Discrete-Bidding Games

• Discrete-bidding games are combinatorial and graph-theoretic games where players bid discrete resources to determine move order and outcomes.
• They incorporate various auction models—such as Richman, poorman, and all-pay bidding—to simulate strategic market design and resource allocation.
• Research emphasizes threshold budgets, equilibrium strategies, and algorithmic computation via fixed-point iterations and game reduction techniques.

Discrete-bidding games are a class of combinatorial and graph-theoretic games in which two or more players compete for control over successive moves, or for the acquisition of goods, through a sequence of auctions or bidding rounds, each subject to a discrete (granular) bidding mechanism. In contrast to classic alternating or turn-based paradigms, discrete-bidding dynamics determine move order or outcomes via explicit allocation of indivisible and finitely valued resources (chips, tickets, budgets, etc.), thus introducing distinct strategic, algorithmic, and equilibrium phenomena shaped by granularity constraints, tie-breaking rules, and resource transfer or depletion. The mathematical and algorithmic paper of discrete-bidding games spans strategic market design, combinatorial game theory, formal systems analysis, online auction mechanisms, and resource allocation with complex objectives.

1. Core Structure and Typology

Discrete-bidding games generalize several paradigms, including Richman games, all-pay and first-price auction games, scoring and combinatorial games with bidding move order, and auction-derived multi-agent control mechanisms.

Generic discrete-bidding setup:

• Two players (generalizations to n players are possible) begin with budgets expressed as non-negative integers (e.g., chips or coins).
• At each decision point (vertex, game state, lot), both players simultaneously choose integer-valued bids, each between 0 and their current budget.
• The highest bidder wins the privilege: in graph games, moving the token to a successor vertex; in market games, acquiring a specific good; in contest-based games, scoring a point or advantage.
• The bid is handled according to the payment scheme:
  • Richman bidding: The winning bid is transferred to the opponent, leaving total budget unchanged.
  • Poorman bidding: The winning bid is lost (transferred to the bank), decreasing the total available budget.
  • All-pay bidding: All players pay their bids, introducing cost/effort regardless of outcome.
• Ties are resolved by a specified tie-breaking mechanism (advantage marker, deterministic rotation, probabilistic coin tosses, or rule-based automata).
• Budgets and bids are always restricted to discrete (e.g., unit, cent, integer) granularity.

This setup applies equivalently to graph games with finite or infinite horizon and to single-shot markets (e.g. auctions for indivisible goods). Variant forms include games with vertex-dependent "charging" (where budgets are refilled at specific states) (Avni et al., 8 Jul 2024), multi-stage auctions, and portfolio or resource allocation contests.

2. Mathematical Foundations and Winning Thresholds

A defining feature of discrete-bidding games is the existence (and structure) of threshold budgets (or, in normalization, threshold ratios): critical values of the initial discrete budget split that delineate the regions of winnability for each player.

• For a given vertex (game state) $v$, the threshold function $T(v)$ or $\mathrm{Th}(v)$ satisfies that Player 1 can force a win if and only if their budget exceeds $T(v)$; similarly for Player 2 and $k-T(v)$, where $k$ is the total budget (Avni et al., 2022, Avni et al., 30 Aug 2025, Aghajohari et al., 2019).
• In discrete-bidding games with qualitative (reachability, parity, safety) objectives, thresholds are characterized as (possibly unique) fixed points of nonlinear discrete "average property" equations, for example:

$$T(v) = \left\lfloor \frac{|T(v^+)| + |T(v^-)|}{2} \right\rfloor + \delta(v)$$

where $v^+, v^-$ correspond to maximizing and minimizing successors for the relevant objective, $|B|$ drops tie-breaking tags, and $\delta$ accounts for parity or tie-breaking contingencies (Avni et al., 2022, Avni et al., 30 Aug 2025, Avni et al., 8 Jul 2024).

• For mean-payoff and energy objectives in discrete-bidding games, the energy threshold function $\mu(v,B)$ is defined via value-iteration, providing the minimal initial energy required given a starting configuration $(v,B)$. The Pres threshold at $v$ is then $$T_{\mathrm{pre}}(v) := \min\{\; B \;|\; \mu(v,B) < \infty\}$$ (Avni et al., 30 Aug 2025).
• In games with vertex-dependent charging, threshold ratios become solutions to nonlinear fixed-point systems reflecting both competitive dynamics and local budget refreshment, and may fail to be unique (Avni et al., 8 Jul 2024).
• The mathematical structure is deeply influenced by the discrete granularity and by tie-breaking; local determinacy and the structure of the "bidding matrix" are central tools for establishing global determinacy properties (Aghajohari et al., 2019).

Significantly, the existence and computability of winning thresholds is a non-trivial property, particularly when discrete granularity implies the absence of smooth averaging or monotonicity otherwise available in the continuous case.

3. Strategy Synthesis and Algorithmic Properties

Discrete-bidding games require the construction of optimal (often budget-agnostic) strategies that guarantee victory if the initial budget is above threshold, with particular care for the limitations and artifacts introduced by granularity.

• In Richman-type discrete-bidding games, optimal strategies are often budget-agnostic: the player acts as if they hold precisely the threshold budget, trimming any excess (Avni et al., 30 Aug 2025, Avni et al., 2022).
• For all-pay bidding and contest games, Nash equilibria may require randomized strategies with possibly infinite support, given that mixed strategies are necessary to avoid exploitation and to guarantee minimal win probabilities (Menz et al., 2015, Avni et al., 2019).
• Algorithmic computation of threshold budgets and strategy synthesis employs recursive value-iteration, dynamic programming, and reductions to structured turn-based (or parity) games. Two principal approaches are:
  • Fixed-point iteration: Applies a recursive update reflecting the discrete average property or minimax recurrence until thresholds stabilize (Avni et al., 2022, Avni et al., 30 Aug 2025, Avni et al., 2023).
  • Reductions to turn-based games: Candidate threshold functions are validated via reduction to a polynomial-size turn-based parity (or mean-payoff) game, enabling the establishment that the threshold computation is in NP ∩ coNP (Avni et al., 2022, Avni et al., 30 Aug 2025).
• In auctions and market settings, equilibrium computation may require Bayesian or strategic learning algorithms—e.g., discretization plus dual-averaging or mirror ascent—combined with offline or online LP solvers (Bichler et al., 2022).
• For games with charging, iterative fixed-point algorithms or mathematical programming formulations (MILP) are necessary to handle the nonlinearities and potential non-uniqueness of the thresholds (Avni et al., 8 Jul 2024).
• Complexity varies by setting: threshold decision is in NP ∩ coNP for qualitative, energy, and mean-payoff objectives (Avni et al., 2022, Avni et al., 30 Aug 2025); for richer objective classes (Rabin/Streett), computing thresholds is NP-hard/coNP-hard (Avni et al., 8 Jul 2024); for repair problems with charging, reachability falls in 2EXPTIME (Avni et al., 8 Jul 2024).

A summary of algorithmic results for discrete-bidding games:

Objective type	Threshold computation	Strategy memory
Reachability/Parity	NP ∩ coNP (Avni et al., 2022)	Linear in
Mean-payoff/Energy	NP ∩ coNP (Avni et al., 30 Aug 2025)	Linear in
Rabin/Streett/Repair	NP/coNP/2EXPTIME (Avni et al., 8 Jul 2024)	As above

4. Discrete-vs-Continuous and Game-theoretic Phenomena

Distinct discontinuities, inefficiencies, and strategic phenomena emerge due to the enforced discreteness:

• Non-existence/discontinuity of pure Nash equilibria: In first-price and all-pay auctions with three or more bidders, or in all-pay auctions even with two bidders, pure-strategy equilibria may vanish for arbitrarily fine discretizations (Rasooly et al., 2020).
• Robustness of symmetric equilibria: In two-bidder first-price auctions, symmetric equilibrium strategies (e.g., $\beta(v) = \lfloor v/2 \rfloor$) converge to their continuous analogues as discretization vanishes (Rasooly et al., 2020).
• Mixed-strategy complexity: To compensate for the absence of pure equilibria, players may be forced to use highly mixed strategies, sometimes randomizing over nearly the entire support of available bids (Menz et al., 2015, Avni et al., 2019, Rasooly et al., 2020).
• Welfare analysis and inefficiency: In discrete combinatorial auctions, while pure Nash equilibria (when they exist) are efficient (first welfare theorem), mixed equilibria may exhibit welfare loss bounded by factors related to the class of valuations—up to $O(\log m)$ or $O(m)$ in subadditive settings (Hassidim et al., 2011).
• Sensitivity to tie-breaking: Determinacy and equilibrium structure depend critically on the tie-breaking rule, with alternating or path-dependent tie-breaking mechanisms leading to non-determined or qualitatively distinct games (Aghajohari et al., 2019).

These phenomena call into question the adequacy of continuous approximations for experimental or real-world mechanisms and motivate explicit consideration of discrete models in analysis and design.

5. Applications and Model Variants

Discrete-bidding games have broad application domains and can be adapted to encompass multiple variant mechanisms.

Applications:

• Resource allocation and scheduling: Auction-based scheduling for tasks or agents, e.g., in robotics, embedded systems, or GPU time allocation (Avni et al., 30 Aug 2025).
• Cyber-physical and blockchain systems: Formal models of system–and–environment competition, transaction fee-based bid selection, and scrip mechanisms (Aghajohari et al., 2019, Avni et al., 2022).
• Combinatorial auctions and market design: Allocation of indivisible goods, spectrum auctions, and ticket-based mechanisms (e.g., Chinese auctions) (Brânzei et al., 2012, Hassidim et al., 2011).
• Combinatorial game theory and recreational games: Extensions of classic heap, subtraction, or chess games with bidding move order (Larsson et al., 2020, Kant et al., 2022).
• Mechanism design and contest theory: All-pay bidding models for rent-seeking contests and Tullock competitions (Avni et al., 2019, Bichler et al., 2022).

Variants:

• Games with charging: Budgets are periodically refilled at vertices according to fixed charge rates, producing nontrivial recurrent strategy patterns and complicating threshold uniqueness and algorithmics (Avni et al., 8 Jul 2024).
• Scoring play: Cumulative scoring or energy objectives, where outcome is not simple reachability but a sum or mean over transitions (Larsson et al., 2020, Avni et al., 30 Aug 2025).
• All-pay, poorman, and taxman bidding: Distinct payment rules reflecting various resource transfer regimes (Menz et al., 2015, Avni et al., 2023, Avni et al., 2020, Avni et al., 8 Jul 2024).
• Complex auctions and procurement contests: Discrete-bidding generalizations to multi-unit, combinatorial, and Bayesian auction settings via convergence and learning algorithms (Bichler et al., 2022).

Discrete granularities are an essential element in all these models, necessitating careful analysis of "integer effects," periodicities, and the direct design for real-world market or systems environments.

6. Recent Theoretical Developments and Open Problems

Recent research has engaged deeply with the algebraic and structural properties of discrete-bidding games, exploring analogue and divergence from classical alternating-play theory.

• Lattice and semilattice structures: The set of possible perfect-play outcomes forms a lattice under budget monotonicity and marker-worth constraints (Kant et al., 2022).
• Generalization of classical play and arithmetics: The extension of integers, dyadics, and numeric games to bidding settings, with subgroup closure properties, invertibility, and infinitesimal elements analyzed (Kant et al., 2022).
• Algorithmic constructive comparison: Play-solution methods for game comparison and orderings, including algorithmic tests for outcome dominance via recursive 0-bid strategies (Kant et al., 2022).
• Open conjectures: Non-uniqueness of fixed points in threshold equations with charging, invertibility of games, and the existence of positive infinitesimals in strict bidding settings (Kant et al., 2022, Avni et al., 8 Jul 2024).
• Complexity bounds for generalized objectives: PSPACE and 2EXPTIME completeness for richer objective and repair problems in the presence of charging (Avni et al., 8 Jul 2024).

These theoretical advances point to a rich and rapidly evolving research landscape, marked by new algebraic, algorithmic, and structural questions.

7. Significance, Implications, and Future Directions

The rigorous paper of discrete-bidding games has revealed both deep mathematical structure and significant practical implications:

• Discrete granularity—ubiquitous in real-world markets, computational systems, and cyber-physical mechanisms—induces sharp departures from classical (continuous) theory; equilibrium selection, welfare, determinacy, and strategy implementation all depend in subtle ways on the nature of the discretization.
• The existence of compact (linear-memory), budget-agnostic winning strategies aligns theoretical tractability with implementability, particularly in resource-constrained or high-stakes contexts (Avni et al., 2022, Avni et al., 30 Aug 2025).
• The sensitivity of thresholds and strategy structure to payment mechanism and local features (charging, tie-breaking) offers a flexible toolkit for mechanism design and automated synthesis with built-in guarantees.
• As new application domains—such as blockchain incentives, decentralized scheduling, and adaptive market protocols—continue to require fine-grained and robust strategic tools, the continued development of discrete-bidding game theory promises to remain both foundational and highly relevant.
• Open problems on structural invertibility, repair mechanisms, and algebraic properties suggest that combinatorial and algebraic game theory, economics, and formal verification will remain strongly interconnected through the lens of discrete bidding and auction mechanisms.

In summary, discrete-bidding games form a unifying and fertile framework for modeling, analyzing, and synthesizing adversarial and market interactions where resource granularity, strategic bidding, and dynamic control are primary features.