We report on a failure to elicit extreme demands in modified ultimatum
games where player 1 is either structurally weaker or stronger than s
he or he would be in the standard ultimatum game. These games are obta
ined by extending the ultimatum game to costless two-period games; in
one type of game player 1 is given two opportunities to propose, and i
n the other, the players alternate playing the proposer role. Our resu
lts coupled with results reported by other experimenters suggest that
ultimatum ourcomes are in inherent contrast with those suggested by st
andard non-cooperative game theory. We argue that as long as one playe
r can veto any offer by the other, shares tend to be more even than is
predicted by the standard game-theoretic model.