Incomplete-Information Ultimatum Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Incomplete-Information Ultimatum Game

• Players: Three players, Sender, Receiver, and Nature, who selects the receiver’s type.
• Strategies: Sender chooses whether to offer $0$ or $5$ from a pie of size $10$; Rational receiver accepts any non-negative offer; Crazy receiver accepts only offers of at least $5$.

Rules

• Start with Nature selecting the receiver’s type with probability $p$; Players move sequentially: the sender offers $0$ or $5$, then the receiver accepts or rejects.
• The player who chooses optimally given beliefs maximises expected payoff.
• With probability $p$, the receiver is crazy.
• With probability $1-p$, the receiver is rational; The sender does not observe the receiver’s type.
• Payoffs are ordered as (Sender, Receiver).

Game Tree

Derivation (Best Response Analysis)

• If Sender offers $0$, only the rational receiver accepts.
• Sender’s expected payoff from offering $0$ is:

• If Sender offers $5$, all receivers accept.
• Sender’s expected payoff from offering $5$ is:

Derivation (Nash Equilibrium)

• Offering $5$ is optimal when:

• Therefore:

• If $p>\frac{1}{2}$, the sender offers $5$.
• If $p<\frac{1}{2}$, the sender offers $0$.
• If $p=\frac{1}{2}$, the sender is indifferent.

Nash Equilibrium

Result:

Offering $5$ is a Perfect Bayesian Equilibrium strategy if and only if:

Insights

Insight:

• A reputation for irrational toughness can shift bargaining power.
• The receiver benefits when the sender assigns enough probability to the crazy type.
• Generating sufficient doubt about rationality can be a strategic move.