Game Theory Note - week 2

This week’s Game Theory is dedicated to Mixed-Strategy Nash Equilibrium.

Mixed strategy, different from pure strategy, means that players can choose an action according to a specific probability distribution (among all possible actions). The following concepts and definitions all derives from this idea:

Strategy $s_i$ : any probability distribution over the actions $A_i$ for agent i.

Pure strategy : only one action is played with positive probability.

Mixed strategy : more than one action is played with positive probability.

Support (of mixed strategy) : all the actions

We denote $s\_i \in S\_i$ as $S_i$ is the set of all strategies for user i. All strategies $S = S\_1 \times S\_2 \times \ldots \times S\_n$

Expected Utility is defined as follows:

In the equations above, a means a possible action profile from A. $a_j$ does not mean each of the action but the player j’s corresponding action in the corresponding profile.

Best response

$s\_{i}^{*} \in BR(s\_{-i}) iff \forall s\_i \in S\_i u\_{i}(s\_{i}^{*}, s\_{-i}) \ge u\_{i}(s\_i, s\_{-i})$

Nash Equilibrium

s=\<s\_1, s\_2,="" \ldots,="" s\_n=""\> \mbox( is a Nash Equilibrium iff }\forall i, s\_i \in BR(s\_{-i})</s_1,>

Theorem : Every finite game has a Nash Equilibrium. (While comparing to pure strategy games!)

It is often very hard to compute the Nash Equilibrium of a game, but in simple cases, in which we know the support, we can get the Nash Equilibrium by being acknowledged that a player will act indifferently facing a mixed strategy.