Defector-responsive switching from unconditional cooperation to Tit-for-Tat has been proposed as a rescue mechanism in the repeated Prisoner’s Dilemma. The response rule itself was presented by Pal and coauthors; here we characterize analytical and finite-population consequences that were not established there. We derive an exact break-even readiness cost for general benefit and cooperation-cost parameters and state the existence and invasion conditions under which the comparison is valid. We then formulate a continuous-time finite-population process with separate birth-death and individual strategy-transition events. Its infinitesimal drift converges to the deterministic replicator-switching equations, resolving the timescale and interpretation problems created when switching is attached to reproduction. Simulations show that constant switching produces Tit-for-Tat rapidly but continues to remove unconditional cooperators after defectors disappear. Responsive switching can rescue slightly less reliably at low intensity, but it shuts off after rescue and preserves unconditional cooperation over 10 to 100 demographic generations. This preservation advantage remains positive across 27 combinations of population size, selection strength, and benefit-to-cost ratio. Adding the readiness cost to the stochastic process reduces the responsive advantage continuously but does not erase it over the tested finite horizons. The results provide a deterministic-stochastic correspondence, a quantitative cost constraint, and a robust finite-population tradeoff for an existing feedback mechanism.
In flight control theory, numerous studies have investigated the optimality of flight trajectories in a gravitational field under various control constraints. This paper considers two inertial objects operating under conflict conditions in a constant gravitational field. Lawden-type constraints are imposed on the objects’ controls, meaning that the engine thrust depends on the rate of change of the object’s (aircraft’s) mass. The objective of the first object (the pursuer) is to capture the second object (the evader) precisely, while the objective of the second object is to avoid capture. The problem is formulated and solved exclusively from the pursuer’s perspective as a differential pursuit game. Depending on the initial positions and initial velocity vectors of the objects, the problem is divided into two cases. In each case, the parallel approach strategy plays a central role. By employing this strategy, solvability conditions for the pursuit problem are derived for both cases. This paper may be of interest to researchers and specialists in flight control theory, particularly those working on conflict-control and differential game problems.
We consider whether the evolution of inequality in a large population game generates a Kuznets inverted-U curve. There are different types of agents, based on the cost of using a technology. Agents migrate and change their type, either by moving to a favorable location or by adopting a better version of the technology. This generates the replicator dynamic. We apply indirect evolution by assuming agents always play the Nash equilibrium. All agents migrate to the lowest-cost type, causing inequality to approach zero eventually. However, if initial access to the technology is limited, inequality increases before decreasing. In that case, we obtain the Kuznets curve. Otherwise, inequality may decline monotonically. As a robustness check, we also consider a multilevel selection model in which agents revise strategies based on the best-response dynamic and migrate based on the replicator dynamic. The conclusions are similar.
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
