Solution
The Rational Ultimatum
Show the problem again
A proposer offers a split of $100 in whole dollars to a responder, who accepts (the split happens) or rejects (both get $0). Both players are rational and care only about money. In the subgame perfect equilibrium where the responder accepts only strictly positive offers, how many dollars does the proposer offer?
Worked solution
The answer is $1. Work backwards: a rational responder accepts any positive amount, since something beats nothing. Knowing this, the proposer offers the minimum positive amount, $1, keeping $99. (In lab experiments, humans routinely reject offers below ~30%, making this a classic gap between game theory and behavior.)
Source: The ultimatum game, introduced by Werner Guth, Rolf Schmittberger and Bernd Schwarze (1982). Statement written for AxiomIQ.