Social preferences: fairness

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Offer someone $2 out of$ 10, take it or leave it. Pure self-interest says they should take it: $2 beats nothing. Around half the time, they refuse, burning their own money to deny you yours. That single fact reshapes what we mean by a payoff.

The subgame-perfect prediction for the ultimatum game is stark: offer a penny, and it will be accepted. Real responders reject lowball offers, and dictators give money away with nothing forcing them to. People care about fairness, and modeling that as part of the payoff explains the data.

From earlier lessons

Two quick ones before we start.

Subgame perfection predicts the ultimatum responder accepts...

A payoff in game theory represents a player's...

Predict, then play

Split $10 with a responder who has fairness preferences: they accept or reject, and rejection leaves both empty-handed. Find how low you can go.

Why did that happen?

Take a guess, then reveal the reason

Standard theory assumes the responder cares only about their own dollars, so any positive amount beats zero. But people also dislike being treated unfairly, and that distaste is real enough to outweigh a small payout. Rejecting a stingy offer is not irrational once you include fairness in the utility: the responder is trading a little money for the satisfaction of not being exploited. Proposers anticipate this, which is why real offers cluster near an even split, not at a penny.

The math

Definition, inequity aversion (Fehr-Schmidt)

A responder who keeps $s$ (proposer keeps $\text{pie} - s$ ) has utility $s - \alpha\max(\text{other} - s, 0) - \beta\max(s - \text{other}, 0)$ : the envy term $\alpha$ penalizes being behind, the guilt term $\beta$ (smaller) penalizes being ahead. Rejecting yields $0$ , so the responder rejects any offer whose utility is negative. This turns fairness into part of the payoff, restoring rational choice.

When behind ( $s < \text{pie}/2$ ), utility is $s - \alpha(\text{pie} - 2s)$ , which is negative when $s < \dfrac{\alpha \cdot \text{pie}}{1 + 2\alpha}$ . With $\alpha = 0.5$ and a pie of $10$ , the threshold is $\dfrac{0.5 \cdot 10}{2} = 2.5$ : the responder rejects offers below $2.50, matching the empirical rejection spike. Dictator games (where the responder cannot reject) show smaller but still positive giving, and trust games show investors sending money that trustees partly return, all evidence that people value more than their own cash.

Work it out

Find the rejection threshold

Suppose the responder is offered $s$ with envy weight $\alpha = 0.5$ on a $10 pie. Being behind, their utility is $s - 0.5\,(\,(10 - s) - s\,) = s - 0.5(10 - 2s) = s - 5 + s = 2s - 5$ .

Set this to zero: $2s - 5 = 0$ , so $s = 2.5$ . Below $2.50 the utility is negative, so the responder rejects, preferring the $0$ of walking away to the sting of a lopsided split. Above $2.50 they accept. A stronger envy weight raises this threshold, and a proposer who understands it offers close to half to avoid rejection, exactly what the experiments show.

Key idea

People reject unfair offers and give unprompted, so payoffs are utility, not just money. Modeling fairness as inequity aversion turns “irrational” rejections into rational choices and predicts near-even offers.

Check yourself

Check yourselfapply

A responder with envy weight 0.5 faces a $10 pie. What is the lowest offer they accept?

Check yourselfanalyse

How does inequity aversion make rejecting a lowball offer rational?

Check yourselfbrings back: subgame perfection

Why does the subgame-perfect 'offer a penny' prediction fail?

Sources

Güth, W., Schmittberger, R., and Schwarze, B. (1982). An experimental analysis of ultimatum bargaining. Journal of Economic Behavior and Organization, 3(4), 367-388.
Fehr, E., and Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. Quarterly Journal of Economics, 114(3), 817-868.
Berg, J., Dickhaut, J., and McCabe, K. (1995). Trust, reciprocity, and social history. Games and Economic Behavior, 10(1), 122-142.