Borel stay-in-a-set games (Q1434481)

The authors consider an \(n\)-person stochastic game with a Borel state space and compact metric action sets. Under some measurability and continuity conditions, the following holds: If the payoff to each player \(i\) is 1 or 0 according to whether or not the stochastic process stays forever in a given Borel set \(G_i\) then there exists a Nash \(\varepsilon\)-equilibrium for every \(\varepsilon>0\). The case \(n=1\) leads to a gambling problem and is treated in greater detail.