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If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player ...
A Bayesian Nash Equilibrium (BNE) is a Nash equilibrium for a Bayesian game, which is derived from the ex-ante normal form game associated with the Bayesian framework. In a traditional (non-Bayesian) game, a strategy profile is a Nash equilibrium if every player's strategy is a best response to the other players' strategies.
The pure Nash equilibria are the points in the bottom left and top right corners of the strategy space, while the mixed Nash equilibrium lies in the middle, at the intersection of the dashed lines. Unlike the pure Nash equilibria, the mixed equilibrium is not an evolutionarily stable strategy (ESS).
Strategies per player: In a game each player chooses from a set of possible actions, known as pure strategies. If the number is the same for all players, it is listed here. Number of pure strategy Nash equilibria: A Nash equilibrium is a set of strategies which represents mutual best responses to the other strategies. In other words, if every ...
Example 1: Two-Stage Repeated Game with Multiple Nash Equilibria Example 1 shows a two-stage repeated game with multiple pure strategy Nash equilibria. Because these equilibria differ markedly in terms of payoffs for Player 2, Player 1 can propose a strategy over multiple stages of the game that incorporates the possibility for punishment or ...
The solution for player 2 is equivalent. Using numerical values b 1 = 2, b 2 = −1, w = 3, this example transforms into a simple battle of the sexes, as shown in Figure 2. The game has two pure Nash equilibria, (+1, +1) and (−1, −1). These are also the local maxima of the potential function (Figure 3).
It should be clear that mixed PoA ≥ pure PoA, because any pure Nash equilibrium is also a mixed Nash equilibrium (this inequality can be strict: e.g. when =, = =, =, and = =, the mixed strategies = = (/, /) achieve an average makespan of 1.5, while any pure-strategy PoA in this setting is /). First we need to argue that there exist pure Nash ...
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.