Sequential Extensive Form Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Sequential Extensive Form Game

• Players: Two players, Player 1 and Player 2.
• Strategies: Player 1: Up / Down; Player 2: contingent strategies $(LL)$, $(LR)$, $(RL)$, $(RR)$.

Rules

• Start with Player 1 at the initial decision node.
• Players move sequentially: Player 1 chooses Up or Down, then Player 2 chooses Left or Right at the reached node.
• The player who chooses optimally by backward induction reaches the subgame-perfect outcome.
• Player 1 moves first
• Player 2 observes and responds

Game Tree

Payoff Matrix

• In Player 2’s contingent strategy, the first action is after Up and the second action is after Down.

Derivation (Best Response Analysis)

• After Up, Player 2 compares Left and Right:
  • Left gives $4$
  • Right gives $1$
  • So Player 2 chooses Left after Up.
• After Down, Player 2 compares Left and Right:
  • Left gives $3$
  • Right gives $2$
  • So Player 2 chooses Left after Down.
• Hence Player 2’s sequentially rational contingent strategy is $LL$.
• Given $LL$, Player 1 compares:
  • Up gives $2$
  • Down gives $3$
  • So Player 1 chooses Down.

Derivation (Nash Equilibrium)

• Player 1’s best responses in the strategic form are:
  • to $LL$: Down, since $3>2$
  • to $LR$: Up, since $2>1$
  • to $RL$: Up, since $4>3$
  • to $RR$: Up, since $4>1$
• Player 2’s best responses are:
  • to Up: $LL$ and $LR$, since Left gives $4>1$
  • to Down: $LL$ and $RL$, since Left gives $3>2$
• Strategic-form Nash equilibria are (Down,LL) and (Up,LR).
• Only (Down,LL) is subgame perfect because after Down, Right is not optimal for Player 2.

Nash Equilibrium

Result:

Nash Equilibria: (Down,LL) and (Up,LR)

Subgame Perfect Nash Equilibrium: (Down,LL)

Equilibrium Path:

Down then Left, with payoff $(3,3)$.

Social Optimum

• Joint payoff is $6$ at (Up,L) and (Down,L).
• So both Left outcomes are socially efficient.

Insights

Insight:

• Strategies must specify actions at all nodes
• Off-path actions matter for equilibrium