Abstract
This paper proposes a formal theoretical model that integrates classical strategic correlation, quantum game theory, and evolutionary dynamics in the context of institutional corruption. I introduce a classical parametric correlator K(λ), defined as a convex combination of strategic independence and the Fréchet–Hoeffding bounds (upper for positive λ and lower for negative λ), into a two-population Hawk–Dove game with enforcement. The correlator modifies both encounter probabilities and the effective transaction costs of legal compliance. The closed-form analysis and the baseline simulations are conducted on the symmetric invariant manifold x = y = z, while the general asymmetric system remains piecewise. A reduced-form alignment-calibration bridge maps the normalized analytical selection differential into material-payoff units and aligns its local stability root with the frequency-dependent replicator integrated numerically; the resulting baseline threshold is λ* = 1/3. For 0 ≤ λ < λ*, the baseline calibration exhibits a stable interior corruption branch, whereas above λ* the honest corner is locally asymptotically stable. Because convergence slows close to the bifurcation, the finite-horizon numerical honesty threshold is reached only when λ is sufficiently above λ*. I derive a restricted Eisert–Wilkens–Lewenstein I/X benchmark and use λ = sin2(γ), the squared concurrence (tangle), solely as a scalar calibration of entanglement strength. I show that, on the symmetric manifold, this restricted I/X distribution collapses to the independent product distribution for every γ, whereas the classical correlator remains concordant for λ > 0. The exact lower Fréchet–Hoeffding branch is used for λ < 0, preserving non-negativity and the stated marginals. The simulations identify a valid adversarial high-risk region under negative dependence and a welfare-improving cooperation corridor under sufficiently positive alignment. Policy implications focus on avoiding adversarial dependence and combining institutional alignment with conventional enforcement.
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1 Introduction
Corruption is considered one of the most pervasive forms of institutional failure, capable of undermining collective trust, distorting resource allocation, and compromising institutional legitimacy. Classical economic theory has provided robust models of its diffusion, interpreting corrupt conduct as the rational outcome of distorted incentives and informational asymmetries [1, 23], but such models often neglect the relational and systemic dimension of the phenomenon. The decision to engage in or resist corruption depends not only on individual payoffs but also on the degree of mutual trust, transparency, and informational coherence that characterizes the institutional environment. In the absence of such social correlations, Nash equilibria may be inefficient and self-reinforcing: honest conduct, although socially desirable, need not be evolutionarily stable. From this perspective, Aumann’s theory of correlated equilibrium [3, 4], together with the existence result of Hart and Schmeidler [17], provides a conceptual basis for interpreting anti-corruption governance as a problem of informational correlation rather than deterrence alone. Quantum game theory extends this intuition by allowing non-classical correlations based on entanglement. The seminal contributions of Eisert, Wilkens and Lewenstein [12], Marinatto and Weber [27], and Iqbal and Toor [19, 20] show how quantum correlations can alter equilibrium and evolutionary-stability properties even without direct communication. This is relevant to governance settings in which distrust and information manipulation act as sources of institutional noise analogous to a corrupted source in a quantum game [22]. Despite these advances, two gaps remain. First, there is no systematic framework for placing classical governance correlations and quantum entanglement on a common scalar benchmark while keeping their probability distributions and strategy spaces conceptually distinct. Existing studies typically treat quantum correlation as an exogenous feature and do not map intermediate correlation intensities or the thresholds governing transitions in evolutionary dynamics. Second, the interaction between strategic correlation and conventional enforcement has not been analysed systematically: the policy value of governing correlation λ relative to monetary incentives and sanctions remains an open question.
This paper addresses both gaps by proposing a hybrid model in which a classical parametric correlator K(λ), capable of varying continuously from independence to perfect concordance, is introduced into an evolutionary Hawk–Dove framework. The parameter λ works through two discrete and clearly delineated channels: (i) a matching-frequency channel that modifies the probability distribution over strategic encounters, and (ii) an institutional-alignment channel that mitigates the effective transaction costs of legal compliance under positive correlation. I show that the matching-frequency channel alone is insufficient to generate a phase transition under standard Hawk–Dove payoffs, and I further show that the bifurcation requires the institutional-alignment mechanism. I also identify the precise boundary of the restricted quantum comparison: on the symmetric manifold used by the analytical model, the mixed I/X benchmark reduces to independent randomization, whereas the copula-based correlator continues to generate concordance. There are two research questions: the first is to assess how a classical correlator can be compared with a tangle-based entanglement scale while delimiting the effects that cannot be reproduced by a classical device (RQ1); the second is to analyse the policy value of acting on λ, compared with traditional enforcement tools, by characterizing how correlation and enforcement jointly affect the stability threshold, expected corruption, and simulated social welfare (RQ2).
The paper makes three contributions. First, I formalize K(λ) as a copula-based correlator and establish a unique interior transcritical threshold λ* for the aligned symmetric replicator (Proposition 1), with explicit comparative statics. The numerical implementation is constructed so that its local stability root is exactly the analytical λ*, while retaining the frequency-dependent Hawk–Dove matching component. Second, I establish a carefully delimited scalar calibration between λ and the entanglement parameter γ in a restricted Eisert–Wilkens–Lewenstein benchmark, setting λ = sin2(γ), and I prove an operational separation result: on x = y = z the restricted EWL distribution is independent of γ, while the classical Fréchet–Hoeffding coupling remains λ-dependent (Propositions 3–4). Third, I state and prove the conditions under which Pλ constitutes a correlated equilibrium in Aumann’s sense (Proposition 2). The numerical analysis supports the local threshold structure, quantifies the slow finite-horizon adjustment near λ*, and reports sensitivity to enforcement and institutional parameters.
The remainder of the paper is organized as follows. Section 2 reviews the relevant literature and identifies the specific gaps addressed. Section 3 develops the theoretical model. Section 4 presents simulation results. Section 5 discusses the findings and their policy implications. Section 6 concludes.
2 Background and Literature Gap
The analysis of corruption through the lens of game theory has a well-established tradition. Corruption between public officials and firms can be represented as a game of imperfect coordination in which agents decide whether to act opportunistically or cooperate with control institutions under asymmetric incentives and incomplete information [1, 25, 35]. Shared information and strategic expectations are central: corruption can be reinforced when agents expect others to act corruptly, generating an inefficient multiple-equilibrium trap [7]. Empirical work documents the importance of auditing [11, 13, 31], information transparency [34], and leniency programmes [16, 36] as instruments for shifting equilibria. Marino [28] models corruption as a cyclical dynamical process driven by feedback between enforcement intensity and corruption and reports evidence from spectral analysis and machine-learning classification across 112 countries. In the classical models of Aumann [3, 4] and Hart and Schmeidler [17], strategic correlation is formalized as a mechanism through which players condition their actions on common signals. A correlated equilibrium may support outcomes that are unavailable under independent randomization. In actual institutional systems, however, information may be noisy or manipulated, and the mediator may be ineffective or corrupt, limiting coordination and reinforcing strategic inefficiency. Benndorf, Martínez-Martínez and Normann [6] show that population coupling interacts non-trivially with equilibrium selection in Hawk–Dove games. Quantum game theory provides a further perspective by introducing non-classical correlations based on entanglement. Eisert, Wilkens and Lewenstein [12] show that quantum strategies can change the equilibrium structure of the Prisoner’s Dilemma. Later studies extend the framework to multiplayer settings [5] and to alternative two-player games [27, 29], while van Enk and Pike [37] clarify the boundary between quantized games and classical reformulations. Evolutionary analyses of quantum games, including Hawk–Dove-type environments, show that stability can depend on entanglement and on the available strategy set [19, 20]. The corrupted-source formulation of Johnson [22] is especially relevant to institutional applications because it studies a quantum game in which the initial system is prepared incorrectly. Work on decoherence and noisy quantum games [14, 21] likewise shows that quantum advantages are sensitive to noise. Brandenburger [8] studies the relationship between quantum and classical correlation in games, while Phoenix and Khan [33] identify correlation as a key operative mechanism behind many apparent quantum advantages. Brunner and Linden [9] connect Bell nonlocality with Bayesian game theory, and Pappa et al. [32] show that nonlocal correlations can help resolve conflicting interests in incomplete-information games. Deckelbaum [10] formalizes quantum correlated equilibria for complete-information games, and Auletta et al. [2] extend the analysis to incomplete information through belief-invariant equilibrium concepts. An explicit debate concerns whether quantum Hawk–Dove outcomes are reducible to an extended classical game. Jabir, Vyas and Benjamin [39] argue that a randomized quantum strategy can generate an equilibrium and payoff unavailable in the original classical Hawk–Dove game. Groisman [40] contests that conclusion on the ground that the proposed construction changes the rules of the original game, while Frąckiewicz [41] identifies nonclassical features of the EWL scheme more generally. This unresolved debate motivates the explicit classical–quantum boundary result developed in Sect. 3.6.
The existing literature has not yet jointly examined the following four dimensions within a unified framework:
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(a)
a continuous parametric correlator K(λ) that modulates strategic dependence and separates the matching and institutional channels through which correlation affects behaviour;
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(b)
critical thresholds λ* in evolutionary coordination and anti-coordination environments, together with comparative statics across structural parameters;
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(c)
a carefully delimited comparison between a classical correlator and the EWL entanglement scale, including an explicit result on the boundary between scalar calibration, distributional equivalence and genuinely quantum strategic effects;
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(d)
a systematic analysis of how correlation structure and enforcement jointly affect local stability, expected corruption, convergence and simulated welfare.
This study contributes to addressing these four gaps by constructing a generalized strategic correlation model K(λ) that includes as limiting cases both classical independence (λ = 0) and perfect concordance (λ = 1), and that allows estimation of institutional stability thresholds and cooperative sustainability in noisy or partially corrupt environments.
3 Theoretical Model
This section develops the formal framework in six steps: (3.1) the game environment and payoff structure; (3.2) the classical parametric correlator K(λ); (3.3) the two channels through which correlation operates; (3.4) the main stability result and its proof; (3.5) comparative statics and correlated-equilibrium conditions; and (3.6) the quantum Hawk–Dove benchmark and the classical–quantum boundary.
3.1 Game Environment
Consider a two-population Hawk–Dove (HD) game; see Kohli and Haslam [24] for the general replicator dynamics of this class of games. Population A (firms) chooses strategy H (corruption) with frequency x ∈ [0, 1], while Population B (officials) chooses H with frequency y ∈ [0, 1]. The general model permits x ≠ y. The closed-form threshold analysis and the baseline numerical experiment below use identical payoff parameters and the symmetric invariant manifold x = y = z. Payoffs are written as matrices: rows correspond to the focal player’s strategies (H,D), while columns correspond to the opponent’s strategies (H,D). The baseline payoff matrix is
where V0 > 0 is the value of the contested good/legal channel, c > 0 is the bilateral conflict cost with c > V0, and k ≥ 0 is the moral or reputational cost of deviation.
With enforcement, the effective payoff matrix becomes
where p ∈ [0, 1] is the audit probability and F > 0 is the sanction. The four entries aHH, aHD, aDH and aDD are used throughout the paper to keep the notation compact.
Assumption 1 (matching-only benchmark). In the absence of correlation, cooperation dominates unilateral corruption in the legal channel: V0/2 > (1 − p)(V0 − k) − pF. This assumption is used only for the auxiliary matching-frequency result in Lemma 2; it is not imposed on the combined-channel baseline calibration.
3.2 The Classical Parametric Correlator
Let πx = (x,1 − x) and πy = (y,1 − y) denote the marginal strategy distributions. The standard HD game assumes strategic independence:
3.2.1 Definition 1 (Parametric Correlator)
For λ ≥ 0, the correlator K(λ) maps the marginals (πx,πy) into the joint distribution
where Φ⁺ is the joint distribution induced by the upper Fréchet–Hoeffding bound [18]:
For λ < 0, the correlator is defined symmetrically by mixing independence with the lower Fréchet–Hoeffding bound:
where Φ⁻ is induced by W(u,v) = max(u + v − 1,0). The parameter therefore spans λ ∈ [− 1,1], from countermonotonic (adversarial) to comonotonic (aligned) matching.
3.2.2 Lemma 1 (Copula Representation)
For every λ ∈ [− 1,1], Pλ is a valid probability distribution with marginals πx and πy. It is generated by the copula
Proof. The boundary conditions Cλ(u,0) = Cλ(0,v) = 0, Cλ(u,1) = u, and Cλ(1,v) = v hold by construction. Since the product copula Π, the upper Fréchet–Hoeffding bound M, and the lower Fréchet–Hoeffding bound W are copulas, and the class of copulas is closed under convex combinations [30], Cλ is a copula for every λ ∈ [− 1, 1]. Therefore, Pλ is a valid joint distribution with marginals πx and πy. For the broader optimal-transport framework underlying such couplings, see [38].
Remark. As λ increases, probability mass shifts from discordant profiles (H,D),(D,H) to concordant profiles (H,H),(D,D). At λ = 1, the coupling attains the maximum concordance compatible with the given marginals; when x = y, players always choose the same strategy; at λ < 0, matching becomes adversarial or countermonotonic.
For later numerical use, the negative branch must retain the complete lower-bound term: for λ < 0, Pλ(H,H) = (1 + λ)xy − λ max(x + y − 1,0). Omitting the second term when x + y > 1 would violate non-negativity and would no longer define a copula with the stated marginals.
3.3 Two Channels of Correlation: Matching Frequency and Institutional Alignment
A central contribution of the paper is the separation of two channels through which K(λ) affects evolutionary dynamics. The first channel changes the distribution of encounters; the second changes the effective transaction cost of legal compliance.
3.3.1 Channel 1: Matching Frequency
For the closed-form analysis, impose the symmetric invariant manifold x = y = z, corresponding to identical payoff parameters, a common initial condition, and a common stochastic perturbation in the baseline simulations. For λ ≥ 0, a Hawk player then faces another Hawk with probability λ + (1 − λ)z, while a Dove player faces another Dove with probability λ + (1 − λ)(1 − z):
This shifts the frequency of same-strategy encounters without altering the payoff entries themselves. Outside x = y, the conditional probabilities must be computed directly from Pλ: Pr(B = H|A = H) = Pλ(H,H)/x and Pr(B = D|A = D) = Pλ(D,D)/(1 − x), with the analogous expressions for Population B. Because min and max enter the copula, the fully asymmetric system is piecewise.
3.3.1.1 Lemma 2 (Insufficiency of Matching Frequency).
Under the standard HD payoffs with c > V0, the matching-frequency channel alone cannot stabilize the collusive equilibrium (1,1) for any λ ∈ [0,1].
Proof
At (1,1), a Dove mutant earns aDH = 0, which exceeds the incumbent Hawk payoff aHH < 0 because c > V0 and enforcement further reduces the corrupt payoff. Thus, the Dove mutant can invade regardless of λ. A symmetric argument shows that the honest corner remains locally stable under Assumption 1. The detailed Jacobian calculation is reported in Appendix A.2.
This result is important because it shows that the phase transition documented below is not a mechanical artefact of altered matching frequencies. It requires the institutional-alignment channel. Assumption 1 is confined to this matching-only comparison and is not required by Proposition 1 or by the weak-enforcement baseline simulation.
3.3.2 Microfoundation of the Institutional-Alignment Channel
The institutional-alignment channel is microfounded through coordination costs in the legal channel. Legal interaction is not frictionless: firms must provide correct documentation, timing and disclosure, while officials must process the request according to formal rules. Let q(λ) be the probability of successful procedural coordination. Higher institutional alignment raises this probability. Let m > 0 be the administrative loss incurred when legal coordination fails. The expected payoff from legal compliance is
Define the transaction-cost term
For tractability, take the first-order approximation
which implies
In the normalized specification used for the threshold analysis and simulations, this is written as
where αN is a reduced-form scale parameter. To avoid confusion in the comparative statics, I distinguish the primitive responsiveness ρ from the normalized threshold-scale αN. Greater primitive responsiveness ρ lowers transaction costs more rapidly as λ increases. By contrast, αN enters the closed-form threshold as a normalization parameter; its sign in comparative statics must therefore be interpreted within that normalization rather than as a universal ranking of policy effectiveness.
3.3.3 Why Asymmetry in the Surplus?
A natural question is whether correlation should also modify the payoff of corruption. The baseline specification treats the legal channel and the corrupt channel asymmetrically. The return to corruption depends primarily on evasion of detection, captured by p and F, whereas the return to legal compliance depends on the efficient operation of standardized procedures. The asymmetry is therefore a modelling choice: alignment directly reduces the transaction cost of legal compliance, while corrupt returns are affected through matching and enforcement.
Appendix B reports a symmetric robustness specification in which both corrupt and legal payoffs respond to correlation. The qualitative results are preserved provided that legal compliance benefits more from alignment than corruption does.
3.4 Main Stability Result
Combining the two channels, the payoff of the corrupt strategy is
for the stability proposition, the normalized local corrupt advantage ΔN(λ) is written as
Let M(z,λ) = fH(z,λ) − fD(z,λ) denote the frequency-dependent matching differential derived in Appendix A.1. The normalized term ΔN(λ) fixes the analytical local threshold but is not expressed in the same payoff units as M. I therefore introduce a reduced-form alignment-calibration bridge ω. It maps ΔN into material-payoff units and calibrates the strength of the institutional-alignment component while preserving the classical matching dynamics at λ = 0. This is not a neutral change of units: ω affects the interior dynamics and convergence paths, although it does not change the root of ΔN.
here ω = [aHD − aDD]/ΔN(0) > 0. Hence G(z,0) = M(z,0) and G(0,λ) = ωΔN(λ). For λ < 0, the institutional-alignment channel is inactive and G(z,λ) = M(z,λ). The stochastic symmetric baseline integrates
The selection intensity s controls adjustment speed. The reduced-form bridge ω calibrates the strength of the normalized institutional-alignment component in material-payoff units; it affects interior dynamics and convergence paths but not the root λ*. In the baseline, s = 0.03 and ω = 4.8333.
3.4.1 Assumption 2 (Interior Threshold)
The institutional-alignment effect is sufficiently strong to generate an interior critical value: 0 < k + pF < αN(V0 − k). Equivalently, the value λ* defined in Eq. (16) belongs to (0,1).
Proposition 1
Critical correlation threshold.
Under Assumption 2 and the standard non-degeneracy condition ∂M(0,λ*)/∂z ≠ 0, the symmetric combined-channel replicator defined by Eqs. (15a)–15b) has a unique critical threshold λ* ∈ (0,1) such that:
(i)For λ < λ*, the honest corner is locally unstable and a stable interior branch with positive corruption propensity may persist. (ii) For λ > λ*, the honest corner z = 0 is locally asymptotically stable and defines the analytical cooperation corridor. (iii) The critical value is
(iv) The boundary equilibrium and the interior branch exchange local stability transcritically at λ*.
Proof. Equation (15) implies ΔN(λ) = αN(V0 − k)(λ* − λ), so ΔN has a unique interior root under Assumption 2. By construction, G(0,λ) = ωΔN(λ); therefore the local eigenvalue at z = 0 has the sign of λ* − λ. At (0,λ*), G(0,λ*) = 0 and ∂G(0,λ*)/∂λ = − ωαN(V0 − k) ≠ 0. Moreover, ∂G(0,λ*)/∂z = ∂M(0,λ*)/∂z = (1 − λ*)(aHH − aHD − aDH + aDD) ≠ 0 under the stated non-degeneracy condition. The one-dimensional symmetric replicator therefore satisfies the standard transcritical conditions: the boundary branch z = 0 and the interior branch intersect at λ* and exchange local stability [15]. The diffusion term in Eq. (15b) is used only in the finite-horizon simulations and does not alter the deterministic local threshold.
3.5 Corollaries and Classical Extensions
3.5.1 Corollary 1 (Comparative Statics of λ*)
From Eq. (16), the direct comparative statics are
Interpretation. Higher audit probability p, heavier sanctions F, and higher moral cost k lower λ*, expanding the range in which the honest equilibrium is stable. The positive derivative with respect to αN should not be read as saying that institutional design is counterproductive. It follows from the chosen normalization: αN is a threshold-scale parameter, not the primitive effectiveness of the alignment technology. If the same channel is parameterized by the primitive responsiveness ρ in Eq. (11), stronger responsiveness reduces transaction costs faster and lowers the effective threshold. The substantive comparative static is therefore unambiguous: more effective institutional alignment lowers the amount of positive correlation required for honesty, whereas a larger residual threshold scale αN raises it.
3.5.2 Corollary 2 (Enforcement-Correlation Complementarity)
Enforcement and correlation operate through different margins. Enforcement shifts the analytical threshold, while institutional alignment moves the system across it. The two levers are complementary around λ*. In the finite-horizon baseline, adjustment is slow immediately above the analytical threshold: the honest corner is locally stable for λ > λ*, but the numerical threshold z ≤ 0.05 is reached within 100 generations only when λ is sufficiently above λ*.
Proposition 2
Correlated-equilibrium conditions.
The joint distribution Pλ constitutes a correlated equilibrium in Aumann’s sense [3] if and only if the following incentive-compatibility constraints hold simultaneously for both players. For player A recommended H:
For player A recommended D:
On the same symmetric manifold x = y = z, substituting the expressions from Sect. 3.2 reduces the first constraint to
The two analogous constraints for player B are obtained by interchanging players A and B; in the symmetric reduction they coincide with Eqs. (19)–(21). In the general asymmetric model, all four constraints must instead be evaluated directly from Pλ with the distinct marginals x and y. The correlated-equilibrium region need not coincide with the evolutionary-stability region because correlated equilibrium is an incentive-compatibility concept, whereas evolutionary stability is a dynamic refinement.
Proof
Aumann’s obedience requirement states that, conditional on each recommendation, a player cannot gain by unilaterally switching actions. With two actions, this produces exactly two inequalities for each player. The displayed conditions are therefore jointly necessary and sufficient. Substitution of the symmetric copula probabilities yields the reduced expressions stated above; in the asymmetric case the four inequalities remain distinct.
3.6 The Quantum Hawk–Dove Benchmark
The correspondence developed below is meaningful only against an explicit quantum benchmark. In the Eisert–Wilkens–Lewenstein (EWL) protocol applied to the Hawk–Dove game, each player controls one qubit whose computational-basis states |H⟩ and |D⟩ encode the two classical strategies. The protocol prepares the reference state |DD⟩, entangles it through the operator J(γ), lets the players apply local operators, applies the disentangling gate J†(γ), and measures in the computational basis. The literature disagrees on how far such quantum outcomes can be reduced to extended classical rules: Jabir, Vyas and Benjamin [39] argue that randomized quantum Hawk–Dove strategies can generate classically unavailable equilibria, Groisman [40] disputes that interpretation, and Frąckiewicz [41] identifies nonclassical features of the EWL rules. The analysis below therefore states both the scalar comparison and its exact boundary.
The entanglement parameter γ ∈ [0, π/2] interpolates between unentangled and maximally entangled preparations. Prior quantum Hawk–Dove results show that evolutionary stability can depend on γ [26, 39]. The purpose of the benchmark here is narrower: it places λ on the same numerical scale as squared concurrence while proving that the restricted EWL and classical copula mechanisms act on different features of the joint distribution.
Proposition 3
Tangle-based calibration of the restricted EWL benchmark.
In the EWL quantum Hawk–Dove benchmark, the entanglement parameter γ ∈ [0, π/2] determines the squared concurrence (tangle) sin2(γ) of the prepared two-qubit state. When local play is restricted to the unitary set I/X, the identity λ = sin2(γ) provides a scalar calibration between the intensity parameter of the classical correlator and entanglement strength. This is a calibration of one scalar measure only: it does not identify the EWL post-measurement distribution with Pλ, it does not make I/X equivalent to the standard EWL classical embedding, and it does not extend to the full SU(2) strategy space.
In the EWL protocol, the initial state is prepared by the entangling gate
where D = iσγis the EWL flip operator. Here I is the identity operator and X = σxis the Pauli bit-flip operator. Although D and X both exchange the computational-basis states \(\left| H \right\rangle {\mkern 1mu} and{\mkern 1mu} \left| D \right\rangle\), they carry different relative phases and are not interchangeable in the presence of entanglement. The restricted I/X benchmark used below is therefore a deliberately limited unitary strategy set, distinct from the standard EWL classical embedding \(\left\{ {I,D} \right\}\).
After the players apply local operators  and B̂, the final state is
Expected payoffs are obtained by weighting the classical payoff matrix with the measurement probabilities:
Restrict now to the I/X benchmark. Player A applies X with probability x and I with probability 1 − x; player B applies X with probability y and I with probability 1 − y. After the disentangling operation J†(γ), the four measurement probabilities are
together with
and, under the same scalar calibration,
Equations (30) and (31) show how the tangle-based scalar λ = sin2(γ) reweights the two discordant measurement outcomes inside the restricted I/X benchmark. They do not identify that EWL distribution with the Fréchet–Hoeffding copula K(λ): in particular, the quantum benchmark leaves the concordant I/I and X/X probabilities unchanged, whereas the classical copula redistributes probability mass according to its fixed marginals. The comparison is therefore scalar and interpretative, not distributional.
Proof
The prepared state in Eq. (23) has concurrence C = 2|cos(γ/2)sin(γ/2)|= sin(γ), so its tangle is C2 = sin2(γ). Applying the local I/X mixtures and the disentangling operator yields Eqs. (26)–(29); substituting λ = sin2(γ) gives Eqs. (30)–(31). The identification is therefore between scalar intensity measures only and does not imply equality of distributions or strategy spaces.
Under the I/X restriction, Fig. 1a plots the four post-disentangling probabilities in Eqs. (26)–(29), evaluated at the deliberately asymmetric point x = 0.3 and y = 0.6 so that the reallocation between discordant outcomes is visible. The concordant probabilities P(H,H) and P(D,D) remain fixed, while entanglement reallocates mass between the two discordant outcomes. Figure 1b maps γ into the scalar calibration λ = sin2(γ) and marks γ* = arcsin(√λ*)≈35.3° implied by the baseline classical threshold λ* = 1/3. The shaded interval identifies only the values of the scalar calibration corresponding to the classical cooperation corridor.
3.6.1 What Is Genuinely Quantum in the Restricted Benchmark?
Proposition 4
Operational separation of the restricted EWL benchmark and the classical correlator).
Under the mixed I/X benchmark and the scalar identification λ = sin2(γ), entanglement changes the measurement marginals whenever x ≠ y and induces weakly negative covariance. On the symmetric manifold x = y = z, however, the restricted EWL distribution reduces to independent randomization for every γ, whereas K(λ) remains positively concordant for λ > 0.
Proof
Summing the joint probabilities in Eqs. (26)–(29) gives the marginals in Eq. (32). Subtracting their product from P(H,H) = xy yields Eq. (33). Setting x = y = z eliminates every γ-dependent term and gives the product probabilities in Eqs. (34)–(35). By contrast, the upper Fréchet–Hoeffding mixture gives Pλ(H,H) = z[λ + (1 − λ)z], which exceeds z2 for 0 < λ ≤ 1 and 0 < z < 1. Thus the two devices operate on distinct features of the joint law: the restricted EWL mechanism mixes asymmetric marginals and discordant outcomes, whereas K(λ) preserves the marginals and controls concordance.
Proposition 4 turns the limitation of the I/X benchmark into an explicit boundary result. Figure 1 is evaluated off the diagonal because on x = y = z its probability curves would be constant in γ. Genuinely quantum Hawk–Dove effects therefore require a richer unitary strategy set, phase-sensitive interference, or equilibrium opportunities that are unavailable under the restricted mixed I/X benchmark. With general SU(2) operations, the strategy set and deviation opportunities expand further, and equilibrium payoff profiles may arise that no correlated equilibrium of the underlying classical game supports [39, 41]. Those effects require quantum resources and lie beyond the organizational interpretation of the present model, while the contrary extended-classical interpretation remains part of the ongoing debate [37, 40].
4 Results and Simulations
4.1 Simulation Design
The simulation implements the two-channel evolutionary Hawk–Dove model of Sect. 3. Two populations—firms (Population A) and officials (Population B)—interact under the parametric correlator K(λ) with replicator dynamics and small stochastic noise (σ = 10⁻3). Both the matching-frequency channel (Sect. 3.3.1) and the institutional-alignment channel (Sect. 3.3.2) operate simultaneously.
Computational implementation. All simulations were implemented in Python 3.11 using NumPy 1.26, pandas 2.1 and matplotlib 3.8. The baseline is simulated on the symmetric invariant manifold x = y = z and integrated with an Euler–Maruyama scheme using Δt = 1 over T = 100 generations. The deterministic drift is s z(1 − z)G(z,λ), with selection intensity s = 0.03 and reduced-form alignment-calibration bridge ω = 4.8333. The same Gaussian perturbation, with standard deviation σ = 10⁻3, is applied to both populations in each replication, so symmetry is preserved path by path; propensities are projected back to [0,1]. For λ < 0, the code uses the complete lower Fréchet–Hoeffding expression Pλ(H,H) = (1 + λ)z2 − λ max(2z − 1,0). Automated tests verify non-negativity, unit mass, marginal preservation, recovery of the classical matching differential at λ = 0, and equality between the local numerical and analytical roots. The λ grid contains 51 equally spaced values over [− 1,1] and 25 refined values over [λ* − 0.12,λ* + 0.12] at increments of 0.01, with the central point set exactly to λ*; their union contains 76 distinct values. Each configuration is replicated 50 times using fixed seeds 1000–1049. The replication script, parameter file, validation tests and figure-generation routines are supplied as supplementary material.
Calibration. The baseline parameters are V0 = 10, c = 15, k = 1, p = 0.10, F = 2, αN = 0.20 and τ0 = 0.5. They satisfy Assumption 2 and imply λ* = 1 − (k + pF)/[αN(V0 − k)] = 1/3. The baseline does not satisfy Assumption 1, because the unilateral corrupt payoff is 7.90 whereas V0/2 = 5.00; this is intentional because Assumption 1 applies only to the auxiliary matching-only Lemma 2. The reduced-form bridge ω = [aHD − aDD]/ΔN(0) = 2.90/0.60 = 4.8333 calibrates the normalized alignment component in material-payoff units, preserves the classical matching differential at independence, and ensures that the integrated local stability root is λ* = 1/3. It is not merely a unit conversion and therefore also affects the interior paths and convergence speeds.
Under this calibration, the analytical cooperation corridor λ > λ* covers two thirds of the positive-λ domain. This is a local stability statement. Because adjustment slows near the bifurcation, finite-horizon convergence from z0 = 0.50 is evaluated separately through the first-passage threshold z ≤ 0.05.
The aligned exact-copula simulation identifies four calibration-specific regimes across the λ-spectrum:
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Regime I—Adversarial high-risk region (− 1 ≤ λ < 0). The common corruption propensity remains elevated, declining from approximately 0.77 at λ = − 1 to 0.46 at independence. The exact lower-bound coupling yields valid joint-corruption probabilities, with P(CC) decreasing from approximately 0.54 at λ = − 1 to 0.22 at λ = 0. Net social welfare rises from about − 5.4 to + 3.2 across this interval.
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Regime II—Stable interior branch (0 ≤ λ < λ*). The common propensity decreases continuously from approximately 0.46 at independence to about 0.12 at λ = 0.30, while P(CC) falls from approximately 0.22 to 0.045. The honest corner remains locally unstable throughout this region.
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Regime III—Near-threshold slow adjustment (λ* ≤ λ < 0.40). The honest corner is locally stable above λ*, but convergence from z0 = 0.50 is slow. At λ* = 1/3 the mean propensity is still about 0.081 at t = 100; at λ = 0.36 it is about 0.058 and only 8% of replications cross z ≤ 0.05 within the horizon.
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Regime IV—Finite-horizon honest region (λ ≥ 0.40 in the baseline). Every replication reaches z ≤ 0.05 by t = 100 at λ = 0.40 and above. Mean first passage is about 79 generations at λ = 0.40, 28 generations at λ = 0.68 and 17 generations at λ = 1. The analytical cooperation corridor nevertheless begins at λ > λ*.
4.1.1 Propensity Dynamics (Fig. 2)
Figure 2 displays the temporal evolution of the common corruption propensity z = cA = cB. Because a common stochastic perturbation is used, the two population frequencies coincide in every baseline path.
Panel A (λ = − 0.80, − 0.30, 0.00). Under the exact lower Fréchet–Hoeffding branch, the common propensity rises from 0.50 and settles near 0.75 at λ = − 0.80 and near 0.65 at λ = − 0.30. At independence it converges to approximately 0.46. Negative correlation therefore produces an adversarial high-risk region, but not the previously reported near-universal corruption generated by the incomplete lower-branch expression.
Panel B (λ = + 0.12, + 0.20, + 0.30). Positive alignment progressively shifts the stable interior branch toward the honest corner. Mean propensities at t = 100 are approximately 0.341, 0.243 and 0.117, respectively. The trajectories descend smoothly but remain above the numerical honesty threshold within the 100-generation horizon.
Panel C (λ = + 0.40, + 0.68, + 1.00). All three values lie above λ*. At λ = 0.40 the mean trajectory first crosses z ≤ 0.05 after about 79 generations; convergence accelerates to about 28 generations at λ = 0.68 and 17 generations at λ = 1.
4.1.2 Collusion Dynamics (Fig. 3)
Figure 3 translates the propensity dynamics into the joint probability P(CC). In Panel A, the complete lower-bound expression preserves valid marginals and non-negative probabilities. At λ = − 0.80, P(CC) rises from approximately 0.05 at z = 0.50 to about 0.51 as the common propensity increases; at λ = − 0.30 it settles near 0.39, while at independence it approaches 0.22. Panel B shows P(CC) of approximately 0.143 at λ = 0.12, 0.096 at λ = 0.20 and 0.045 at λ = 0.30. In Panel C it falls to about 0.013 at λ = 0.40 and to roughly 0.001 at stronger alignment.
4.1.3 Finite-Horizon Joint-Corruption Profile (Fig. 4)
Figure 4 presents the finite-horizon joint-corruption profile at t = 100. With the complete lower Fréchet–Hoeffding term, mean P(CC) decreases monotonically from approximately 0.54 at λ = − 1 to 0.22 at independence. In the positive domain it falls to about 0.096 at λ = 0.20, 0.045 at λ = 0.30, 0.032 at λ* = 1/3 and 0.013 at λ = 0.40. The analytical threshold lies inside the steep descent rather than coinciding with a finite-horizon cutoff.
-
The deterministic local stability exchange occurs at λ* = 1/3. In the finite-horizon stochastic simulation, the numerical threshold z ≤ 0.05 is not reached at λ* and is reached reliably only by approximately λ = 0.40.
-
The strongest adversarial risk occurs toward the lower end of the negative branch, where both marginal corruption and P(CC) are highest under the exact coupling.
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For λ > λ*, the honest corner is locally stable. This local statement should be distinguished from finite-horizon convergence: at λ* = 1/3 the mean propensity remains about 0.081 at t = 100, whereas at λ = 0.40 all replications cross z ≤ 0.05.
4.1.4 Steady-State Propensities (Fig. 5)
Figure 5 plots the end-of-simulation common propensity across the full λ-spectrum. Symmetry is exact in the baseline because the populations have identical parameters, the same initial condition and a common stochastic perturbation. The propensity declines from about 0.77 at λ = − 1 to 0.46 at independence, follows a stable interior branch below λ*, and continues toward zero above the local threshold. The vertical line marks the analytical stability boundary, not a finite-horizon first-passage cutoff.
4.1.5 Steady-State Metrics (Table 1)
The data summarize the aligned exact-copula results. Private welfare is evaluated on the material payoff matrix in Eq. (2), excluding the reduced-form alignment adjustment to avoid double counting; net social welfare is Wsocial = Wprivate − 10P(CC). The adversarial region has the lowest net welfare, while sufficiently strong positive alignment moves welfare toward the material upper bound of 10.
4.1.6 Convergence Dynamics (Table 2 and Fig. 6)
The convergence measure is based on the marginal propensity z rather than P(CC), because strong negative dependence can make joint corruption relatively low even when marginal corruption is high. No replication crosses a pure-regime threshold by t = 100 for λ ≤ λ*. At λ = 0.36, 8% of replications cross z ≤ 0.05, with a conditional mean first-passage time of about 99 generations. At λ = 0.40 all replications cross after an average of about 79 generations; the corresponding means are 28 generations at λ = 0.68 and 17 generations at λ = 1.
4.1.7 Phase Portrait (Fig. 7)
Figure 7 shows mean trajectories for the exact grid values λ = 0.16, 0.24, 0.33, 0.36 and 0.44. All trajectories begin at z = 0.50, while their initial P(CC) values differ because the copula depends on λ. The two sub-threshold paths approach positive interior states. At λ* = 1/3 the path moves slowly toward the boundary; at λ = 0.36 and 0.44 the honest corner is locally attracting, with visibly faster convergence at the larger value. The collusive corner is not reached under this calibration.
4.1.8 Welfare Decomposition (Fig. 8)
Figure 8 shows net social welfare rising from about − 5.4 at λ = − 1 to 3.2 at independence, 8.95 at λ* = 1/3, 9.57 at λ = 0.40 and 9.97 under strong alignment. Relative to independence, the simulated gain is about 5.75 points at λ* and 6.38 points at λ = 0.40, with enforcement parameters unchanged.
4.1.9 Sensitivity Analysis and Enforcement Interaction (Fig. 9 and 10)
Figure 9 maps the sensitivity of λ* to audit probability p and the normalized threshold-scale parameter αN. The baseline sits in the lower-left corner of the heatmap, where λ* = 0.33. Increasing p lowers the threshold and makes the honest regime attainable at lower levels of positive alignment. The effect of αN is different: because αN is defined as a threshold-scale parameter rather than as primitive institutional effectiveness, larger αN raises the transition boundary. If the same channel is written in terms of the primitive responsiveness ρ in Eq. (11), stronger responsiveness lowers the effective threshold. The heatmap is therefore interpreted as a sensitivity map, not as a universal monotonic ranking of policy instruments.
Figure 10 provides a three-dimensional view of the interaction between correlation and enforcement over the interior-threshold range p ∈ [0.05, 0.35]. The strongest risk remains on the negative-λ, low-enforcement side. For positive λ, the surface falls toward zero across a transition band that shifts with p. The base-plane curve λ*(p) is the local stability boundary of the same differential integrated in the simulation.
The simulation results, taken together, support three overarching conclusions that refine and extend the theoretical framework of Sect. 3.
First, the aligned simulations support the local threshold structure of Proposition 1. The local numerical eigenvalue changes sign exactly at λ* = 1/3 by construction of the reduced-form alignment-calibration bridge. Finite-horizon first passage is slower: at λ* = 1/3 the mean path remains above z = 0.05 at t = 100, while all replications cross the threshold by λ = 0.40. This distinction is consistent with critical slowing near a local bifurcation and is not evidence of a different numerical stability root.
Second, the exact negative branch produces an adversarial high-risk region without violating the probability axioms. The most adverse baseline outcome occurs near λ = − 1, where the common corruption propensity is about 0.77, P(CC) is about 0.54, and net social welfare is about − 5.4. The policy interpretation should therefore remain model-based: institutional design should avoid adversarial dependence, but the simulations do not establish that any specific staff-rotation policy necessarily generates λ < 0.
Third, the analytical cooperation corridor is welfare-improving, but local stability and finite-horizon arrival should be distinguished. The honest corner is locally stable for λ > 1/3; from the simulated initial condition, all replications reach z ≤ 0.05 within 100 generations at λ = 0.40 and above. Institutional alignment does not replace enforcement, but can lower the enforcement intensity required for local honesty and accelerate convergence once the system is sufficiently inside the corridor.
A final methodological observation follows from the sensitivity analysis (Fig. 9). The robust result is not that one parameter mechanically dominates all others, but that institutional design and enforcement interact around λ*: organizational alignment moves the system toward the cooperation corridor, while enforcement can lower the alignment required for honesty to become stable.
5 Discussion
This work began with a specific question: how can a classical parametric correlator be compared, in an evolutionary setting, with the correlation intensity associated with quantum entanglement (RQ1)? The answer is deliberately limited. The classical correlator K(λ) does not use quantum resources, does not involve superposition of actions and cannot reproduce the full EWL strategy space. In the restricted I/X benchmark, squared concurrence sin2(γ) provides a scalar calibration for λ, but the EWL post-measurement distribution is not the Fréchet–Hoeffding coupling. Proposition 4 sharpens this distinction: on x = y = z, the mixed I/X distribution is the independent product for every γ, whereas K(λ) remains λ-dependent and concordant. RQ2 concerns the policy value of acting on the correlation structure relative to conventional enforcement. The aligned simulations show that strong negative dependence creates an adversarial high-risk region, while positive alignment shifts the interior branch toward the honest corner. The analytical stability boundary is λ* = 1/3; finite-horizon convergence from the specified initial condition requires a larger margin above the threshold.
Formally, the model combines a valid copula-based joint distribution with a frequency-dependent matching differential and a normalized institutional-alignment differential. The reduced-form alignment-calibration bridge ensures that the numerical replicator and Proposition 1 have the same local stability root while also affecting the interior dynamics and convergence paths. Below λ*, the honest corner is unstable and an interior branch persists; above λ*, the honest corner is locally asymptotically stable. This is a local deterministic statement, whereas the reported first-passage times depend on the finite horizon, selection intensity, initial condition and stochastic perturbations.
This phenomenon is a threshold transition in an evolutionary game. Its main implication is that long-run behaviour depends not only on payoffs but also on the correlation structure that governs strategic encounters. Models that assume strategic independence may therefore miss an important institutional channel through which corruption is either reinforced or contained.
5.1 Limitations
Several limitations deserve discussion. The closed-form analysis and baseline numerical experiment use the symmetric invariant manifold x = y = z, identical payoff parameters and a common stochastic perturbation; the general asymmetric system is piecewise and should be analysed separately. Assumption 1 is an auxiliary matching-only condition and is not satisfied by the weak-enforcement baseline. The bridge ω is a reduced-form alignment calibration chosen to preserve the classical matching dynamics at independence and align the local numerical root with the analytical threshold. It is not a neutral conversion of units: it affects the interior dynamics and convergence paths, and alternative microfoundations and calibrations should be studied. The selection intensity s = 0.03, the horizon T = 100, the initial condition z0 = 0.50 and the noise level affect convergence times but not the deterministic local threshold. The model assumes a single institutional environment, and the linear transaction-cost specification is adopted for tractability. The analytical result establishes local stability and a transcritical exchange; finite-horizon paths do not prove global stability. Whistleblowing is not modelled and no empirical calibration of λ is attempted. The restricted I/X benchmark is deliberately asymmetric in Fig. 1; on the symmetric manifold it reduces to independent randomization for every γ. It therefore neither reproduces K(λ) nor covers the full SU(2) quantum strategy space.
6 Conclusions
The present study suggests that the architecture of strategic correlations is an important, and often neglected, determinant of anti-corruption dynamics. The formal contribution is the construction of a classical parametric correlator K(λ) that links correlated equilibrium and evolutionary Hawk–Dove dynamics, together with a restricted quantum benchmark expressed on the same scalar intensity scale. The model does not claim that institutional systems are quantum systems, that a classical copula reproduces the EWL measurement distribution, or that it reproduces the full EWL strategy space. Its narrower claim is that squared concurrence sin2(γ) can be used as a transparent scalar calibration for λ while the classical and quantum mechanisms remain formally distinct. Proposition 4 makes that distinction exact: on the symmetric analytical manifold the restricted I/X distribution is independent, whereas the classical correlator continues to alter concordance.
Within this framework, corruption depends not only on sanctions and monetary payoffs but also on the dependence structure governing strategic encounters. The exact lower Fréchet–Hoeffding branch produces a valid adversarial high-risk region. On the positive branch, the aligned analytical and numerical model places the local stability transition at λ* = 1/3; convergence within a finite horizon becomes reliable only farther inside the cooperation corridor. Enforcement shifts the boundary, while organizational design changes both the correlation environment and the speed with which the honest regime is approached.
Several extensions remain open. Future work should test the threshold structure in asymmetric games with more than two strategies, heterogeneous populations and networked interactions. A second avenue concerns mechanism design: if λ is treated as a control variable, the relevant problem is the design of an informational and organizational mediator that maximizes social welfare subject to incentive-compatibility constraints. On the empirical side, procurement records, mobility data and administrative network data could make it possible to estimate implicit correlation structures and test the model’s predictions outside the purely theoretical setting.
Data Availability
No empirical data were used in this study. The paper is based on formal theoretical modelling and numerical simulations. The supplementary package contains one parameter file, the exact-copula and aligned-selection code, fixed random seeds, automated probability and threshold-alignment tests, and the routines used to generate Figs. 1–10 and Tables 1–2.
References
Andvig JC, Moene KO (1990) How corruption may corrupt. J Econ Behav Organ 13(1):63–76
Auletta V, Ferraioli D, Rai A, Scarpa G, Winter A (2021) Belief-invariant and quantum equilibria in games of incomplete information. Theor Comput Sci 895:151–177
Aumann RJ (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1(1):67–96
Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55(1):1–18
Benjamin SC, Hayden PM (2001) Multiplayer quantum games. Phys Rev A 64(3):030301
Benndorf V, Martínez-Martínez I, Normann H-T (2016) Equilibrium selection with coupled populations in Hawk-Dove games: theory and experiment in continuous time. J Econ Theory 165:472–486
Bose, G (2010) Aspects of bureaucratic corruption. UNSW Australian School of Business Research Paper No. 2010ECON14.
Brandenburger A (2010) The relationship between quantum and classical correlation in games. Games Econ Behav 69(1):175–183
Brunner N, Linden N (2013) Connection between Bell nonlocality and Bayesian game theory. Nat Commun 4:2057
Deckelbaum, A (2011) Quantum correlated equilibria in classical complete-information games. arXiv:1101.3380.
Djankov S, La Porta R, Lopez-de-Silanes F, Shleifer A (2002) The regulation of entry. Q J Econ 117(1):1–37
Eisert J, Wilkens M, Lewenstein M (1999) Quantum games and quantum strategies. Phys Rev Lett 83(15):3077–3080
Ferraz C, Finan F (2008) Exposing corrupt politicians: the effects of Brazil’s publicly released audits on electoral outcomes. Q J Econ 123(2):703–745
Flitney AP, Abbott D (2005) Quantum games with decoherence. J Phys A Math Gen 38(2):449–459
Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer
Harrington JE (2008) Optimal corporate leniency programs. J Ind Econ 56(2):215–246
Hart S, Schmeidler D (1989) Existence of correlated equilibria. Math Oper Res 14(1):18–25
Hoeffding W (1940) Maßstabinvariante Korrelationstheorie. Schr Math Inst Univ Berl 5:179–233
Iqbal A, Toor AH (2001) Evolutionarily stable strategies in quantum games. Phys Lett A 280(5–6):249–256
Iqbal A, Toor AH (2002) Quantum mechanics gives stability to a Nash equilibrium. Phys Rev A 65(2):022306
Iqbal A, Toor AH (2002) Darwinism in quantum systems? Phys Lett A 294(5–6):261–270
Johnson NF (2001) Playing a quantum game with a corrupted source. Phys Rev A 63(2):020302(R)
Klitgaard R (1988) Controlling corruption. University of California Press
Kohli IS, Haslam MC (2017) An analysis of the replicator dynamics for an asymmetric Hawk-Dove game. Int J Differ Eq 2017:8781570
Lambsdorff JG (2002) Corruption and rent-seeking. Public Choice 113(1–2):97–125
Lindsell, E. A. B., & Wiesner, K (2014) Entanglement gives rise to Pareto optimality in the generalised quantum Hawk–Dove game. arXiv:1410.3631.
Marinatto L, Weber T (2000) A quantum approach to static games of complete information. Phys Lett A 272(5–6):291–303
Marino D (2025) Dynamics of corruption: theoretical explanatory model and empirical results. Phys A Stat Mech Appl 658:130288
Nawaz A, Toor AH (2004) Dilemma and quantum battle of sexes. J Phys A Math Gen 37(15):4437–4443
Nelsen RB (2006) An introduction to copulas, 2nd ed. Springer
Olken BA (2007) Monitoring corruption: evidence from a field experiment in Indonesia. J Polit Econ 115(2):200–249
Pappa A, Chailloux A, Wehner S, Diamanti E, Kerenidis I (2015) Nonlocality and conflicting interests in Bayesian games. Phys Rev Lett 114(2):020401
Phoenix SJD, Khan FS (2013) The role of correlation in quantum and classical games. Fluct Noise Lett 12(3):1350011
Reinikka R, Svensson J (2011) The power of information in public services: evidence from education in Uganda. J Public Econ 95(7–8):956–966
Rose-Ackerman S (1999) Corruption and government: causes, consequences, and reform. Cambridge University Press
Spagnolo G (2008) Leniency and whistleblowers in antitrust. In: Buccirossi P (ed) Handbook of antitrust economics. MIT Press, pp 259–304
van Enk SJ, Pike R (2002) Classical rules in quantum games. Phys Rev A 66(2):024306
Villani C (2009) Optimal transport: old and new. Springer
Jabir MT, Vyas N, Benjamin C (2021) Is the essence of a quantum game captured completely in the original classical game? Phys A Stat Mech Appl 584:126360
Groisman, B (2018) When quantum games can be played classically: in support of van Enk–Pike’s assertion. arXiv:1802.00260.
Frąckiewicz P (2021) Nonclassical rules in quantum games. Entropy (Basel) 23(5):604
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Appendices
Appendix A—Detailed Derivations
1.1 A.1 Conditional Payoffs and Fitness Differential
On the symmetric invariant manifold x = y = z and under the positive-correlation branch of K(λ), the conditional probabilities of the opponent’s strategy given the focal player’s choice are
The corresponding material matching fitnesses are
Write M(z,λ) = fH(z,λ) − fD(z,λ). For λ ≥ 0, the combined differential is G(z,λ) = M(z,λ) − M(0,λ) + ωΔN(λ), while G = M for λ < 0. The reduced-form alignment-calibration bridge ω = [aHD − aDD]/ΔN(0) maps the normalized institutional component into material-payoff units, preserves the classical matching differential at λ = 0 and gives G(0,λ) = ωΔN(λ). It is not a neutral change of units and affects the interior dynamics and convergence paths. Under the baseline calibration, ω = 4.8333.
1.2 A.2 Local Bifurcation and Matching-Only Corners
For the auxiliary matching-only benchmark, the collusive corner is unstable when c > V0 because a Dove mutant earns more than an incumbent Hawk; under Assumption 1, the honest corner is locally stable. These statements are separate from the weak-enforcement combined-channel baseline. For the aligned symmetric model, the drift is F(z,λ) = s z(1 − z)G(z,λ). At (0,λ*), Fz = 0, while ∂G(0,λ*)/∂λ = − ωαN(V0 − k) ≠ 0 and ∂G(0,λ*)/∂z = (1 − λ*)(aHH − aHD − aDH + aDD) ≠ 0. The standard non-degeneracy conditions for a transcritical exchange are therefore satisfied: the boundary branch z = 0 intersects the interior branch at λ* and they exchange local stability.
1.3 A.3 Logit-Response Extension
Under bounded rationality, the probability of legal/cooperative play can be written as
where β ≥ 0 measures response precision. The limiting case β → 0 gives random choice, while β → ∞ gives best-response dynamics.
Appendix B—Robustness: Symmetric Surplus Specification
To address the concern that the asymmetric surplus specification may drive the result, I consider a symmetric extension in which both legal compliance and corruption respond to correlation:
where ηD ≥ ηH ≥ 0. The threshold becomes
provided ηD > ηH. Thus, the qualitative result is preserved whenever legal compliance benefits more from institutional alignment than corruption does. When ηH = 0, the baseline logic is recovered.
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Marino, D. Quantum-Inspired Correlated Governance: A Parametric Correlator for Anti-Corruption in Evolutionary Hawk–Dove Games. Dyn Games Appl (2026). https://doi.org/10.1007/s13235-026-00714-1
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DOI: https://doi.org/10.1007/s13235-026-00714-1
Keywords
- Correlated equilibrium
- Quantum game theory
- Evolutionary dynamics
- Corruption
- Hawk–Dove game
- Parametric correlator
- Entanglement
- Institutional design