their opponent chooses ‘Safe’in an earlier piece of their sequence. When F . 800/T, the unique equilibrium strategies require

1sequences consisting of ‘Risky’, and give both players their maximum possible payoff.2In Sefton and Yavas (1996) subjects played 14 rounds of one-shot Mechanism Games , allowing

them to gain experience with the game without introducing repeated game effects. The effect ofexperience was not very marked, and the theoretical prediction fared poorly in later as well as earlierrounds. Subjects overwhelmingly did not play dominated strategies, but failed to drive the logic ofiterated dominance to its conclusion, supporting an argument made by Glazer and Rosenthal (1992).

As with the Coordination Game, an explanation for these results in terms of strategic uncertaintycan be offered. Subjects may believe their opponent will use only a limited number of applications ofiterated dominance, and against these beliefs may themselves only apply a limited number of

1In fact this is the only strategy surviving iterated elimination of strictly dominated strategies.2The comparable sessions of their experiment used F 5 160, two values of T, T 5 8 and 12, and the underlying game

featured the payoffs of Fig. 1 scaled up by a factor of 1.2.

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iterations. Differing beliefs across subjects would then induce a distribution of observed outcomes thatcontrasts with the equilibrium prediction.

3. Experimental design and procedures

Each of 160 subjects, recruited from the undergraduate population at the University of Manchester,took part in exactly one of eight 20-subject sessions. The sessions followed a 2 3 2 design, with twosessions conducted under each treatment combination. The Game treatment used either theCoordination Game of Fig. 1, or the Mechanism Game with T 5 10 and F 5 130. The Repetitiontreatment implemented either 10-fold repeated games or sequences of 10 one-shot games within a

3session.In all sessions 20 subjects were randomly assigned to one of two connecting rooms. The one-shot

sessions consisted of 10 rounds. In each round each subject played a game against a subject in theother room, with the pairings constructed so that in each round it was impossible for a subject to havereceived any information from their opponent either directly or indirectly. This design was used inorder to focus on how the relevant game is played as a sequence of one-shot games, rather than as arepeated game. Kamecke (1997) shows that this matching procedure efficiently preserves the strategicindependence of rounds. The repeated game sessions used exactly the same procedures as ourone-shot sessions except that subjects were matched with the same opponent for all 10 rounds. Clarkand Sefton (1999) contains the written instructions given to subjects and a full description of theprocedures used.

The repeated Coordination Game, like the stage game, has multiple equilibria and so the sameselection problem applies. The repeated Mechanism Game has a unique subgame perfect Nashequilibrium that consists of repeating the stage game equilibrium strategies, i.e., sequences consistingentirely of ‘Risky’ in each of the 10 rounds. Equilibrium arguments do not, therefore, give a clearbasis for predicting differences in outcomes across Repetition treatment. However, differentiallearning and signaling opportunities in one-shot and repeated games may have important effects onoutcomes.

Standard learning models, applied to a repeated game, imply less than fully rational behavior:subject behavior is conditioned on observed choices, but subjects only react to observed choices in apassive manner. A more sophisticated subject may try and take advantage of the conditioning byacting in such a way as to influence future opponent’s actions. These are precisely the sort ofconsiderations eliminated by one-shot matching, but allowed in the repeated game. Thus, signalingconsiderations may have an implication for the pattern of play across treatments: for a given observedhistory of play, decisions may differ between the repeated and one-shot games.

While a comparison of one-shot and repeated games allow a test for either learning or signalingeffects, it is difficult to separate the two. We test for the existence of signaling effects by focussing oninitial play. Specifically, if initial behavior differs between repeated and one-shot games, we interpret

4this as some form of signaling.

3The two sessions of one-shot coordination games were previously reported in Clark et al. (in press).4In principle one could use final round data to test for pure learning effects but this would involve comparing matching

protocols after conditioning on the observed history of play, and would require much more data.

360 K. Clark, M. Sefton / Economics Letters 70 (2001) 357 –362

Fig. 2. Proportions of ‘Risky’ choices in Coordination Game sessions.

4. Results and conclusions

4.1. Coordination Game

Fig. 2 shows that across rounds there is considerably less ‘Risky’ play in the one-shot sessions thanin the repeated sessions. The efficient outcome is attained in just four of 200 one-shot games, but in116 of 200 stages of the repeated games. The difference in proportions of first-round choices of‘Risky’ (0.3 and 0.6 for the one-shot and repeated games, respectively) is statistically significant (Pvalue 5 0.01).

In addition to increasing the amount of coordination on the efficient equilibrium, repetitionincreased subject earnings by reducing the amount of ‘disequilibrium’ play. In the one-shot games —where ‘Risky’ comprised 24% of choices — players failed to coordinate in 30% of the games. In therepeated games, even though choices were more evenly split — ‘Risky’ comprised 65.5% of choices— only 17% of stage games ended in a disequilibrium outcome.

4.2. Mechanism Game

In the Mechanism Game sessions there is almost no play of dominated strategies (i.e., sequencesbeginning with ‘Safe’). Across all rounds of the one-shot sessions only 3.25% of sequences aredominated (all observed in the first three rounds), and in the repeated game sessions only 5% ofsequences are dominated (half of these occurring in the first three rounds). Thus the behavior of themajority of subjects conforms to a weak form of rationality whereby individuals do not playdominated strategies.

However, in terms of conformance with rationalizability, the results from our repeated game andone-shot game sessions are quite different. Fig. 3 displays the proportions of sequences that are aspredicted across rounds for the two protocols. The results from one-shot sessions are qualitativelyconsistent with those from Sefton and Yavas: few choices correspond to the equilibrium predictionand the proportion of sequences conforming to the prediction decreases across rounds (19% ofsequences in the first five rounds of the one-shot sessions conform to the theoretical prediction,compared with only 10.5% in the last five rounds). In the repeated game sessions, on the other hand,

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Fig. 3. Proportions of sequences as predicted in Mechanism Game sessions.

the majority of sequences, 58.75%, conform to the theoretical prediction, and the percentage ofsequences that match the equilibrium prediction climbs slightly from 50.5% in the first five rounds to67% in the last five rounds.

Based on first round data, the difference in the proportion of sequences that are as predicted (0.2 forthe one-shot sessions versus 0.4 for the repeated game sessions) is significant at the 10% level (Pvalue 5 0.087). Thus, there is more efficient play in repeated than one-shot Mechanism Games, evenin the first round.

4.3. Conclusion

Why should choices conform more closely to the prediction in the repeated game treatment? Seftonand Yavas conjectured that if subjects played repeatedly against the same opponent, subjects wouldget better feedback about their opponent’s behavior, and this would lead to convergence toequilibrium play. However, this is not a complete explanation because it fails to account for thedifference in initial behavior.

We interpret part of the matching protocol effect as due to signaling. We find persuasive theargument that the risk involved in playing the efficient equilibrium strategy is too high to justify itsuse in the one-shot game. By playing it, a subject simply leaves himself vulnerable to getting a zeropayoff, and since he will face a brand new opponent in subsequent games this would constitute anirrevocable loss. In the repeated game, some subjects may find it worthwhile to signal that they desirethe efficient equilibrium outcome. Even if they get a zero payoff in the first round, if they stay withtheir choice their payoff will improve if their opponent adjusts behavior in response to them. If theiropponent eventually best-responds to their choice, they will attain the efficient payoff in later rounds.

Acknowledgements

We thank Dilip Abreu, Daniel Seidmann, Abdullah Yavas and the editor for useful comments. Wealso thank the Faculty of Economics and Social Studies at the University of Manchester, the

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Department of Economics at the University of Newcastle, and the Nuffield Foundation for funding theexperiments.

References

Abreu, D., Matsushima, H., 1992. Virtual implementation in iteratively undominated strategies: complete information.Econometrica 60, 993–1008.

Clark, K., Kay, S., Sefton, M., in press. When are Nash equilibria self-enforcing: an experimental analysis, InternationalJournal of Game Theory.

Clark, K., Sefton, M., 1999. Matching Protocols in Experimental Games, School of Economic Studies Discussion Paper,University of Manchester.

Cooper, R., 1999. In: Coordination Games. Cambridge University Press, Cambridge.Cooper, R., DeJong, D.V., Forsythe, R., Ross, T.W., 1992. Communication in coordination games. Quarterly Journal of

Economics 107, 739–771.Glazer, J., Rosenthal, R.W., 1992. A note on Abreu-Matsushima mechanisms. Econometrica 60, 1435–1438.Harsanyi, J., Selten, R., 1988. In: A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge.Kamecke, U., 1997. Rotations: matching schemes that efficiently preserve the best-reply structure of a one-shot game.

International Journal of Game Theory 26, 409.Sefton, M., Yavas, A., 1996. Abreu-Matsushima mechanisms: experimental evidence. Games and Economic Behavior 16,

280–302.