Abstract
This study investigates the moral hazard problem inherent in mudarabah (profit-sharing) contracts through the lens of evolutionary game theory, introducing taqwa (God-consciousness) as an exogenous moral-cost parameter that is monetized within the payoff structure. I construct a two-population asymmetric evolutionary game between the mudarib (entrepreneur) and the rabb al-mal (capital provider), each choosing between cooperative and opportunistic strategies. The replicator dynamics are solved numerically using a fourth-order Runge–Kutta integrator implemented in Stata 19.5/Mata, and every result is independently reproduced in Python 3, for a total of more than 1,200 trajectory integrations across five analytical stages. The central finding identifies a critical taqwa threshold at τ* = 1—α (where α is the profit-sharing ratio), below which the system exhibits persistent oscillatory behavior around a center-type interior equilibrium, and above which the interior equilibrium exits the state space through a boundary bifurcation and the system converges globally to full cooperation; both regimes are established analytically in Appendix A. This threshold effect is robust across three layers of sensitivity analysis: one-at-a-time parameter variation reveals that the profit-sharing ratio dominates system behavior (total effect = 0.238); Latin Hypercube Sampling with 500 draws confirms this finding globally (SRC = 0.624, PRCC = 0.819, R-squared = 0.743); and an initial-condition analysis across 256 starting points demonstrates that long-run time averages are independent of the starting state (CV = 0.15%). These results offer a formal mathematical framework bridging Islamic moral philosophy and institutional economics, suggesting that intrinsic ethical motivation can serve as a functional substitute for costly external monitoring in profit-sharing arrangements.
Data Availability
This study is based entirely on computational simulations of an evolutionary game-theoretic model; no proprietary or third-party empirical data were used. All quantitative results, figures, and robustness checks reported in the manuscript were generated from the Stata 19.5/Mata and Python 3 replicator-dynamics code developed by the author. The complete replication package, including the Mata routines for the fourth-order Runge–Kutta integration of the replicator system, the parameter sets used for the baseline model, the one-at-a-time sensitivity analysis, the Latin Hypercube global sensitivity analysis, and the initial-conditions experiments, is not deposited in a public repository at this stage. The full code and simulation outputs will be made available by the author upon reasonable request.
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E.A.K. is the sole author of this manuscript and is solely responsible for all aspects of the work. The author conceived and designed the study, developed the theoretical model and analytical derivations, implemented the numerical simulations and robustness analyses in Stata 19.5/Mata and Python 3, prepared all figures and tables, wrote the original draft, and carried out subsequent revisions. The author has read and approved the final version of the manuscript. E.A.K.: Conceptualization, Methodology, Formal analysis, Software, Investigation, Visualization, Writing – original draft, Writing – review and editing.
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Appendix A: Equilibrium Structure, Stability, and Global Dynamics
Appendix A: Equilibrium Structure, Stability, and Global Dynamics
This appendix provides the symbolic equilibrium and stability analysis of the system defined by Eqs. (5) and (6). Write the dynamics as dx/dt = x(1-x)f(y) and dy/dt = y(1-y)g(x), where f(y) = A + (1-y)qK and g(x) = c—(1-x)qK, with A = πδ(τ—(1-α)) = πδ(τ—τ*) and K = (1-α)πδ + F > 0. Two structural features drive everything that follows: f depends only on y and is strictly decreasing, and g depends only on x and is strictly increasing. All parameters are strictly positive and α, δ, q lie in (0, 1).
Proposition 1 (Equilibria and Existence)
The four corners (0, 0), (1, 0), (0, 1), and (1, 1) are equilibria for all parameter values. An interior equilibrium exists if and only if 0 < c < qK and 0 < -A < qK; the second pair of inequalities is equivalent to τ < τ* together with πδ(τ*—τ) < qK. When it exists, the interior equilibrium is unique and given by Eqs. (7) and (8): x* = 1—c/(qK) and y* = 1 + A/(qK). Under these conditions, all four corners are saddle points: the eigenvalue pairs of the linearization are (f(0), g(0)) = (A+qK, c-qK) at (0, 0); (-f(0), g(1)) = (-(A+qK), c) at (1, 0); (f(1), -g(0)) = (A, qK-c) at (0, 1); and (-f(1), -g(1)) = (-A, -c) at (1, 1); in each pair, one entry is positive and the other negative. If instead c ≥ qK while τ < τ*, then g ≥ 0 throughout, monitoring never pays, and trajectories are absorbed at (0, 1); if f(0) = A + qK ≤ 0, honesty never pays and trajectories are absorbed on the edge x = 0, at (0, 0) when additionally c < qK. These two failure modes generate the boundary-dominated regime reported in Sect. 4.4.
Proof.
The corner and interior equilibrium claims follow by direct computation from f(y*) = 0 and g(x*) = 0 and from monotonicity of f and g. At each corner, the Jacobian is diagonal with the stated entries because the off-diagonal terms vanish when x(1-x) = 0 and y(1-y) = 0. Existence requires f to change sign on (0, 1), which holds if and only if f(0) > 0 > f(1), that is, A + qK > 0 > A, and similarly g(0) < 0 < g(1) if and only if 0 < c < qK. For the absorption claims, c ≥ qK gives g ≥ 0 with equality only at x = 0, so y(t) increases to 1 from any interior start; on approach to the edge y = 1, f tends to A < 0, so x(t) tends to 0. The case f(0) ≤ 0 is symmetric. ∎
Proposition 2 (First Integral and Nonlinear Center for τ < τ*)
Assume the existence conditions of Proposition 1. Define H(x, y) = G(x) + Ψ(y) with G(x) = (c-qK) ln x—c ln(1-x) and Ψ(y) = -(A+qK) ln y + A ln(1-y). Then H is constant along every solution in the open unit square, H is strictly convex with a unique minimum at (x*, y*), and H tends to infinity on the boundary of the square. Consequently, every interior orbit other than the equilibrium itself is a closed curve encircling (x*, y*), which is therefore a genuine nonlinear center. At the equilibrium, the Jacobian has zero trace and determinant x*(1-x*)y*(1-y*)(qK)2 > 0; under the base parameters, this equals 20.43, giving eigenvalues ± 4.52i.
Proof.
Differentiating along trajectories, dH/dt = G’(x) dx/dt + Ψ’(y) dy/dt. Since G’(x) = g(x)/[x(1-x)] and Ψ’(y) = -f(y)/[y(1-y)], substitution gives dH/dt = gf—fg = 0. Convexity follows from G’’(x) = (qK-c)/×2 + c/(1-x)2 > 0 and Ψ’’(y) = (A+qK)/y2—A/(1-y)2 > 0, both strict under the existence conditions. The boundary behavior follows from the signs of the logarithmic coefficients: c-qK < 0, -c < 0, -(A+qK) < 0, and A < 0 each force the corresponding term to +∞ at the relevant edge. A strictly convex, proper function has closed convex level sets around its unique minimizer, and each orbit lies in one level set, so each nonstationary interior orbit is periodic. The critical point of H coincides with the equilibrium because G’ and Ψ’ vanish exactly where g and f do. ∎
Proposition 3 (Exact Time Averages)
Along any periodic orbit of period P in the interior, the time averages of x and y equal exactly x* and y*.
Proof.
Integrating d ln[y/(1-y)]/dt = g(x(t)) over one period gives 0 = cP—qK ∫(1-x)dt, so the mean of 1-x is c/(qK) = 1-x*. Similarly, integrating d ln[x/(1-x)]/dt = f(y(t)) gives 0 = AP + qK ∫(1-y)dt, so the mean of 1-y is -A/(qK) = 1-y*. Both steps use only the linearity of f and g. This explains the near-zero coefficient of variation reported in Sect. 4.5: deviations arise solely because the finite averaging window is not an integer number of periods. ∎
Proposition 4 (Global Convergence for τ > τ*)
If τ > τ*, then every trajectory starting in the open unit square converges to (1, 1).
Proof.
If τ > τ*, then A > 0 and f(y) = A + (1-y)qK ≥ A > 0 for all y in [0, 1]. Hence d ln[x/(1-x)]/dt ≥ A, the log-odds of x grow without bound, and x(t) → 1. Choose x̃ with g(x̃) = c/2; since x(t) → 1, there is a time after which x > x̃, so d ln[y/(1-y)]/dt ≥ c/2 from that time on and y(t) → 1. Thus (1, 1) attracts the entire open square. ∎
Proposition 5 (Degenerate Threshold τ = τ*)
At τ = τ*, A = 0 and f(y) = (1-y)qK ≥ 0, vanishing only at y = 1. Every point of the edge y = 1 is an equilibrium, so the threshold case features a continuum of boundary equilibria rather than a single attractor. From any interior initial condition, x(t) is nondecreasing and the trajectory converges to the equilibrium set on the edge y = 1; in the simulations of Sect. 4.2, trajectories accumulate near (1, 1). The transition at τ* is therefore a boundary bifurcation: as τ increases through τ*, the interior center collides with the edge y = 1 and exits the state space, and the qualitative dynamics switch from conservative cycles to global convergence.
Proof.
On the edge y = 1, dy/dt = 0 and f(1) = 0, so dx/dt = 0 as well; hence every point of the edge is stationary. In the interior, f ≥ 0 implies that ln[x/(1-x)] is nondecreasing, so x(t) converges to a limit. If y(t) remained bounded away from 1 along a subsequence with lim sup y < 1, then f(y) would be bounded below by a positive constant, forcing x(t) → 1; but then g(x(t)) → c > 0, forcing y(t) → 1, a contradiction. Hence y(t) → 1 and the trajectory converges to the equilibrium edge. ∎
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Kaplan, E.A. The Taqwa Threshold: From Persistent Cycles to Global Cooperation in Mudarabah Contracts Under Evolutionary Replicator Dynamics. Dyn Games Appl (2026). https://doi.org/10.1007/s13235-026-00717-y
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DOI: https://doi.org/10.1007/s13235-026-00717-y