1 Economics 142: Behavioral Game Theory Winter 2008 Vincent Crawford Many strategic situations in business, international relations, politics, or war are well approximated by games. A game is a multi-person decision situation, defined by its structure: the players, its “rules” (the order of players’ decisions, their feasible decisions at each point, and the information they have when making them); how players’ decisions determine the outcome; and players’ preferences over outcomes. Behavioral game theory is a blend of traditional game theory and empirical knowledge whose goal is the understanding of strategic behavior needed for applications. Such understanding includes topics from behavioral decision theory plus two topics that are specific to multi-person settings: (1) preference interdependence (such as altruism, envy, spite, or reciprocity); and (2) players’ models of other players. Here I narrow the focus to (2), assuming that behavior is (mostly) rational in the decision-theoretic sense and self-interested. I further subdivide (2) into: (2a) how players model others’ decisions in initial responses to games with no clear precedents; and (2b) how players learn to predict others’ decisions in repeated play of analogous games. We’ll start with a Game Survey designed to highlight some of the issues studied in behavioral game theory. The results will be anonymously tabulated and used as empirical support for some of the ideas developed below.2 As just noted, analyses of behavior in games must confront all the issues that arise with individual decisions, plus one that is unique to games: Because the outcome is influenced by others’ decisions as well as your own, to do well in a game you need to predict others’ decisions, taking their incentives into account. This may require a mental model of other players (including a model of their models of you!). Game theory has a standard model of how people decide what to do in games, which rests on the assumption that people can perfectly predict others’ decisions: Nash equilibrium (often shortened to equilibrium) in which each player chooses a decision that is best for himself, given correct expectations about others’ decisions Equilibrium makes clear predictions of game outcomes, which are often accurate when players have learned to predict others’ decisions from experience with analogous games (for example, Walker and Wooders, “Minimax Play at Wimbledon,” 2001 AER). But in novel situations there may be no analogous games, and equilibrium must then come from sophisticated strategic thinking rather than learning from direct experience. This makes equilibrium a less plausible assumption, and equilibrium predictions are often much less reliable for initial responses to games than when learning is possible.3 In these notes I will briefly discuss the strategic issues addressed by game theory and how the standard theory addresses them, using equilibrium and related notions. I then outline a behavioral game theory synthesis of models of thinking and learning. I then compare equilibrium predictions in some simple situations with history, experimental data, or intuitions regarding initial responses to games, highlighting situations where there are systematic deviations from equilibrium predictions. I then describe a structural but non-equilibrium model of initial responses to games that has emerged from recent experimental work, based on something called “level-k” thinking, and compare its predictions with intuition and experimental data. In simple games a level-k model’s predictions tend to coincide with equilibrium, so equilibrium predictions rest on a broader and more plausible set of behavioral assumptions, and are correspondingly more reliable. In more complex games a level-k model’s predictions can deviate systematically from equilibrium, but in predictable ways. These deviations often bring the model’s predictions closer to evidence and intuition, resolving puzzles left open by equilibrium analysis. I conclude with a brief introduction to learning in games.4 Ideas and Issues Something is mutual knowledge if all players know it; and common knowledge if all know it, all know that all know it, and so on. I focus on players’ problem of predicting others’ decisions by assuming that they have common knowledge of the structure of the game. This allows game outcomes to be uncertain if their distributions are common knowledge. The theory does not require common knowledge of the structure, but is easier to explain with it. Rationality and dominance L R Confess Don’t T 2 2 1 2 Confess -5 -5 -10 -1 B 2 1 1 1 Don’t -1 -10 -2 -2 Crusoe vs Crusoe Prisoner’s Dilemma Crusoe vs. Crusoe is just two decision problems “traveling together,” not really a game; each player has a best decision independent of the other’s (a dominant decision or strategy; the dominant decision dominates (strictly, in this case) the other decision). In Prisoner’s Dilemma, players’ decisions affect each other’s payoffs but each player still has a dominant decision. The game is interesting because individually optimal decisions yield a Pareto-inefficient outcome, highlighting an important distinction between individual and group rationality when there are payoff interactions, even when there are no interactions between players’ choices.5 Iterated knowledge of rationality and iterated dominance Push Wait Push 1 5 5 3 Wait -1 9 0 0 Pigs in a Box In Pigs in a Box, think of Row (R) as a big pig and Column (C) as a little pig. (The box is a Skinner box, named for the famous behavioral psychologist B.F. Skinner.) Pushing a lever at one end yields 10 units of grain at the other. Pushing “costs” either pig the equivalent of 2 units of grain. (That is, a pig’s utility is his grain consumption, minus 2 if he pushed the lever and minus 0 otherwise.) If R pushes while C waits, C can eat 5 units before R runs down and shoves C aside. If C pushes while R waits, C cannot shove R aside, so R gets all but one unit of grain. If both push and then arrive at the grain together, C gets 3 units and R gets 7. If both wait, both get 0 units of grain.6 Push Wait Push 1 5 5 3 Wait -1 9 0 0 Pigs in a Box In experiments with real pigs playing the game over and over, if a stable behavior pattern emerges it tends to be at (R Push, C Wait), the equilibrium outcome. Like