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:

ui(s)=∑a∈Aui(a)P(a∣s)\ P(a∣s)=∏j∈Nsj(aj) u_{i}(s) = \sum_{a \in A} u_{i}(a) P(a|s) \ P(a|s) = \prod_{j \in N} s_j(a_j)

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.


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