# Nash Equilibrium

Multi-agent learning in continuous general-sum games has long struggled with non-stationarity, because agents maximizing their own payoffs make the learning target move. DRAC shifts the objective from payoff maximization to regret minimization, fixing opponents' actions and sampling unilateral deviations to build a stationary regret target, with a dual-head critic predicting best-response utility and regret separately. In a three-agent, five-round sequential Cournot oligopoly and a two-agent, five-stage proportional prize contest, DRAC's exploitability converged to 1.37×10⁻⁴ and 7.45×10⁻⁴, with mean relative absolute deviations of 0.011 and 0.030, both below MADDPG at the same training budget.

Multi-agent learning in continuous general-sum games has long struggled with non-stationarity, because agents maximizing their own payoffs make the learning target move. DRAC shifts the objective from payoff maximization to regret minimization, fixing opponents' actions and sampling unilateral deviations to build a stationary regret target, with a dual-head critic predicting best-response utility and regret separately. In a three-agent, five-round sequential Cournot oligopoly and a two-agent, five-stage proportional prize contest, DRAC's exploitability converged to 1.37×10⁻⁴ and 7.45×10⁻⁴, with mean relative absolute deviations of 0.011 and 0.030, both below MADDPG at the same training budget.

The study models two competing hybrid DC pension fund managers as a stochastic differential game in which liability risk follows a jump-diffusion process with truncated exponential jump amplitudes and the exact asset-to-liability ratio formulation is used, and it derives closed-form Nash equilibrium portfolio strategies and value functions via Hamilton–Jacobi–Bellman dynamic programming, finding that equilibrium weights are independent of liability jump parameters, reduce to K=(μ−r)/σ² and are independent of both managers' risk aversion in the symmetric liability-correlation case, that competition strictly amplifies risk-taking relative to the single-agent benchmark, and that a manager's own liability jumps reduce welfare while the competitor's jumps improve it.

The study models two competing hybrid DC pension fund managers as a stochastic differential game in which liability risk follows a jump-diffusion process with truncated exponential jump amplitudes and the exact asset-to-liability ratio formulation is used, and it derives closed-form Nash equilibrium portfolio strategies and value functions via Hamilton–Jacobi–Bellman dynamic programming, finding that equilibrium weights are independent of liability jump parameters, reduce to K=(μ−r)/σ² and are independent of both managers' risk aversion in the symmetric liability-correlation case, that competition strictly amplifies risk-taking relative to the single-agent benchmark, and that a manager's own liability jumps reduce welfare while the competitor's jumps improve it.