# Strategies in Game Theory

payoff: 即為在對方某一行動下，自己在某一個特定策略所獲得的 utility。在 pure strategy 中會是 metrix 中的一個數值。在 mixed strategy 中需要乘以 player 1 行動的機率。

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Best Response (BR)

Dominant Strategies (DS)

A weakly dominant strategy

Pure strategy Nash Equilibrium
- In any Nash equilibrium each player receives a payoff that is equal to both his maximin value and his minimax value. (Problem Set 3.4)

Pareto Optimal

Mixed strategy Nash Equilibrium

Elimination of strictly dominated strategies

Elimination of weakly dominated strategies
- 只要有一個 action 被另一個 action dominated，就能夠將它移除

Maxmin strategy

Minimax strategy

Correlated Equilibrium

### Subgame Perfection (4-4)

**def. subgame of G rooted at h**
The subgame of G rooted at h is the restriction of G to the desendents of H.

**def. subgames of G**
The set of subgames of G is defined by the subgames of G rooted at each of the nodes in G.

### Subgame Perfect Equilibrium (SPE)
$s$ is a subgame perfect equilibrium of $G$ if for any subgame $G'$ of $G$, the restriction of $s$ to $G'$ is a Nash equilibrium of $G'$

**Notes**
- since G is its own subgame, every SPE is a NE.
- this definition rules out "non-credible threats"