Capital Games: Dynamics & Growth Equilibria

• Capital games are finite strategic models with deterministic capital dynamics that transform observed capital payoffs into induced, time-average growth utilities.
• They replace traditional von Neumann–Morgenstern utilities by using dynamics linearization, accommodating both additive and multiplicative capital evolutions.
• The framework bridges standard game theory with capital growth analysis, showing that growth equilibria correspond directly with Nash equilibria under specified conditions.

Capital games are games in which action-profile payoffs are specified in units of capital, while utility is not taken as primitive. In the formal framework of “Capital Games and Growth Equilibria,” a finite capital game is a tuple $(N, W, A, x, D, f)$, where $N$ is a finite player set, $A$ is a profile of finite action sets, $x_i$ gives player $i$’s payoff in capital for each action profile, $W$ is the vector of initial capital endowments, $D$ gives game durations, and $f_i(x_i(a), w_i, \delta_i)$ specifies player-specific capital dynamics over the duration $\delta_i$ (Abramowitz, 1 Oct 2025). The central claim of the framework is that payoffs measured in capital are not automatically von Neumann–Morgenstern utilities; rather, utility is induced by the dynamics through which capital evolves. In adjacent literatures, the phrase has also been used more loosely for stochastic games in which capital is the central state variable, including Parrondo-type redistribution models, network games for inequality control, and multiplicative human-capital models with taxation (Zappalà et al., 2014, Miszczak, 2022, Lorenz et al., 2012).

1. Formal structure and departure from standard utility-based games

A standard finite normal-form game is written as $(N, A, u)$, with $N$0 directly interpreted as von Neumann–Morgenstern utility. In that representation, best responses maximize expected utility, $N$1. Capital games replace this primitive-utility specification with a capital specification: observable consequences are given by $N$2, each player has an initial endowment $N$3, and the effect of an action profile is mediated by a deterministic capital-dynamics function $N$4 (Abramowitz, 1 Oct 2025).

For most of the analysis, durations are normalized so that $N$5 for all players, and the game is written as $N$6. This normalization does not remove the temporal content of the model; it compresses each decision into a representative time step while preserving the claim that players evaluate strategies through long-run capital growth rather than through one-shot capital levels. The framework assumes complete information: players know others’ action sets, initial capital, capital dynamics, and durations, because those objects are needed to infer equilibrium strategies (Abramowitz, 1 Oct 2025).

The paper also distinguishes positive capital games, defined by $N$7 and $N$8 for all players and all action profiles. Positivity is not merely a technical refinement. It is required for multiplicative dynamics and logarithmic linearizations, and it underpins the main equilibrium-correspondence theorem (Abramowitz, 1 Oct 2025).

2. Capital dynamics, linearization, and induced utilities

The core technical device is dynamics linearization. A function $N$9 is a linearization of player $A$0’s capital dynamics if

$A$1

This condition identifies a transformed scale on which the dynamics become linear in time, so that the relevant object for choice is a growth rate in the transformed variable rather than raw capital itself (Abramowitz, 1 Oct 2025).

Two canonical cases organize the theory. Under additive dynamics,

$A$2

a linearization is the identity, $A$3. Under multiplicative dynamics,

$A$4

and for positive capital a linearization is $A$5, because

$A$6

The framework allows other deterministic dynamics as long as a monotonically increasing $A$7 exists and satisfies the linearization condition (Abramowitz, 1 Oct 2025).

The decision axiom is that players seek to maximize the time-average growth rate $A$8 of their capital. Because the growth rate defined through the linearization is treated as ergodic, its time average equals its expectation. With $A$9,

$x_i$0

where the expectation is taken with respect to the mixed-strategy distribution $x_i$1 (Abramowitz, 1 Oct 2025).

This produces an induced utility function

$x_i$2

Hence

$x_i$3

The construction

$x_i$4

turns capital-valued payoffs into a von Neumann–Morgenstern utility function whose expectation coincides with the player’s time-average growth rate (Abramowitz, 1 Oct 2025).

3. Growth equilibrium and correspondence with Nash equilibrium

Best response in a capital game is defined in growth terms. Given opponents’ strategy profile $x_i$5, player $x_i$6’s best response is

$x_i$7

With $x_i$8, this becomes

$x_i$9

A growth equilibrium is a strategy profile $i$0 such that every $i$1 is a best response to $i$2 in this sense. Equivalently, no player can improve their time-average growth rate by a unilateral deviation (Abramowitz, 1 Oct 2025).

The paper’s central theorem states that if $i$3 is a positive capital game with $i$4 and linearizable dynamics, and if $i$5 is the associated standard game with

$i$6

then the Nash equilibria of $i$7 are exactly the growth equilibria of $i$8 (Abramowitz, 1 Oct 2025). The proof is direct: for every fixed $i$9, the argmax set defining player $W$0’s best responses in the capital game is identical to the argmax set in the derived standard game.

Several consequences follow immediately. Every positive capital game with linearizable dynamics has at least one growth equilibrium, because the associated finite standard game has a Nash equilibrium by Nash’s theorem. Computing a growth equilibrium is PPAD-complete, because the transformation to the derived normal-form game preserves the equilibrium-computation problem (Abramowitz, 1 Oct 2025).

The framework also isolates a useful boundary case: pure growth equilibria. For pure strategies, the growth rate is defined as

$W$1

In deterministic pure-strategy settings, any strictly monotone transformation preserves preference orderings, so the best response does not depend on the particular choice of $W$2. The paper therefore proves that if an action profile is a pure growth equilibrium for one capital dynamics, then it is a pure growth equilibrium for all other capital dynamics. A common misconception is that the choice of additive versus multiplicative dynamics always changes equilibrium behavior; the result shows that this dependence is specific to environments with randomization and expected transformed payoffs, not to deterministic pure-strategy comparisons (Abramowitz, 1 Oct 2025).

4. Canonical dynamics, explicit constructions, and the utility-conflation problem

Under additive dynamics, $W$3 and $W$4, so the derived utility is

$W$5

Since $W$6 is constant, best responses maximize expected capital. In this case, growth equilibria coincide with Nash equilibria in a game whose utilities are capital shifts. The paper also gives the reverse construction: from any standard game $W$7, one may choose any $W$8, define $W$9, and recover an equivalent capital game with additive dynamics (Abramowitz, 1 Oct 2025).

Under multiplicative dynamics,

$D$0

and therefore

$D$1

Best responses satisfy

$D$2

Players therefore maximize the geometric mean of capital payoffs, yielding a Kelly-type objective. Again there is a reverse construction: from any standard game $D$3, choose $D$4, define $D$5, and obtain a positive capital game whose growth equilibria match the Nash equilibria of the original standard game (Abramowitz, 1 Oct 2025).

The paper’s one-player illustration makes the dependence on dynamics explicit. With initial capital $D$6 and action payoffs $D$7, $D$8, additive dynamics imply utilities $D$9 and $f_i(x_i(a), w_i, \delta_i)$0, whereas multiplicative dynamics imply utilities $f_i(x_i(a), w_i, \delta_i)$1 and $f_i(x_i(a), w_i, \delta_i)$2. The ordinal ranking is unchanged, but the cardinal utilities differ. This is used to motivate what the paper calls the “fallacy of utility conflation”: capital-valued payoffs and initial wealth do not by themselves determine utility; the dynamics under which capital evolves must also be specified (Abramowitz, 1 Oct 2025).

5. Broader capital-centric game models in adjacent literature

Outside the equilibrium-correspondence framework, several strands of work treat capital as the state variable of a stochastic or strategic process. These models are not identical to the formal definition above, but they illuminate the wider landscape in which “capital games” operate.

In collective Parrondo models, capital evolves through gambling outcomes and redistributive transfers. “Selective altruism in collective games” studies populations of altruistic and selfish players under Toral’s version of collective Parrondo games, where redistribution through game $f_i(x_i(a), w_i, \delta_i)$3 and casino play through capital-dependent game $f_i(x_i(a), w_i, \delta_i)$4 jointly shape capital dynamics. The paper shows that naive altruism can improve aggregate capital while making altruists poorer and selfish players richer, whereas selective altruism restricts transfers to altruists and makes altruism individually profitable and evolutionarily stable under imitation (Zappalà et al., 2014).

Network inequality models reinterpret local game outcomes as inequality-changing transfers rather than direct gains. “Constructing games on networks for controlling the inequalities in the capital distribution” places agents with non-negative capital on a $f_i(x_i(a), w_i, \delta_i)$5 periodic grid, uses capital-conserving pairwise interactions, and studies Janosik, Matthew, strong Janosik, and strong Matthew policies through the Gini index

$f_i(x_i(a), w_i, \delta_i)$6

A central conclusion is that Parrondo’s paradox in mean capital does not translate straightforwardly into a paradox in inequality reduction; in some parameter regimes, playing the losing game $f_i(x_i(a), w_i, \delta_i)$7 alone reduces inequality more than paradoxical mixtures (Miszczak, 2022).

A second major cluster concerns Parrondo games as random walks of capital. “Exact probability distribution functions for Parrondo’s games” derives exact finite-time probability distributions for capital-dependent and history-dependent Parrondo games via Fourier transforms and eigenvalue decompositions of the transition matrix $f_i(x_i(a), w_i, \delta_i)$8, identifying oscillations near the maximum of the probability distribution and distinct odd/even limiting distributions after many rounds (Zadourian et al., 2016). “Parrondo games as disordered systems” reformulates these models through products of non-commuting Markov matrices and a transfer-matrix analogy with one-dimensional disordered systems, showing how weak-contrast regimes, temporal patterns, and rule correlations determine gain (Luck, 2019).

A different but conceptually related literature studies multiplicative growth and redistribution. “Redistribution spurs growth by using a portfolio effect on human capital” models human capital as a stochastic multiplicative process, $f_i(x_i(a), w_i, \delta_i)$9, and shows that equal-share redistribution after taxation can transform individually destructive dynamics into sustainable aggregate growth through a portfolio effect. In that setting, the growth factor lies between the geometric-mean benchmark with no redistribution and the scaled arithmetic-mean benchmark under full taxation and infinitely many agents (Lorenz et al., 2012). A further extension appears in “Does Capital Dream of Artificial Labour?”, which treats capital and labour as strategic resources in a multi-agent reinforcement-learning environment with Cobb–Douglas production functions and finds that learning agents disproportionately gravitate toward capital-intensive processes because capital is accumulative while labour is non-accumulable (Korecki et al., 16 Oct 2025).

Taken together, these literatures suggest a broader research program in which capital is not merely a payoff label but an evolving state whose dynamics, redistribution rules, and compounding properties alter strategic behavior. The formal contribution of the 2025 capital-games framework is to make that intuition explicit at the level of equilibrium theory (Abramowitz, 1 Oct 2025).

6. Assumptions, limitations, and theoretical significance

The formal theory is deliberately narrow. Its main assumptions are finite player sets and finite action sets, deterministic capital dynamics, linearizable dynamics for every player, positivity for the main correspondence theorem, complete information, and an ergodicity assumption equating time-average growth rates with expectations of transformed payoffs (Abramowitz, 1 Oct 2025). The model treats the game as a single representative time step of a repeated or ongoing process, but it does not explicitly formulate repeated games with histories or stochastic games.

These restrictions delimit the scope of the results. The framework does not analyze stochastic dynamics in which $\delta_i$0 itself is random, does not model uncertainty about dynamics or learning of dynamics, and does not study multiplicity, stability, or equilibrium selection beyond existence and correspondence. The theory is strongest precisely where it is most static: it shows how to map a dynamics-based growth criterion into an ordinary finite normal-form game without yet confronting richer intertemporal complications (Abramowitz, 1 Oct 2025).

Its significance lies in the bridge it establishes. The framework shows that when players care about the growth of wealth over time rather than about one-shot expected capital, their behavior can still be analyzed with standard Nash-equilibrium tools once utilities are derived from capital dynamics. It also sharpens a methodological point that recurs across the broader literature: observable capital, money, or wealth should not be conflated with utility. In the formal model, that warning appears as the “fallacy of utility conflation”; in the surrounding literatures, it reappears whenever redistribution, multiplicative compounding, network interaction, or capital-dependent gambling changes the relevant decision criterion away from raw expected capital (Abramowitz, 1 Oct 2025).

In the strict technical sense, then, capital games are finite games $\delta_i$1 in which utilities are induced by capital dynamics and equilibria are defined by time-average growth maximization. In the broader literature, the term also points to a wider class of models in which capital is the evolving strategic state. The unifying theme is that capital dynamics—not merely capital levels—determine the meaningful objective of the game.