Permissive Equilibria in Multiplayer Reachability Games

TL;DR

It is shown that the existence of a multi-strategy that is a Nash equilibrium or a subgame perfect equilibrium can be decided, while satisfying some upper-bound constraints on the penalties in PSPACE, if the upper-bound penalties are given in unary.

Abstract

We study multi-strategies in multiplayer reachability games played on finite graphs. A multi-strategy prescribes a set of possible actions, instead of a single action as usual strategies: it represents a set of all strategies that are consistent with it. We aim for profiles of multi-strategies (a multi-strategy per player), where each profile of consistent strategies is a Nash equilibrium, or a subgame perfect equilibrium. The permissiveness of two multi-strategies can be compared with penalties, as already used in the two-player zero-sum setting by Bouyer, Duflot, Markey and Renault. We show that we can decide the existence of a multi-strategy that is a Nash equilibrium or a subgame perfect equilibrium, while satisfying some upper-bound constraints on the penalties in PSPACE, if the upper-bound penalties are given in unary. The same holds when we search for multi-strategies where certain players are asked to win in at least one play or in all plays

Figures (6)

• Figure 1: In this game, player $1$ owns all vertices and wants to reach $v_1$. For all $k \in \mathbb{N}$, we define the multi-strategy $\mathop{\mathrm{\Theta}}\nolimits^k_1$ such that for all $h \in \mathop{\mathrm{Hist}}\nolimits(v_0)$, $\mathop{\mathrm{\Theta}}\nolimits^k_1(h) = \{v_0,v_1 \}$ if $\mathop{\mathrm{Last}}\nolimits(h) = v_0$ and $|\{ n\in \mathbb{N} \mid h_n = v_0 \}| \leq k$, and $\mathop{\mathrm{\Theta}}\nolimits^k_1(h) = \{ v_1 \}$ otherwise. We have that for all $k \in \mathbb{N}$, for all $\sigma_1 \lesssim \mathop{\mathrm{\Theta}}\nolimits^k_1$, $\mathop{\mathrm{Gain}}\nolimits_1( \langle \sigma_1 \rangle_{v_0})=1$ (and thus $\mathop{\mathrm{\Theta}}\nolimits^k_1$ is a permissive SPE), but for all $k \in \mathbb{N}$, $\langle \mathop{\mathrm{\Theta}}\nolimits^{k}_1 \rangle_{v_0} \subseteq \langle \mathop{\mathrm{\Theta}}\nolimits^{k+1}_1 \rangle_{v_0}$.
• Figure 2: An example of a reachability game where player 1 (resp. player 2) owns circle (resp. rectangle) vertices. The initial vertex is $v_0$. Target vertices $F_{1} = \{ v_3,v_6,v_8,v_9 \}$ of player $1$ and $F_{2} = \{v_4,v_6 \}$ of player $2$ are drawn with gray vertices and double-bordered vertices respectively.
• Figure 3: Examples of permissive equilibria: (a) a permissive NE and (b) a permissive SPE
• Figure 4: Examples of trees that do not respect: (a) the resistance to internal deviations since $\mathop{\mathrm{Gain}}\nolimits_i(\rho')=0$ but $\mathop{\mathrm{Gain}}\nolimits_i(\rho) = 1$; (b) the resistance to external deviations since $\mathop{\mathrm{Gain}}\nolimits_i(\rho) = 0$ but Player $i$ can win from $u'$.
• Figure 5: Example of forest that does not respect the resistance to constrained external deviations since $\mathop{\mathrm{Gain}}\nolimits_i(\rho)=0$ but $\mathop{\mathrm{Gain}}\nolimits_i(\rho')=1$.

Theorems & Definitions (20)

• Definition 1: Penalties
• Definition 2: Weakly and strongly winning
• Example 3
• Definition 4
• Theorem 5
• Remark 6
• Definition 7: Good forest
• Theorem 8
• Definition 9
• Remark 10