WISHFUL THINKING IN STRATEGIC ENVIRONMENTSMUHAMET YILDIZAbstract. Towards developing a theory of systematic biases about strategies,I analyze strategic implications of a particular bias: wishful thinking aboutthe strategies. I identify a player as a wishful thinker if she hopes to enjoythe highest payoff that is consistent with her information about the others’strategies. I develop a straightforward elimination process that characterizes thestrategy profiles that are consisten t with wishful thinking, mutual knowledge ofwishful thinking, and so on. Every pure-strategy Nash equilibrium is consistentwith common knowledge of wishful thinking. For generic t wo-person games, Ifurther sho w that the pure Nash equilibrium strategies are the only strategiesthat are consistent with common knowledge of wishful thinking. My analysisalso illustrates ho w one can characterize the strategic implications of generaldecision rules using the tools of game theory.Keywords: optimism, strategic uncertainty, wishful thinkingJEL Classifications : C72, D80Date: July, 2004.MIT Economics Department, 50 Memorial Drive, Cambridge, MA 02142, USA.E-mail: [email protected] thank Daron Acemoglu, Glenn Ellison, Haluk Ergin, Casey Rothschild, Marzena Rostek,Jonathan Weinstein, Robert Wilson, two anonymous referees, and an editor for detailed,though tful comments.1Forthcoming in Review of Economic Studies21. IntroductionTraditiona lly, game theory assumes rationality and focuses on Nash equilibriu m .In real life and in experiments, how e ver, economic agents have systematically bi-ased beliefs1and use decision rules that are inconsistent with rationalit y. T his hasinspired many non-rationa l theories of individual decision making. Clearly, in or-der for these theories to be useful, we need to know ho w they can be incorporatedinto strategic environments with m u ltiple play ers. For the most part, the papersthat consider such decision rules in strategic environments resort to equilibriumanalysis. But equilibrium analysis assumes that playe rs correctly guess the otherpla yers’ strategies.2This assumption is problematic in the presence of irrational-it y: if there are enough repetitions for pla yers to learn each others’ strategies,then they would also learn to play a best response. Moreov er, if a play er uses anon-rational rule of thumb as a decision rule, then she may also use some rule ofth u mb in predicting other pla yers’ strategies. Suc h rules of thumb ma y in troducesystematic biases, and the resulting beliefs ma y not correspond to an equilibrium.Similarly, the papers that investigate the role of optim istic beliefs in economicapplications assume that pla y ers hold optimistic beliefs about some underlyingparameters, and then apply equilibrium analysis.3The equilibrium assumptionis again problem atic. Firstly, it is hard to int erpr et such equilibria as outcom esof a learning process (Dekel, Fuden berg, and Levine (2004)). More importan tly,most observers agree that it is often harder to predict the outcom e of strategicin tera ctio ns than to predict aspects of ph ysica l reality. This suggests that players1For example, there is a large empirical literature on self-serving biases (e.g., Larw ood andWittaker (1977), Weinstein (1980), and Babcock and Loew enstein (1997)). The common-priorassumption rules out such biases (Aumann (1976,1987) and Feinberg (2000)).2Even when there is uncertainty about strategies in the form of mixed strategies, in order tojustify equilibrium analysis, one needs to make similar assumptions, such as mutual knowledgeof players’ conjectures–which implies that players guess other players’ conjectures correctly–ora common prior about strategies (Aumann and Brandenburger (1995)).3There are applications on fi nancial markets (Harrison and Kreps (1978), Morris (1996)),bargaining (Posner (1972), Landes (1971), Yildiz (2003,2004), Ali (2003) Watanabe (2004)),collective action (Wilson (1968), Banerjee and Somanathan (2001)), lending (Manove and Padilla(1999)), and theory of the firm (van den Steen (2005)).WISHFUL THINKING 3have more substantial self-serving biases about the strategies of others than theydo regardin g phy sical reality. Hence, it would be methodologically preferable tomodel players ha ving biased beliefs about strategic uncertainty whenev er theyha v e biased beliefs about the underlying parameters of the real w orld.In this paper, I take an epistemic approach to in vestigate the strategic implica-tions of non-rational decision rules and systematic biases, allow ing the play ers tohold systematic biases about the other play ers’ strategies. I focus on the exam -ple of wishful thinking, i.e., the extreme form of optimism, and characterize thestrategy profiles that are consistent with this decision rule, mutual knowledge ofthis decision rule and so on. Similar analyses hav e been done for rationality andsome v ariations of it (see e.g., Bernheim (1984), P earce (1984), Tan and Werlang(1988), and Epstein (1997), who considers maximizing behavior without expectedutilit y). These studies allowed heterogeno us beliefs about stra tegies, but theyha v e not considered system a tic biases, the focus of the presen t w ork.In m y analysis, I use standa rd models of strategic uncertaint y, as in Au m ann(1976, 1987). Any such m odel consists of several componen ts. First, there arestates, eac h of which is the complete description of a strategic situation, includingwhat strategy each player pla y s, what each pla yer believe s about these strategies,and so on. Players do not necessarily kno w the state. A t each state, eac h playerhas an information cell consisting of the states that she cannot rule out at thatstate. This cell represents the set of correct assump tions that she takes as giv en,which is som etim es referred to as her information. She also has a (conditional)probab ility distribution on this cell, whic h is tak en to represent her “subjective”beliefs (about other pla yers’ strategies, etc.).Idefine wishful thinking as follows. Consider an information cell of a player.This player tak es the set of strategy profiles play ed by the other pla yers on thiscell as given. She also know s that she can choose any strategy from the set of hero w n strategies. The product of these t wo sets is the set of all possible outcom esaccording to the information cell. I iden tify a pla yer as a wishful thinker at a stateif her expected pa yoff (according to