Luis Schmitz

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Ultimatum Game

Explore fairness vs. rationality in offer-acceptance dynamics with heterogeneous players.

Setup Parameters

Endowment (Total Pie)100
Proposer Offer40
Minimum Acceptable Offer30

Metrics

Acceptance Rate
0.0%
Avg Proposer Payoff
0.0/100
Total Welfare
0.0/100

Results Distribution

Run simulation to see results distribution

What to Notice

  • Set your offer amount and responder threshold, then run the simulation to see how fairness considerations affect acceptance rates and payoffs.
  • Try the Monte Carlo mode to see how heterogeneous populations respond to different offers.
  • The ultimatum game reveals tension between economic rationality and fairness norms.
  • Low offers are often rejected even when acceptance would benefit both parties.

Math & Strategy

1. Game Structure

The ultimatum game is a two-player sequential game where:

  • Proposer: Offers a split (x, S-x) of endowment S
  • Responder: Accepts or rejects the offer
  • Payoffs: If accepted: (S-x, x); If rejected: (0, 0)

2. Subgame Perfect Equilibrium (SPE)

Under standard rationality assumptions:

  • Responder strategy: Accept any offer x ≥ ε (where ε is the smallest monetary unit)
  • Proposer strategy: Offer x = ε (minimum positive amount)
  • SPE outcome: (S-ε, ε) ≈ (S, 0)

3. Behavioral Deviations

Empirical evidence shows systematic deviations from SPE:

3.1 Proposer Behavior

Modal offers are typically 40-50% of the endowment, not the theoretical minimum. This suggests:

  • Fairness preferences: Utility U_P = π_P - α|π_P - π_R|
  • Strategic anticipation: Expecting rejection of unfair offers
3.2 Responder Behavior

Rejection rates increase as offers become more unequal. Common models:

Threshold model: Accept if x ≥ θ where θ is individual fairness threshold

Inequity aversion: U_R = π_R - β \max(π_P - π_R, 0)

4. Population Heterogeneity

In heterogeneous populations, let F(θ) be the distribution of responder thresholds.

For offer x, acceptance probability is:

P(\text{accept}) = F(x) = \Pr(\theta ≤ x)

Proposer's expected payoff:

E[π_P(x)] = (S - x) \cdot F(x)

Optimal offer x^* satisfies first-order condition:

\frac{dE[π_P]}{dx} = -F(x) + (S - x) \cdot f(x) = 0

Where f(x) = F'(x) is the density function.

5. Common Distributions

5.1 Uniform Distribution

If θ ~ U[0, S], then F(x) = x/S and optimal offer is:

x^* = S/2
5.2 Normal Distribution

If θ ~ N(μ, σ²), the optimal offer requires numerical solution of:

Φ\left(\frac{x-μ}{σ}\right) = (S-x) \cdot \frac{1}{σ}φ\left(\frac{x-μ}{σ}\right)

Where Φ and φ are the CDF and PDF of standard normal distribution.

6. Welfare Analysis

Total expected welfare:

E[W] = S \cdot F(x)

Efficiency loss from rejections:

\text{Loss} = S \cdot (1 - F(x))

References:Wikipedia - Ultimatum Game,Güth et al. (1982)