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Multi-Player Discrete-Bidding Games; Determinacy, Equilibria, and Complexity

T0 review · 0 major / 5 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Under linear tie-breaking, multi-player discrete-bidding games are determined: one of two coalitions always has a winning strategy from every configuration.

desk verdict First solid multi-player discrete-bidding theory: determinacy under linear ties, pure NE, mean-payoff values, and a clean PSPACE separation from the two-player case. read the letter →

arxiv 2607.27456 v1 pith:7I7H6EBW submitted 2026-07-29 cs.GT cs.FL

Guy Avni , Fatima Murra This is my paper
classification cs.GTcs.FL
keywords biddinggamesmulti-playerdiscretedeterminacyNashequilibriummean-payoffconcurrentauction-basedscheduling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Bidding games let players spend budgets in auctions that decide who moves a token on a graph. Two-player versions are well understood; this paper treats the multi-player case for the first time, under discrete (integer) budgets and bids. It proves that when players are split into two coalitions and ties are broken by a fixed total order at each vertex, the game is determined: from any starting configuration exactly one coalition has a winning strategy for Müller objectives. The same determinacy yields pure Nash equilibria in qualitative non-zero-sum games and existence of a value in mean-payoff games. Deciding the winner is already PSPACE-hard for reachability with unary budgets, sharply harder than the two-player setting. The result carves out a determined fragment of concurrent multi-player games and supplies the first rigorous foundation for auction-based multi-objective scheduling with more than two policies.

What carries the argument

The bidding matrix of a configuration, together with the local-to-global determinacy principle: if every bidding matrix has an X-row or a Y-column (proved by induction on tie sub-matrices under linear tie-breaking), then the concurrent game is determined for Müller objectives.

What would settle it

Exhibit a concrete multi-player discrete-bidding reachability instance whose tie-breaking orders are linear at every vertex, yet from some configuration neither coalition possesses a winning strategy (equivalently, produce a bidding matrix that has neither an all-X row nor an all-Y column).

Watch

Extended reading notes

Core claim

Müller multi-player discrete-bidding games are determined whenever tie-breaking (and, for presentation, rounding) mechanisms are linear orders at every vertex: for every game, every Müller objective, and every configuration, either Coalition X has a strategy that forces the objective or Coalition Y has a strategy that violates it. Local determinacy of the bidding matrix—every matrix contains an all-X row or an all-Y column—lifts to global determinacy, and the same argument yields values for mean-payoff objectives and pure Nash equilibria for qualitative individual objectives.

Load-bearing premise

Tie-breaking at every vertex must be a total linear order on the players; without it the bidding matrix need not contain a pure winning row or column and determinacy can fail.

Editorial extensions

If this is right

  • Pure-strategy Nash equilibria exist in every qualitative multi-player discrete-bidding game with Müller objectives.
  • Mean-payoff multi-player discrete-bidding games possess a value for every initial configuration.
  • Auction-based scheduling can now decouple more than two omega-regular objectives by solving one zero-sum coalition game per objective.
  • Deciding the winner remains PSPACE-hard already for reachability and unary budgets, so exponential configuration graphs cannot be avoided in general.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The hardness gap suggests that succinct certificates based on threshold budgets, which work for two players, are unlikely to extend; new algorithmic ideas will be needed for practical multi-player solvers.
  • If subgame-perfect equilibria can be shown to exist by a similar local-to-global argument, auction-based multi-agent controllers would gain a stronger solution concept than plain Nash.
  • Relaxing linearity of rounding while keeping linear tie-breaking already works in the proof; fully general tie-breaking remains the natural next boundary to map.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper initiates the study of multi-player discrete-bidding games on graphs. Under linear (total-order) tie-breaking and rounding mechanisms at each vertex, it proves that games between two coalitions are determined for Müller objectives (Theorem 7): from every configuration exactly one coalition has a winning strategy. The argument proceeds by establishing local determinacy of every bidding matrix (Definition 10, Proposition 16) via an induction on tie sub-matrices that maintains revealed-player sets and precedence invariants (Lemmas 12–15), then lifting to global determinacy by a cited concurrent-game framework (Theorem 11). Determinacy is used to obtain pure Nash equilibria in qualitative non-zero-sum games (Theorem 19) and existence of values in mean-payoff games (Theorem 24). Complexity-wise, deciding the winner is shown PSPACE-hard already for reachability and unary budgets (Theorem 20), in contrast to the NP∩coNP membership of the two-player case.

Significance. Multi-player bidding games had been essentially unexplored; continuous-bidding variants were known to fail convexity of winning regions, and concurrent games are not determined in general. Establishing determinacy for a natural discrete subclass under a mild, necessary tie-breaking hypothesis is therefore a genuine advance. The pure-NE and mean-payoff-value corollaries, together with the auction-based scheduling application, give the result immediate modelling value. The PSPACE-hardness lower bound cleanly separates the multi-player setting from the two-player theory and is obtained by an explicit, checkable gadget. The local-determinacy induction is fully spelled out and elementary, so the central claim is verifiable by a careful reader.

minor comments (5)
  1. [§4.3, Proposition 16] In the inductive step of Proposition 16 the phrase “if neither an X-row nor a Y-column is not found” contains a double negative; a single “is found” would improve readability.
  2. [§4.2, Figure 4] Figure 4 is referenced as illustrating Lemmas 13–15, yet the concrete matrix entries that realise the equalities and arrows are never written out; a short caption listing one or two concrete equalities would help the reader.
  3. [Appendix B] The reduction sketch for TQBF in Appendix B is only a few lines; expanding the neighbour sets and the ownership of the x0_i players for universal quantifiers would make the PSPACE claim self-contained.
  4. [§4.1, Theorem 11] Several self-citations appear as “[1]”, “[12]”, etc.; expanding the first occurrence of each key reference (especially the local-to-global lift) would aid readers unfamiliar with the two-player literature.
  5. Typographical slips: “G. Avni and F. Murra” header, “envel⌢pe”, and occasional missing spaces around mathematical operators appear throughout; a final copy-edit pass is warranted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: determinacy is proved by an explicit induction on bidding matrices under the linear-tie axiom, not assumed or fitted.

full rationale

The central claim (Theorem 7) is obtained from local determinacy of every bidding matrix (Definition 10, Proposition 16) via a standard local-to-global lift (Theorem 11 citing [1]). Proposition 16 is a self-contained combinatorial induction: base case is the pure turn-based S0 matrix; the inductive step maintains finite revealed-player sets PX/PY with the precedence invariant supplied by Lemmas 13–15 and terminates because the player sets are finite. Example 2 correctly shows that the linear-tie hypothesis is necessary. Nash-equilibrium existence (Theorem 19) and mean-payoff value existence (Theorem 24) are routine corollaries that invoke the newly proved determinacy; they do not feed the target result back into the hypothesis. The PSPACE-hardness reduction is an explicit gadget construction. Self-citations to the authors’ prior two-player work are used only for contrast and for the already-published local-to-global framework; they are not load-bearing for the multi-player induction. No fitted parameters, self-definitional loops, or uniqueness theorems imported from the same authors appear.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central determinacy theorem rests on Martin’s determinacy for turn-based Müller games, the discrete Richman bidding model with integer budgets, and the paper’s standing assumption that tie-breaking (and for presentation rounding) are given by linear orders at each vertex. No free parameters are fitted. Technical devices (bidding matrix, revealed-player sets) are proof machinery, not new physical or ontological entities.

assumptions (5)
  • standard math Two-player turn-based games with Müller objectives are determined (Martin’s theorem).
    Invoked as Theorem 5 and used via the GX/GY turn-based unfoldings (Lemma 6) to transfer local matrix properties to global coalition winning strategies.
  • domain assumption Bids and budgets are natural numbers; the winner pays the bid evenly to the other players under Richman rules, with a fixed total budget k conserved.
    Defines the discrete-bidding arena (Section 3). Finiteness of the configuration space and the structure of budget updates depend on it.
  • domain assumption Tie-breaking at each vertex is given by a linear order Tv among players (and rounding by a linear order Rv).
    Stated in Section 3 and used throughout Section 4. Example 2 shows determinacy fails without it; Proposition 16’s induction relies on a total precedence to maintain the revealed-player invariant.
  • domain assumption If a multi-player Müller discrete-bidding game is locally determined (every bidding matrix has an X-row or Y-column), then it is globally determined.
    Theorem 11, cited from prior concurrent-game work [1]. The paper proves local determinacy and imports the lift; the lift itself is not re-proved in full.
  • ad hoc to paper Coalitions may be restricted to single-bidder joint bids without loss of winning power (Lemma 9).
    Used to shrink the bidding matrix to rows/columns indexed by one positive bid per coalition member; essential for the matrix induction but specific to this model’s payment and move rules.
invented entities (1)
  • Bidding matrix Mc with tie sub-matrices Sb
    purpose: Finite combinatorial object that encodes all single-bidder auction outcomes at a configuration and on which local determinacy is stated and proved.
    Standard proof device adapted from two-player discrete bidding; not an ontological postulate. independent_evidence is false only in the sense that it is internal machinery.

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Pith. "Pith review of Multi-Player Discrete-Bidding Games; Determinacy, Equilibria, and Complexity." pith.science (2026). https://pith.science/paper/7I7H6EBW

@misc{pith2026260727456,
  author       = {Pith},
  title        = {Pith review of: Multi-Player Discrete-Bidding Games; Determinacy, Equilibria, and Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7I7H6EBW}},
  note         = {Machine review of arXiv:2607.27456}
}
read the original abstract

Games on graphs constitute a fundamental model. Applications include reactive synthesis, which reduces to solving a zero-sum two-player game, and reasoning about multi-agent systems by modeling them as a multi-player game. We study a class of graph games called bidding games in which the players are allocated a budget, and in each turn, an auction determines which player moves the token. Two-player bidding games have been extensively studied. We study, for the first time, multi-player bidding games. We focus on discrete bidding, which imposes granularity restrictions on the players' budgets and bids. The original motivation for discrete bidding is practical applications, and technically, it is appealing that the game has only finitely-many configurations. We initiate our study by considering a game between two coalitions of players. We show that under mild assumptions on the mechanism that is applied to break bidding ties, bidding games are determined: from every initial configuration, one of the coalitions has a winning strategy. Thus, we identify a sub-class of multi-player concurrent games that is determined. We extend the result to existence of a value in mean-payoff games. Using determinacy, we show that a pure Nash equilibrium always exists in qualitative games. Finally, we show that the complexity of deciding which coalition wins from a given configuration is PSPACE-hard already in reachability games and already for budgets given in unary. This is in stark contrast to two-player games, which are known to be in NP and coNP even for budgets given in binary.

Figures

Figures reproduced from arXiv: 2607.27456 by the authors.

Figure 4
Figure 4. the entry represented with a point equals [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.