Eric Budish

The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes

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 combinatorial assignment problem: approximate competitive equilibrium from equal incomes

Impossibility theorems suggest that the only efficient and strategyproof mechanisms for the problem of combinatorial assignment - e.g., assigning schedules of courses to students - are dictatorships. Dictatorships are mostly rejected as unfair: for any two agents, one chooses all their objects before the other chooses any. Any solution will involve compromise amongst efficiency, incentive and fairness considerations. This paper proposes a solution to the combinatorial assignment problem. It is developed in four steps. First, I propose two new criteria of outcome fairness, the maximin share guarantee and envy bounded by a single good, which weaken well-known criteria to accommodate indivisibilities; the criteria formalize why dictatorships are unfair. Second, I prove existence of an approximation to Competitive Equilibrium from Equal Incomes in which (i) incomes are unequal but arbitrarily close together; (ii) the market clears with error, which approaches zero in the limit and is small for realistic problems. Third, I show that this Approximate CEEI satisfies the fairness criteria. Last, I define a mechanism based on Approximate CEEI that is strategyproof for the zero-measure agents economists traditionally regard as price takers. The proposed mechanism is calibrated on real data and is compared to alternatives from theory and practice: all other known mechanisms are either manipulable by zero-measure agents or unfair ex-post, and most are both manipulable and unfair.

Strategy-Proofness in the Large

We propose a criterion of approximate incentive compatibility, strategy-proofness in the large (SP-L), and argue that it is a useful second-best to exact strategy-proofness (SP) for market design. Conceptually, SP-L requires that an agent who regards a mechanism’s “prices” as exogenous to her report—be they traditional prices as in an auction mechanism, or price-like statistics in an assignment or matching mechanism—has a dominant strategy to report truthfully. Mathematically, SP-L weakens SP in two ways: (1) truth-telling is required to be approximately optimal (within epsilon in a large enough market) rather than exactly optimal, and (2) incentive compatibility is evaluated ex interim, with respect to all full-support i.i.d. probability distributions of play, rather than ex post with respect to all possible realizations of play. This places SP-L in between the traditional notion of approximate SP, which evaluates incentives to manipulate ex post and as a result is too strong to obtain our main results in support of SP-L, and the traditional notion of approximate Bayes-Nash incentive compatibility, which, like SP-L, evaluates incentives to manipulate ex interim, but which imposes common knowledge and strategic sophistication assumptions that are often viewed as unrealistic.

Course Match: A Large-Scale Implementation of Approximate Competitive Equilibrium from Equal Incomes for Combinatorial Allocation

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 new mechanism proposed by Budish [2011] is attractive in theory, but has several features that limit its feasibility for practice: reporting complexity, computational complexity, and approximations that can lead to violations of capacity constraints. This paper reports on the design and implementation of a new course allocation mechanism, Course Match, that enhances the Budish [2011] mechanism in various ways to make it suitable for 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 (Wharton, 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.

Can Agents 'Report Their Types'? An Experiment that Changed the Course Allocation Mechanism at Wharton

This paper reports on an experimental test of a new market design that is attractive in theory but makes the common and potentially unrealistic assumption that “agents report their type”; that is, that market participants can perfectly report their preferences to the mechanism. Concerns about preference reporting led to a novel experimental design that brought real market participants’ real preferences into the lab, as opposed to endowing experimental subjects with artificial preferences as is typical in market design. The experiment found that market participants were able to report their preferences “accurately enough” to realize efficiency and fairness benefits of the mechanism even while preference reporting mistakes meaningfully harmed mechanism performance. The experimental results persuaded the Wharton School to adopt the new mechanism and helped guide its practical implementation.This paper reports on an experiment conducted at the Wharton School of the University of Pennsylvania, testing a new mechanism for matching students to schedules of courses. The experiment compared Budish’s (2011) approximate competitive equilibrium from equal incomes (CEEI) to the incumbent, a fake-money auction used by Wharton and numerous other professional schools. CEEI outperformed the auction on quantitative measures of efficiency and fairness and qualitative measures of perceived strategic simplicity and student satisfaction. The experiment succeeded in the Roth (1986) sense of “whispering in the ears of princes”, persuading the Wharton administration to adopt CEEI and guiding real-world implementation.

Primary-market auctions for event tickets: eliminating the rents of "Bob the broker"

Economists have long been puzzled by event-ticket underpricing: underpricing reduces revenue for the performer, and encourages socially wasteful rent-seeking by ticket brokers. Why not use an auction to set price correctly? This paper studies the recent introduction of auctions into the event-ticket market by Ticketmaster. By combining primary-market data from Ticketmaster with secondary-market resale value data from eBay, we show that Ticketmasters auctions work: the auctions substantially improve price discovery, roughly double performer revenues, and, on average, nearly eliminate the arbitrage profits associated with underpriced tickets. The data thus suggest that auctions can eliminate the speculator rent-seeking that has been associated with this market since the 19th century, and that seems to have exploded in volume in the 21st century.

The Economic Limits of Bitcoin and the Blockchain

The amount of computational power devoted to anonymous, decentralized blockchains such as Bitcoins must simultaneously satisfy two conditions in equilibrium: (1) a zero-profit condition among miners, who engage in a rent-seeking competition for the prize associated with adding the next block to the chain; and (2) an incentive compatibility condition on the systems vulnerability to a “majority attack�?, namely that the computational costs of such an attack must exceed the benefits. Together, these two equations imply that (3) the recurring, “flow�?, payments to miners for running the blockchain must be large relative to the one-off, “stock�?, benefits of attacking it. This is very expensive! The constraint is softer (i.e., stock versus stock) if both (i) the mining technology used to run the blockchain is both scarce and non-repurposable, and (ii) any majority attack is a “sabotage�? in that it causes a collapse in the economic value of the blockchain; however, reliance on non-repurposable technology for security and vulnerability to sabotage each raise their own concerns, and point to specific collapse scenarios. In particular, the model suggests that Bitcoin would be majority attacked if it became sufficiently economically important — e.g., if it became a “store of value�? akin to gold — which suggests that there are intrinsic economic limits to how economically important it can become in the first place.

Can “Complex” Market Designs Make it from Theory to Practice? Changing the Course Allocation Mechanism at Wharton

Budish (2011) proposes a new mechanism for the problem of combinatorial assignment -e.g., assigning students to schedules of courses -called approximate competitive equilibrium from equal incomes (CEEI). While the CEEI mechanism satisfies attractive properties of efficiency, fairness, and incentives, it is “complicated” in several ways that one might reasonably wonder whether the theory could actually be implemented in the real world. To give just a few examples, agents are assumed to report their complete preferences over all possible schedules of courses, the mechanism is assumed to solve a high-dimensional approximate Kakutani fixed point problem, and all of the economic properties the mechanism satisfies involve approximations. While there is no perfect definition of a mechanism’s complexity, for a contrast consider the famous GaleShapley deferred acceptance algorithm, which is sufficiently simple to imagine implementing in practice that the medical profession actually did so some fifteen years before Gale and Shapley’s paper was even published. This talk reports on two papers (Budish and Kessler, 2015; Budish et al., 2015) that helped bring this complex market design theory to successful implementation in practice, at the Wharton School at the University of Pennsylvania. The first paper, joint with Judd Kessler, reports on experiments conducted at Wharton to test the CEEI mechanism. In addition to showing that the CEEI mechanism improved the efficiency and fairness of the allocation, the experiment also serves as a roadmap for other market design researchers seeking to test complex mechanisms in practice. The second paper, joint with Gerard Cachon, Judd Kessler, and Abe Othman, reports on the computational and economic engineering work involved in actually implementing the mechanism in practice. This involved modifications of the CEEI mechanism to deal with some of the issues caused by approximations in the theory, and a computational procedure that performs a massive parallel heuristic search, solving billions of mixed-integer programs along the way, to output an approximate competitive equilibrium in the fake-money economy for courses.

Strategyproofness for “Price Takers” as a Desideratum for Market Design

We distinguish between two ways a mechanism can fail to be strategyproof. A mechanism may have manipulations that persist with market size (first-order manipulations); and, a mechanism may have manipulations that vanish with market size (second-order manipulations). We say that a non-strategyproof mechanism is strategyproof in the large (SP-L) if all of its manipulations vanish with market size; that is, if it is strategyproof for “price takers”. We put “price takers” in quotes because our notion is not limited to mechanisms that explicitly use prices. Our main result is that, given a mechanism with Bayes-Nash or complete information Nash equilibria, there exists a prior free mechanism that is SP-L and that coincides exactly with the original mechanism in the limit. It coincides approximately in large finite markets, with exponential rate of convergence. Thus, while strategyproofness often severely limits what kinds of mechanisms are possible, for our class of problems SP-L does not, and hence may be a useful second-best. We illustrate our concepts with examples from single-unit assignment, multi-unit assignment, matching and auctions.