Sequential Centipede Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Sequential Centipede Game

• Players: Player 1 and Player 2.
• Strategies: Player 1 acts at nodes $1$ and $3$, so a pure strategy specifies one action at each node; Player 2 acts at nodes $2$ and $4$, so a pure strategy specifies one action at each node.
• Rules:
  • Start at the first decision node with a growing surplus path.
  • Players alternate choosing Stop or Continue.
  • The player who stops ends the game at the current terminal payoff.

Game Tree

Payoff Details

• In the tree, $S=\text{Stop}$ and $C=\text{Continue}$.
• Let the decision nodes be:

• Terminal payoffs if the mover stops:
  • Stop at 1: $(2,0)$
  • Stop at 2: $(1,3)$
  • Stop at 3: $(4,2)$
  • Stop at 4: $(3,5)$
• If P2 continues at 4 (i.e. the game reaches the end), payoffs are $(6,4)$.

Derivation (Backward Induction)

• At node N4 (P2): compare Stop $(3,5)$ vs Continue $(6,4)$. P2 prefers Stop since $5>4$.

• At node N3 (P1): if Continue, P2 stops at 4 $\Rightarrow$ P1 gets $3$; stopping now gives $4$. Hence P1 stops at 3.

• At node N2 (P2): if Continue, P1 stops at 3 $\Rightarrow$ P2 gets $2$; stopping now gives $3$. Hence P2 stops at 2.

• At node N1 (P1): if Continue, P2 stops at 2 $\Rightarrow$ P1 gets $1$; stopping now gives $2$. Hence P1 stops immediately.

Nash Equilibrium

Result:

The unique subgame perfect Nash equilibrium strategy profile is:

Hence, Player 1 stops at the first move and the game ends.

Insights

Insight:

Backward induction can predict early termination even when mutual continuation would increase total surplus.