  Toral introduced so-called cooperative Parrondo games, in which there are N
players (3 or more) arranged in a circle. At each turn one player is randomly
chosen to play. He plays either game A or game B. Game A results in a win or
loss of one unit based on the toss of a fair coin. Game B results in a win or
loss of one unit based on the toss of a biased coin, with the amount of the
bias depending on whether none, one, or two of the player's two nearest
neighbors have won their most recent games. Game A is fair, so the games are
said to exhibit the Parrondo effect if game B is losing or fair and the random
mixture (1/2)(A+B) is winning. With the parameter space being the unit cube, we
investigate the region in which the Parrondo effect appears. Explicit formulas
can be found if N=3,4,5,6 and exact computations can be carried out if
N=7,8,9,...,19, at least. We provide numerical evidence suggesting that the
Parrondo region has nonzero volume in the limit as N approaches infinity.
