For a cost-minimization game with positive social optimum and a nonempty set of the specified Nash equilibria, the price of stability isIt measures the cost of requiring stability even when the best equilibrium can be selected. The equilibrium class, such as pure or mixed, and social-cost convention must be stated.
In a symmetric two-player game with actions , let each player's cost be one at , two at and three when the actions differ. Both coordinated choices are Nash equilibria; the best equilibrium has total cost two, matching the optimum, so the price of stability is one. The worst equilibrium instead costs four, giving price of anarchy two. The distinction matters when a designer can recommend an outcome but cannot force anyone to remain at an unstable one.[ref]
The Price of Stability for Network Design with Fair Cost Allocation by Elliot Anshelevich and others. Introduction, best equilibrium and stability interpretation
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The Price of Stability (PoS) is a concept in game theory and algorithmic social choice that measures the efficiency of equilibria in games, particularly in the context of strategic interactions among multiple agents or players. Specifically, it quantifies how much the performance of the best Nash equilibrium (a stable state where no player has anything to gain by changing only their own strategy) deviates from the optimal outcome that could be achieved with cooperation.