22  The Ultimatum Game

The Ultimatum Game is a foundational experiment in behavioral economics and game theory that challenges traditional assumptions about rational choice. It was first introduced by Güth, Schmittberger, and Schwarze (1982).

The Ultimatum Game involves two players who must divide a windfall of \(\$r\). The game proceeds sequentially:

  1. The Proposer suggests a division of the total amount \(r\)

  2. The Responder observes the proposed division and chooses to either:

    • Accept: Both players receive the proposed amounts
    • Reject: Both players receive nothing

The Proposer can offer any division from keeping everything \((r, 0)\) to giving everything away \((0, r)\). The most common proposals involve the Proposer keeping \(r-d\) and offering \(d\) to the Responder for some \(d>0\).

Game theory predicts a stark outcome through backward induction:

  1. If the Proposer offers any positive amount, the Responder does strictly better by accepting than rejecting (which yields zero).
  2. Therefore, the Proposer will not reject any positive offers.
  3. Knowing this, the Proposer should offer the smallest positive amount possible.

This analysis suggests that the Proposer should keep nearly everything, offering only a token amount to ensure acceptance.

However, this theoretical prediction fails dramatically in practice:

“…offers typically average about 30-40 percent of the total, with a 50-50 split often the mode. Offers of less than 20 percent are frequently rejected. These facts are not now in question. What remains controversial is how to interpret the facts and how best to incorporate what we have learned into a more descriptive version of game theory.” (p. 210, Camerer and Thaler 1995)

Two key findings contradict the standard game-theoretic prediction:

  1. Rejections: Low offers (under \(20\%\)) are frequently rejected, which is impossible to reconcile with payoff maximization since something is better than nothing.

  2. Generous offers: While proposing larger than token amounts could be rational if Proposers fear rejection, offers are typically larger than the amount proposers believe would result in acceptance (Henrich et al. 2001).

To isolate the role of strategic considerations versus other motivations, researchers developed the Dictator Game (Kahneman, Knetsch, and Thaler 1986):

In experiments involving this game, a significant number of Allocators give some money. The distribution of donations tends to be bimodal, with peaks at zero and at half the total (Engel 2011).

These findings challenge the assumption of non-tuism in game theory (and more broadly in economics). Non-tuism assumes that individuals make choices based solely on their own utilities, not on others’ utilities. A non-tuist need not be selfish—their utility function may include altruistic preferences, deriving satisfaction from helping others. But crucially, they maximize only their own utility function, even when that function includes concern for others.

Yet the Ultimatum and Dictator games demonstrate that people systematically violate non-tuism even in anonymous, one-shot interactions. Players reject non-zero offers and give away money with no strategic benefit, suggesting they value fairness and others’ welfare directly—not merely their own satisfaction from acting fairly. These behaviors indicate that people’s utility functions genuinely depend on others’ outcomes (Bicchieri and Zhang 2012; Capraro, Halpern, and Perc 2024).