  In the standard minority game, each agent in the minority group receives the
same payoff regardless of the size of the minority group. Of great interest for
real social and biological systems are cases in which the payoffs to members of
the minority group depend on the size of the minority group. This latter
includes the fixed sum game. We find, remarkably, that the phase structure and
general scaling behavior of the standard minority game persists when the payoff
function depends on the size of the minority group. there is still a phase
transition at the same value of z, the ratio of the dimension of the strategy
space to the number of agents playing the game. We explain the persistence of
the phase structure and argue that it is due to the absence of temporal
cooperation in the dynamics of the minority game. We also discuss the behavior
of average agent wealth and the wealth distribution in these variable payoff
games.
