📈 economics

Capital Games and Growth Equilibria

This paper introduces "capital games," a dynamic framework where payoffs are measured in capital rather than utility, allowing researchers to infer player utilities by assuming they aim to maximize the time-average growth rate of their capital.

Original authors: Ben Abramowitz

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ben Abramowitz

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

Imagine you are playing a board game. In the classic version of game theory (the kind invented by von Neumann and Morgenstern), the game ends, you count your points, and you say, "I won 10 points, you won 5." The theory assumes that 10 points is always twice as good as 5 points, no matter what. It treats "points" as a direct measure of happiness or "utility."

But Ben Abramowitz's paper, Capital Games and Growth Equilibria, argues that this assumption is often wrong in the real world. In real life, points aren't just abstract numbers; they are capital (like money, resources, or time), and how you use them changes how much they are actually worth to you.

Here is the paper's core idea, broken down with simple analogies.

1. The Problem: Points vs. Growth

The paper starts with a famous puzzle: The "Coin Flip."
Imagine you have $100. You can play a game where you flip a coin:

Heads: You win 50% (you now have $150).
Tails: You lose 40% (you now have $60).

If you treat this like a standard math problem, the average outcome is $105. Since $105 is more than your starting $100, a standard game theorist would say, "Play it! It's a good bet."

But here is the catch: What if you have to play this game over and over again, using your winnings as the bet for the next round?

If you win then lose: $100 $\to$ $150 $\to$ $90.
If you lose then win: $100 $\to$ $60 $\to$ $90.

Even though the average of the two outcomes is positive, your actual money shrinks every time you play a mix of wins and losses. If you play this game 1,000 times in a row, you will likely go broke.

The paper says: Standard game theory fails here because it ignores time and dynamics. It assumes you can reset your wealth to $100 after every game. But in "Capital Games," your wealth carries over.

2. The Solution: Capital Games

The author introduces a new type of game called a Capital Game.

Standard Game: Payoffs are "Utility" (happiness). You just maximize the average number.
Capital Game: Payoffs are "Capital" (stuff you own). You don't just maximize the average; you maximize your growth rate over time.

Think of it like this:

Standard Game: You are a tourist visiting a theme park. You want to ride the most rides possible. Each ride is independent.
Capital Game: You are a farmer planting seeds. If you eat your seeds today, you have no seeds for tomorrow. The "payoff" of eating a seed depends on whether it helps your farm grow for the next season.

3. The Secret Sauce: Linearization

The paper's biggest breakthrough is a mathematical trick called Linearization.

In the real world, money grows in two main ways:

Additive: You win or lose a fixed amount (e.g., +$10 or -$10).
Multiplicative: You win or lose a percentage (e.g., +50% or -40%).

The paper shows that if you know how your capital grows (the "dynamics"), you can translate those real-world numbers into a "Utility Score" that standard game theory can understand.

If your money grows additively: The "Utility Score" is just the money itself. (Winning $10 is twice as good as winning $5).
If your money grows multiplicatively: The "Utility Score" is the logarithm of the money. (This is why the coin flip game is a bad bet; the "log score" of losing 40% hurts much more than the "log score" of winning 50% helps).

The Analogy:
Imagine you are trying to compare two different languages.

Capital is the raw text (e.g., "150" and "60").
Dynamics is the grammar rule (e.g., "Is this a percentage or a flat number?").
Linearization is the translator. It takes the raw text and the grammar rule and converts it into "Utility" (a language standard game theory understands).

Once you have this translation, you can use the standard tools of game theory (like finding a Nash Equilibrium) to solve the Capital Game.

4. The Result: Growth Equilibria

The paper defines a new kind of winning strategy called a Growth Equilibrium.

In a standard game, a Nash Equilibrium is where no one wants to change their move because they can't get more points.
In a Capital Game, a Growth Equilibrium is where no one wants to change their move because they can't get better long-term growth.

The paper proves a beautiful symmetry: If you translate a Capital Game into a Standard Game using the right "translator" (linearization), the winning strategies are exactly the same.

5. Why This Matters (According to the Paper)

The paper doesn't claim to solve every problem in the world. It specifically claims:

We can figure out what people actually want (their utility) if we know how their resources grow over time.
We can use old, familiar math tools to solve new, complex problems involving time and wealth growth.
The "payoff" isn't always the utility. Sometimes, a big win in dollars is actually a loss in "growth potential" if it puts you at risk of going broke.

Summary

Think of the paper as a manual for upgrading the rules of a board game.

Old Rule: "Count your points. The highest number wins."
New Rule (Capital Games): "Count your points, but remember that points today become your starting money for tomorrow. If you lose too much, you can't play anymore. So, the best move isn't always the one with the highest average points; it's the one that keeps your game going the longest."

The author shows us how to mathematically translate this "keep the game going" logic into the language of standard game theory, allowing us to solve these complex, time-based puzzles using familiar methods.

Technical Summary: Capital Games and Growth Equilibria

Problem Statement

Standard Game Theory, rooted in the von Neumann–Morgenstern (vNM) framework, assumes that players maximize the expected value of a utility function. However, a fundamental practical challenge remains: in many real-world scenarios, outcome payoffs are given in units of observable quantities (e.g., capital, money, resources) rather than utility. The vNM theorem establishes that a utility function exists for any preference relation satisfying specific axioms, but it does not prescribe how to derive that utility function from raw payoffs when the underlying dynamics of the system are unknown.

The core problem addressed by Abramowitz is the indeterminacy of player preferences when payoffs are not inherently utility. As illustrated by the Peters and Gell-Mann (PGM) coin flip example, the same monetary payoff can represent a desirable gamble or a losing proposition depending on whether the game is played in parallel (additive dynamics) or sequentially (multiplicative dynamics). Without specifying the temporal dynamics of how wealth evolves, one cannot uniquely determine the utility function or the rational strategy for a player.

Methodology

The paper introduces a generalized framework called Capital Games to incorporate dynamics directly into the game-theoretic model. The methodology proceeds through the following steps:

Generalization of Game Definitions:
- Capital Games: A capital game is defined by a tuple $(N, W, A, x, f)$ , where $N$ is the set of players, $A$ is the action space, $x$ represents payoffs in units of "capital" (not utility), $W$ is the initial capital endowment, and $f$ represents deterministic capital dynamics functions over a duration $\delta t$ .
- Dynamics Linearization: The paper assumes that capital dynamics are linearizable. A function $v_i$ is a linearization of dynamics $f_i$ if $v_i(f_i(x, w)) = \frac{v_i(x) - v_i(w)}{\delta t}$ . This allows the transformation of non-linear dynamics (like multiplicative growth) into additive forms.
Decision Criterion (The Decision Axiom):
- Drawing from Ergodicity Economics, the paper posits that players aim to maximize the time-average growth rate of their capital, denoted as $\bar{g}_i$ .
- Under the assumption of linearizable dynamics, maximizing the time-average growth rate is equivalent to maximizing the expected value of the linearized payoff: $E[v_i(f_i(x, w))]$ .
Equilibrium Definition:
- The paper defines a Growth Equilibrium as a strategy profile where every player maximizes their time-average growth rate given the strategies of others.
- This is contrasted with the standard Nash Equilibrium, where players maximize expected utility.
Mapping to Standard Games:
- The central methodological contribution is the construction of a corresponding standard game $G' = (N, A, u)$ from a capital game $G$ .
- The utility function for the standard game is derived directly from the capital dynamics: $u_i(a) = v_i(f_i(x_i(a), w_i))$ .
- The paper proves that the Growth Equilibrium of the capital game $G$ is exactly the Nash Equilibrium of the derived standard game $G'$ .

Key Contributions

1. Formalization of Capital Games

The paper generalizes the standard game definition to include initial endowments and capital dynamics. This shifts the focus from abstract utility to observable capital and its evolution over time, acknowledging that payoffs are not utility until processed through a specific dynamic lens.

2. Derivation of Utility Functions from Dynamics

The paper demonstrates that utility functions can be inferred from observables (payoffs) and dynamics.

Additive Dynamics: If capital changes additively ( $x_{new} = x_{old} + \Delta$ ), the linearization is the identity function ( $v(x)=x$ ). Here, capital is utility.
Multiplicative Dynamics: If capital changes multiplicatively ( $x_{new} = x_{old} \cdot r$ ), the linearization is the logarithmic function ( $v(x)=\ln x$ ). Here, utility is in "log-dollars."
This resolves the ambiguity in the PGM coin flip: the decision to play depends entirely on whether the dynamics are additive (play) or multiplicative (do not play).

3. Equivalence Theorem (Theorem 3)

The paper establishes a rigorous correspondence: For any positive capital game with linearizable dynamics, the set of Growth Equilibria is identical to the set of Nash Equilibria of a corresponding standard game where utilities are defined by the linearized capital dynamics.

Implication: Computing a growth equilibrium in a complex dynamic setting is mathematically equivalent to computing a Nash equilibrium in a transformed static game.
Complexity: Consequently, computing growth equilibria for positive capital games with linearizable dynamics is PPAD-complete, inheriting the complexity of finding Nash equilibria in standard games.

4. Pure Growth Equilibria

The paper defines pure growth equilibria (where players use pure strategies) and proves that if a pure growth equilibrium exists for any capital dynamics, it is also a pure growth equilibrium for all other capital dynamics. This is because, in deterministic pure strategies, the preference ordering over outcomes is independent of the specific dynamic transformation, provided the transformation is monotonic.

Results and Claims

Reversibility: The mapping between capital games and standard games is reversible. Any standard game can be viewed as a capital game with additive dynamics, and any positive capital game with linearizable dynamics can be mapped to a standard game.
Uniqueness of Preference: By specifying the dynamics, the paper claims to uniquely determine the preference ordering over lotteries for players maximizing time-average growth. Without specifying dynamics, no unique utility function can be derived from payoffs alone.
Limitations: The paper explicitly notes that it does not address non-linearizable dynamics or uncertainty regarding the dynamics themselves. It assumes deterministic dynamics where the only source of uncertainty is the mixed strategies of players.

Significance

The paper claims to bridge the gap between Ergodicity Economics and Game Theory. Its significance lies in providing a mechanism to:

Generalize Game Theory: Move beyond the assumption that payoffs are utility, allowing for the analysis of games where payoffs are physical quantities (capital) subject to specific temporal dynamics.
Resolve Indeterminacy: Offer a method to derive utility functions from observable payoffs by incorporating the time dimension and the nature of wealth evolution (additive vs. multiplicative).
Unify Equilibria: Show that the concept of "Growth Equilibrium" is not a new, intractable class of problems but is mathematically isomorphic to the well-studied Nash Equilibrium, provided the dynamics are linearizable.

The paper concludes that determining the appropriate dynamics for a specific application remains a critical problem for Mechanism Design and Decision Theory, particularly in settings like infinitely repeated games where endowments evolve from stage to stage.