Expectation Conformity in Strategic Cognition * Alessandro Pavan † and Jean Tirole ‡ October 2, 2021 Abstract The paper studies “cognitive games,” that is, games in which the players can influence their understanding of a strategic situation before playing the primitive (normal- or extensive-form) game. The analysis covers both the case of self-directed cognition (as when a player controls her own information structure) and the case of manipulative cognition (as when a player influences her opponents’ understanding of the game). We introduce the concept of expectation conformity and show how the latter, together with its decomposition into unilateral expectation conformity and increasing differences, sheds light on the choice of the cognitive structures (both on and off the equilibrium path) and on the sensitivity of the cognitive postures to the type of strategic interaction (e.g., complements vs substitutes). We show that constant-sum games never give rise to self-fulfilling cognition. By contrast, the latter emerges in many non-constant-sum games, both when cognition is self-directed and takes the form of “sparsity,” noisy information acquisition, or “espionage,” (i.e., learning about others’ beliefs), and when it is manipulative and takes the form of framing, signal jamming, noisy disclosures, and counter-intelligence. Finally, we discuss the role that expectation conformity plays in games with boundedly-rational players such as those considered in the level-k literature. Keywords : cognition, expectation conformity, cognitive traps, sparsity, framing, memory man- agement, endogenous depth of reasoning. JEL numbers : C72; C78; D82; D83; D86. * Research support from European Research Council advanced grant (European Community’s Seventh Framework Programme (FP7/2007-2013) Grant Agreement no. 249429 and European Union’s Horizon 2020 research and innovation programme, Grant Agreement no. 669217) is gratefully acknowledged. We are grateful to Matteo Camboni for excellent research assistance. † Department of Economics, Northwestern University, and CEPR. Email: alepavan@northwestern.edu. ‡ Toulouse School of Economics (TSE) and Institute for Advanced Study in Toulouse (IAST), University of Toulouse Capitole. Email: jean.tirole@tse-fr.eu. 1