Gap Maximality Principle in Stochastic Games

• Gap Maximality Principle is the maximality inheritance concept that identifies a state s* maximizing the gap between an upper fixed point and the true reachability value.
• It underpins the 2WP-BVI algorithm by proving that optimal action choices propagate this maximal gap, thereby establishing the uniqueness of the fixed point without explicit end-component detection.
• The technique bridges proof theory and algorithmic strategy in finite stochastic games, ensuring convergence by eliminating spurious upper bounds in end-component regions.

to=arxiv_search.search 山大发cript 天天中彩票能json {"^^^^^^^^11query11 inheritance principle^^^^^^^^11\11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^^^^^", "^^^^^^^^11max_results11 ^^^^11\11^^^^} to=arxiv_search.search 񹚂ാസcript 尚度քայինjson {"^^^^^^^^11query11 Maximality Principle^^^^^^^^11\11 OR ^^^^^^^^11\11 maximizer^^^^^^^^11\11 arXiv11^^^^^^^^", "^^^^^^^^11max_results11 ^^^^11\11query11^^^^} to=arxiv_search.search anasiyana 微信天天彩票ңизjson {"^^^^^^^^11query11 Path Games and Maximality Inheritance in Bounded Value Iteration for Stochastic Games^^^^^^^^11\11 "^^^^^^^^11max_results11 ^^^^11\11^^^^} “Gap Maximality Principle” is best understood as an Editor’s term for the argument pattern called the maximality inheritance principle in the analysis of bounded value iteration for reachability in finite stochastic games. In that setting, the central object is a state PRESERVED_PLACEHOLDER_^^^^11query11^^^^ that maximizes the pointwise gap between an upper fixed point and the reachability value, typically PRESERVED_PLACEHOLDER_^^^^11\11^^^^. The principle asserts, informally, that under suitable action choices this maximality is inherited by successors, and then by all states reachable under an induced memoryless strategy profile. The resulting max-gap region yields a contradiction with target reachability in the widest-path semantics, establishing uniqueness of the relevant fixed point and convergence of the upper bound without explicit end-component detection (&&&^^^^11query11^^^^&&&).

^^^^11\11^^^^. Terminological scope and basic idea

The term “Gap Maximality Principle” is not formally introduced in the literature under that name. In the relevant stochastic-game source, the actual term is maximality inheritance principle, and the repeatedly used technical notion is the gap maximizer. The intended object is a state whose gap

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^

is globally maximal among all states, with the paper’s main instances taking PRESERVED_PLACEHOLDER_^^^^11max_results11^^^^ and PRESERVED_PLACEHOLDER_^^^^11query11^^^^.

The proof pattern has four characteristic features. First, the goal is to show that two fixed points on PRESERVED_PLACEHOLDER_^^^^11\11^^^^ coincide, typically the least fixed point of one operator and the greatest fixed point of another. Second, one argues by contradiction and, using finiteness of the state space, selects a gap maximizer PRESERVED_PLACEHOLDER_^^^^11 OR \11^^^^. Third, one proves that maximality is inherited along suitable transitions. Fourth, one derives a contradiction from the existence of the induced max-gap reachable region. In this sense, the “gap” in the informal label refers not to a metric gap in state space, but to the numerical discrepancy between an upper approximation or greatest fixed point and the true reachability value.

The principle arose in a setting where ordinary bounded value iteration can stagnate in the presence of end components. Its significance is therefore algorithmic as well as proof-theoretic: it provides a route to convergence arguments that does not rely on explicit end-component decomposition.

^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^. Stochastic games, bounded value iteration, and the gap

The underlying model is a finite stochastic game

PRESERVED_PLACEHOLDER_^^^^11 arXiv11^^^^

where PRESERVED_PLACEHOLDER_^^^^11max_results11^^^^ is partitioned into Maximizer and Minimizer states, PRESERVED_PLACEHOLDER_^^^^11query11^^^^ is the initial state, PRESERVED_PLACEHOLDER_^^^^11\11query11^^^^ is a finite action set, PRESERVED_PLACEHOLDER_^^^^11\11\11^^^^ is the available-action set, PRESERVED_PLACEHOLDER_^^^^11\11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^ is the transition function, and PRESERVED_PLACEHOLDER_^^^^11\11max_results11^^^^ is the target set. For an action PRESERVED_PLACEHOLDER_^^^^11\11query11^^^^ at PRESERVED_PLACEHOLDER_^^^^11\11\11^^^^, the successor support is

PRESERVED_PLACEHOLDER_^^^^11\11 OR \11^^^^

Strategies are memoryless and pure. For a strategy pair PRESERVED_PLACEHOLDER_^^^^11\11 arXiv11^^^^, the induced Markov chain has reachability probability

PRESERVED_PLACEHOLDER_^^^^11\11max_results11^^^^

and the reachability value is

PRESERVED_PLACEHOLDER_^^^^11\11query11^^^^

The Bellman operator is

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11query11^^^^

with

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11\11^^^^

and the value function is the least fixed point

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^

Bounded value iteration maintains a lower sequence

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11max_results11^^^^

and an upper sequence

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11query11^^^^

The pointwise gap is then PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11\11^^^^, or in fixed-point uniqueness arguments the gap is PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11 OR \11^^^^, where PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11 arXiv11^^^^ is a greatest fixed point of an upper-bound operator. The stopping criterion is

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11max_results11^^^^

which guarantees that

PRESERVED_PLACEHOLDER_^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11query11^^^^

is PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^-close to PRESERVED_PLACEHOLDER_^^^^11max_results11\11^^^^.

The obstruction comes from end components. Ordinary Bellman iteration may satisfy

PRESERVED_PLACEHOLDER_^^^^11max_results11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^

Then the lower sequence converges to PRESERVED_PLACEHOLDER_^^^^11max_results11max_results11^^^^, but the upper sequence converges only to PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^, which may remain strictly above PRESERVED_PLACEHOLDER_^^^^11max_results11\11^^^^. Informally, non-target end components can sustain spurious upper values, so the gap does not vanish. This failure mode is exactly the context in which the max-gap argument becomes relevant.

^^^^11max_results11^^^^. Formal content of gap maximality

A gap maximizer for PRESERVED_PLACEHOLDER_^^^^11max_results11 OR \11^^^^ is a state PRESERVED_PLACEHOLDER_^^^^11max_results11 arXiv11^^^^ such that

PRESERVED_PLACEHOLDER_^^^^11max_results11max_results11^^^^

In the stochastic-game proof and in the motivating Markov-chain proof, the choice is

PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^

The technical engine is the paper’s “max. vs. average” lemma. If PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^ is finite, PRESERVED_PLACEHOLDER_^^^^11query11\11^^^^, and PRESERVED_PLACEHOLDER_^^^^11query11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^ is a distribution on PRESERVED_PLACEHOLDER_^^^^11query11max_results11^^^^, then for an PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^-maximizer PRESERVED_PLACEHOLDER_^^^^11query11\11^^^^,

PRESERVED_PLACEHOLDER_^^^^11query11 OR \11^^^^

Moreover, if equality holds,

PRESERVED_PLACEHOLDER_^^^^11query11 arXiv11^^^^

then every state in the support of PRESERVED_PLACEHOLDER_^^^^11query11max_results11^^^^ has the same value: PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^ This is the exact inheritance mechanism. Once the value at a maximizer equals the average over a successor distribution, every positive-probability successor is itself a maximizer.

The paper isolates this pattern from a classical Markov-chain uniqueness proof. There the modified Bellman operator is

PRESERVED_PLACEHOLDER_^^^^11\11query11^^^^

and the result is

PRESERVED_PLACEHOLDER_^^^^11\11\11^^^^

The key equality in that proof is

PRESERVED_PLACEHOLDER_^^^^11\11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^

Because the left side is the maximum of the gap and the right side is its average over successors, all successors inherit maximality. The stochastic-game argument generalizes this prototype from a single probabilistic transition structure to a two-player Bellman/game setting.

^^^^11query11^^^^. Maximality inheritance in the correctness proof of ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI

The principle becomes central in the analysis of ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI, whose updates are

PRESERVED_PLACEHOLDER_^^^^11\11max_results11^^^^

with termination when

PRESERVED_PLACEHOLDER_^^^^11\11query11^^^^

Its main fixed-point theorem is

PRESERVED_PLACEHOLDER_^^^^11\11\11^^^^

so PRESERVED_PLACEHOLDER_^^^^11\11 OR \11^^^^ has a unique fixed point (&&&^^^^11query11^^^^&&&).

The proof has two phases. First, if PRESERVED_PLACEHOLDER_^^^^11\11 arXiv11^^^^, then PRESERVED_PLACEHOLDER_^^^^11\11max_results11^^^^ is also a fixed point of the ordinary Bellman operator PRESERVED_PLACEHOLDER_^^^^11\11query11^^^^, because

PRESERVED_PLACEHOLDER_^^^^11 OR \11query11^^^^

so the widest-path Bellman term

PRESERVED_PLACEHOLDER_^^^^11 OR \11\11^^^^

collapses to PRESERVED_PLACEHOLDER_^^^^11 OR \11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^. Since PRESERVED_PLACEHOLDER_^^^^11 OR \11max_results11^^^^ is itself a fixed point of PRESERVED_PLACEHOLDER_^^^^11 OR \11query11^^^^, this yields

PRESERVED_PLACEHOLDER_^^^^11 OR \11\11^^^^

Second, to prove

PRESERVED_PLACEHOLDER_^^^^11 OR \11 OR \11^^^^

one assumes PRESERVED_PLACEHOLDER_^^^^11 OR \11 arXiv11^^^^ and chooses a gap maximizer PRESERVED_PLACEHOLDER_^^^^11 OR \11max_results11^^^^ with

PRESERVED_PLACEHOLDER_^^^^11 OR \11query11^^^^

Because targets satisfy PRESERVED_PLACEHOLDER_^^^^11 arXiv11query11^^^^, such a state cannot lie in PRESERVED_PLACEHOLDER_^^^^11 arXiv11\11^^^^.

The argument then splits by controller. If PRESERVED_PLACEHOLDER_^^^^11 arXiv11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^, an action PRESERVED_PLACEHOLDER_^^^^11 arXiv11max_results11^^^^ is chosen optimal for the widest-path operator, yielding

PRESERVED_PLACEHOLDER_^^^^11 arXiv11query11^^^^

If PRESERVED_PLACEHOLDER_^^^^11 arXiv11\11^^^^, an action PRESERVED_PLACEHOLDER_^^^^11 arXiv11 OR \11^^^^ is chosen optimal for the reachability Bellman operator, and one obtains similarly

PRESERVED_PLACEHOLDER_^^^^11 arXiv11 arXiv11^^^^

In either case, all inequalities are equalities, and the max-vs.-average lemma gives

PRESERVED_PLACEHOLDER_^^^^11 arXiv11max_results11^^^^

Because the same reasoning applies at every gap-maximizing state, the selected actions define memoryless strategies PRESERVED_PLACEHOLDER_^^^^11 arXiv11query11^^^^ such that every state reachable from PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^ under PRESERVED_PLACEHOLDER_^^^^11max_results11\11^^^^ is also a gap maximizer. Since targets are not gap maximizers, PRESERVED_PLACEHOLDER_^^^^11max_results11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^ is unreachable from PRESERVED_PLACEHOLDER_^^^^11max_results11max_results11^^^^ under these strategies.

The contradiction comes from the greatest-fixed-point semantics of PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^. Under PRESERVED_PLACEHOLDER_^^^^11max_results11\11^^^^, Maximizer plays optimally for the widest-path objective, so the widest path width from PRESERVED_PLACEHOLDER_^^^^11max_results11 OR \11^^^^ is at least PRESERVED_PLACEHOLDER_^^^^11max_results11 arXiv11^^^^. Because PRESERVED_PLACEHOLDER_^^^^11max_results11max_results11^^^^, one has PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^, hence there must exist a path from PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^ to PRESERVED_PLACEHOLDER_^^^^11query11\11^^^^ of width at least PRESERVED_PLACEHOLDER_^^^^11query11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^. That makes PRESERVED_PLACEHOLDER_^^^^11query11max_results11^^^^ reachable from PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^ under PRESERVED_PLACEHOLDER_^^^^11query11\11^^^^, contradicting the inherited max-gap region. The contradiction eliminates the possibility of a positive persistent gap.

This proof avoids explicit end-component detection. Instead of computing end components and then repairing the upper bound, it shows that any putative positive-gap region would have to be closed under inherited maximality and simultaneously support a positive-width target-reaching path, which is impossible.

^^^^11\11^^^^. Widest-path games and algorithmic significance

The upper-bound operator of ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI is built from widest path games. Given an SG and PRESERVED_PLACEHOLDER_^^^^11query11 OR \11^^^^, the state-action width is

PRESERVED_PLACEHOLDER_^^^^11query11 arXiv11^^^^

For an infinite path PRESERVED_PLACEHOLDER_^^^^11query11max_results11^^^^, the path width is

PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^

For strategies PRESERVED_PLACEHOLDER_^^^^11\11query11query11^^^^,

PRESERVED_PLACEHOLDER_^^^^11\11query11\11^^^^

and the widest-path value is

PRESERVED_PLACEHOLDER_^^^^11\11query11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^

This induces the global operator

PRESERVED_PLACEHOLDER_^^^^11\11query11max_results11^^^^

The associated widest-path Bellman operator is

PRESERVED_PLACEHOLDER_^^^^11\11query11query11^^^^

The paper proves

PRESERVED_PLACEHOLDER_^^^^11\11query11\11^^^^

This construction over-approximates reachability because, after an action choice, Maximizer effectively receives the best successor compatible with the bottleneck width. At the same time, PRESERVED_PLACEHOLDER_^^^^11\11query11 OR \11^^^^ preserves the stochastic expectation that still matters for Bellman comparison. This mixture is what makes the contradiction proof with inherited gap maximality possible.

The paper contrasts ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI with the earlier ^^^^11\11^^^^WP-BVI. In ^^^^11\11^^^^WP-BVI, player reduction is performed first: at Minimizer states, only actions minimizing PRESERVED_PLACEHOLDER_^^^^11\11query11 arXiv11^^^^ with respect to the current lower bound are retained, reducing the game to an MDP. A ^^^^11\11^^^^-player widest-path problem is then solved on that reduced structure. By contrast, ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI computes

PRESERVED_PLACEHOLDER_^^^^11\11query11max_results11^^^^

directly on the original two-player structure. The resulting proof is correspondingly cleaner: the paper emphasizes that ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI replaces the earlier infinitary pigeonhole-style reasoning with a direct maximality-inheritance contradiction (&&&^^^^11query11^^^^&&&).

The algorithmic effect is visible in the running example. Ordinary BVI keeps the upper sequence at

PRESERVED_PLACEHOLDER_^^^^11\11query11query11^^^^

while the true value is

PRESERVED_PLACEHOLDER_^^^^11\11\11query11^^^^

Under ^^^^11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^WP-BVI, the upper sequence descends: PRESERVED_PLACEHOLDER_^^^^11\11\11\11^^^^ and converges to

PRESERVED_PLACEHOLDER_^^^^11\11\11 stochastic games bounded value iteration (Phalakarn et al., 8 Aug 2025)11^^^^

This suggests that widest-path propagation removes unsupported optimistic values in end-component-like regions without explicitly computing those regions.

^^^^11 OR \11^^^^. Relation to other maximality principles

The phrase “maximality principle” has multiple unrelated meanings across mathematics and logic, and “Gap Maximality Principle” is therefore terminologically ambiguous. In modal and intuitionistic logic, maximality principles include Fine’s and Esakia’s principles; the relevant distinction there is a strength gap between weaker clopen or algebraic formulations and stronger closed-set formulations, with the former tied to the Boolean Prime Ideal Theorem and the latter to the Axiom of Choice (&&&^^^^11query11^^^^&&&). In forcing theory, Hamkins-style principles take the modal form PRESERVED_PLACEHOLDER_^^^^11\11\11max_results11^^^^ relative to a forcing class PRESERVED_PLACEHOLDER_^^^^11\11\11query11^^^^, yielding schemes such as PRESERVED_PLACEHOLDER_^^^^11\11\11\11^^^^ rather than numerical gap propagation (&&&^^^^11\11^^^^&&&). In optimal stopping, the “maximality principle” refers to selecting a free boundary as the maximal solution of a nonlinear ODE below an admissibility curve, again a distinct mechanism (&&&^^^^11 OR \11^^^^&&&).

Against that background, the stochastic-game notion discussed here is specific. It does not assert existence of maximal points, maximal ideals, or maximal forcing persistence. It concerns a state maximizing a pointwise value gap, and the crucial claim is that this maximality is inherited along suitable transitions. The inherited-max-gap region is then used to refute the existence of a spurious greatest fixed point above the reachability value. Accordingly, the most precise encyclopedia description is that “Gap Maximality Principle” is an informal label for the maximality inheritance principle in widest-path-based bounded value iteration, where the maximal quantity is the upper-versus-exact gap PRESERVED_PLACEHOLDER_^^^^11\11\11 OR \11^^^^.