Discrete vs Continuous Bidding Games

• Discrete/continuous bidding games are strategic frameworks where players allocate resources through integral or real-valued bids to influence outcomes in competitive settings.
• Recent advances reveal that continuous games typically yield pure Nash equilibria via convex optimization, whereas discrete games often require mixed strategies and are sensitive to bid granularity.
• Analytical and algorithmic techniques such as threshold budgets and tie-breaking mechanisms provide practical insights for auction design, resource allocation, and fair division problems.

Discrete and continuous bidding games constitute a central framework in algorithmic game theory, economics, combinatorial optimization, and formal verification for modeling sequential and simultaneous competitions wherein agents strategically expend resources to influence the evolution or outcome of a system. These games exhibit rich mathematical structure and diverse phenomenology, especially as rule variants and payoff objectives (e.g., reachability, mean-payoff, energy) are considered under discrete or continuous bidding paradigms. Theoretical developments in recent years have resolved foundational questions regarding equilibrium existence, strategy structure, complexity, and welfare guarantees in these models, exposing deep differences between discrete and continuous bidding as well as their connections to classical market, auction, and combinatorial game theory.

1. Foundational Models and Bidding Mechanisms

Discrete and continuous bidding games generalize both classic turn-based games and auction environments by introducing a resource-based control protocol over moves or allocations. At each decision point (graph vertex, item, or combinatorial game node), two (or more) players submit bids—either integral (discrete) or real-valued (continuous), subject to budget constraints—with rules for resolving ties and assigning payments that critically shape strategic behavior.

A canonical archetype is the Richman bidding game: at each round, both players simultaneously bid, and the higher bidder moves the token (in a graph or game position), transferring their bid to the opponent (budget-preserving). Alternate mechanisms include poorman (winner pays the bank), all-pay (all bidders pay their bids), and taxman (winner pays a fraction $\tau$ to the loser, rest to the bank) rules. The setting may be zero-sum (e.g., reachability, mean-payoff objectives) or admit general utilities (e.g., combinatorial allocation). The granularity—discrete (integer bids) vs. continuous (arbitrary precision)—is pivotal, as it determines the existence, uniqueness, and structure of equilibria.

2. Existence and Structure of Equilibria: Discrete vs. Continuous

2.1 Existence in Continuous Bidding Games

Continuous bidding games often admit pure Nash equilibria under fairly weak conditions, owing to the convexity and continuity of utility/best-reply correspondences. In Chinese auctions for multiple items with strictly positive valuations, symmetric and asymmetric budget settings, equilibrium always exists; equilibrium strategies are symmetric when players and values are symmetric, and allocations are proportional to the valuations: $x_{i,j} = \frac{v_j}{\sum_\ell v_\ell} w_i$ Convexity ensures unique minimizers, and the existence of equilibrium is guaranteed by convex-compact machinery or Reny's better-reply security argument in non-symmetric cases (Brânzei et al., 2012).

2.2 Non-Existence and Pathologies in Discrete Bidding Games

In discrete bidding games, the atomicity and indivisibility of bids produces a far more intricate picture. Existence of pure equilibria is conditional:

• In symmetric, indivisible ticket settings (e.g., one ticket per player in Chinese auction), pure Nash equilibrium exists and can be constructed sequentially by maximizing marginal utilities (Brânzei et al., 2012).
• In asymmetric cases (budgets or valuations), equilibria may not exist, even with as few as two items and two players. Counterexamples demonstrate the impossibility of simultaneously satisfying all players' best responses in the integral allocation space.

This non-existence extends to classical first-price and all-pay auctions with multiple bidders: arbitrarily small granularity destroys pure equilibria that always exist in the continuous case (Rasooly et al., 2020). Mixed-strategy equilibria persist by Nash's theorem in finite games, but are often combinatorially complex, require wide randomization support, and may be qualitatively unlike their continuous counterparts.

2.3 Stabilization Mechanisms

In continuous Chinese auctions and costly ticket scenarios, pure Nash equilibrium existence fails with zero valuations; this degeneration can be cured by auctioneer stabilization: introducing a strictly positive ticket in each basket regularizes utility functions and restores continuity and equilibrium existence (Brânzei et al., 2012).

2.4 Structure and Uniqueness of Equilibria

Game-theoretic refinements in discrete settings reveal a rich lattice-theoretic structure. In discrete Richman-bidding combinatorial games, the set of PSPEs forms a meet-semilattice with a unique, monotone, bottom equilibrium that is Pareto-efficient and covers all efficient outcomes in binary games (1311.0913), while in combinatorial auctions with indivisible items, the set of pure Nash equilibria coincides exactly with price-based Walrasian equilibria (Hassidim et al., 2011).

The discrete case may have non-uniqueness of equilibria, with entire continuums of mixed Nash equilibria for certain symmetric parameters in all-pay auctions (Dziubiński et al., 2023), but payoff uniqueness is restored except at symmetry-induced measure-zero points.

3. Analytical and Algorithmic Techniques

3.1 Threshold Budgets and Value Recurrences

A unifying concept in graph-based and infinite-duration bidding games is the threshold budget: the minimal initial budget required for a player to guarantee a win (e.g., reachability or parity objectives) or a target mean-payoff. Recursive value functions and average properties underpin their computation: $T(v) = \frac{T(v^+)+T(v^-)}{2}$ for continuous settings, and a piecewise discretized version for discrete bidding (Avni et al., 2022, Avni et al., 30 Aug 2025). For energy and mean-payoff, value iterations and min-max recursions synthesize winning strategies and verify threshold properties, with adjustments for tie-breaking rules and bid granularity.

3.2 Determinacy and the Role of Tie-Breaking

Determination of which player can force a win (determinacy) in infinite-duration bidding games—unlike general concurrent games—is deeply sensitive to the tie-breaking rule: mechanisms that ignore ties ("unaware" transducers), random tie-breaks, or advantage-based systems restore determinacy (strong/weak), while alternating or stateful tie-breaking can induce indeterminacy (e.g., matching pennies effect) (Aghajohari et al., 2019). Discrete bidding games, thus, are not mere $\epsilon \to 0$ limits of their continuous analogs.

3.3 Algorithmic Complexity

The computation of threshold budgets in parity, mean-payoff, and energy discrete-bidding games can be reduced to polynomial (NP ∩ coNP) time in graphs, despite exponential worst-cases under naive recursion when budgets are binary (Avni et al., 2022, Avni et al., 30 Aug 2025). Critical use of the threshold average property, constructive fixed points, and reductions to succinctly-represented turn-based games yield efficient, linear-memory strategies.

4. Strategic and Welfare Properties

4.1 Mixed Strategy Equilibria

Where pure Nash equilibria fail in discrete games, mixed-strategy equilibria emerge. In all-pay auctions with granular bid spaces, optimal play involves randomizing over nearly the entire feasible interval for high-valuation bidders, and the support can be much broader than what is observed in continuous or experimental approximations (Rasooly et al., 2020, Dziubiński et al., 2023).

4.2 Welfare and Efficiency

Pure Nash equilibria (when they exist) in discrete combinatorial auctions are perfectly efficient (Price of Anarchy = 1). In the absence of such equilibria, the welfare can degrade, with bounds scaling with the degree of subadditivity of bidder valuations and multiplicity of items (Hassidim et al., 2011). For submodular/subadditive bidders, welfare is preserved within modest factors; for unrestricted settings, anarchy can grow rapidly with problem size.

In combinatorial bargaining, the monotonicity and Pareto-efficiency of the bottom equilibrium ensures that any desired outcome in the Pareto set can be attained for some budget split; notably, a player with $X\%$ of the budget can guarantee an allocation at least as good as $X\%$ of all possibilities (1311.0913).

5. Extensions: Objective Variants, Richman/Tullock Games, and Combinatorial Bidding

5.1 Mean-Payoff, Energy, and Infinite-Horizon Games

Discrete-bidding games with mean-payoff and energy objectives display properties fundamentally distinct from their continuous counterparts. In the discrete case, the achievable mean-payoff may depend on the initial budgets, with thresholds governed by novel, discretized average properties and dynamic min-max recursions (Avni et al., 30 Aug 2025). In continuous games, the mean-payoff is purely a function of the structural parameters—budget ratios do not influence the long-run value (Avni et al., 2019).

5.2 Bidding Combinatorial Game Theory

Discrete bidding generalizes alternating normal play in combinatorial game theory, generating outcome classes characterized by monotonicity and marker worth, organized as finite lattices for any fixed budget (Kant et al., 2022). Constructive comparison and outcome testing extend classical Sprague-Grundy and Conway-Berlekamp techniques to the bidding regime, supported by poset structures, invertibility results for numerical games, and explicit group properties for numbers/dyadics (Kant et al., 2022).

5.3 Discrete All-Pay Bidding in Games and On Graphs

Discrete all-pay bidding games require gap-free mixed strategies, with explicit support, structure theorems (notably the Reverse Theorem), and computational methods based on Toeplitz matrix inversion and dynamic programming. Precise handling of chip valuation enables unique equilibria even as games are perturbed to imprecise settings. Combinatorial graph-based all-pay games admit threshold ratio analyses, stepwise/continuous value functions, and FPTAS implementations for DAGs (Menz et al., 2015, Avni et al., 2019).

6. Distinctions, Implications, and Practical Relevance

The distinctions between discrete and continuous bidding games are stark:

Dimension	Discrete Bidding	Continuous Bidding
Bid space	Integral/granular	Real-valued/arbitrary granularity
Equilibrium	Existence conditional; often mixed/unstable	Broad existence; often pure and symmetric
Welfare	Can be inefficient (mixed NE); PoA unbounded	Typically efficient with price/proportional splits
Determinacy	Dependent on tie-breaking, non-monotone	Robust, tie-breaking often irrelevant
Strategic form	Randomization, combinatorial complexity	Simple proportional/convex best-replies
Algorithmics	NP ∩ coNP, value iteration on thresholds	LP, convex minimization algorithms
Applications	High in auctions, real-world events	Theoretical, idealized contexts

These differences are not simply limit behaviors: small amounts of atomicity can qualitatively alter equilibria, strategic structure, and efficiency (Rasooly et al., 2020). Practical implications pervade market design (auctions with minimal increments), resource allocation, fair division, and automated reasoning in verification contexts.

7. Synthesis and Ongoing Directions

Research in discrete and continuous bidding games is rapidly evolving, with new results on algorithmic efficiency (Avni et al., 2022, Avni et al., 30 Aug 2025), expressive power in combinatorial game theory (Kant et al., 2022, Kant et al., 2022), extensions to all-pay and scoring objectives (Menz et al., 2015, Larsson et al., 2020), systematic exploration of poorman/discrete mechanisms (Avni et al., 2023), and empirical paper of discrete phenomena in economic experiments (Rasooly et al., 2020). Open problems include tighter welfare bounds in mixed equilibria, deeper integration of discrete bidding into mechanism design, and broader applications in formal verification and combinatorial computation.

The field demonstrates that the granularity of resource transfer in competitive dynamics is not a technical afterthought, but a fundamental determinant of mathematical behavior, strategic complexity, and societal outcomes.