Optimization of Two-Stage Military Product Revenue-Sharing Game Model Based on Particle Swarm Algorithm

To address three major industry pain points—the lack of quantified profit-sharing standards in the two-stage pricing under the separation model of military research and production, the absence of stable Nash equilibrium in single-layer synchronous optimization, and insufficient incentives for full-cycle process optimization in design units—this paper constructs a two-level Stackelberg leader–follower game model with the general contracting unit as the leader and design and general contracting units as followers. This aligns with current prototype incentives and phased pricing policies for production rewards and penalties. At the theoretical level, it improves the complete proof system for the two-stage concave profit two-level Stackelberg Nash equilibrium, distinguishes the mathematical differences in equilibrium existence between sequential decision-making and synchronous optimization, and extracts general rules for phased differentiated profit sharing: high-innovation segments should be allocated more profit weight; simply maximizing total alliance profit may cause imbalanced interests, while introducing a minimum net profit-weighted objective can achieve Pareto improvements without profit loss. This conclusion can be applied to multi-stage general contracting scenarios across industries, such as EPC and military–civil collaborative innovation, enriching the basic theory of profit sharing and hierarchical games. Theoretically, the existence of the lower-level Nash equilibrium is proven using Brouwer’s fixed-point theorem, and combining it with the strictly monotonically decreasing feature of the best response function, uniqueness of the equilibrium is derived. Multiple sets of differentiated initial values are simulated to rule out multi-equilibrium bifurcation risk. The model incorporates the military’s reward and penalty policies as rigid exogenous constraints, sets dual individual rationality constraints of ‘cooperative profit greater than baseline profit with no allocation, and both parties’ net profit non-negative,’ and introduces differentiated cost-reduction efficiency and quadratic increasing effort costs to characterize the heterogeneous input of the two types of development entities. For models with piecewise nonlinearity and multi-constraint nonconvex structures, this paper modifies the standard PSO into a Bi-PSO solving framework through hierarchical temporal adaptation. It does not innovate the underlying particle update mechanism and is only used to match the sequential decision order of the leader–follower game. By comparing five algorithms—IPM, GA, SA, DE, and adaptive PSO—through 20 repeated simulations: gradient-based interior point methods easily get stuck in locally invalid solutions that violate cooperation thresholds; differential evolution has the best numerical global search performance, but all general evolutionary algorithms optimize allocation and effort variables simultaneously, disrupting the Stackelberg hierarchical timing. Only Bi-PSO maintains consistent game logic. Using a pricing case for a certain type of equipment and jointly calibrating all parameters with policy documents, three simulation scenarios were set up: no allocation, equal 50/50 split, and single-layer profit maximization. Under the no-allocation mode, R&D investment from the design unit drops to zero and alliance benefits plummet; a blanket equal split ignores differences in technical contributions across two stages, leading to clear efficiency losses; single-layer optimization only pursues total profit maximization, causing a severe imbalance in profit distribution. The two-layer basic framework can achieve the upper limit of alliance benefits, and by adding a weighted optimization goal that considers both total profit and cooperation fairness, it can achieve equal net profits for both parties without reducing overall profit. Through single-parameter sweeps and two-factor heatmap simulations, the study further revealed the coupled effects of main party efficiency and mass production rewards and penalties on equilibrium input and optimal sharing ranges. A robust check was performed by replacing the logarithmic concave output function, producing a standardized allocation range resilient to parameter perturbations: optimal split for the prototype stage is 0.4–0.6 for the design unit, and for mass production stage 0.7–0.9. The findings suggest that high-contribution stages in multi-phase collaboration contracts should receive more benefits, and a weighted fairness objective can achieve Pareto improvements. These conclusions can extend to multi-stage collaboration scenarios such as EPC and military–civilian cooperation. Theoretically, this research further completes the equilibrium proof system for two-party concave payoff two-layer games, providing a new reference for the theory of phased differentiated benefit-sharing contracts in the military sector. Methodologically, it proposes a two-layer intelligent solving tool adapted to leader–follower sequential decisions, effectively mitigating issues where single-layer model equilibria fail or analytical algorithms struggle with multi-constraint nonconvex games. The results can provide quantitative support for the military, general contracting unit, and design unit in drafting equipment incentive pricing contracts and managing full-cycle cost collaboration.