Extensive games with possibly unaware players

Abstract

Standard game theory assumes that the structure of the game is common knowledge among players. We relax this assumption by considering extensive games where agents may be unaware of the complete structure of the game. In particular, they may not be aware of moves that they and other agents can make. We show how such games can be represented; the key idea is to describe the game from the point of view of every agent at every node of the game tree. We provide a generalization of Nash equilibrium and show that every game with awareness has a generalized Nash equilibrium.Finally, we extend these results to games with awareness of unawareness, where a player i may be aware that a player j can make moves that i is not aware of, and to subjective games, where payers may have no common knowledge regarding the actual game and their beliefs are incompatible with a common prior.

Unawareness—A gentle introduction to both the literature and the special issue

“…The same critique applies to the interpretation of information sets in Grant and Quiggin [2013] as for Feinberg [2012] and Halpern and Rêgo [2014]. The information sets model implicit knowledge and not necessarily explicit knowledge; see our discussion in Subsection 3.1.…”

“…Figure 5: Unforeseen Roadworkà la Halpern and Rêgo [2014] 3.2 The Approach by Halpern and Rêgo [2014] Halpern and Rêgo [2014]'s approach to extensive-form games with unawareness also involves a forest of game trees. Figure 5 shows the three games of the roadwork example.…”

“…Not surprisingly, the three trees are the same as in the the previous setting. Yet, instead of modeling explicitly each player's view at any of her decision nodes about views of others etc., Halpern and Rêgo [2014] use an awareness correspondence F that maps each game and history into a game and information set in that game. We view this mapping somewhat analogous to awareness correspondences in awareness structures.…”

“…Apart from the awareness correspondence, the the forest of trees in Halpern and Rêgo [2014] looks pretty standard in the sense that each game tree is a standard extensive-form game. Yet, as in Feinberg [2012]'s approach the interpretation of some information sets becomes nonstandard as they actually model implicit knowledge and not necessarily explicit knowledge; see our discussion in Subsection 3.1.…”

“…As in the other approaches, it uses a forest of game trees. Yet, instead of the awareness correspondence in Halpern and Rêgo [2014], the authors use what they call a perception mapping Z that maps each tree and history to the set of terminal histories in a weakly less expressive tree. The interpretation is that the value of the perception map at a player's history is the set of (terminal) histories that this player is aware of.…”