Let's talk it over: Coordination via preplay communication with level-k thinking

This paper reconsiders Joseph Farrell's (1987) and Matthew Rabin's (1994) analyses of coordination via preplay communication, focusing on Farrell's analysis of Battle of the Sexes. Replacing their equilibrium and rationalizability assumptions with a structural nonequilibrium model based on level-k thinking, I reevaluate FR's assumptions on how players use language and their conclusions on the limits of communication in bringing about coordination. The analysis partly supports their assumptions about how players use language, but suggests that their “agreements” do not reflect a full meeting of the minds. A level-k analysis also yields very different conclusions about the effectiveness of communication. & 2016 The Authors. Published by Elsevier Ltd. on behalf of University of Venice. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Tacit coordination is ubiquitous in the animal kingdom, but explicit coordination—the use of preplay communication to structure relationships via non-binding agreements—may be fully realized only in human societies. Although explicit coordination is an essential part of our lives, and there is now a substantial body of experimental evidence on it (Crawford (1998) surveys early work), our theoretical understanding remains incomplete. This paper proposes and analyzes a model of coordination via pre-play communication that seeks to narrow the gaps between theory, evidence, and intuition, building on the work of Joseph Farrell (1987) and Matthew Rabin (1994) (see also Farrell, 1988 and Rabin, 1991), henceforth collectively “FR”. FR's analyses address two conjectures regarding complete-information games that are still widely held despite FR's partly negative conclusions: that preplay communication will yield an effective agreement to play an equilibrium in the underlying game; and that the agreed-upon equilibrium will be Pareto-efficient within that game's set of equilibria (henceforth “efficient”). FR assume that communication takes the form of one or more two-sided, simultaneous exchanges of messages about players’ intended actions in the underlying game. The messages are in a pre-existing common language and they are nonbinding and costless. FR also assume equilibrium, sometimes weakened to rationalizability. They further restrict er Ltd. on behalf of University of Venice. This is an open access article under the CC BY-NC-ND license . e 26th Arne Ryde Symposium, “Communication in Games and Experiments,” 24–25 August 2007, geeb Ali, Tore Ellingsen, Nagore Iriberri, Navin Kartik, Robert Östling, Adam Sanjurjo, and Joel Sobel for uls College and the University of California, San Diego provided research support. The research leading ean Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / t only the author’s views and not the views of the ERC or the European Commission, and the European the information contained therein. ac.uk V.P. Crawford / Research in Economics 71 (2017) 20–31 21 attention to outcomes that satisfy plausible behavioral restrictions defining which combinations of messages create agreements, and whether and how agreements can be changed. Under these assumptions FR show that rationalizable preplay communication need not assure equilibrium; and that, although communication enhances coordination, even equilibrium with “abundant” (Rabin's term for “unbounded”) communication does not assure that the outcome will be Pareto-efficient. In the part of FR's analyses that is most important for this paper, Farrell (1987) uses Battle of the Sexes to study symmetry-breaking via one or more rounds of two-sided preplay communication with conflicting preferences about how to coordinate. He focuses on the symmetric mixed-strategy equilibrium in the entire game, including the communication phase, in which the first pair of messages in the same communication round that identify a pure-strategy equilibrium in Battle of the Sexes are treated as an agreement to play that equilibrium, ignoring all previous messages. He calculates the equilibrium rate of efficient coordination with one or more rounds of communication, showing that the rate increases steadily with the number of rounds but converges to a limit less than one even with abundant communication. Because Farrell's analysis is specific to Battle of the Sexes and assumes equilibrium, it is reasonable to ask how general his insights are. Rabin (1994) extends Farrell's analysis to a wide class of underlying games while dropping his symmetry restriction; augmenting his restrictions on how players use language to allow them to make interim agreements, which can be improved upon in subsequent agreements; and considering the implications of rationalizability as well as equilibrium. Rabin defines notions called negotiated equilibrium and negotiated rationalizability that combine the standard notions with his restrictions on how players use language. He shows that with abundant communication, each player's negotiated equilibrium expected payoff is at least as high as in his worst efficient equilibrium in the underlying game. He then shows, replacing negotiated equilibrium by negotiated rationalizability, that even without equilibrium, each player expects (perhaps incorrectly) a payoff at least as high as in his worst efficient equilibrium. Thus Farrell's insights are quite general: “...the potential efficiency gains from communication illustrated by [Farrell, 1987] do not rely on ad hoc assumptions of symmetry or on selecting a particular type of mixed-strategy equilibrium. Rather, the efficiency gains...inhere in the basic assumptions about how players use language.” (Rabin, p. 373). Although equilibrium and rationalizability are the natural places to start in analyses like FR's, recent experiments suggest that in settings without clear precedents people often deviate systematically from equilibrium, especially when the reasoning behind it is not straightforward. The evidence also suggests that in such settings a structural non-equilibrium model can often out-predict equilibrium. While the existence of an empirically successful alternative to treating deviations as errors makes equilibrium seem too strong an assumption, rationalizability may be too weak. This paper takes a middle course, reconsidering FR's analyses with particular attention to Farrell's analysis of Battle of the Sexes, but replacing equilibrium or rationalizability with a non-equilibrium model based on level-k thinking. Although level-k models have not yet been thoroughly tested in this kind of setting, they explain much of the predictable part of subjects’ deviations from equilibrium in experiments that elicit initial responses to games in other settings, and their strong experimental support makes them a natural candidate. A level-k analysis allows a unified treatment of players’ messages and actions and how messages create agreements, deriving all three from simple assumptions that explain behavior in other settings. The analysis also allows a reevaluation of FR's plausible but ad hoc restrictions on how players use language. With one round of communication, the analysis justifies FR's assumption that a message pair that identifies an equilibrium leads to that equilibrium. However, the resulting “agreements” do not fully reflect the meeting of the minds that FR sought to model. Instead they reflect either one player's perceived credibility as a sender or the other's perceived credulity as a receiver, never both at the same time. As a result, a level-k analysis may not fully support the assumptions about agreements in Rabin's analysis of negotiated rationalizability. Turning to abundant communication, I assume in the spirit (but not the letter) of Rabin's analysis that players always have the option of an additional round of communication by mutual consent, but that in any round either player can 1 These restrictions make subgame-perfection superfluous. Rabin (1994, pp. 389–390) discusses the rationale for studying models in which two-sided messages are simultaneous rather than sequential. As he notes (and as Schelling (1960) noted), if there are no delay costs, as in FR's and my analyses, with sequential messages the outcome may be arbitrarily determined by assumptions about who can speak last or how players form their beliefs. 2 Farrell's analysis also sheds light on the symmetry-breaking role of communication in the pure coordination games studied by Schelling (1960) and others. 3 Symmetry is a natural restriction when players cannot distinguish their roles, and avoids begging the question of symmetry-breaking. Crawford and Haller (1990, p. 580) provide a justification for the symmetry assumption. 4 Rabin's model of communication is similar to Kalai and Samet's (1985). They assume agreements are binding, though renegotiable; but this difference is unimportant here because in coordination games the potential agreements are equilibria, and Rabin's assumptions make agreements to play them effectively binding, though renegotiable. Costa-Gomes (2002) extends Rabin's theory and tests it with experimental data. 5 With enough clear precedents, equilibrium is more reliable; but explicit agreements may then be unnecessary. 6 Level-k models, described in Section 1, also tend to out-predict equilibrium models with payoff-sensitive error distributions such as quantal response equilibrium. They were introduced to explain experimental data by Stahl and Paul (1994) and Nagel (1995) and further developed by Ho et al. (1998); Costa-Gomes et al. (2001); Camerer et al. (2004; “CHC”); Costa-Gomes and Weizsäcker (2008); Costa–Gomes and Crawford (2006; “CGC”); Crawford and Iriberri (2007a, 2007b); and Crawford et al. (2008). Crawford et al. (2013, Section 3) review the evidence. 7 Negotiated rationalizability is potentially relevant here because level-k types choose k–rationalizable strategies (Bernheim (1984); CGC, Section 1) and k–rationalizability, even for moderate k, is close to rationalizability in this setting. However, Rabin's analysis is not conclusive here because it requires levels of k higher than those that are realistic, and negotiated rationalizability builds the assumption that agreements are effective into players’ beliefs, but in the level-k analysis agreements reflect weaker restrictions that may not always satisfy Rabin's assumption. V.P. Crawford / Research in Economics 71 (2017) 20–31 22 unilaterally shut off communication and force play of the underlying game. As Rabin's analysis of negotiated rationalizability suggests, level-k players need not keep communicating until an agreement is reached as in Farrell's equilibrium. Finally, a level-k analysis implies very different conclusions about the effectiveness of communication than Farrell's equilibrium analysis. A level-k analysis suggests that coordination rates in Battle of the Sexes, with or without communication, will be largely independent of the difference in players’ preferences, while in Farrell's equilibrium analysis coordination rates are highly sensitive to this difference. A level-k analysis already has surprising implications for tacit coordination: Evenwith moderate differences in preferences, the level-k coordination rate without communication is likely to be higher, for empirically plausible type distributions, than the mixed-strategy equilibrium rate. Further, with one round of communication, the level-k rate is well above the rate without communication, and is likely to be higher than the equilibrium rate with one round of communication unless preferences are very close. Finally, with abundant communication, the level-k coordination rate is likely to be higher than the equilibrium rate unless preferences are moderately close. The model's predictions with abundant communication are consistent with Rabin's bounds based on negotiated rationalizability, but their precision yields additional insight into the causes and consequences of breakdowns in negotiations. This paper's closest relatives other than FR are Crawford (2003) and Ellingsen and Östling (2010; henceforth “EÖ”). Crawford (2003) introduces a level-k model of one-sided communication of intentions and uses it to study deception in zero-sum games. EÖ generalize Crawford's model to allow two-sided communication and use it to study two central issues in coordination: symmetry-breaking in games like Battle of the Sexes; and reassurance in games like Stag Hunt, where there is a tension between the higher payoffs and greater fragility of the Pareto-dominant equilibrium. They show that in a level-k model, as in equilibrium with suitable refinements, one-sided communication almost trivially solves the coordination problem in Battle of the Sexes, and is therefore more effective than two-sided communication, as is usually found in experiments (Crawford, 1998, Section 3). They also show that, unlike equilibrium with suitable refinements, a level-k model can also explain why two-sided communication is more effective than one-sided in Stag Hunt, as is also found in experiments. EÖ focus on the implications of these results for organizational design. This paper adapts EÖ's generalized model of two-sided communication to a different purpose: reevaluating FR's assumptions about how players use language and providing a more realistic characterization of the effectiveness of communication in bringing about coordination via symmetry-breaking. The paper is organized as follows. Section 1 introduces the level-k model by using it to analyze Battle of the Sexes without preplay communication, following CHC's (Section 3.3) level-k (or as they call it, “cognitive hierarchy”) analysis of closely related market-entry games. It has long been noted that subjects in market-entry experiments (Rapoport et al., 1998 and Rapoport and Seale, 2002) regularly achieve better ex post coordination (number of entrants closer to market capacity) than in the symmetric mixed-strategy equilibrium, the natural equilibrium benchmark. Earlier versions of this result led Daniel Kahneman (1988) to remark, “...to a psychologist, it looks like magic.” CHC show that Kahneman's “magic” can be explained by a level-k model, in which the predictable heterogeneity of strategic thinking allows some players to mentally simulate others’ entry decisions and accommodate them. The more sophisticated players become like Stackelberg followers, with coordination benefits for all. Section 1's analysis adapts CHC's analysis to Battle of the Sexes, showing that level-k thinking yields similar symmetrybreaking benefits there. The analysis suggests a view of tacit coordination profoundly different from the traditional view: With level-k thinking, equilibrium and, a fortiori, selection principles such as riskor payoff-dominance (Harsanyi and Selten, 1988) play no direct role in players’ strategic thinking. Coordination, when it occurs, is an almost accidental (though predictable) by-product of the use of non-equilibrium decision rules. These striking differences motivate a level-k analysis of explicit coordination: At the very least, a level-k analysis will shift the equilibrium benchmarks in Farrell's analysis. Section 2 reviews Farrell's equilibrium analysis of communication in Battle of the Sexes and the implications of Rabin's analysis of negotiated rationalizability in this setting. Section 3 presents a level-k analysis of Battle of the Sexes with one round of communication. It then compares the resulting coordination outcomes with Section 1's level-k outcomes for Battle of the Sexes without communication, and with Section 2's equilibrium outcomes with one round. Finally, it uses the level-k model to reevaluate Farrell's assumptions regarding which combinations of messages create agreements. Section 4 extends Section 3's analysis to allow abundant communication, modeled as allowing players the option, at the end of any communication round, of an additional round by mutual consent. It then compares the resulting coordination outcomes with the level-k outcomes with one round of communication, and with the outcomes in Farrell's equilibrium characterization of the limits of abundant communication. Section 5 is the conclusion. 8 I do not consider one-sided communication because it begs the question of symmetry-breaking that is at the heart of the coordination problem in Battle of the Sexes. Nonetheless, as EÖ show, the model used here has the “right” implications to explain experimental results with one-sided as well as two-sided communication. Kartik et al. (2007) introduce level-k models of one-sided strategic information transmission, in the limited sense of credulous receivers; see also Kawagoe and Tazikawa (2009). V.P. Crawford / Research in Economics 71 (2017) 20–31 23 1. A level-k model of tacit coordination This section introduces the level-k model by using it to analyze Battle of the Sexes without communication, following CHC's (Section 3.3) analysis of market-entry games. Level-k models allow behavior to be heterogeneous, but they assume that each player follows a rule drawn from a common distribution over a particular hierarchy of decision rules or types. I assume throughout that both player roles are filled from the same distribution of types, which restricts attention to symmetric outcome distributions, paralleling Farrell's restriction to the symmetric mixed-strategy equilibrium. As implemented here, type Lk anchors its beliefs in a nonstrategic L0 type and adjusts them via thought-experiments with iterated best responses: L1 best responds to L0, L2 to L1, and so on. L1 and higher types have accurate models of the game and are rational in that they choose best responses to beliefs. Their only departure from equilibrium is in replacing its assumed perfect model of others with simplified models that avoid the complexity of equilibrium analysis. In applications the type frequencies are treated as behavioral parameters (or in CHC's cognitive hierarchy model, a parameterized distribution) to be estimated or translated from previous analyses. The estimated distribution is fairly stable across games, with most weight on L1, L2, and L3. The estimated frequency of the anchoring L0 type is usually 0 or very small; thus L0 exists mainly as L1's model of others, L2's model of L1's model, and so on. Even so, the specification of L0 is the main issue in defining a level-k model and the key to its explanatory power. L0 often needs to be adapted to the setting; but the definition of higher types via iterated best responses allows an empirically plausible explanation of behavior in most settings. In CHC's market-entry games, n risk-neutral firms simultaneously decide whether to enter a market with capacity m o n. If m or fewer firms enter, the entrants all earn a profit; but if more than m enter they all earn a loss. Staying out yields zero. Like Battle of the Sexes, this game has a unique symmetric equilibrium in mixed strategies, in which the expected number of entrants is approximatelym, but there are significant probabilities of overor under-entry. Yet in Rapoport et al.’s (1998) and Rapoport and Seale's (2002) experiments with closely related games, the numbers of entrants ex post were systematically closer to m than in the symmetric equilibrium. How can subjects do systematically better than in the symmetric equilibrium? CHC show that this can be explained by a level-k model with an empirically plausible type distribution. In their model, L0 is uniformly random, the usual assumption for normal-form games. L1s mentally simulate L0s’ random entry decisions and accommodate them, entering only if they expect enough L0s to stay out. L2s accommodate L1s’ (and in CHC's model, unlike in mine, L0s’) entry decisions; and so on. Even though players’ decisions are simultaneous and there is no communication, the heterogeneity of strategic thinking allows more sophisticated types to accommodate less sophisticated types’ decisions, just as (noisy) Stackelberg followers would. Now consider the closely related Battle of the Sexes game in Fig. 1, where a41 without loss of generality. Two players choose simultaneously between two pure actions, H for Hawk or D for Dove, using the standard labeling of the strategies from evolutionary game theory to emphasize the symmetry of actions and payoffs across player roles. The unique symmetric equilibrium is in mixed strategies, with p Pr{H}1⁄4a/(1þa) for both players. The expected coordination rate is 2p(1– p)1⁄42a/(1þa), and players’ expected payoffs are a/(1þa)o1, worse for each player than his worst pure-strategy equilibrium. In the level-kmodel, each player follows one of four types, L1, L2, L3, or L4, with each player role filled by a draw from the same distribution. I assume, as in most previous analyses, that L0 chooses its action randomly, with Pr{H}1⁄4Pr{D}1⁄41⁄2. Higher types’ best responses are easily calculated: L1 chooses H, L2 chooses D, L3 chooses H, and L4 chooses D (Table 1). Although L3 behaves like L1 here, and L4 behaves like L2, I retain all four for comparability with the analysis below. But I assume for simplicity, from now on, that the frequency of L0 is 0. The model's predicted outcome distribution is determined by the outcomes of the possible type pairings in Table 1 and the type frequencies. The type frequencies are assumed to be independent of payoffs, in keeping with the fact that, like equilibrium, they are intended as general models of strategic behavior. Because in Battle of the Sexes, the outcomes of the possible type pairings are independent of a as long as a41, the payoff-independence of the type frequencies implies that the model's predicted outcome distribution is independent of a. By contrast, a has a strong influence on the equilibrium coordination rate, so this independence is important in the comparison between level-k and equilibrium rates.