We study a class of semi-discrete variational problems that arise in economic matching and game theory, where agents with continuous attributes are matched to a finite set of outcomes with a one dimensional structure. Such problems appear in applications including Cournot-Nash equilibria, and hedonic pricing, and can be formulated as problems involving optimal transport between spaces of unequal dimensions. In our discrete strategy space setting, we establish analogues of results developed for a continuum of strategies in \cite{nenna2020variational}, ensuring solutions have a particularly simple structure under certain conditions. This has important numerical consequences, as it is natural to discretize when numerically computing solutions. We leverage our results to develop efficient algorithms for these problems which scale significantly better than standard optimal transport solvers, particularly when the number of discrete outcomes is large, provided our conditions are satisfied. We also establish rigorous convergence guarantees for these algorithms. We illustrate the advantages of our approach by solving a range of numerical examples; in many of these our new solvers outperform alternatives by a considerable margin.

翻译：我们研究一类源于经济匹配与博弈论的半离散变分问题，其中具有连续属性的智能体被匹配到具有一维结构的有限结果集。此类问题出现在包括古诺-纳什均衡与享乐定价等应用中，可表述为涉及不等维空间间最优传输的问题。在离散策略空间设定下，我们建立了与\cite{nenna2020variational}中连续策略情形相关结果的类比，确保在特定条件下解具有特别简单的结构。这具有重要的数值计算意义，因为在数值求解时进行离散化是自然的选择。我们利用所得结果开发了针对此类问题的高效算法，在满足条件时其计算规模显著优于标准最优传输求解器，尤其在离散结果数量较大时表现突出。同时，我们为这些算法建立了严格的收敛性保证。通过求解一系列数值算例，我们展示了所提方法的优势；在多数案例中，新求解器的性能显著优于其他方法。