Commitment Game / Sequential Decision Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Commitment Game / Sequential Decision Game

• Players: Two players, Ulysses before approaching the Sirens and Ulysses under the influence of the Sirens.
• Strategies: Player 1: Get Close or Stay Away; Player 2: Get Closer or Stay Safe if Player 1 gets close; Commitment version: Player 1 can remove Player 2’s option to Get Closer.

Rules

• Start with Ulysses choosing whether to approach the Sirens; Players move sequentially: pre-temptation Ulysses chooses access, then tempted Ulysses chooses whether to get closer.
• The player who commits before temptation avoids the dynamically inconsistent outcome.
• Player 1 first chooses whether to approach the coast.
• If Player 1 gets close, Player 2 chooses whether to get closer or stay safe; Payoffs are ordered as (Player 1, Player 2).

Game Tree

• $C=$ Get Close, $A=$ Stay Away, $G=$ Get Closer, $S=$ Stay Safe.

Derivation (Best Response Analysis)

• If Player 1 gets close, Player 2 compares $10$ from Get Closer with $-10$ from Stay Safe.
• Player 2 chooses Get Closer.
• Anticipating this, Player 1 compares Stay Away payoff $0$ with Get Close payoff $-10$.
• Player 1 chooses Stay Away.

Derivation (Nash Equilibrium)

• Backward induction gives Player 2’s continuation action:

• Player 1’s optimal initial action is:

• The SGPE is therefore Stay Away, with the off-path continuation Get Closer.

Nash Equilibrium

Result:

Without commitment, the SGPE is: (Stay Away,Get Closer) with outcome $(0,0)$. \newline With commitment, Ulysses ties himself and fills the crew’s ears with wax, removing Get Closer. The outcome becomes: (Get Close,Stay Safe) with payoff $(10,-10)$.

Social Optimum

• For Player 1, the best feasible outcome is Get Close and Stay Safe.
• Commitment makes that outcome reachable by removing the future temptation.

Insights

Insight:

• Commitment can solve a conflict between present and future selves.
• Tying oneself removes the future self’s harmful action.
• The strategic move changes the SGPE outcome.