By Lucrecia Crider
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Additional info for Introducing game theory and its applications
More formally, consider a subset A of Baire space; recall that the latter consists of all ωsequences of natural numbers. Then in the game GA, I plays a natural number a0, then II plays a1, then I plays a2, and so on. Then I wins the game if and only if and otherwise II wins. A is then called the payoff set of GA. It is assumed that each player can see all moves preceding each of his moves, and also knows the winning condition. Strategies Informally, a strategy for a player is a way of playing in which his plays are entirely determined by the foregoing plays.
The Nash equilibrium can be found by solving the following linear program to find a vector u: ∑u Minimize: i i Subject to the constraints: u≥0 Mu ≥ 1. The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the Mu vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
If instead, for some player, there is exact equality between and some other strategy in the set S, then the equilibrium is classified as a weak Nash equilibrium. Examples Coordination game In game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies. Coordination games are a formalization of the idea of a coordination problem, which is widespread in the social sciences, including economics, meaning situations in which all parties can realize mutual gains, but only by making mutually consistent decisions.