Constrained Existence in Graph Games

• The constrained existence problem determines if a weak subgame perfect equilibrium exists in multiplayer turn-based graph games with ω-regular Boolean objectives by checking threshold payoff conditions.
• The methodology employs a fixpoint computation with alternating Remove and Adjust steps to label vertices and synthesize compact symbolic witnesses for equilibrium outcomes.
• Complexity results vary by objective class, ranging from P-complete for Explicit Muller to PSPACE-complete for Reachability and Safety, highlighting distinct computational tradeoffs.

Searching arXiv for the specified paper and closely related work on weak subgame perfect equilibria in graph games. The constrained existence problem is the decision problem associated with weak subgame perfect equilibria in multiplayer turn-based games played on a finite directed graph with $\omega$-regular Boolean objectives. Given an initialized game $(G,v_0)$ and threshold vectors $x,y\in\{0,1\}^n$ with $x\le y$ componentwise, it asks whether there exists a weak subgame perfect equilibrium whose outcome payoff $p$ satisfies $x\le p\le y$. The problem is studied for objective classes including Reachability, Safety, Büchi, co-Büchi, Parity, Explicit Muller, Muller, Rabin, and Streett, and its analysis is organized around a fixpoint computation that determines the payoff profiles compatible with weak SPE behavior (Brihaye et al., 2018).

1. Formal game model

A multiplayer turn-based graph game with Boolean objectives is a tuple

$G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$

where $\Pi=\{1,\dots,n\}$ is a finite set of players, $(V,E)$ is a finite directed graph in which every vertex has at least one outgoing edge, and $V=\bigsqcup_{i\in\Pi}V_i$ is a partition of the vertices among the players. When the token is on a vertex $(G,v_0)$0, Player $(G,v_0)$1 chooses the successor. For each player $(G,v_0)$2, the gain function

$(G,v_0)$3

is induced by an $(G,v_0)$4-regular objective $(G,v_0)$5 (Brihaye et al., 2018).

A play is an infinite path $(G,v_0)$6 with $(G,v_0)$7, and its payoff vector is

$(G,v_0)$8

The standard vertex-occurrence operators are

$(G,v_0)$9

The objective classes considered in the complexity classification are the following:

• Reachability$x,y\in\{0,1\}^n$0: $x,y\in\{0,1\}^n$1
• Safety$x,y\in\{0,1\}^n$2: $x,y\in\{0,1\}^n$3
• Büchi$x,y\in\{0,1\}^n$4: $x,y\in\{0,1\}^n$5
• co-Büchi$x,y\in\{0,1\}^n$6: $x,y\in\{0,1\}^n$7
• Parity$x,y\in\{0,1\}^n$8: $x,y\in\{0,1\}^n$9 is even
• Explicit Muller$x\le y$0: $x\le y$1
• Muller, Rabin, and Streett in the usual way

All these objectives are prefix-independent except Reachability and Safety.

2. Weak subgame perfect equilibrium

A strategy $x\le y$2 for Player $x\le y$3 maps each history ending in a vertex of $x\le y$4 to a successor vertex. The equilibrium concept studied here is not Nash equilibrium or subgame perfect equilibrium in the usual sense, but weak subgame perfect equilibrium. The defining restriction is on the space of admissible deviations: a deviating strategy $x\le y$5 must be finitely deviating from $x\le y$6, meaning that the two strategies differ on only finitely many histories (Brihaye et al., 2018).

In an initialized game $x\le y$7, a strategy profile $x\le y$8 is a weak Nash equilibrium if no player $x\le y$9 has a profitable finitely deviating deviation, that is, no $p$0 with

$p$1

A profile is a weak SPE if, in every subgame reachable by a history $p$2 ending in a vertex $p$3, the restriction of the profile is a weak Nash equilibrium.

A basic structural fact is that one-shot deviations suffice: Proposition 2 states that the weak SPE condition can be checked against deviations that differ only at the first decision. This equivalence is algorithmically central because it converts a seemingly global refinement into a local deviation test.

The weak SPE notion refines SPE in the sense adopted by the paper, but it does so under a restricted deviation model. The resulting equilibrium concept is particularly suited to infinite-duration graph games, where deviations that require infinitely many changes of behavior are excluded from the admissible counterfactuals.

3. Decision problem and fixpoint characterization

The constrained existence problem is defined as follows. Given an initialized game $p$4 and thresholds $p$5 with $p$6 componentwise, decide whether there exists a weak SPE $p$7 whose outcome payoff $p$8 satisfies

$p$9

The decision procedure is based on a labeling of each vertex by tentative payoff sets. For every vertex $x\le p\le y$0, the initialization is

$x\le p\le y$1

The algorithm then alternates two operations until a fixpoint is reached (Brihaye et al., 2018).

For odd $x\le p\le y$2, the Remove step deletes a payoff $x\le p\le y$3 from a vertex $x\le p\le y$4 whenever

$x\le p\le y$5

This captures the idea that if Player $x\le p\le y$6 can move from $x\le p\le y$7 only to successor labels that are strictly better in component $x\le p\le y$8, then the payoff $x\le p\le y$9 cannot persist as weak-SPE-compatible at $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$0.

For even $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$1, the Adjust step propagates the consequences of previous removals. If a payoff $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$2 was removed from some vertex, then $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$3 is purged from any vertex $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$4 for which there no longer exists a $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$5-labeled play with payoff $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$6, meaning a play all of whose vertices still carry the label $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$7.

At the fixpoint $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$8, the labels stabilize. The central characterization theorem states that there exists a weak SPE of payoff $G=(\Pi,V,(V_i)_{i\in\Pi},E,(Gain_i)_{i\in\Pi}),$9 in $\Pi=\{1,\dots,n\}$0 if and only if

• $\Pi=\{1,\dots,n\}$1 for every reachable vertex $\Pi=\{1,\dots,n\}$2, and
• $\Pi=\{1,\dots,n\}$3.

The full version further states that one can extract a polynomial-size good symbolic witness, namely a family of up to $\Pi=\{1,\dots,n\}$4 lassoes, from which a finite-memory weak SPE can be synthesized (Brihaye et al., 2018). This suggests that the fixpoint is not only a decision device but also a compact certificate for equilibrium existence.

4. Complexity landscape

The paper gives a complete complexity classification for the constrained existence problem across the standard classes of $\Pi=\{1,\dots,n\}$5-regular Boolean objectives (Brihaye et al., 2018).

Objective class	Complexity
Explicit Muller	P-complete
co-Büchi	NP-complete
Parity	NP-complete
Muller	NP-complete
Rabin	NP-complete
Streett	NP-complete
Büchi	in NP
Reachability	PSPACE-complete
Safety	PSPACE-complete

Two uniform tractability results accompany this classification. The problem is fixed-parameter tractable when parameterized by the number of players, and for Rabin, Streett, and Muller objectives also by the size of the objective description. Moreover, if the number of players $\Pi=\{1,\dots,n\}$6 is fixed, then the constrained existence problem is solvable in polynomial time.

The complexity bounds are derived from a fine-grained analysis of the fixpoint procedure. Each Remove/Adjust step can be implemented in

$\Pi=\{1,\dots,n\}$7

time, where $\Pi=\{1,\dots,n\}$8 is the maximum size of $\Pi=\{1,\dots,n\}$9, $(V,E)$0 is the cost of building $(V,E)$1, and $(V,E)$2 is the cost of testing existence of a single play with a given payoff. The classification then follows by showing, for each objective class, whether $(V,E)$3, $(V,E)$4, and $(V,E)$5 are polynomial, NP-bounded, or PSPACE-relevant.

A notable boundary case is Büchi objectives: the paper proves NP-membership, while hardness remains open.

5. Running example and interpretive value

The running example in the paper is a two-player game with Büchi objectives. Player 1 wants to visit $(V,E)$6 infinitely often, and Player 2 wants to visit $(V,E)$7 infinitely often. A positional strategy profile yields the outcome

$(V,E)$8

with payoff $(V,E)$9 (Brihaye et al., 2018).

This profile is not a Nash equilibrium, because Player 1 could deviate at every occurrence of $V=\bigsqcup_{i\in\Pi}V_i$0. It is nevertheless a weak SPE, because any profitable deviation is infinitely deviating. The example isolates the conceptual difference between ordinary deviation stability and weak deviation stability: the failure of NE does not preclude weak SPE when the profitable deviation requires infinitely many departures from the prescribed strategy.

Applying the Remove-Adjust procedure to the example produces a fixpoint labeling in which only payoff $V=\bigsqcup_{i\in\Pi}V_i$1 survives at $V=\bigsqcup_{i\in\Pi}V_i$2. By the characterization theorem, there exists a weak SPE with payoff $V=\bigsqcup_{i\in\Pi}V_i$3, and this is exactly the positional profile exhibited in the example. In the full version, the example is also used to illustrate witness extraction through a family of lassoes (Brihaye et al., 2018).

6. Position within equilibrium theory and remaining issues

The constrained existence problem sits at the intersection of algorithmic game theory, infinite-duration verification games, and $V=\bigsqcup_{i\in\Pi}V_i$4-regular objective synthesis. Its specificity lies in combining three ingredients: multiplayer turn-based graph games, Boolean $V=\bigsqcup_{i\in\Pi}V_i$5-regular winning conditions, and the weak SPE deviation model. The results show that the existence question does not admit a uniform complexity profile across objective classes: Explicit Muller objectives yield a P-complete problem, co-Büchi/Parity/Muller/Rabin/Streett objectives yield NP-complete problems, and Reachability/Safety objectives yield PSPACE-complete problems (Brihaye et al., 2018).

The dependence on objective type is structurally meaningful. Prefix-independent objectives fit especially well with payoff propagation through the fixpoint labels, whereas Reachability and Safety require more costly reasoning. This suggests that the deviation-restricted equilibrium concept preserves enough global structure for a uniform symbolic treatment, but not enough to erase the distinctions between objective classes.

Two additional points delimit the scope of the theory. First, the problem remains polynomial when the number of players is fixed, which sharply separates combinatorial blow-up in payoffs from graph size. Second, the Büchi case is only partially classified: NP-membership is established, but hardness is open. In that sense, the classification is complete everywhere except at one remaining frontier.

Taken together, these results turn the constrained existence problem into a benchmark decision problem for weak SPEs. The fixpoint computation of payoff labels $V=\bigsqcup_{i\in\Pi}V_i$6 provides both the semantic characterization of equilibrium outcomes and the algorithmic basis for exact complexity bounds, while the symbolic-witness construction shows that equilibrium existence can be certified without enumerating the full strategy space (Brihaye et al., 2018).