On Equilibria in Quantitative Games with Reachability/Safety Objectives

On (Subgame Perfect) Secure Equilibrium in Quantitative Reachability Games

“…Unfortunately the proof does not seem to extend to the multiplayer case. Indeed we face the same kind of problems encountered in [BBDP10,BBDP11], where the existence of secure equilibria is proved for two-player games and left open for multiplayer games.…”

“…The proof uses techniques completely different from the ones given in [BBDP10,BBDP11] for the existence of Nash equilibria, and secure equilibria in two-player games.…”

“…As an extension, we consider multiplayer quantitative reachability games with tuples of costs on edges (as in [BBDP11]). In these games, we assume that edges are labelled with tuples of strictly positive costs (one cost for each player).…”

“…The proof of Theorem 4.1 is inspired from ideas developed in [BBDP10,BBDP11]. The keypoint is to show that the existence of a secure equilibrium in a game (G, v 0 ) is equivalent to the existence of a secure equilibrium (with two additional properties) in the finite game Trunc d (T ) for a well-chosen depth d. The existence of the latter equilibrium is decidable.…”

“…For each player i = j, let ν v i,j be the memoryless strategy of player i in G induced by ν v −j . We define a finite-memory secure equilibrium in the game T using the same idea as in the proof of [BBDP11,Proposition 25]. The idea is to specify the required secure equilibrium as follows: each player i plays according to αβ ω (which is the outcome of this equilibrium) and punishes player j = i if he deviates from αβ ω , by playing according to τ i until depth |α|, and after that, by playing arbitrarily if α visits Goal j , and according to ν v i,j otherwise (where v is the vertex visited at depth |α| when deviating).…”