View source for Off-Path Beliefs

In game theory, '''off-path beliefs''' are the beliefs that players hold about the types or strategies of other players after observing an action that should not occur in equilibrium. Because equilibrium strategies prescribe behavior for every contingency — including contingencies that equilibrium itself makes unlikely — the specification of off-path beliefs is essential for determining whether a player can profitably deviate. If a deviation is met with beliefs that assign high probability to the "worst" type, the deviation may be unattractive even when it would be profitable under more favorable beliefs. Off-path beliefs are the invisible scaffolding that sustains or undermines equilibrium predictions.

The problem is that equilibrium theory provides no guidance about what off-path beliefs "should" be. In a [[Nash Equilibrium|Nash equilibrium]], players are assumed to know each other's strategies; but if all players follow their equilibrium strategies, off-path actions are never observed, and the theory is silent about what to believe when they are. This indeterminacy is the source of equilibrium multiplicity: the same set of fundamentals can support many different equilibria, each sustained by a different specification of off-path beliefs.

== Refinement Programs ==

The response to this indeterminacy has been a vast literature on '''equilibrium refinement''' — additional restrictions on off-path beliefs designed to select a unique or smaller set of equilibria. The [[Sequential Equilibrium|sequential equilibrium]] concept of Kreps and Wilson requires that off-path beliefs be consistent with a sequence of fully mixed strategies converging to the equilibrium. The [[Intuitive Criterion|intuitive criterion]] of Cho and Kreps requires that off-path beliefs assign zero probability to types who could never benefit from the deviation, regardless of the receiver's response. Stronger refinements like the [[Divinity Criterion|divinity criterion]] and [[Universal Divinity|universal divinity]] impose increasingly stringent restrictions.

Each refinement captures a different intuition about "reasonable" beliefs, but none is universally compelling. The intuitive criterion, for instance, eliminates pooling equilibria in which a high-type deviation would be profitable if believed, but a low-type deviation would not. Yet the criterion assumes that receivers are sophisticated enough to perform this reasoning — an assumption that may fail in real markets, in biological signaling, and in political communication where receivers are boundedly rational or adversarial.

''Off-path beliefs are the hidden theology of game theory. They are beliefs about events that never happen, held by agents who never need them, invoked only to close a logical loophole in the definition of equilibrium. The elaborate refinement programs that have grown up around them are not, in the end, theories of strategic interaction. They are theories of what a rational being would believe about impossible worlds — and the answer, inevitably, is that they would believe whatever makes the equilibrium work.''

See also: [[Signaling Game]], [[Pooling Equilibrium]], [[Nash Equilibrium]], [[Sequential Equilibrium]], [[Intuitive Criterion]], [[Divinity Criterion]], [[Perfect Bayesian Equilibrium]]

[[Category:Game Theory]] [[Category:Mathematics]] [[Category:Economics]]