Going with the group in a competitive game of iterated reasoning

Going with the group in a competitive game of iterated reasoning Seth Frey (sethfrey@indiana.edu) Psychology Building, 1101 E. 10th St. Indiana University, Bloomington, IN 47405 Robert L. Goldstone (rgoldsto@indiana.edu) Psychology Building, 1101 E. 10th St. Indiana University, Bloomington, IN 47405 Abstract In some strategic games, thinking ahead about other players’ reasoning can lead to better predictions about what they will do. In other games, infinitely iterated reasoning ultimately prescribes random play. In an online experiment of strategic thinking in groups, we tested participants in a game with the formal structure of a random game, but the superficial struc- ture of a game that rewards iterated reasoning. We found that participants conformed to the superficial structure of the game, and earned more than they would have by playing randomly. We estimated how many steps participants thought ahead in the game and discovered implicit coordination at the group level. Participants unexpectedly “matched” their degree of iterated thinking to each other. Keywords: experiment; strategic thinking; thinking steps; col- lective behavior; beauty pageant; matching pennies; rock pa- per scissors; n-player games Introduction Reflecting on how people learn to think ahead, behavioral economist Colin Camerer muses that . . . strategic thinking seems to be more like learning to windsurf, ski, or fly an airplane, activities that require people to learn skills which are unnatural but teach- able, and less like weight-lifting or dunking a basket- ball, where performance is constrained by physical lim- its. (Camerer, 2003, p.[249]) Our work challenges this analogy to individual activity. We propose that thinking strategically—thinking through the an- ticipated actions of others—may be a group activity: less like skiing or dunking and more like hockey or basketball. We report an experiment in which participants set their level of strategic thinking to match that of their peers, and earned more than economic theory predicts. The Nash equilibrium is a cornerstone of game theory. It describe those outcomes of an economic game for which no player can gain by changing their strategy individually. John Nash proved that every game has at least one such equilib- rium. Mapped onto the real outcomes it models, the reason- ing process behind the Nash equilibrium invokes an infinite regress of thoughts about thoughts. But human limits to this ideal reasoning process constrain the applicability of theory to real decisions. This concern has made experiments about iterated reasoning important. Ho, Camerer, and Weigelt (1998) have shown demographic and motivational effects on the number of steps that partici- pants think ahead through the thoughts of others’ thoughts. In unpublished data, they report that participants playing for higher stakes ($28 vs. $7) thought further ahead by about half of a thinking-step. Other work has shown that training can also improve iterated reasoning. In the project that mo- tivated the introductory analogy to individual sports, Costa- Gomes, Crawford, and Broseta (2001) elicited near-perfect equilibrium play from participants who had been exposed to the relevant aspects of game theory before the experiment. Matching Pennies The novel game that we tested in this experiment has some neighbors among familiar economic games. The near- est neighbors are Matching Pennies—a game that rewards randomness—and the Beauty Pageant, a classic domain for studying the levels of iterated reasoning. In Matching Pennies, two players select between two strategies: heads and tails. One player earns a point if the two selected strategies are the same (e.g. both heads), and the other player earns the point if they are different (e.g. one heads and one tails). Matching Pennies can be considered a two-player version of the children’s game Rock Paper Scissors. In these games there is no single best strategy: every choice can be defeated by a choice that can also be defeated. Rock Paper Scissors and Matching Pennies are the archetypal examples of games with mixed-strategy equilibria. The best approach in each of them is to pick strategies randomly (rather than playing “pure” strategies). Mixed strategies draw randomly from the possible strategies to maximize expected payoff. Behavioral results from the simplest two- and three-person Matching Pennies games are consistent with the hypothesis that peo- ple play mixed strategies (Goeree & Holt, 2001; McCabe, Mukherji, & Runkle, 2000). The Beauty Pageant Though the game that we report is formally related to Match- ing Pennies, it has a superficial resemblance to another popu- lar economic game, the Beauty Pageant. In the Beauty Pageant game, competing players are told to select a number 1–100. A player wins if they guessed the value that is closest to two-thirds the average of all other sub- mitted values. Reasoning that (a) no player would choose a value over sixty-six, (b) all players would realize this, (c) therefore, no player would chose a value over 2/3 ∗ 66, (d) all players would recognize this, and (e) so on, the player who iteratively