Non-Archimedean Games

• Non-Archimedean games are a class of combinatorial games characterized by violating the classical Archimedean property using ordinal-indexed spaces, p-adic fields, and ultrametrics.
• They extend traditional game theory by incorporating infinite, hierarchical, and fractal-like strategy spaces that yield novel equilibria and deterministic outcomes.
• Applications span algorithmic game solutions, evolutionary dynamics, and quantum extensions, offering fresh computational and theoretical insights.

Non-Archimedean games constitute a broad and active area of research in combinatorial and algorithmic game theory in which the underlying objects, payoffs, strategy spaces, or order relations violate the classical Archimedean property. The archetypal examples involve games defined on ordinal-indexed spaces, games modeled via non-Archimedean fields (including $p$-adic numbers or hyperreal extensions), and infinite games on non-Archimedean ordered structures. These models reveal new phenomena in strategy, determinacy, and equilibrium—often yielding results unattainable in their Archimedean counterparts. Non-Archimedean games also provide new mathematical foundations for uniform processes, hierarchical strategy games, and extensions of quantum and evolutionary game theory.

1. Theoretical Foundations: Non-Archimedean Structures in Game Theory

A space $(\mathbb{K}, |\cdot|)$ is non-Archimedean if its valuation satisfies the ultrametric inequality (for valuations), or the order relation does not admit a scale such that $n x > 1$ for all nonzero $x$ in $\mathbb{K}$ and $n \in \mathbb{N}$ (for ordered groups/fields). Core non-Archimedean objects in game theory include:

• Ordinal numbers as in games played on well-ordered sets of arbitrary transfinite length. Such orderings are well-founded, unbounded above, and lack linear Archimedean comparability.
• $p$-adic numbers ($\mathbb{Q}_p$) and fields of hyperreals ($^*\mathbb{R}$), which have infinitesimal and infinitely large elements, with “distances” governed by an ultrametric instead of a Euclidean metric.
• Lexicographic products and partially ordered sets with blocks, as in lexicographically ordered trees and fibers, producing discrete stratification that prevents dense embedding of the reals.

The non-Archimedean property manifests in both the combinatorial structure (e.g., lack of finite upper bounds, infinitesimals) and the topology (ultrametric balls, disconnectedness, absence of Archimedean “scaling”).

2. Transfinite and Ordinal Games

Games defined on ordinal-indexed spaces are among the earliest and most canonical non-Archimedean games. The transfinite version of Welter's Game (Abuku, 2017) exemplifies this approach:

• The board is an ordinal-indexed belt (e.g., positions labeled by ordinals), with coins placed on distinct ordinal squares.
• Each move consists of relocating a coin leftward to a strictly smaller, unoccupied ordinal.
• Terminal positions arise when no further moves are possible due to well-foundedness.

The game is solved using an extension of Grundy values and nim-sum operations to ordinals, employing Cantor normal form. The Grundy number of a position $(a_1, \ldots, a_n)$, where $a_i = \omega\lambda_i + m_i$, is given as

$$\mathcal{G}(a_1, \ldots, a_n) = \omega \cdot (\lambda_1 \oplus \ldots \oplus \lambda_n) + \bigoplus_{\lambda} \left[ m_{i_1} \mid \cdots \mid m_{i_{k_\lambda}} \right]$$

where the direct sum runs over all finite parts sharing their infinite coefficient. This decomposition leverages the non-Archimedean nature of ordinals: moves always decrease some component, preventing infinite descent. The analysis generalizes the classical Sprague-Grundy theory to transfinite domains, showing that classical impartial game solution methods adapt to non-Archimedean settings so long as the underlying move relations are well-founded. Such games clarify the structural meaning of infinite divisibility and “inaccessible” positions (Abuku, 2017).

3. p-adic and Non-Archimedean Game-Theoretic Frameworks

Non-Archimedean game theory extends classical frameworks by formulating game objects, payoffs, or strategy probabilities in $p$-adic or other ultrametric fields (Toni, 15 Apr 2025). Key features:

• Strategic games with $p$-adic payoffs: Utilities $u_i : S \to \mathbb{Q}_p$ permit ultrametric modeling of hierarchical or abrupt (mutational) strategy change, supporting multi-layered modeling absent in standard $\mathbb{R}$-valued frameworks.
• Probability measures: A $p$-adic “probability” is an element of $\mathbb{Q}_p$ assigned to outcomes. Non-Archimedean models (e.g., hyperfinite counting on large but “finite” sets $\mathcal{H}$) enable regular, positive probability assignment to every possible outcome, including infinitesimals, unattainable in Archimedean setups (Bottazzi et al., 25 Feb 2025).
• Evolutionary game theory and dynamics: Replicator and evolutionary processes can be defined on $p$-adic simplices to model hierarchical frequencies and cascaded mutations, capturing phenomena that would be masked or collapsed by real-valued averaging.

A fundamental difference is the ultrametric (strong) triangle inequality:

$|x + y|_p \leq \max(|x|_p, |y|_p)$

which leads to fractal-like clustering of solutions, layered equilibria, and non-uniform convergence properties in strategy space.

4. Infinite Games on Non-Archimedean Ordered Sets

Non-Archimedean linear and partially ordered sets serve as the basis for a range of infinite games with distinct properties, often violating classical dichotomies.

• In Baker’s Game and generalizations (Matos-Wiederhold et al., 26 Aug 2024), the separation between "large" and "small" sets (e.g., countable vs uncountable) is radically altered on non-Archimedean orders such as lexicographic products $\kappa^* \times \mathbb{Q}$. For these, Player II can have a winning strategy on every payoff set, regardless of cardinality, in contrast to the real line, where II's winning is equivalent to the payoff set's countability.
• Dense lexicographic products present "block" structures, with fibers acting as discrete levels: a player’s winning strategy can proceed independently level by level, which is impossible in Archimedean, densely ordered sets (Matos-Wiederhold et al., 26 Aug 2024).
• Poset versions of topological games (e.g., Banach-Mazur games on partial orders (Kubiś, 2015)) admit non-Archimedean branching and antichain structures, further generalizing universality results in model theory, forcing, and category theory.

The general trend is that “size” and “density” may fail to characterize determinacy or strategic strength as in the Archimedean case; instead, structural decomposability and non-linear orderings enable more robust or even “omnipotent” player strategies.

5. Non-Archimedean Models in Uniform Processes and Critique

Uniform probability processes exhibit fundamental differences when modeled non-Archimedeanly (Bottazzi et al., 25 Feb 2025). In hyperfinite or $p$-adic models:

• Every possible outcome can receive positive, often infinitesimal, probability.
• Constraints such as full rotational symmetry (invariance under arbitrary real rotations) are generally relaxed: non-Archimedean models can maintain invariance under rational steps or cyclic permutations but not full $\mathbb{R}$-symmetry.
• Non-uniqueness and underdetermination: Physically plausible axioms (regularity, uniformity, some symmetry) are insufficient to select a unique probability model in either paradigm. Multiple Archimedean and non-Archimedean extensions exist, each reflecting different aspects of the underlying process.

Debate on these grounds has focused on technical and philosophical issues:

• Symmetry concerns: Critics assert that non-Archimedean models violate intuitive isomorphism invariance, but this relies on a prior commitment to $\mathbb{N}$-indexed bases, which non-Archimedean models eschew.
• Haecceity and excess structure: Objections regarding label dependence and existential flexibility are addressed by distinguishing internal vs external objects in nonstandard analysis.
• Regularity, minimality, and model selection: Both frameworks yield minimal models for appropriate subalgebras, and regular, highly symmetric hyperfinite models often approximate Lebesgue measure as closely as required within their domain.

Non-Archimedean models thus expand the arsenal for modeling stochastic or combinatorial uniformity and are increasingly recognized as legitimate, particularly for large-$N$ stochastic processes or foundational exploration (Bottazzi et al., 25 Feb 2025).

6. Algorithmic and Applied Non-Archimedean Games

Recent advances demonstrate the computational impact of non-Archimedean concepts for infinite-duration, multi-dimensional, and direction-sensitive games:

• Perfect half space games (Colcombet et al., 2017) generalize escape games in $\mathbb{Z}^d$ by requiring Player 2 to drive the sum of weights to $-\infty$ in directions consistent with a dynamically chosen sequence of perfect half spaces—a formalization of nested escape directions akin to non-Archimedean priority structures.
• These models admit optimal positional strategies, strong determinacy, and efficient algorithmic solutions for multi-dimensional energy parity games—closing complexity gaps and providing pseudo-polynomial or 2-EXPTIME algorithms in fixed dimensions.
• The translation chain

$$\text{Energy Parity Games} \to \text{Extended Energy Games} \to \text{Bounding Games} \to \text{Perfect Half Space Games} \to \text{Lexicographic Energy Games} \to \text{Mean-Payoff Games}$$

demonstrates how non-Archimedean (directional, priority-based) game perspectives can be systematically integrated into algorithmic game analysis (Colcombet et al., 2017).

Applications span controller synthesis for multi-resource systems, hierarchical strategy optimization, and formal verification.

7. Non-Archimedean Games and Quantum/Game-Theoretic Extensions

Recent efforts couple non-Archimedean structures with quantum information and extended equilibrium theories (Toni, 15 Apr 2025):

• p-adic quantum games extend the quantum game-theoretic paradigm by formulating the Hilbert space over quadratic extensions of $\mathbb{Q}_p$, utilizing ultrametric probability and facilitating entanglement and superposition in enriched (non-Archimedean) state spaces.
• The resulting models permit the definition of p-adic qubits, non-Archimedean superpositions, and the analysis of joint strategies in layered or fractal environments. Equilibrium and Pareto-optimality are analyzed with respect to ultrametric topologies, yielding new efficiency or coordination properties out of reach in the classical or Archimedean quantum framework.
• Evolutionary models (p-adic EGT) benefit from the ability to model mutation and adaptation not just linearly, but cascaded via ultrametric distances, capturing multi-level social, biological, or computational dynamics.

A summary of key distinctions is provided below.

Concept	Archimedean Model	Non-Archimedean Model
Payoff Space	$\mathbb{R}$	$\mathbb{Q}_p$, $^*\mathbb{R}$
Probability Distribution	Real, $[0,1]$	$p$-adic, not necessarily in $[0,1]$
Metric	Euclidean	Ultrametric (strong)
Quantum Base Field	$\mathbb{C}$	Quadratic extension of $\mathbb{Q}_p$
Hierarchy/Evolution	Flat, linear	Hierarchical, multi-level, fractal

8. Algebraic and Combinatorial Extensions: Affine Normal Play

Recent developments in algebraic combinatorial game theory have led to the analysis of affine normal play, an extension of classical Conway games to include infinitesimal and infinite elements—explicitly constructing absorbing values ($\infty$, $\overline{\infty}$) and "pathetic" minimal infinitesimals (Larsson et al., 8 Feb 2024).

• Affine games allow for termination by checkmate (absorbing elements), reflect true non-Archimedean behavior in their order and sum structure, and produce lattices with multiple atoms.
• The algebra accommodates games outside the classical group, with non-unique reduced forms and a proper embedding of the classical normal play set as the invertible substructure.
• Comparison and reduction involve handling of forced sequences and checks, not just direct options.
• The presence of minimal nonzero elements (e.g., $\#1\infty$), strictly less than any positive classical game, extends the combinatorial landscape in a non-Archimedean direction.
• This structure supports systematic CGT analysis of rulesets involving forced terminations and layered threats, such as chess or nimstring (Larsson et al., 8 Feb 2024).

9. Open Problems and Future Directions

Non-Archimedean game theory remains fertile for foundational and applied investigation. Key open problems include:

• Classifying which non-Archimedean ordered structures admit player omnipotence or dichotomies absent from Archimedean environments (e.g., dense Aronszajn lines, Souslin continua) (Matos-Wiederhold et al., 26 Aug 2024).
• Extending algorithmic frameworks for games on more general ultrametric or ordinal-indexed spaces.
• Further integration of non-Archimedean models with quantum, stochastic, and evolutionary theories, particularly for systems exhibiting hierarchical, multi-scale, or infinite-component dynamics (Toni, 15 Apr 2025).
• Formalization and application of non-Archimedean probability models to the simulation and analysis of large-$N$ processes, infinite lotteries, and statistical regularity (Bottazzi et al., 25 Feb 2025).

10. Significance

Non-Archimedean games transcend limitations of classical game-theoretic models by providing rigorously analyzable, functionally rich, and computationally tractable frameworks for infinite, hierarchical, fractal, and directionally-sensitive scenarios. They bridge combinatorial, algebraic, probability-theoretic, and quantum paradigms, enabling solution methods and conceptual understanding tailored to complex, layered, or infinite-dimensional systems.