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Current automated market makers over binary events suffer from two problems that make them impractical. First, they are unable to adapt to liquidity, so trades cause prices to move the same amount in both thick and thin markets. Second, under normal circumstances, the market maker runs at a deficit. In this paper, we construct a market maker that is both sensitive to liquidity and can run at a profit. Our market maker has bounded loss for any initial level of liquidity and, as the initial level of liquidity approaches zero, worstcase loss approaches zero. For any level of initial liquidity we can establish a boundary in market state space such that, if the market terminates within that boundary, the market maker books a profit regardless of the realized outcome. Furthermore, we provide guidance as to how our market maker can be implemented over very large event spaces through a novel cost-function-based sampling method.

Current automated market makers over binary events suffer from two problems that make them impractical. First, they are unable to adapt to liquidity, so trades cause prices to move the same amount in both thick and thin markets. Second, under normal circumstances, the market maker runs at a deficit. In this paper, we construct a market maker that is both sensitive to liquidity and can run at a profit. Our market maker has bounded loss for any initial level of liquidity and, as the initial level of liquidity approaches zero, worstcase loss approaches zero. For any level of initial liquidity we can establish a boundary in market state space such that, if the market terminates within that boundary, the market maker books a profit regardless of the realized outcome. Furthermore, we provide guidance as to how our market maker can be implemented over very large event spaces through a novel cost-function-based sampling method.

Competitive equilibrium with equal incomes (CEEI) is a wellknown fair allocation mechanism [Foley, 1967, Varian, 1974, Thomson and Varian, 1985; however, for indivisible resources a CEEI may not exist. It was shown in Budish [2011] that in the case of indivisible resources there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient, for some favorable approximation parameters. This approximation is used in practice to assign business school students to classes. In this paper we show that finding the A-CEEI allocation guaranteed to exist by Budish's theorem is PPAD-complete. We further show that finding an approximate equilibrium with better approximation guarantees is even harder: NP-complete.

Abstract. Combinatorial allocation involves assigning bundles of items to agents when the use of money is not allowed. Course allocation is one common application of combinatorial allocation, in which the bundles are schedules of courses and the assignees are students. Existing mechanisms used in practice have been shown to have serious flaws, which lead to allocations that are inefficient, unfair, or both. A recently developed mechanism is attractive in theory but has several features that limit its feasibility for practice. This paper reports on the design and implementation of a new course allocation mechanism, Course Match, that is suitable in practice. To find allocations, Course Match performs a massive parallel heuristic search that solves billions of mixed-integer programs to output an approximate competitive equilibrium in a fake-money economy for courses. Quantitative summary statistics for two semesters of full-scale use at a large business school (the Wharton School of Business, which has about 1,700 students and up to 350 courses in each semester) demonstrate that Course Match is both fair and efficient, a finding reinforced by student surveys showing large gains in satisfaction and perceived fairness.