📈 economics

All Games Have Equilibria

This paper establishes that every game with a nonempty set of players, nonempty action sets, and bounded utility functions admits a Nash equilibrium in finitely additive mixed strategies, thereby unifying equilibrium theory for infinite games and overcoming previous technical limitations associated with countable additivity.

Original authors: M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Game of "What If?"

Imagine a world where every decision you make is part of a giant, invisible game. You choose what to wear, what to eat, or how to drive to school, but your outcome depends not just on your choice, but on what everyone else chooses too. This is the realm of Game Theory, a branch of mathematics that studies how people (or computers, or animals) make decisions when their fates are intertwined. For a long time, mathematicians had a golden rule for these games: if the number of choices is finite (like Rock, Paper, Scissors), there is always a "perfect" balance called a Nash Equilibrium. It's a state where no one wants to change their move because they'd only get a worse result.

But what happens when the game gets infinite? What if you can choose any number between 0 and 1, or if there are infinite players? In the real world, many situations feel infinite. You can't always count the grains of sand on a beach or the exact speed of a car. For decades, when mathematicians tried to apply the "perfect balance" rule to these infinite games, the math broke. The tools they used, which relied on counting things in a very strict, "countable" way, would suddenly vanish or produce impossible results. It was like trying to measure the ocean with a teaspoon; the more you tried, the more the water seemed to disappear. This left a huge gap in our understanding: do these infinite games actually have a stable solution, or are they just chaotic messes?

The Paper's Big Discovery: "All Games Have Equilibria"

This paper, titled "All Games Have Equilibria," by M. Ali Khan, Arthur Paul Pedersen, and Maxwell B. Stinchcombe, steps in to fix the broken tools. The authors argue that the problem wasn't the games themselves, but the way mathematicians were trying to measure them. They propose a new, more flexible way of thinking about "mixed strategies"—which are basically random choices, like flipping a coin to decide your move.

In the old school of thought, if you wanted to mix your strategies, you had to follow strict rules of "countable additivity." Imagine you have a jar of marbles. If you pick a red one, then a blue one, then a green one, the total probability of picking one of those three is just the sum of their individual chances. This works great for a finite jar. But in an infinite game, this rule is like trying to add up an infinite list of numbers that keeps changing the answer depending on the order you add them. The authors say, "Let's drop that strict rule." Instead, they use finitely additive probabilities. Think of this as a super-powerful magnifying glass that can see the "just barely" moments that the old tools missed. It allows the math to handle the infinite without losing the tiny, crucial details that determine who wins.

The Main Finding:
The paper proves a stunningly simple but powerful theorem: Every single game with a bounded payoff (meaning the rewards aren't infinite) has a Nash Equilibrium. It doesn't matter if the game has infinite players, infinite choices, or messy, jagged payoffs that jump around. If you use these new, flexible "finitely additive" strategies, a stable balance always exists. The authors show that the set of these equilibria is not just empty or chaotic; it is "well-behaved," meaning it's stable and predictable. If you tweak the game slightly, the equilibrium shifts slightly, rather than vanishing into thin air.

What They Rule Out:
The authors are very clear about what doesn't work. They explicitly argue against the idea that we can always force these infinite games to fit into the old, strict "countably additive" box. They show that trying to do so often leads to "equilibria" that are actually nonsense—like a game where both players win money in a zero-sum game (where one person's gain is the other's loss), which is impossible. They also rule out the idea that we can just ignore the "just under" or "just over" details. In many of these games, the difference between choosing 0.4999 and 0.5000 is everything. The old math would treat them as the same point and lose the information; the new math keeps that distinction alive.

How Sure Are They?
This isn't a guess or a simulation. The authors provide rigorous mathematical proofs. They don't just suggest that these equilibria exist; they demonstrate that they must exist under the conditions they define. They also prove that these equilibria are "finitely approximable," meaning you can find them by looking at smaller, finite versions of the game and seeing where they lead. This gives the theory a practical, operational feel: it's not just a theoretical ghost; it's something you can actually approach and understand.

The "Just Under" Magic

To understand why this matters, imagine a game of "Just Under." Two players are trying to pick a number just below 0.5.

Player A picks 0.49.
Player B picks 0.499.
Player C picks 0.4999.

In the old math, as you get closer and closer to 0.5, the numbers all blur together into the single point 0.5. If both players pick 0.5, the game might break or have no solution. But in the real world, there is always a "winner" of the "just under" contest. The player who picked 0.4999 is closer to the goal than the one who picked 0.49.

The authors show that their new math preserves this "just under" information. It's like having a camera that never loses focus, even as the numbers get infinitely small. They use a clever trick involving "nets" (a fancy way of organizing infinite lists) and "hyperfinite sets" (imaginary sets that are huge but still finite in a special way) to show that you can always find a stable balance.

Why This Changes Everything

The paper tackles famous, stubborn problems that mathematicians have struggled with for decades, like the Sion and Wolfe game and Wald's Largest Integer Game. In these games, the old math said, "No equilibrium exists!" or "The answer depends on how you count!" The new math says, "Here is the equilibrium, and here is exactly why it works."

For example, in a game where you try to pick the largest integer, the old math got stuck because there is no "largest" integer. The new math shows that the equilibrium involves a specific kind of "finitely additive" probability that captures the spirit of "trying to be the biggest" without getting stuck on the fact that you can always go bigger.

The authors also show that this approach doesn't break the games we already know. If a game is simple and finite, their new math gives the exact same answer as the old math. It's a superset: it includes all the old solutions and adds new ones for the infinite cases.

The Bottom Line

This paper is a unifying program. It takes a patchwork of confusing rules and counterexamples and replaces them with a single, clean framework. It tells us that infinite games are not broken; our tools were just too rigid. By relaxing the rules of how we count probabilities, we can see that stability and balance are always possible, even in the most chaotic, infinite scenarios. The authors have built a bridge between the finite world we can count and the infinite world we can only imagine, showing that the laws of game theory hold true all the way to the edge of infinity.

Technical Summary: "All Games Have Equilibria"

Problem Statement
The paper addresses a fundamental gap in game theory: while Nash's existence theorem guarantees equilibria for finite games, the literature on infinite games (where player sets or action sets are infinite) lacks a unified existence theory. The prevailing framework relies on countably additive mixed strategies and requires strong topological assumptions, specifically compact action spaces and joint continuity of utility functions. Without these, classical results fail; seminal counterexamples by Wald (1945) and Sion and Wolfe (1957) demonstrate that zero-sum games with discontinuities or non-compactness often lack countably additive equilibria, or even approximate ( $\epsilon$ -) equilibria. Previous attempts to resolve this have resulted in a "patchwork" of technical preconditions, restrictive conditions on discontinuities, or the redefinition of utility functions via sharing rules, often at the cost of strategic integrity (e.g., allowing strictly dominated strategies or violating the antagonistic nature of zero-sum games).

Methodology
The authors propose a unified program that revises the model of mixed strategies by replacing countable additivity with finite additivity.

Game Specification: A game is defined by a nonempty set of players, nonempty action sets (with no topological or measure-theoretic structure assumed), and bounded von Neumann-Morgenstern utility functions.
Strategy Space: Mixed strategies are modeled as finitely additive probabilities defined on the class of all subsets of the action space (denoted $\Delta(X)$ ). This space is compact in the product topology (Tychonoff's theorem), where convergence is defined by the convergence of integrals against all bounded functions.
Equilibrium Concept: An equilibrium is defined as a vector of independent mixed strategies such that no player can gain by deviating to a pure strategy. Independence is maintained via product extensions of individual strategies.
Approximation: The authors utilize exhaustive nets of finite approximations rather than simple sequences. An exhaustive net ensures that every action in the infinite set is eventually included in the finite approximations. This allows for the analysis of limits that preserve payoff-relevant information lost in standard countably additive limits.
Refinement: The paper isolates a subset of equilibria that arise as limits of equilibria in finite approximations where iteratively weakly dominated strategies have been deleted.

Key Contributions and Results

Universal Existence (Theorem A):
The paper proves that every game with bounded utilities admits a Nash equilibrium in finitely additive mixed strategies. This result holds regardless of the cardinality of the player set, the topology of action spaces, or the continuity of utility functions.
Stability and Continuity (Theorem B):
The equilibrium correspondence (mapping utility functions to the set of equilibria) is shown to be nonempty-valued, compact-valued, and upper hemicontinuous. This establishes that the theory is stable under perturbations of payoffs, a property often lacking in classical infinite game theory.
Operationalizability and Finite Approximability (Theorem C):
The authors demonstrate that the set of finitely approximable equilibria (limits of equilibria from exhaustive finite approximations) is nonempty and well-behaved. Furthermore, the subset of these equilibria that survive the iterated deletion of weakly dominated strategies is also nonempty, compact-valued, and upper hemicontinuous. Crucially, these limits do not place mass on iteratively weakly dominated strategies.
Resolution of Classical Pathologies:
- Fubini's Theorem: The paper argues that the failure of Fubini's theorem for finitely additive measures is not a defect for equilibrium analysis. For the specific class of equilibria identified in Theorem C (limits of finitely supported product probabilities), the limit remains a product extension, preserving independence.
- Discontinuities: In games like the Sion-Wolfe game or the asymmetric location game, the finitely additive framework captures "just under" or "just over" behaviors (e.g., concentrating mass infinitesimally close to a discontinuity point) that countably additive limits collapse into a single point, thereby destroying the equilibrium property.
- Sharing Rules: Unlike the Simon-Zame (1990) sharing rule approach, which redefines utilities at discontinuities and may select strictly dominated strategies, the finitely additive approach preserves the original utility functions and strategic logic without arbitrary tie-breaking rules.
Utility Equivalence (Theorems D & E):
The paper delineates the boundary between the new theory and the classical one. It identifies conditions (related to the continuity of deviation payoffs and the mass of the equilibrium on discontinuity sets) under which finitely additive equilibria yield the same utility predictions as countably additive ones. When these conditions fail, the divergence reflects payoff-relevant distinctions present in finite approximations that the classical framework cannot represent.

Significance and Claims
The authors claim that their approach provides a single overarching program for equilibrium theory that restores the foundational properties of finite games (existence, stability, and operationalizability) to the infinite case.

Naturalness: They argue that relaxing countable additivity is not merely a technical maneuver but a necessary step to retain information encoded in finite approximations. In discontinuous problems, the countably additive framework systematically discards information relevant to strategic responses.
Non-Revisionist: The theory is not indiscriminately revisionist; it coincides with classical results (e.g., Glicksberg, 1952) for compact Hausdorff spaces with jointly continuous payoffs.
Operationalizability: By grounding equilibria in limits of finite approximations and iterated dominance, the theory connects infinite games to the strategic logic of finite models, making previously "intractable" games amenable to direct analysis.
Future Directions: The paper suggests that this framework is particularly well-suited for continuum extensive-form games and infinite population games, where the "disappearance" of information in the limit (a known issue in countably additive models) is central to the equilibrium problem.

In summary, the paper asserts that by adopting finitely additive mixed strategies and exhaustive finite approximations, one can construct a robust, general, and strategically faithful theory of equilibrium for all games with bounded utilities, effectively unifying the treatment of finite and infinite games.