This paper proposes a new mechanism for combinatorial assignment—for example, assigning schedules of courses to students—based on an approximation to competitive equilibrium from equal incomes (CEEI) in which incomes are unequal but arbitrarily close together. The main technical result is an existence theorem for approximate CEEI. The mechanism is approximately efficient, satisfies two new criteria of outcome fairness, and is strategyproof in large markets. Its performance is explored on real data, and it is compared to alternatives from theory and practice: all other known mechanisms are either unfair ex post or manipulable even in large markets, and most are both manipulable and unfair.

The high-frequency trading arms race is a symptom of flawed market design. Instead of the continuous limit order book market design that is currently predominant, we argue that financial exchanges should use frequent batch

Randomization is commonplace in everyday resource allocation. We generalize the theory of randomized assignment to accommodate multi-unit allocations and various real-world constraints, such as group-specific quotas ("controlled choice") in school choice and house allocation, and scheduling and curriculum constraints in course allocation. We develop new mechanisms that are ex-ante efficient and fair in these environments, and that incorporate certain non-additive substitutable preferences. We also develop a "utility guarantee" technique that limits ex-post unfairness in random allocations, supplementing the ex-ante fairness promoted by randomization. This can be applied to multi-unit assignment problems and certain two-sided matching problems.

The high-frequency trading arms race is a symptom of flawed market design. Instead of the continuous limit order book market design that is currently predominant, we argue that financial exchanges should use frequent batch auctions: uniform price double auctions conducted, for example, every tenth of a second. That is, time should be treated as discrete instead of continuous, and orders should be processed in a batch auction instead of serially. Our argument has three parts. First, we use millisecond-level direct-feed data from exchanges to document a series of stylized facts about how the continuous market works at high-frequency time horizons: (i) correlations completely break down; which (ii) leads to obvious mechanical arbitrage opportunities; and (iii) competition has not affected the size or frequency of the arbitrage opportunities, it has only raised the bar for how fast one has to be to capture them. Second, we introduce a simple theory model which is motivated by and helps explain the empirical facts. The key insight is that obvious mechanical arbitrage opportunities, like those observed in the data, are built into the market design—continuous-time serial-processing implies that even symmetrically observed public information creates arbitrage rents. These rents harm liquidity provision and induce a never-ending socially wasteful arms race for speed. Last, we show that frequent batch auctions directly address the flaws of the continuous limit order book. Discrete time reduces the value of tiny speed advantages, and the auction transforms competition on speed into competition on price. Consequently, frequent batch auctions eliminate the mechanical arbitrage rents, enhance liquidity for investors, and stop the high-frequency trading arms race.

This paper uses data consisting of students' strategically reported preferences and their underlying true preferences to study the course allocation mechanism used at Harvard Business School. We show that the mechanism is manipulable in theory, manipulated in practice, and that these manipulations cause meaningful welfare losses. However, we also …nd that ex-ante welfare is higher than under the strategyproof and ex-post e¢ cient alternative, the Random Serial Dictatorship. We trace the poor ex-ante performance of RSD to a phenomenon speci…c to multi-unit assignment, "callousness". We draw lessons for the design of multi-unit assignment mechanisms and for market design more broadly.