Intuitive criterion

The intuitive criterion is a technique for equilibrium refinement in signaling games. It aims to reduce possible outcome scenarios by first restricting the type group to types of agents who could obtain higher utility levels by deviating to off-the-equilibrium messages and second by considering in this sub-set of types the types for which the off-the-equilibrium message is not equilibrium dominated.[1]

In economics, signaling games are games in which a player with private information moves first. Private information generally refers to the player's hidden or unobservable type. Signaling games typically have many perfect Bayesian equilibria. Equilibrium refinement techniques are ways of reducing the set of equilibria. Most refinement techniques are broadly based on restricting beliefs off the equilibrium path. Off equilibrium actions or outcomes are those that are different from what is predicted in a Nash equilibrium. The intuitive criterion was presented by In-Koo Cho and David M. Kreps in a 1987 article.[2] Their idea was to try to reduce the set of equilibria by requiring off-equilibrium beliefs to be reasonable in some sense. This refinement of the solution concept allows the modeller to choose among multiple perfect Bayesian equilibria.

Intuitively, we can eliminate a perfect bayesian equilibrium if there is a type of player who wants to deviate, assuming that other players are reasonable. What does it mean to be reasonable? It is reasonable to believe the deviating player is of a type who would benefit from the deviation in at least the best-case scenario. If a type of player could not benefit from the deviation even if the other players changed their beliefs in the best possible way for him, they should put zero probability on the player being of that type. Altering the words of the original Kreps-Cho article slightly, if ${\displaystyle t'}$ is the best possible type, the deviating player could persuasively tell the other players to interpret his signal ${\displaystyle m'}$ favorably:

I am sending the message ${\displaystyle m',}$ which ought to convince you that my type is ${\displaystyle t'.}$ For I would never wish to send ${\displaystyle m'}$ if my type were ${\displaystyle t,}$ while if it is ${\displaystyle t',}$ and if sending this message so convinces you, then, as you can see, it is in my interest to send it.

Formally, we can eliminate a particular perfect Bayesian equilibrium by using the intuitive criterion if there is some type θ who could benefit from a deviation that is assured of yielding them a payoff above their equilibrium payoff as long as other players do not assign a positive probability to the deviation having been made by any type θ for whom this action is equilibrium dominated.

An equilibrium strategy ${\displaystyle (m^{*},a^{*})}$ violates the intuitive criterion if there is a type ${\displaystyle \theta }$ and a message they can send ${\displaystyle m}$ such that:

${\displaystyle \min _{a\in A^{*}(\Theta ^{**}(m),m)}\left[u_{i}(m,a,\theta )\right]>u_{i}^{*}(\theta )}$ for some ${\displaystyle \theta \in \Theta ^{**}(m)}$

where ${\displaystyle m^{*}}$ is the equilibrium message, ${\displaystyle a^{*}}$ is the equilibrium response (action) of the receiver, and ${\displaystyle \Theta ^{**}(m)}$ is the set of types for which that message cannot be equilibrium dominated.[3]