Common Knowledge and State-Dependent Equilibria

Abstract

Abstract. Many puzzling social behaviors, such as avoiding eye contact, using innuendos, and insignificant events that trigger revolutions, seem to relate to common knowledge and coordination, but the exact relationship has yet to be formalized. Herein, we present such a formalization. We state necessary and sufficient conditions for what we call state-dependent equilibria -equilibria where players play different strategies in different states of the world. In particular, if everybody behaves a certain way (e.g. does not revolt) in the usual state of the world, then in order for players to be able to behave a different way (e.g. revolt) in another state of the world, it is both necessary and sufficient for it to be common p-believed that it is not the usual state of the world, where common p-belief is a relaxation of common knowledge introduced by Monderer and Samet [16]. Our framework applies to many player r-coordination games -a generalization of coordination games that we introduce -and common (r, p)-beliefs -a generalization of common p-beliefs that we introduce. We then apply these theorems to two particular signaling structures to obtain novel results.

Common knowledge, coordination, and strategic mentalizing in human social life

“…When expectations are mismatched (such as a citizen bribing a police officer, a supervisor soliciting a sexual favor, or a friend selling a car to another friend), the clash could be damaging for everyone (see refs. 9 and 49 for game-theoretic analyses). The choice of a Relational Model, like other coordination games, is ratified by common knowledge: each of 2 partners knows that the other considers herself a friend (or a lover, or a boss, or a customer), knows that the other knows she knows this, and so on.…”

The Evolution of Morality

“…If observers were to know one purposely chose to defect or chose the less cooperative act, but they do not know that others know this, then observers think others will think punishment is not warranted, and observers will not punish. The argument is analogous to the discussion of higher-order beliefs in the omission-commission subsection and formalized in Dalkiran et al ( 2012 ) and .…”

“…The explanation again uses "higher-order beliefs" and is based on the key insight in and formalized in Dalkiran et al ( 2012 ) and : When the harm is done as a by-product, the harm is not usually anticipated.…”

“…Neither does it require that the distribution is uniform nor that the Coordination Game is not affected by the behavior of Assad. The only crucial assumptions are that the distribution is not too skewed and that the payoffs are not too dependent on the behavior of Assad (for details, see Dalkiran et al, 2012 ;Hoffman, Yoeli, & Dalkiran, 2015 ).…”

“…For the above result to hold, all that is needed is the following: (1) The more the costs of punishment decrease, the more others punish and (2) omissions are usually unintended (Dalkiran, Hoffman, Paturi, Ricketts, & Vattani, 2012 ;. 4 4 In fact, even if one knows that others know that the transgression was intended, omission will still be judged as less wrong, since the transgression still won't create what game theorists call common p -belief, which is required for an event to infl uence behavior in a game with multiple equilibria.…”