Other meanings of Proper equilibrium
Game theory
Proper equilibrium is a refinement of Nash equilibrium introduced by Roger Myerson in which inferior strategies must be played with vanishingly smaller probabilities than strategies that yield higher payoffs, providing a stringent test for whether an equilibrium survives carefully structured mistakes.
Proper equilibrium strengthens Nash equilibrium by ranking the likelihood of mistakes according to their strategic cost. In a finite normal-form game, an ε-proper strategy profile is completely mixed and requires that, for each player, a pure strategy yielding a lower expected payoff than another receive no more than ε times the probability assigned to the better strategy. A proper equilibrium is a limit of ε-proper equilibria as ε approaches zero.
The condition does more than require every player's main choice to be optimal. It says that mistakes may occur, but that more damaging mistakes must be much rarer than less damaging ones. Because probabilities of inferior choices disappear relative to better alternatives, every proper equilibrium is a Nash equilibrium, while the converse need not hold.
Proper equilibrium belongs to the family of Nash-equilibrium refinements designed to eliminate outcomes that depend on implausible off-equilibrium behavior. It is stronger than trembling-hand perfect equilibrium: every proper equilibrium is perfect, but some perfect equilibria fail the properness requirement.1
The distinction matters when several actions are best responses or when weakly dominated actions appear in equilibrium supports. Perfect equilibrium asks whether an outcome can survive small, unrestricted trembles; proper equilibrium adds an ordering of tremble probabilities based on payoff losses. It is therefore closely related to, but distinct from, sequential equilibrium, which specifies consistent beliefs at information sets in extensive-form games. Refinement theory treats these concepts as different ways of imposing discipline on behavior outside the equilibrium path.
Every finite game has at least one proper equilibrium, although identifying all of them can be difficult. Existence follows from constructing suitable ε-proper profiles for positive ε and taking a convergent subsequence as ε tends to zero; compactness of mixed-strategy spaces supplies the limiting argument.
Computational work commonly represents the requirement through perturbed payoff or probability conditions and studies equilibria of a sequence of increasingly stringent perturbations. Numerical methods must distinguish genuine limiting behavior from artifacts of finite precision, especially when probabilities differ by many orders of magnitude. The refinement is consequently most useful as a theoretical selection principle, while practical calculations often combine it with homotopy or continuation methods developed for equilibrium computation.2
Properness is sensitive to payoff comparisons that ordinary Nash analysis treats as behaviorally irrelevant. Two equilibria can have the same on-path payoffs yet differ in whether their off-equilibrium actions can be supported by mistakes whose relative frequencies are economically credible.
The refinement is defined for strategic-form games, but its interpretation often concerns threats and responses in dynamic settings after those games are converted into normal form. It does not by itself impose a unique prediction: a game may retain several proper equilibria, and properness does not replace beliefs, information, or an equilibrium-selection argument. Myerson introduced the concept as part of a broader program of refining Nash equilibrium, alongside related ideas such as sequential rationality and stability rather than as a universal solution to strategic indeterminacy.
The term “proper equilibrium” is used here exclusively in Myerson’s game-theoretic sense, not for physical, statistical, or informal notions of equilibrium.
Help improve the encyclopedia. Reports go straight to the site manager.