  We consider a collective version of Parrondo's games with probabilities
parametrized by rho in (0,1) in which a fraction phi in (0,1] of an infinite
number of players collectively choose and individually play at each turn the
game that yields the maximum average profit at that turn. Dinis and Parrondo
(2003) and Van den Broeck and Cleuren (2004) studied the asymptotic behavior of
this greedy strategy, which corresponds to a piecewise-linear discrete
dynamical system in a subset of the plane, for rho=1/3 and three choices of
phi. We study its asymptotic behavior for all (rho,phi) in (0,1)x(0,1], finding
that there is a globally asymptotically stable equilibrium if phi<=2/3 and,
typically, a unique (asymptotically stable) limit cycle if phi>2/3 ("typically"
because there are rare cases with two limit cycles). Asymptotic stability
results for phi>2/3 are partly conjectural.
