Ultimatum Game

The rules, in one paragraph

Two strangers are seated. The experimenter hands one of them — the Proposer — a fixed amount of money, say $100. The Proposer must offer the other player, the Responder, some share of the pie: any amount from $0 to $100, in integer dollars. The Responder sees the offer and makes a single decision: accept, in which case both players pocket their shares of the split; or reject, in which case both players walk away with nothing. The game is played exactly once, anonymously, with no future interaction. That is the entire experiment.

It looks trivial. It is not. Stripped of every confound — repeated play, reputation, communication, threats of future retaliation — the ultimatum game presents the cleanest possible test of whether human decisions reduce to material payoff maximisation. They do not. Decades of evidence say so, and the question of why has driven the design of dozens of theories of social preferences.

The textbook prediction

Apply backward induction. Start at the Responder's decision node. Faced with an offer of ε > 0, the Responder chooses between ε (accept) and 0 (reject). For any self-interested Responder, ε > 0, so accept. Knowing this, the Proposer at the previous node chooses the offer that maximises their own payoff subject to the Responder accepting — which, by the above, is the smallest positive amount the rules permit. The subgame-perfect equilibrium (SPE) of the ultimatum game is therefore:

Proposer: offer = ε       (smallest positive unit)
Responder: accept any offer > 0

For a $100 pie split in $1 units, the SPE is "offer $1, Responder accepts". The Proposer keeps $99. This is not an obscure or contested prediction — it is what game theory textbooks have predicted since Selten formalised subgame perfection in the 1960s.

What actually happens

Güth, Schmittberger and Schwarze ran the first experiment in 1982 at the University of Cologne, with German student subjects and modest Deutsche Mark stakes. They found:

• Mean offer ≈ 37 percent of the pie. Far higher than the SPE-predicted minimum.
• Modal offer ≈ 50 percent. A surprisingly large fraction of Proposers offered exactly half.
• Substantial rejection rates for low offers. Offers below ~20 percent were rejected roughly half the time.

Forty years and thousands of replications later, the result is rock solid. In a 2011 meta-analysis covering more than 30 countries and 10,000 subjects, Oosterbeek, Sloof and van de Kuilen found mean offers of 40 percent and rejection rates near 16 percent overall, climbing sharply for offers below 25 percent. The ultimatum game is one of the most replicable findings in all of behavioral science.

The empirical rejection curve

The clearest single picture of the ultimatum game is the rejection curve — the probability that the Responder rejects, as a function of the percentage of the pie offered. Pooled across roughly two dozen Western lab studies:

Offer (% of pie)	Approximate rejection rate	Interpretation
50 %	~ 0 – 5 %	Almost always accepted; the fair split.
40 %	~ 5 – 10 %	Accepted by most; faintly suboptimal.
30 %	~ 20 – 30 %	Inflection point; rejection becomes common.
20 %	~ 45 – 60 %	Coin-flip; "unfair" signal dominates.
10 %	~ 60 – 80 %	Usually rejected; Responder pays to punish.
1 % (ε)	~ 90 %+	Almost always rejected; the SPE prediction fails completely.

The 30-percent inflection is the operationally important number. Below it, rejection is the typical response; above it, acceptance is. Proposers know this — they offer 40–50 percent in part because rationally-self-interested-but-aware-of-fairness Proposers anticipate the rejection curve and price their offer at the lowest level that comfortably clears it.

Cross-cultural variation

For two decades after Güth et al., critics could (and did) argue that the result was an artefact of the WEIRD subject pool: Western, Educated, Industrialised, Rich and Democratic university students with hyper-internalised fairness norms. Joseph Henrich and an international consortium (Henrich et al., Science 2001; Behavioral and Brain Sciences 2005) ran the ultimatum game in fifteen small-scale societies — Amazonian foragers, African pastoralists, Indonesian whale-hunters, Mongolian herders. The result demolished the artefact hypothesis but in a more interesting way: the WEIRD pool turned out to be an outlier on the generous end.

Society	Region	Mean offer	Rejection pattern
Machiguenga	Peruvian Amazon (foragers)	~ 26 %	Low offers rarely rejected
Hadza	Tanzania (foragers)	~ 33 %	Frequent rejection of low offers
Au & Gnau	Papua New Guinea	~ 38 %	Hyper-fair offers rejected (gift-obligation)
Tsimané	Bolivian Amazon	~ 37 %	Low rejection rates
WEIRD students	U.S. / Europe	~ 45 %	Reject below ~ 30 %
Orma	Kenya (pastoralists)	~ 44 %	Norms of harambee (collective effort) invoked
Lamalera	Indonesia (whale-hunters)	~ 57 %	Hyper-fair; rare rejections

Two robust patterns emerged. First, mean offers correlated with the extent of market integration and the importance of cooperation outside the household — the more a society's productive life depended on coordinating with non-kin (whaling, organised pastoralism, market exchange), the higher its offers. Second, the Au and Gnau presented a genuine puzzle: they routinely rejected offers that were too generous, because in their culture a large gift creates a reciprocal debt the recipient may not want to incur. Fairness, in other words, is real but its content is culturally calibrated.

Dictator game — decomposing the offer

Why are Proposers generous? Two reasons could be at work: pure altruism (they care about the Responder's welfare) and strategic fear (a low offer might be rejected). The dictator game, introduced by Kahneman, Knetsch and Thaler in 1986 and formalised by Forsythe, Horowitz, Savin and Sefton in 1994, separates them. In the dictator game, the Responder has no choice — the Proposer (now "Dictator") simply unilaterally splits the pie. There is no rejection option, no strategic concern, no game in the technical sense; only a one-shot allocation.

Stripped of the rejection threat, average Dictator allocations fall to roughly 20–30 percent, and a substantial minority of Dictators keep the entire pie. The difference between ultimatum offers and dictator offers — typically 15–20 percentage points — is the strategic component: how much Proposers give purely because they fear rejection. The dictator residual (~25 percent on average) is the pure-altruism baseline. Two-thirds, or so, of generous ultimatum offers are strategic; one-third reflects genuine other-regarding preferences.

Inequity aversion: the Fehr-Schmidt model

Ernst Fehr and Klaus Schmidt's 1999 Quarterly Journal of Economics paper proposed a single utility function that fits ultimatum, dictator, public-goods and trust-game data simultaneously. Each player's utility from outcome (x_i, x_j) is

U_i(x_i, x_j) = x_i − α_i · max(x_j − x_i, 0) − β_i · max(x_i − x_j, 0)

with 0 ≤ β_i ≤ α_i  and  β_i < 1.

In words: a player's own payoff x_i enters linearly; a penalty α applies when the player has less than the other (disadvantageous inequity, the "envy" or "spite" parameter); a smaller penalty β applies when the player has more (advantageous inequity, "guilt"). With population-distributions of (α, β) estimated from ultimatum-game data, the same parameters predict dictator offers, public-goods contributions and trust-game returns within a few percentage points. The Fehr-Schmidt model has become the workhorse formalism of behavioral game theory.

An alternative class of models, due to Matthew Rabin (1993) and elaborated by Dufwenberg and Kirchsteiger (2004), formalises reciprocity: utility depends on whether the other player is perceived as kind or unkind, and the response is symmetric in kind. Rabin-style models predict the same broad ultimatum patterns but better capture asymmetries in second-order belief (the Responder cares not just about what they got, but about what the Proposer could have done).

The brain on an unfair offer

Sanfey, Rilling, Aronson, Nystrom and Cohen's 2003 Science paper "The Neural Basis of Economic Decision-Making in the Ultimatum Game" scanned Responders with fMRI while they evaluated offers. Three regions tracked offer fairness:

• Bilateral anterior insula. Activation scaled with the unfairness of the offer and predicted rejection on a trial-by-trial basis. The anterior insula is the same region that lights up during physical disgust, social pain, and unpleasant taste — strongly suggesting the affective signature of unfairness is literally a visceral aversive response.
• Dorsolateral prefrontal cortex (DLPFC). Activation scaled with deliberative weighing — Responders who eventually accepted unfair offers showed higher DLPFC activity, consistent with this region's role in cognitive control overriding the prepotent affective response.
• Anterior cingulate cortex (ACC). Conflict monitoring — activation rose when the two systems disagreed.

Follow-ups (Knoch et al. 2006) used repetitive transcranial magnetic stimulation (rTMS) to disrupt right DLPFC — and subjects became less willing to reject unfair offers despite still finding them unfair. The deliberative override of the affective response had been knocked out, leaving raw self-interest behind. Rejection is not just instrumentally irrational; it is the deliberative system intentionally overriding the rational-self-interest response, costing money to enforce a fairness norm.

Does it survive high stakes?

A standard objection: maybe people reject only because the lab stakes are negligible. Slonim and Roth (1998) ran the ultimatum game in the Slovak Republic at stakes equal to about three months' average salary. Cameron (1999) ran it in Indonesia at stakes up to several months' wages. List and Cherry (2008) ran high-stakes versions in the United States. The result: stakes matter a little, but not nearly enough to rescue the SPE prediction.

At very low stakes ($10 pie), rejection of a 20-percent offer runs around 45 percent. At three-months-salary stakes, it falls to maybe 30 percent. The marginal effect is real and in the predicted direction (people fight harder for fairness when fairness is cheaper). But the effect is small. Pure income-maximisation predicts essentially zero rejections at any positive stake — the deviation from theory is what the entire literature is studying.

Key variants of the game

• Dictator game. Proposer allocates unilaterally; no rejection. Pure altruism baseline. Mean offer ~ 25 percent.
• Best-shot game (Harrison & Hirshleifer 1989). Multi-round version with proposer/responder roles alternating; tests reciprocity over time.
• Multi-stage / alternating-offer (Rubinstein 1982). Both parties can counter-offer; introduces discount factors. The Rubinstein-Ståhl bargaining game; SPE depends sensitively on patience.
• Mini-ultimatum game (Falk, Fehr & Fischbacher 2003). Proposer chooses between just two pre-specified splits; isolates intention from outcome — when the same unfair offer is forced (no fair alternative), rejection rates drop, showing perceived intention matters.
• Third-party punishment. A third observer can pay to reduce the Proposer's payoff. Substantial third-party punishment occurs even with no direct stake in the outcome — pure norm enforcement.
• Hyper-fair offers. Au and Gnau village data; some cultures reject offers above 50 percent because the gift creates a reciprocal obligation.
• Children & primates. Children develop "fair" offering behaviour by age 7–8 (Blake et al. 2015). Chimpanzees, by contrast, behave closer to the SPE prediction — they accept any nonzero offer (Jensen et al. 2007), suggesting strong-form fairness norms may be a human-distinctive social technology.

Where it matters outside the lab

• Wage bargaining. Final wage offers in non-union sectors are functionally ultimatum games. Employers offer above the worker's reservation wage in part because workers reject "insulting" offers — quit, slack, sabotage — even when accepting would be income-maximising. Akerlof's "fair wage-effort hypothesis" (1990) is essentially ultimatum-game logic in a labour market.
• Plea bargaining. Prosecutors propose a sentence; defendants accept or face trial. About 90 percent of U.S. felony convictions come from plea deals. Defendants reject lowball offers at rates that look like ultimatum-game rejections — and prosecutorial offers in repeat-player jurisdictions cluster around perceived-fair shares of the expected trial outcome.
• Divorce settlements. When one spouse has set a "final" number, the other faces an ultimatum-game choice: accept or litigate. Settlement rates fall sharply when the offer crosses a perceived fairness threshold.
• International negotiations. Trade deadlines, debt restructurings, climate-treaty haircuts — all force last-mover decisions that closely resemble ultimatum responses. Greece's repeated rejections of EU/IMF terms in 2010–2015 are a textbook example of ultimatum-game refusal at extreme stakes.
• Surge pricing & dynamic platforms. Uber and similar platforms post take-it-or-leave-it prices. Acceptance rates fall non-linearly when the displayed multiplier exceeds a perceived-fair threshold (typically 2×), even when the absolute price is still below the rider's reservation value.
• Tipping norms. Restaurant tipping is essentially a dictator-game decision (the server cannot retaliate) — and yet tip rates cluster near a customary 15–20 percent, well above zero. The dictator residual at work.

What it implies for economic theory

The ultimatum game is a single, cheap, replicable experiment that decisively rules out the strong-form "homo economicus" assumption — that human decisions can be modelled as maximisation of own material payoff. The implication is not that people are irrational; it is that their utility function includes terms beyond own payoff. Behavioral economists call these social preferences: inequity aversion, reciprocity, fairness-based norms, intentional kindness. Each can be formalised, parameterised and estimated; the resulting models often outperform the pure-payoff baseline on out-of-sample tasks.

The methodological lesson is at least as important. A well-designed minimal experiment can adjudicate between rival theories of preferences far more cleanly than observational data — which always confounds preferences with constraints, beliefs and information. The ultimatum game inaugurated a now-standard toolkit: dictator, trust, public-goods, gift-exchange, third-party punishment. Together they map out the shape of human social preferences with the kind of empirical resolution that, in 1980, behavioral economists did not have at all.

Common pitfalls

• Conflating ultimatum with dictator. The presence of the rejection option is what makes the ultimatum game a game; without it, the Proposer's behaviour reflects pure preferences, not strategic anticipation. Comparing the two is what lets you decompose generosity.
• Treating rejections as "irrational". They are payoff-suboptimal under the assumption that payoff is the only thing in the utility function. Under utility functions with fairness terms, rejections are perfectly rational. The lesson is about the model, not about the subject.
• Assuming the cross-cultural variation cancels out. It does not. Ignoring culture-specific norms (gift-obligation, market integration, kinship structure) loses the variance that identifies what fairness preferences are made of.
• Extrapolating to repeated games. Single-shot ultimatum behaviour is one thing; the same subjects in repeated play, with reputation and information, behave very differently. The clean theoretical prediction is for the one-shot game only.
• Using the SPE as a benchmark. It is the wrong benchmark — almost no Proposer plays SPE. The empirically grounded prediction is "offer around 40 percent, expect rare rejections". Modelling exercises that compare to SPE rather than to the empirical distribution will misattribute the gap.