Nash Equilibrium

• In this article, we are going to discuss about Nash equilibrium.

Motivation

• previously, we tried to determine and predict the course of game by using iterative dominance.
• However, we noticed that even with iterative dominance some games remained inconclusive.
• We want to investigate more on these games by using nash equilibrium.

Definition of Nash equilibrium

• A Nash equilibrium is the case in which all strategic uncertainty has been removed, so players behave exactly as others believe they will behave.
• A strategy profile $s \in S$ is a Nash equilibrium if and only if $s_i \in BR_i(s_{-i})$ for each player i. That is, $u_i(s_i, s_{-i}) \geq u_i(s_i', s_{-i})$ for every $s_i' \in S_i$ and i.

Finding nash equilibrium

• Let the normal form of game be,

• In order to find nash equilibrium, we want to find coordinates in this table where both players are best responding.
• Let's bold player 1's best response.

• and then, bold player 2's best response.

• Recall that both players must be best responding to each other.
• Thus, two strategy profiles are in nash equilibrium, (y, a) and (x, c)
• Note that one of these strategy profile (y, a) is not efficient, i.e. there is another strategy profile (x, c) that is more efficient.

Intuition behind Nash equilibrium

• A strategy profile (state) is a Nash equilibrium if and only if no player in the game has any incentive to unilaterally deviate from current strategy profile.
• This means that no one in the game actually wants to change their stategy, unless they can talk to each other.
• This statement above gives a reason why nash equilibrium is useful and stable.

When Nash equilibrium is enforced

• iterative game
• mediation
• pre-play communication

Strict Nash Equilibrium

• A nash equlilbrium is a strict nash equilibrium if and only if for every player i, thier best response is thier only best response.
• i.e. when the payoff is strctly larger than any other options.