Ultimatum Game with a Non-Credible Threat

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Ultimatum Game with a Non-Credible Threat

• Players: Two players, Sender / proposer and Receiver / responder.
• Strategies: Sender chooses an offer $x$ to give the receiver from a pie of size $10$; Receiver chooses Accept or Reject after every possible offer; Threat strategy: reject any offer with $x<5$.

Rules

• Start with a fixed pie and a threatened rejection rule; Players move sequentially: the sender offers $x$, then the receiver accepts or rejects.
• The player who ignores non-credible off-path threats reaches the subgame-perfect outcome.
• If the receiver accepts offer $x$, payoffs are $(10-x,x)$; If the receiver rejects, payoffs are $(0,0)$.
• Payoffs are ordered as (Sender, Receiver).

Game Tree

Derivation (Best Response Analysis)

• If $x>0$, the receiver strictly prefers Accept because $x>0$.
• If $x=0$, the receiver is indifferent between Accept and Reject.
• Rejecting $0<x<5$ is therefore not sequentially rational.
• If the sender believes the receiver rejects all $x<5$, the sender’s best accepted offer is $x=5$.

Derivation (Nash Equilibrium)

• Strategy profile:

• Given the receiver’s strategy, offering less than $5$ gives sender payoff $0$.
• Offering $5$ gives sender payoff $5$.
• Offering more than $5$ gives sender payoff below $5$.
• Thus offering $5$ is a best response to the receiver’s threat strategy.
• Given offer $5$, the receiver accepts and receives $5$, so the receiver does not gain by deviating on the equilibrium path.
• Hence the profile is a Nash equilibrium.

Nash Equilibrium

Result:

The profile

is a Nash equilibrium, but it is not subgame perfect because rejecting $0<x<5$ is not credible.

Insights

Insight:

• A threat can support a Nash equilibrium without being credible.
• SGPE removes the threat because the receiver would not carry it out after a positive offer below $5$.
• Strategic moves are needed to make such threats credible.