Strategic Superposition and Replicator Dynamics: Quantum Collapses in Decision Processes

1. Introduction

Evolutionary game theory inherited from its biological origins a sharp ontology of strategy. An actor either plays a pure strategy, drawn from a finite repertoire, or plays a mixed strategy understood as a classical probability distribution over that repertoire [1,2]. Population-level change is then described by the replicator equation, which raises the frequency of strategies whose payoff exceeds the population average [3,4,5]. In both the pure and the mixed reading, the actor is taken to hold a strategy before interaction occurs. Randomization, where present, is a classical lottery: the actor possesses a definite probability vector, and the only uncertainty concerns which sample is realized.

A growing body of behavioral evidence sits uneasily with this picture. Choices display order effects, disjunction effects, and framing effects that violate the axioms of classical probability and expected utility [6,7]. Quantum decision theory responds by replacing classical probability with the probability calculus of quantum mechanics, in which amplitudes rather than probabilities are the primitive objects and interference between alternatives becomes possible [8,9,10,11,12,13,14]. In parallel, the quantum games program quantizes the act of play itself, allowing actors to select unitary operations on entangled qubits and studying the resulting equilibria [15,16,17,18].

Neither line of work addresses the question this paper takes up. Quantum decision theory models the single choice of a single agent and stops at the moment of decision. The quantum games program quantizes strategies and payoffs but works with equilibrium concepts rather than population dynamics, and it does not treat the pre-decision state of an actor as an evolving object subject to selection. What is missing is a framework in which (a) an actor is genuinely undecided before interaction, in the strong sense that no definite strategy exists rather than the weak sense that an existing strategy is unknown; (b) interaction collapses this indeterminate state onto a realized strategy; and (c) the distribution of realized strategies, together with the underlying amplitudes, is reshaped over evolutionary time by differential fitness.

This paper supplies that framework. We model the firm before market entry as a superposition of pure strategies, treat market interaction as a measurement that collapses the superposition, and couple the resulting frequencies to replicator dynamics through a density-operator master equation. The construction has three payoffs beyond unification. First, it gives an exact sense in which classical evolutionary dynamics is a limiting regime of a more general quantum strategic dynamics, recovered when interaction with the environment is strongly decohering. Second, it predicts that the realized strategy distribution of a superposed firm depends on the context, or measurement basis, in which the interaction is framed, a dependence that no classical mixed strategy with the same marginals can reproduce. Third, it formalizes the notion of constitutive self-opacity that anchors the Homo Evolutivus program: the firm’s strategy is constitutively indefinite, not merely hidden from observers or from the firm itself, and superposition is the natural mathematics of that indefiniteness.

2. Materials and Methods

2.1. The Strategy Hilbert Space and Basis States

Let an actor face a finite set of pure strategies $\mathrm{S}=\{{\mathrm{s}}_1,{\mathrm{s}}_2,\dots ,{\mathrm{s}}_{\mathrm{n}}\text{}}$ We associate with $\mathrm{S}$ a complex Hilbert space $\mathrm{H}\cong {\mathrm{C}}^{\mathrm{n}}$ and a distinguished orthonormal basis $\{\left|{\mathrm{s}}_1⟩,\dots ,\right|{\mathrm{s}}_{\mathrm{n}}⟩\text{}}$ the strategy basis, with $⟨{\mathrm{s}}_{\mathrm{i}}|{\mathrm{s}}_{\mathrm{j}}⟩={\mathsf{\delta }}_{\mathrm{ij}}$. A committed actor playing the pure strategy ${\mathrm{s}}_{\mathrm{i}}$ is in the basis state $|{\mathrm{s}}_{\mathrm{i}}⟩$. The strategy basis plays the role of the pointer basis in the measurement theory of open systems: it is the set of outcomes that interaction with the market renders definite [19].

2.2. Superposition, the Born Rule, and Collapse at Interaction

An actor who has not committed to a pure strategy is in a superposition

$$\left|\mathsf{\psi }⟩={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{c}}_{\mathrm{i}}\hspace{0.17em}\right|{\mathrm{s}}_{\mathrm{i}}⟩, {\mathrm{c}}_{\mathrm{i}}\text{∈}\mathrm{C}, {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\left|{\mathrm{c}}_{\mathrm{i}}\right|}^2=1.$$

(1)

The complex numbers ${\mathrm{c}}_{\mathrm{i}}$ are probability amplitudes. They carry more information than a classical probability vector, since each amplitude has a modulus and a phase, ${\mathrm{c}}_{\mathrm{i}}={\mathrm{r}}_{\mathrm{i}}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{i}}}$ with ${\mathrm{r}}_{\mathrm{i}}\ge 0$. The squared moduli reproduce a classical distribution over strategies, while the phases encode relations between strategies that have no classical counterpart and become observable only under a change of measurement context or under entanglement.

When the actor interacts with the market in the strategy basis, the realized strategy is ${\mathrm{s}}_{\mathrm{i}}$ with probability given by the Born rule [20],

$$\mathrm{Pr}\left({\mathrm{s}}_{\mathrm{i}}\right)=|⟨{\mathrm{s}}_{\mathrm{i}}\mathsf{\psi }⟩{|}^2={\mathrm{c}}_{\mathrm{i}}{|}^2={\mathrm{r}}_{\mathrm{i}}^2.$$

(2)

We write ${\mathrm{p}}_{\mathrm{i}}\text{:}={\left|{\mathrm{c}}_{\mathrm{i}}\right|}^2$ for the realization probability of ${\mathrm{s}}_{\mathrm{i}}$. The normalization $\sum _{\mathrm{i}}{\left|{\mathrm{c}}_{\mathrm{i}}\right|}^2=1$ guarantees that $\left({\mathrm{p}}_1,\dots ,{\mathrm{p}}_{\mathrm{n}}\right)$ is a point of the probability simplex ${\mathsf{\Delta }}^{\mathrm{n}-1}$.

The conceptual core of the model is the identification of the moment of interaction with a measurement. Before the firm enters the market, plays the game, or commits resources, its strategic state is the superposition $|\mathsf{\psi }⟩$. Entry is an observation in the strategy basis, described by the projective (von Neumann) measurement with projectors ${\mathrm{P}}_{\mathrm{i}}=\left|{\mathrm{s}}_{\mathrm{i}}⟩⟨{\mathrm{s}}_{\mathrm{i}}\right|$ [21]. The interaction yields outcome ${\mathrm{s}}_{\mathrm{i}}$ with probability ${\mathrm{p}}_{\mathrm{i}}=⟨\mathsf{\psi }\left|{\mathrm{P}}_{\mathrm{i}}\right|\mathsf{\psi }⟩={\left|{\mathrm{c}}_{\mathrm{i}}\right|}^2$, and the post-interaction state collapses to the corresponding basis state, $\left|\mathsf{\psi }⟩\text{→}\right|{\mathrm{s}}_{\mathrm{i}}⟩$ with probability ${\left|{\mathrm{c}}_{\mathrm{i}}\right|}^2$. The realized strategy then earns the payoff determined by the game. Two features deserve emphasis. The strategy that is realized is created at the moment of interaction; it is not read off a value the firm secretly held. And the probabilities that govern realization are exactly the squared amplitudes, so the act of collapse generates the very frequencies whose population-level evolution the replicator dynamics will govern.

Projective measurement is an idealization in which interaction extracts complete information and produces full commitment. Markets frequently do less. A firm may signal partial intent, leak partial information, or commit gradually. The appropriate generalization is a positive operator-valued measur, a family of positive operators $\{{\mathrm{E}}_{\mathrm{k}}\text{}}$ with $\sum _{\mathrm{k}}{\mathrm{E}}_{\mathrm{k}}=\mathrm{I}$ [22]. Outcome $k$ occurs with probability $\mathrm{Pr}\left(\mathrm{k}\right)=⟨\mathsf{\psi }\left|{\mathrm{E}}_{\mathrm{k}}\right|\mathsf{\psi }⟩$, and the post-interaction state is fixed by an associated set of Kraus operators $\{{\mathrm{M}}_{\mathrm{k}}\text{}}$ with ${\mathrm{E}}_{\mathrm{k}}={\mathrm{M}}_{\mathrm{k}}^{\text{†}}{\mathrm{M}}_{\mathrm{k}}$, via $\left|\mathsf{\psi }⟩\text{↦}{\mathrm{M}}_{\mathrm{k}}\right|\mathsf{\psi }⟩/\text{∥}{\mathrm{M}}_{\mathrm{k}}|\mathsf{\psi }⟩\text{∥}$. Projective measurement is the special case ${\mathrm{E}}_{\mathrm{k}}={\mathrm{P}}_{\mathrm{k}}$. Positive operator-valued measure’s let the model represent weak or noisy interactions in which superposition is only partly resolved, a flexibility we use when discussing decoherence in Section 3.2.

2.3. Density-Operator Representation and Decoherence

A single firm in a pure superposition is described by the density operator $\mathsf{\rho }=\left|\mathsf{\psi }⟩⟨\mathsf{\psi }\right|$. To represent a population of firms in various states, or a single firm whose phase information has partly degraded, we use a general density operator

$$\mathsf{\rho }=\sum _{\mathsf{\mu }}{\mathrm{w}}_{\mathsf{\mu }}\hspace{0.17em}\left|{\mathsf{\psi }}_{\mathsf{\mu }}⟩⟨{\mathsf{\psi }}_{\mathsf{\mu }}\right|, {\mathrm{w}}_{\mathsf{\mu }}\ge 0, \sum _{\mathsf{\mu }}{\mathrm{w}}_{\mathsf{\mu }}=1,$$

(3)

a positive semidefinite operator with unit trace. In the strategy basis its entries split into two kinds with distinct economic meaning. The diagonal entries ${\mathrm{p}}_{\mathrm{i}}\text{:}={\mathsf{\rho }}_{\mathrm{ii}}=⟨{\mathrm{s}}_{\mathrm{i}}\left|\mathsf{\rho }\right|{\mathrm{s}}_{\mathrm{i}}⟩$ are the populations: ${\mathrm{p}}_{\mathrm{i}}$ is the share of the population that would realize strategy $s_i$ upon interaction, the quantity that feeds replicator dynamics. The off-diagonal entries ${\mathsf{\rho }}_{\mathrm{ij}}$ with $\mathrm{i}\text{≠}\mathrm{j}$ are the coherences: they measure the degree to which the population maintains genuine superposition between strategies $s_i$ and $s_j$, as opposed to a classical mixture in which firms have already settled. A purely diagonal density operator is mathematically identical to a classical probability distribution over pure strategies. Nonzero coherences are the signature of unresolved strategic superposition at the population level.

Interaction with an environment, including observation by competitors and the partial commitments of a market, suppresses coherences while leaving populations intact. This is decoherence, and in the theory of open quantum systems it is generated by a dissipator of Gorini–Kossakowski–Sudarshan–Lindblad form [23,24,25]. Taking the jump operators to be the strategy projectors ${\mathrm{P}}_{\mathrm{k}}=\left|{\mathrm{s}}_{\mathrm{k}}⟩⟨{\mathrm{s}}_{\mathrm{k}}\right|$ with rates ${\mathsf{\gamma }}_{\mathrm{k}}\ge 0$,

$$\mathrm{D}\left[\mathsf{\rho }\right]={\sum }_{\mathrm{k}}{\mathsf{\gamma }}_{\mathrm{k}}\left({\mathrm{P}}_{\mathrm{k}}\mathsf{\rho }{\mathrm{P}}_{\mathrm{k}}-1/2\{{\mathrm{P}}_{\mathrm{k}},\mathsf{\rho }\text{}}\right)$$

(4)

a direct computation using ${\mathrm{P}}_{\mathrm{k}}^2={\mathrm{P}}_{\mathrm{k}}$ gives $\mathrm{D}{\left[\mathsf{\rho }\right]}_{\mathrm{ij}}=0$ for $\mathrm{i}=\mathrm{j}$ and $\mathrm{D}{\left[\mathsf{\rho }\right]}_{\mathrm{ij}}=-1/2\left({\mathsf{\gamma }}_{\mathrm{i}}+{\mathsf{\gamma }}_{\mathrm{j}}\right){\mathsf{\rho }}_{\mathrm{ij}}$ for $i\ne j$. Decoherence therefore acts as pure dephasing in the strategy basis: populations are untouched and each coherence decays at rate ${\mathsf{\gamma }}_{\mathrm{ij}}\text{:}=1/2\left({\mathsf{\gamma }}_{\mathrm{i}}+{\mathsf{\gamma }}_{\mathrm{j}}\right)$. Under $\mathcal{D}$ alone, $\rho$ relaxes to its diagonal part $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(p_1,\dots ,p_n\right)$, a classical distribution. The quantum-to-classical transition of a market is exactly this loss of strategic superposition: firms that maintain coherence keep their options open across strategies, while firms that have decohered have committed to a definite strategy in all but the sampling. The strategy basis is selected as the pointer basis precisely because it is the basis the environment monitors [19].

2.4. Replicator Preliminaries

Let $\mathrm{A}\text{∈}{\mathrm{R}}^{\mathrm{n}\text{×}\mathrm{n}}$ be the payoff matrix of a symmetric game, with $A_{ij}$ the payoff to strategy $s_i$ against strategy $s_j$. For a population state $p\in {\Delta }^{n-1}$, the fitness of strategy $s_i$ is ${\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)={\left(\mathrm{Ap}\right)}_{\mathrm{i}}=\sum _{\mathrm{j}}{\mathrm{A}}_{\mathrm{ij}}{\mathrm{p}}_{\mathrm{j}}$, and the mean fitness is $\mathsf{\phi }\left(\mathrm{p}\right)=\mathrm{p}\text{⋅}\mathrm{Ap}=\sum _{\mathrm{i}}{\mathrm{p}}_{\mathrm{i}}{\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)$. The replicator equation is

$${\dot{\mathrm{p}}}_{\mathrm{i}}={\mathrm{p}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)-\mathsf{\phi }\left(\mathrm{p}\right)\right), \mathrm{i}=1,\dots ,\mathrm{n}$$

(5)

Summation gives $\sum _{\mathrm{i}}{\dot{\mathrm{p}}}_{\mathrm{i}}=\mathsf{\phi }-\mathsf{\phi }=0$, so the simplex is invariant, and its faces are invariant as well. Interior rest points solve ${\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)=\mathsf{\phi }\left(\mathrm{p}\right)$ for all $i$ with $p_i>0$, equalizing the fitness of all strategies present [4,26,27].

2.5. The Strategic Master Equation

The full dynamics on the density operator combines three processes acting on potentially different timescales [25]. The first is coherent deliberation, an internal weighing of options that preserves superposition and is generated by a Hermitian deliberation Hamiltonian $H$ through the von Neumann term $-\mathrm{i}\left[\mathrm{H},\mathsf{\rho }\right]$. The second is decoherence in the strategy basis, defined in Section 2.3, which suppresses coherences while leaving populations intact. The third is selection, which reshapes the population frequencies according to differential fitness. We represent selection by a replicator superoperator $\mathrm{R}$, defined in the strategy basis by

$$\mathrm{R}{\left[\mathsf{\rho }\right]}_{\mathrm{ij}}=1/2\left({\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)+{\mathrm{f}}_{\mathrm{j}}\left(\mathrm{p}\right)-2\mathsf{\phi }\left(\mathrm{p}\right)\right)\hspace{0.17em}{\mathsf{\rho }}_{\mathrm{ij}}, {\mathrm{p}}_{\mathrm{k}}={\mathsf{\rho }}_{\mathrm{kk}},$$

(6)

with ${\mathrm{f}}_{\mathrm{i}}$ and $\mathsf{\phi }$ the fitness and mean fitness of Section 2.4. The off-diagonal form assigns to each coherence ${\mathsf{\rho }}_{\mathrm{ij}}$ the average of the two selection gradients, $1/2\left[\left({\mathrm{f}}_{\mathrm{i}}-\mathsf{\phi }\right)+\left({\mathrm{f}}_{\mathrm{j}}-\mathsf{\phi }\right)\right]$, which is the Hermitian lift of the replicator field consistent with the amplitude dynamics introduced above.

Combining the three contributions, the strategic master equation governing the joint evolution of populations and coherences is

$$\dot{\mathsf{\rho }}=-\hspace{0.17em}\mathrm{i}\hspace{0.17em}\left[\mathrm{H},\mathsf{\rho }\right]+\mathrm{D}\left[\mathsf{\rho }\right]+\mathrm{R}\left[\mathsf{\rho }\right].$$

(7)

Here $-\mathrm{i}\left[\mathrm{H},\mathsf{\rho }\right]$ generates coherent deliberation and interference, $\mathrm{D}\left[\mathsf{\rho }\right]$ drives the quantum-to-classical transition toward the strategy (pointer) basis, and $\mathrm{R}\left[\mathsf{\rho }\right]$ implements evolutionary selection. The dynamics is nonlinear.

3. Results

3.1. Amplitude Representation of the Replicator Field

The first result places quantum normalization and evolutionary selection on a single geometric footing. Writing the amplitude moduli as ${\mathrm{r}}_{\mathrm{i}}=\sqrt{{\mathrm{p}}_{\mathrm{i}}}$, the replicator equation on the simplex is carried to a flow on the positive orthant of the unit sphere ${\mathrm{S}}_{+}^{\mathrm{n}-1}$,

$${\dot{\mathrm{r}}}_{\mathrm{i}}=1/2\hspace{0.17em}{\mathrm{r}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\left(\mathrm{r}\text{⊙}\mathrm{r}\right)-\mathsf{\phi }\left(\mathrm{r}\text{⊙}\mathrm{r}\right)\right), {\left(\mathrm{r}\text{⊙}\mathrm{r}\right)}_{\mathrm{i}}\text{:}={\mathrm{r}}_{\mathrm{i}}^2,$$

(8)

and the correspondence is a bijection between simplex and sphere trajectories that preserves (Proposition 1). The map ${\mathrm{r}}_{\mathrm{i}}=\sqrt{{\mathrm{p}}_{\mathrm{i}}}$ is, up to a constant factor, the isometry between the simplex with the Shahshahani metric and the sphere with the round metric, since $\sum _{\mathrm{i}}\mathrm{d}{\mathrm{r}}_{\mathrm{i}}^2=1/4\sum _{\mathrm{i}}\mathrm{d}{\mathrm{p}}_{\mathrm{i}}^2/{\mathrm{p}}_{\mathrm{i}}$ [4,28,29]. For games with symmetric payoff matrix the amplitude flow is the Shahshahani gradient ascent. The economic content is that selection acts directly on the quantum amplitudes constrained to the normalization sphere, with fitter strategies acquiring larger amplitude, while the phases remain inert under population-level selection.

3.2. Structural Properties of the Replicator Superoperator

The selection superoperator $\mathrm{R}{\left[\mathsf{\rho }\right]}_{\mathrm{ij}}=1/2\left({\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)+{\mathrm{f}}_{\mathrm{j}}\left(\mathrm{p}\right)-2\mathsf{\phi }\left(\mathrm{p}\right)\right){\mathsf{\rho }}_{\mathrm{ij}}$ reproduces the classical replicator field on the diagonal, $\mathrm{R}{\left[\mathsf{\rho }\right]}_{\mathrm{ii}}=\left({\mathrm{f}}_{\mathrm{i}}-\mathsf{\phi }\right){\mathrm{p}}_{\mathrm{i}}$, preserves Hermiticity, and annihilates the trace, so the density operator retains unit trace. Its action on coherences is the natural Hermitian lift of the replicator field, consistent with the amplitude dynamics of Section 3.1. Physical admissibility holds in the decoherence-present regime and fixes the validity domain of the model, a property confirmed numerically in Section 3.6.

3.3. Emergence of Classical Replicator Dynamics in the Decoherence Limit

The central analytical result is that classical evolutionary dynamics is the strong-decoherence limit of the strategic master equation. With $\mathsf{\Gamma }={\min }_{\mathrm{i}\text{≠}\mathrm{j}}{\mathsf{\gamma }}_{\mathrm{ij}}$, adiabatic elimination of the coherences yields, to leading order, the closed replicator equation for the populations,

$${\dot{\mathrm{p}}}_{\mathrm{i}}={\mathrm{p}}_{\mathrm{i}}\left({\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)-\mathsf{\phi }\left(\mathrm{p}\right)\right)+\mathrm{O}\text{}\left({\displaystyle \frac{\text{∥}\mathrm{H}{\text{∥}}^2}{\mathsf{\Gamma }}}\right)$$

(9)

(Theorem 1).

3.4. Stability of Evolutionarily Stable Strategies

For an interior ESS $x^{\mathrm{*}}$ [2], the cross-entropy $\mathrm{V}\left(\mathrm{p}\right)=\sum _{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}^{*}\ln \left({\mathrm{x}}_{\mathrm{i}}^{*}/{\mathrm{p}}_{\mathrm{i}}\right)$ is a strict local Lyapunov function for the replicator dynamics. By Theorem 1 this Lyapunov function governs the diagonal of the strategic master equation in the strong-decoherence regime, so interior ESS attract the population once decoherence dominates. The natural candidate for the full quantum dynamics is the quantum relative entropy $\mathrm{S}\left({\mathsf{\rho }}^{*}\text{‖}\mathsf{\rho }\right)=\mathrm{Tr}\hspace{0.17em}{\mathsf{\rho }}^{*}\left(\ln {\mathsf{\rho }}^{*}-\ln \mathsf{\rho }\right)$, which reduces to $V$ on diagonal states and is monotone nonincreasing under the decoherence part of the dynamics by Lindblad’s theorem [30]; the nonlinear selection term is left for separate treatment.

3.5. Context Dependence and Interference

Strategy realization is shown to be basis sensitive. For a rotated measurement basis the realized outcome probabilities are

$${\mathrm{q}}_{\mathrm{k}}=\sum _{\mathrm{i}}{\left|{\mathrm{U}}_{\mathrm{ki}}\right|}^2{\mathrm{p}}_{\mathrm{i}}+\sum _{\mathrm{i}\text{≠}\mathrm{j}}{\mathrm{U}}_{\mathrm{ki}}\hspace{0.17em}\overline{{\mathrm{U}}_{\mathrm{kj}}}\overline{{\mathrm{c}}_{\mathrm{i}}}\hspace{0.17em}{\mathrm{c}}_{\mathrm{j}}$$

(10)

where the first sum is the classical prediction and the second is an interference term depending on the relative phases of the amplitudes (Proposition 2). No classical mixture over pure strategies with the same marginals ${\mathrm{p}}_{\mathrm{i}}$ reproduces ${\mathrm{q}}_{\mathrm{k}}$ for all rotations. This is the formal counterpart of framing and context effects [7] and of the order and disjunction effects that motivate quantum decision theory [6,9], and it is a feature classical evolutionary game theory cannot represent.

3.6. Numerical Results for the Two-Strategy Market Game

For the Hawk–Dove game with $\mathrm{V}=2$, $\mathrm{C}=3$ the reduced qubit system was integrated under three settings (Figure 1a). Under coherent deliberation alone the population oscillates and never settles; adding decoherence damps the oscillation to the fitness-blind value $p=1/2$; and the full model is driven toward the evolutionarily stable strategy. Increasing the decoherence rate $\mathsf{\gamma }\text{∈}\{2,8,60\text{}}$ drives the stationary population toward the classical ESS (Figure 1b), confirming Theorem 1.

4. Discussion

Quantum effects are transient and corrective, except in a coherence-sustained regime. The results draw a clean line. In the decoherence-dominated regime, long-run population outcomes are classical evolutionarily stable strategies (Theorem 1 and Section 3.4), and quantumness appears only as transient, individual-level deliberation and as a finite-$\gamma$ correction to equilibria. But the correction is real, and it grows as coherent coupling strengthens relative to decoherence and to the selection gradients. When that ratio is large, the classical ESS can be substantially displaced or destabilized, and the population can sustain coherent, oscillatory behavior rather than relaxing to a classical rest point.

The construction is complementary to, and distinct from, both established lines. The quantum games program quantizes the act of play, with actors choosing unitary operations on entangled states and the analysis centered on quantum equilibria [15,16,17]. The present model keeps payoffs and the act of play classical and locates the novelty earlier, in the pre-commitment strategic state and its collapse at interaction, then couples that collapse to population dynamics. Quantum decision theory models the single choice of a single agent and explains anomalies of individual choice through interference [8,9,10,11,12,13,14]. The present model embeds that machinery in evolutionary dynamics, so that interference at the decision layer propagates into selection at the population layer, as Proposition 2 makes explicit.

Strategic superposition gives a formal model of an indeterminate pre-decision state that does not require positing a hidden optimum the actor approximates. This complements the bounded-rationality tradition [31], which abandons the optimizing ideal without supplying a positive account of the unsettled state that precedes choice. Within the Homo Evolutivus program, the superposition is the mathematical content of constitutive self-opacity: the firm’s strategy is constitutively indefinite, and selection operates on the squared amplitudes that collapse produces, reshaping the distribution of strategic indeterminacy over evolutionary time.

Four limitations are salient. The selection superoperator $\mathcal{R}$ is phenomenological, and a complete characterization of its complete-positivity domain remains open. The model treats a single population; entanglement between interacting firms, which would let strategic choices be correlated in ways no independent mixture allows. The payoff structure is classical, so the framework does not address quantized payoffs. And the empirical identification of the decoherence rate and the deliberation Hamiltonian from market data is an open measurement problem, though the order ${\mathsf{\Delta }}^2/\mathsf{\gamma }$ displacement of equilibria suggests where to look.

5. Conclusions

We have constructed a mathematical foundation that integrates quantum decision theory with evolutionary game theory and replicator dynamics. An economic actor before market entry is modeled as a superposition of pure strategies in a Hilbert space; the moment of interaction is a measurement that collapses the superposition onto a realized strategy with Born-rule probabilities; and these probabilities, carried by the diagonal of a density operator, evolve under a strategic master equation whose three superoperators represent coherent deliberation, decoherence in the strategy basis, and replicator selection. The amplitude representation places quantum normalization and evolutionary selection on a common geometric footing through the square-root map to the sphere. The decoherence-limit theorem shows that classical replicator dynamics is the strong-decoherence limit of the quantum strategic dynamics, with an explicit error bound. Context dependence distinguishes superposed strategies from any classical mixture with the same marginals. And the worked two-strategy game demonstrates that coherent coupling displaces the evolutionary equilibrium from the classical evolutionarily stable strategy by a quantity of order ${\mathsf{\Delta }}^2/\mathsf{\gamma }$, recovering the stable strategy as decoherence dominates. The construction formalizes constitutive self-opacity and connects bounded rationality to the interference structure of quantum probability. The bifurcation between the decohered-classical regime and the coherence-sustained regime, together with entanglement between firms, defines the agenda for the installments that follow.