Price of stability

In game theory, the price of stability (PoS) of a game is the ratio between the best objective function value of one of its equilibria and that of an optimal outcome. The PoS is relevant for games in which there is some objective authority that can influence the players a bit, and maybe help them converge to a good Nash equilibrium. When measuring how efficient a Nash equilibrium is in a specific game we often also talk about the price of anarchy (PoA), which is the ratio between the worst objective function value of one of its equilibria and that of an optimal outcome. Another way of expressing PoS is: In particular, if the optimal solution is a Nash equilibrium, then the PoS is 1. In the following prisoner’s dilemma game, since there is a single equilibrium we have PoS = PoA = 1/2. On this example which is a version of the battle of sexes game, there are two equilibrium points, and , with values 3 and 15, respectively. The optimal value is 15. Thus, PoS = 1 while PoA = 1/5. The price of stability was first studied by A. Schulz and N. Stier-Moses while the term was coined by E. Anshelevich et al. Schulz and Stier-Moses focused on equilibria in a selfish routing game in which edges have capacities. Anshelevich et al. studied network design games and showed that a pure strategy Nash equilibrium always exists and the price of stability of this game is at most the nth harmonic number in directed graphs. For undirected graphs Anshelevich and others presented a tight bound on the price of stability of 4/3 for a single source and two players case. Jian Li has proved that for undirected graphs with a distinguished destination to which all players must connect the price of stability of the Shapely network design game is where is the number of players. On the other hand, the price of anarchy is about in this game. Network design games have a very natural motivation for the Price of Stability. In these games, the Price of Anarchy can be much worse than the Price of Stability. Consider the following game.