Research on Cooperative Dual Equilibria with Asymmetric Uncertainty Strategy
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摘要: 从参与者对手利益出发，研究对手成本最低的双人合作博弈问题：当博弈活动出现非理性现象及竞争者自身策略不确定情形时，假设竞争者自身支付矩阵能准确获知，对手策略落在混合策略集内，但自身策略集为一非对称有界闭集(混合策略集的子集)，提出了合作对偶博弈模型。并采用鲁棒优化技术，研究了双人合作对偶博弈中均衡的求解问题，得到以下结论：当非对称不确定策略集中元素取l1∩∞- 范数时，使对方成本最低的问题可转化成一个线性规划问题，使双方成本同时最低的问题可转化成一个混合互补问题。最后用数值算例验证了模型的合理性和有效性。Abstract: The two-player cooperative game with the lowest cost from the perspective of the player's opponent's interests is considered. Based on irrational phenomena and strategy uncertainty in the game and under the assumption that the payoff matrix for each player is exactly known and each player's own strategy set cannot be evaluated while the mixed strategy set may be estimated at an asymmetric bounded closed set (a subset of the mixed strategy set) and the opponent's strategies can be included in the mixed strategy set, a cooperative dual game model is introduced in a bimatrix game. Then by means of the robust optimization technique, a cooperative dual equilibria with two players is investigated. Some results are obtained as follows: the problem of minimization of the opponent's cost can be converted to a linear programming (LP) and solving the corresponding cooperative dual equilibria can be expressed as solving a mixed complementarity problem (MCP) with l1∩∞-norm uncertainty. Finally, a numerical experiment is provided to illustrate the feasibility and validity of the robust cooperative dual equilibria.
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表 1 策略非对称不确定性下的鲁棒合作对偶均衡
Table 1 Robust cooperative dual equilibria with asymmetric strategy uncertainty
Ω Υ zr yr yTrAzr yrTBzr 0.1 0.1 (0, 1, 0) (0, 0.198 7, 0.801 3) 14.807 8 -9.801 3 0.2 0.2 (0, 1, 0) (0, 0.085 7, 0.914 3) 15.485 8 -9.914 3 0.3 0.3 (0, 1, 0) (0, 0, 1) 16 -10 0.5 0.5 (0, 1, 0) (0, 0, 1) 16 -10 0.8 0.8 (0.033 5, 0.962 1, 0.004 4) (0.005 1, 0.247, 0.747 9) 14.687 1 -9.568 8 0.9 0.9 (0, 0.733 4, 0.266 6) (0, 0.2, 0.8) 22.051 5 -10.813 1 1 1 (0, 1, 0) (0, 0, 1) 16 -10 2 2 (0, 1, 0) (0, 0, 1) 16 -10 
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