Structural Models of Nonequilibrium Strategic Thinking : Theory , Evidence , and Applications

model of a game played on a nonneutral cultural or geographic landscape. The frame has no payoff consequences, but it is non-neutral in that the “B” location is distinguished by its label and the two “end A” locations may be inherently salient as well. Rubinstein’s (1999) hide-and-seek game has a unique equilibrium prediction, whose logic is transparent and which leaves no room for framing to influence the outcome. Even so, framing had a strong and systematic effect in Rubinstein’s (1999) experiments, qualitatively the same in six experiments around the world where labeling created salience without positive or negative connotations, with Central A (or its analogs in other treatments) most prevalent for hiders (37 percent in the aggregate) and Central A even more prevalent for seekers (46 percent).50 These results pose two puzzles. On average hiders are as smart as seekers, so hiders tempted to hide in central A should realize that seekers will be just as tempted to look there. Why then do hiders allow seekers to find them 32 percent of the time when they could hold it down to 25 percent by using the equilibrium mixed strategy? And why are the results role-asymmetric, with seekers choosing central A more often (46 percent) than hiders (37 percent)? Rubinstein (1999) takes the nonequilibrium patterns in the data as evidence that subjects did not think strategically (Crawford and Iriberri 2007a, 1733, footnote 3). But it would be surprising if responses to such simple games were completely nonstrategic, and the fact that subjects’ behavior patterns 49 This gives the “central A” location its own brand of uniqueness as the “least salient” location. Mathematically this uniqueness is no different from the uniqueness of “B”, but Crawford and Iriberri’s (2007a) analysis suggests that its psychological effects are quite different. 50 This statement depends on identifying analogies among Rubinstein’s (1999) treatments as explained in Crawford and Iriberri (2007a, section 1). One might argue that because any strategy, pure or mixed, is a best response to equilibrium beliefs, deviations do not violate the theory. But systematic deviations from equilibrium choice frequencies must (with high probability) have a cause that is partly common across players. They are therefore symptomatic of systematic deviations from equilibrium probabilities. 43 Crawford, Costa-Gomes, and Iriberri: Strategic Thinking were qualitatively the same in six independent treatments suggests that they have a common structure. What kind of model can explain the roleasymmetric patterns in Rubinstein’s (1999) data? Rubinstein’s (1999) game and the roleasymmetric responses it evoked are an interesting test case, because although its payoff structure is asymmetric, models like equilibrium and QRE, which in this game coincides with equilibrium for any distribution and precision, imply role-symmetric responses. Even a level-k model with a uniform random L0 coincides with equilibrium. Crawford and Iriberri (2007a) uses Rubinstein’s (1999) data to compare adaptations of equilibrium, LQRE, and level-k models that allow salience an effect. They use Rubinstein’s (1999) six treatments in which decision labels had neutral connotations, which presumably influenced behavior only via salience, to avoid confounding subjects’ strategic thinking with their responses to positive or negative connotations. Crawford and Iriberri (2007a) adapt equilibrium or LQRE to this setting by adding payoff perturbation parameters that reflect plausible instinctive reactions to salience: a payoff gain for choosing a salient location for seekers, or a payoff loss for hiders. Depending on the flexibility of the specification, LQRE gets the role-asymmetry in the data qualitatively wrong, or estimates an infinite precision and so turns itself back into an equilibrium with perturbations model, which also misses the role-asymmetry, in a less extreme way. Crawford and Iriberri (2007a) adapt the level-k model to settings where salience is important by assuming that L0’s strategically naïve initial assessment of others’ likely responses to the game deviates from uniform randomness by favoring salient locations, either B or one of the end As, with no other changes. Crawford and Iriberri (2007a) take L0 to be determined by the same general principles in each player role, which in this setting means that L0 is the same in each role, although its implications for L1 differ across roles due to hiders’ and seekers’ different payoffs. Crawford and Iriberri (2007a) make no assumption about whether B or the end As are more salient. But the facts that L0 is the same in each role and that all that matters about it are the best responses it yields for hiders’ and seekers’ L1s imply that estimating B’s and the end A’s relative salience adds one binary parameter (Crawford and Iriberri 2007a,  the minimum possible flexibility that allows salience to have an influence. Finally, the idea that level-k rules are meant to generalize across games suggests that they should generalize across player roles within this game, which suggests the restriction that the population distribution of rules is the same for hiders and seekers. That and the assumption that L0 is determined by the same general principles in each role requires the model to explain the role-asymmetric patterns in the data endogenously, rather than via unexplained parameter variation across hiders and seekers. Unrestricted estimates of the level-k rule distribution are almost hump-shaped (0 percent L0, 19 percent L1, 32 percent L2, 24 percent L3, 25 percent L4), as is plausible for a homogeneous population. Counter to our intuition, the end locations are estimated to be more salient than the B location. This L0 and rule distribution tracks the observed prevalence of central A for hiders, its even greater prevalence for seekers, and the other main patterns in Rubinstein’s (1999) data. 51 L1 hiders then choose central A to avoid L0 seekers and L1 seekers avoid central A. L2 hiders choose central A with probability between 0 and 1, breaking payoff ties randomly; and L2 seekers choose it with probability 1. L3 hiders avoid central A and L3 seekers choose it with probability between zero and one. L4 hiders and seekers avoid central A. The heterogeneity of L1, L2, and L3 yields a role-asymmetric aggregate pattern that reconciles the prevalence of central A for hiders with its greater prevalence for seekers. Journal of Economic Literature, Vol. LI (March 2013) 44 Despite Rubinstein’s (1999) intuition, the analysis suggests that their subjects were unusually sophisticated, with an average k a level higher than usual; they just didn’t follow the fixed-point logic of equilibrium.  The level-k model with L0 adapted to respond to salience is the only one of which we are aware that responds to the hide and seek game’s asymmetric payoff structure in a way that can explain the robust role-asymmetric patterns in Rubinstein’s (1999) data. Crawford and Iriberri’s (2007a) model has been criticized for having too much flexibility for its explanation to be credible. But all that matters about its L0 is the best responses it yields for L1s, so the adaptation adds one binary parameter: the minimum flexibility that allows salience an influence, less than the adapted equilibrium or LQRE model. 8.2 Portability to Other Outguessing Games To address the criticism that adapting L0 to respond to salience gives the level-k model too much flexibility, Crawford and Iriberri (2007a) compare the equilibrium with payoff perturbations and adapted level-k models’ portability, the extent to which a model estimated for one game can describe responses to other games.52 They port the models to the two closest relatives of Rubinstein’s (1999) games that have been studied experimentally: O’Neill’s (1987) card-matching game and Rapoport and Boebel’s (1992) closely related outguessing game. Both games raise the same strategic issues as Rubinstein’s (1999) games, but with more complex patterns of wins and losses, 52 Crawford and Iriberri (2007a) also test the models for overfitting by using them to compute estimates separately for each of Rubinstein’s (1999) six treatments and using the estimated models to “predict” the results of the other five. The results moderately favor the level-k model. different framing, and in the latter case five locations. Here we discuss only the results for O’Neill’s (1987) game. In it, players simultaneously choose one of four cards: A, 2, 3, J. One player wins if there is a match on J or a mismatch on A, 2, or 3; otherwise the other wins. The game is like hide and seek, but with each player a hider for some locations and a seeker for others. Without payoff perturbations it has a unique equilibrium, in which each player plays A, 2, and 3 with probability 0.2 and J with probability 0.4. Crawford and Iriberri (2007a) define salience-sensitive versions of equilibrium for O’Neill’s (1987) game by introducing payoff perturbations as for Rubinstein’s (1999) game: a player gains (loses) payoff for a salient location in which he is a seeker (hider). They then use O’Neill’s (1987) data to estimate the perturbations.53 Even with these estimates, equilibrium with perturbations cannot explain subjects’ initial responses better than equilibrium without perturbations, which explains them poorly. Crawford and Iriberri (2007a) adapt the level-k model to O’Neill’s (1987) game by defining a nonstrategic L0 that favors A and J, which (as both face cards and end locations) are intuitively more salient than 2 and 3. With no assumption about which is more salient, the adaptation again adds a single binary parameter. Crawford and Iriberri (2007a) use O’Neill’s (1987) data to estimate that parameter, but instead of reestimating the rule distribution they reused the