Finite Perfect-Information Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Finite Perfect-Information Game

• Players: Player 1 and Player 2.
• Strategies: Player 1 chooses L or R; Player 2 chooses u or d at the reached node.
• Rules: Player 2 observes Player 1’s move, and optimal play is found by backward induction.

Game Tree

Payoff Details

• $(L,u) \mapsto (2,0)$
• $(L,d) \mapsto (0,2)$
• $(R,u) \mapsto (1,4)$
• $(R,d) \mapsto (2,3)$

Derivation (Backward Induction)

• After $L$, Player 2 compares the second payoff from $u$ and $d$: $0$ vs $2$. Hence Player 2 chooses $d$ after $L$.
• After $R$, Player 2 compares the second payoff from $u$ and $d$: $4$ vs $3$. Hence Player 2 chooses $u$ after $R$.

• Anticipating Player 2’s optimal continuation actions, Player 1 compares the first payoff from $L$ and $R$: $0$ vs $1$.
• Hence Player 1 chooses $R$.

Nash Equilibrium

Result:

A subgame perfect Nash equilibrium strategy profile is:

Insights

Insight:

In sequential games, strategies must specify actions at every contingency.

Backward induction selects optimal actions in every subgame.