Nash Equilibrium

General

• A situation where each players behavior is optimal
• As long as nobody changes their behavior, nobody will change the behavior
• Everyone is playing the best response
• there is always at least 1 equilibrium in pure or mixed strategies

Solving

Best Response

• for each player look the best responses to every action of every other player
  • players with the same strategies and payoffs can be grouped

Elimination of Dominated Strategies

• find a dominated strategy which will never be picked

• eliminate

• take another look at now changed strategies for everyone else

• find another dominated strategy which will never be picked

• repeat until only the Nash(es) are present

• normal form: remove a row or column from the matrix

• extensive form: cut off a branch completely

History

• a 10-page paper
• definition, existence, applications
• includes “mixed strategies”
• extremely influential

Math

$$\begin{array}{c}{s}^{\ast }=(s_1^{\ast },s_2^{\ast },\dots ,s_n^{\ast })\text{ is a Nash equilibrium if }\\ \forall i\in \{1,2,\dots ,n\}|u_i(s_i^{\ast },s_{-i}^{\ast })\ge u_i(s_i,s_{-i}^{\ast })\forall s_i\in S_i\end{array}$$

• $s^{\ast }=(s_1^{\ast },s_2^{\ast },\dots ,s_n^{\ast })$ represents the strategy profile, where $s_i^{\ast }$ is the strategy chosen by player $i$.
• $s_{-i}^{\ast }$ denotes the strategies of all players except player $i$.
• $u_i(s_i,s_{-i})$ is the utility (or payoff) function for player $i$, depending on their own strategy $s_i$ and the strategies of the others $s_{-i}$.
• The condition $u_i(s_i^{\ast },s_{-i}^{\ast })\ge u_i(s_i,s_{-i}^{\ast })$ for all $s_i\in S_i$ means that player $i$ cannot achieve a higher utility by deviating to any other strategy $s_i$ given that the other players stick to $s_{-i}^{\ast }$.