Stay-in-a-set games

On Nash Equilibria in Stochastic Games

“…This also generalizes a result of Condon [3]. Open and closed sets form the lowest level of the Borel hierarchy, and together with [23], this paper answers positively the existence of ( ) Nash equilibria in such games. We leave the generalization of these results to higher levels of the Borel hierarchy as an interesting open problem.…”

“…Formally, each player i has a subset of states S i as their safe states, and gets a payoff 1 if the play never leaves the set S i and gets payoff 0 otherwise. This result was generalized to general state and action spaces [23,15], where only -equilibria exist. However, not much more is known: for example, the existence of -Nash equilibria when the payoff for each player is given by an open set ("reachability games") remained open.…”

“…We refer to β as the discount-factor. The proof of the next lemma uses Kakutani's Fixed point theorem to show the existence of Nash equilibria in discounted games [25,23].…”

“…This construction yields an -Nash equilibrium in memoryless strategies. In contrast, the proof for safety games [23] proceeds by induction on the number of players, and does not give a Nash equilibrium in memoryless strategies. It is not known if exact Nash equilibria in memoryless strategies exist for safety games.…”

“…For games with payoffs defined by Borel sets, surprisingly little is known. Secchi and Sudderth [23] showed that exact Nash equilibria do exist when all players have payoffs defined by closed sets ("safety games"), where the objective of each player is to stay within a certain set of good states. Formally, each player i has a subset of states S i as their safe states, and gets a payoff 1 if the play never leaves the set S i and gets payoff 0 otherwise.…”