Summary EC 109 Nash Equilibrium and mixed strategies Review

Nash Equilibrium and Mixed Strategies

Nash equilibrium

● Nash equilibrium distinguishes stable outcomes from unstable ones
○ All players play the best response
○ 𝑎𝑖∗ ∈ 𝑎𝑟𝑔𝑚𝑎𝑥𝜋𝑖 (𝑎𝑖 , 𝑎−𝑖
∗
)
● Dominant strategy equilibria are a subset of Nash equilibria, which is a subset
of rationalisable outcomes
○ Cannot always be reached - (coordination games)
● Can interpret Nash equilibria as:
○ A self-enforcing outcome - no individual has an incentive to change
○ The stable point in a dynamic adjustment process
○ An outcome arrived at by rational introspection
● Players respond to correct expectations about the other’s strategy
○ Their expectations are correct because:
■ They are updated
■ Players have reputations
■ There are focal points due to cultural similarities

Single vs multiple equilibria

● Some games have multiple pure strategy equilibria
● The one that is chosen is determined by
○ Conventions - self-reinforcing
○ Communication - verbally or nonverbally
○ Mediation by 3rd parties to coordinate activity

Mixed strategies

● Pure strategies - players do not randomise their choice
● Mixed strategy - players randomise between pure strategies
● Games which do not have a pure strategy NE have a mixed strategy NE
● Mixed strategies can be interpreted as:
○ Players randomising over strategies
○ Mass action interpretation