Unexpected Moves Are the Most Interesting

Game theory is not limited to what we usually call a game. Indeed, it addresses any situation in which one's actions have consequences over the other participants (players). In principle, this expands the scope of game theory to almost any human interaction. There is a whole literature within game theory that is aimed at mathematically formalizing how players think, both about the others' moves and about their own. In this context, the game of mirrors produced by the I-think-you-think-I-think loop is technically called a hierarchy of beliefs.

In static games – in which all the players make their own move simultaneously, once and for all (e.g. rock-paper-scissors) – such beliefs are formed at the beginning of the game and do not need any updating, since the game is over after just one move. On the contrary, dynamic games – in which the players make their moves sequentially and repeatedly (e.g. card games, chess, etc.) – require to update the players' beliefs about each other after each player's turn, based on what is observed. In a seminal paper in 1999, Bocconi professor Pierpaolo Battigalli, together with Marciano Siniscalchi (now at Northwestern University), provided a theoretical framework for belief updating in dynamic games, resorting to sophisticated mathematical tools that transcend standard probability theory.

However, this paper was still conceived under the assumption that all the players' moves comply with their strategic plans. Hence, if a player makes a move that does not belong to the plan that you ascribed to her, you must update your belief about her plan and conclude that her plan must be different. A new paper by Professor Battigalli and Bocconi Academic Fellow Nicodemo De Vito goes beyond this traditional assumption, decoupling plans from behaviors.

"All the literature on the foundations of game theory, also known as epistemic game theory", explains Battigalli, "tacitly assumed the coincindence of plans and behaviors, admitting no deviation of the players' behaviors from their plans. On the contrary, admitting the possibility that the players' behaviors can occasionally depart from their plans has huge consequences on the belief updating mechanism. Namely, when a player makes a move that does not belong to the plan that you attributed to her, you are not forced to totally discard your belief about her plan: you can consider it a one-off deviation from her plan."

"This produces a whole new theoretical framework that – among other implications – allows a justification of the backward induction algorithm, a procedure used to compute the equilibrium of a specific class of games by starting from the last stage of the game and proceeding backwards. This algorithm has been widely employed, but no satisfactory theoretical foundation was available so far. Decoupling plans from behaviors allows us to justify this algorithm, and opens up new research directions. In particular, it provides a new way to deal with unexpected moves, and how we deal with the unexpected is always the most interesting part, both in game theory and in life."


Find out more

Battigalli, P., & De Vito, N. (2021). "Beliefs, plans, and perceived intentions in dynamic games." Journal of Economic Theory, 105283. DOI: https://doi.org/10.1016/j.jet.2021.105283.

Battigalli, P., & Siniscalchi, M. (1999). "Hierarchies of conditional beliefs and interactive epistemology in dynamic games." Journal of Economic Theory, 88(1), 188-230. DOI: https://doi.org/10.1006/jeth.1999.2555.