Extensive Games with Possibly Unaware Players

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.Comment: 45 pages, 3 figures, a preliminary version was presented at AAMAS0

Dynamic unawareness and rationalizable behavior

We define generalized extensive-form games which allow for mutual unawareness of actions. We extend Pearce's (1984) notion of extensive-form (correlated) rationalizability to this setting, explore its properties and prove existence.Unawareness, extensive-form games, extensive-form rationalizability

Unawareness of theorems

This paper provides a set-theoretic model of knowledge and unawareness. A new property called Awareness Leads to Knowledge shows that unawareness of theorems not only constrains an agent's knowledge, but also, can impair his reasoning about what other agents know. For example, in contrast to Li (2006), Heifetz et al. (2006a) and the standard model of knowledge, it is possible that two agents disagree on whether another agent knows a particular event. The model follows Aumann (1976) in defining common knowledge and characterizing it in terms of a self-evident event, but departs in showing that no-trade theorems do not hold.

Dynamic Unawareness and Rationalizable Behavior

We define generalized extensive-form games which allow for mutual unawareness of actions. We extend Pearce's (1984) notion of extensive-form (correlated) rationalizability to this setting, explore its properties and prove existence. We define also a new variant of this solution concept, prudent rationalizability, which refines the set of outcomes induced by extensive-form rationalizable strategies. Finally, we define the normal form of a generalized extensive-form game, and characterize in it extensive-form rationalizability by iterative conditional dominance.Unawareness, extensive-form games, extensive-form rationalizability, prudent rationalizability, iterative conditional dominance

A Note on Unawareness and Zero Probability

I study how choice behavior given unawareness of an event differs from choice behavior given subjective belief of zero probability on that event. Depending on different types of unawareness the decision-maker suffers, behavior under unawareness is either incomparable with that under zero probability (in the case of pure unawareness), or drastically different (in the case of partial unawareness). The key differences are (1) partial unawareness permits dynamically inconsistent choice, while zero probability beliefs do not; and (2) there are unforeseen options in an unawareness environment that are necessarily modeled as dominated options in zero probability models.unawareness, zero probability, dynamic consistency, unforeseen contingency, unforeseen options

Prudent Rationalizability in Generalized Extensive-Form Games

We define an extensive-form analogue of iterated admissibility, called Prudent Rationalizability (PR). In each round of the procedure, for each information set of a player a surviving strategy of hers is required to be rational vis-a-vis a belief system with a full-support belief on the opponents' previously surviving strategies that reach that information set. Somewhat surprisingly, prudent rationalizable strategies may not refine the set of Extensive-Form Rationalizable (EFR) strategies (Pearce 1984). However, we prove that the paths induced by PR strategy-profiles (weakly) refine the set of paths induced by EFR strategies. PR applies also to generalized extensive-form games which model mutual unawareness of actions (Heifetz, Meier and Schipper, 2011a). We demonstrate the applicability of PR in the analysis of verifiable communication, and show that it yields the same, full information unraveling prediction as does the unique sequential equilibrium singled out by Milgrom and Roberts (1986); yet, we also show that under unawareness full unraveling might fail.Prudent rationalizability, caution, extensive-form rationalizability, extensive-form games, unawareness, verifiable communication

You Better Play 7: Mutual versus Common Knowledge of Advice in a Weak-link Experiment

This paper presents the results of an experiment on mutual versus common knowl- edge of advice in a two-player weak-link game with random matching. Our experimen- tal subjects play in pairs for thirteen rounds. After a brief learning phase common to all treatments, we vary the knowledge levels associated with external advice given in the form of a suggestion to pick the strategy supporting the payoff-dominant equilib- rium. In the mutual knowledge of level 1 treatment, the suggestion appears on every subject's monitor at the beginning of every round, with no common knowledge that everybody sees the same suggestion. In the mutual knowledge of level 2 treatment, the same suggestion appears on each subject's monitor, accompanied by the request to "send" the suggestion to the partner in the round, followed by a notification that the message has been read. Finally, in the common knowledge treatment, the suggestion is read aloud by the experimenter at the end of the learning phase. Our results are somewhat surprising and can be summarized as follows: in all our treatments both the choice of the efficiency-inducing action and the percentage of efficient equilibrium play are higher with respect to the control treatment, revealing that even a condition as weak as mutual knowledge of level 1 is sufficient to significantly increase the salience of the efficient equilibrium with respect to the absence of advice. Furthermore, and contrary to our hypothesis, mutual knowledge of level 2 (as the one occurring in our "message" treatment) induces successful coordination more frequently than common knowledge.Coordination games; experimental philosophy; epistemic attitudes, weak-link game; conventions

You Better Play 7: Mutual versus Common Knowledge of Advice in a Weak-link Experiment

This paper presents the results of an experiment on mutual versus common knowl- edge of advice in a two-player weak-link game with random matching. Our experimen- tal subjects play in pairs for thirteen rounds. After a brief learning phase common to all treatments, we vary the knowledge levels associated with external advice given in the form of a suggestion to pick the strategy supporting the payo-dominant equilib- rium. In the mutual knowledge of level 1 treatment, the suggestion appears on every subject's monitor at the beginning of every round, with no common knowledge that everybody sees the same suggestion. In the mutual knowledge of level 2 treatment, the same suggestion appears on each subject's monitor, accompanied by the request to "send" the suggestion to the partner in the round, followed by a notication that the message has been read. Finally, in the common knowledge treatment, the suggestion is read aloud by the experimenter at the end of the learning phase. Our results are somewhat surprising and can be summarized as follows: in all our treatments both the choice of the efficiency-inducing action and the percentage of e cient equilibrium play are higher with respect to the control treatment, revealing that even a condition as weak as mutual knowledge of level 1 is sufficient to signicantly increase the salience of the e cient equilibrium with respect to the absence of advice. Furthermore, and contrary to our hypothesis, mutual knowledge of level 2 (as the one occurring in our "message" treatment) induces successful coordination more frequently than common knowledge.Coordination games; experimental philosophy; epistemic attitudes, weak-link game; conventions