Computing Core-Stable Outcomes in Combinatorial Exchanges with Financially Constrained Bidders

The computation of market equilibria is a fundamental and practically relevant research question. Advances in computational optimization allow for the organization of large combinatorial markets in the field nowadays. While we know the computational complexity and the types of price functions necessary on combinatorial exchanges with quasi-linear preferences, prior literature did not consider financially constrained buyers. We aim at allocations and competitive equilibrium prices that respect budget constraints. Such constraints are an important concern for the design of real-world markets, but we show that the allocation and pricing problem becomes even Σ2p-hard. Problems in this complexity class are rare, but ignoring budget constraints can lead to significant efficiency losses and instability. We introduce mixed integer bilevel linear programs (MIBLP) to compute core prices, and effective column and constraint generation algorithms to solve the problems. While full core stability becomes quickly intractable, we show that small but realistic problem sizes can actually be solved if the designer limits attention to deviations of small coalitions. This n-coalition stability is a practical approach to tame the computational complexity of the general problem and at the same time provide a reasonable level of stability.