Approximate Competitive Equilibrium Research Articles

Can Market Participants Report Their Preferences Accurately (Enough)?

In mechanism design theory it is common to assume that agents can perfectly report their preferences, even in complex settings in which this assumption strains reality. We experimentally test whether real market participants can report their real preferences for course schedules “accurately enough” for a novel course allocation mechanism, approximate competitive equilibrium from equal incomes (A-CEEI), to realize its theoretical benefits. To use market participants’ real preferences (i.e., rather than artificial “induced preferences” as is typical in market design experiments), we develop a new experimental method. Our method, the “elicited preferences” approach, generates preference data from subjects through a series of binary choices. These binary choices reveal that subjects prefer their schedules constructed under A-CEEI to their schedules constructed under the incumbent mechanism, a bidding points auction, and that A-CEEI reduces envy, suggesting subjects are able to report their preferences accurately enough to realize the efficiency and fairness benefits of A-CEEI. However, preference-reporting mistakes do meaningfully harm mechanism performance. One identifiable pattern of mistakes was that subjects had relatively more difficulty reporting cardinal as opposed to ordinal preference information. The experiment helped to persuade the Wharton School to adopt the new mechanism and helped guide aspects of its practical implementation, especially around preference reporting. This paper was accepted by Yan Chen, decision analysis.

Fair Allocation of Indivisible Goods: Improvement

We study the problem of fair allocation for indivisible goods. We use the maximin share paradigm introduced by Budish [Budish E (2011) The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. J. Political Econom. 119(6):1061–1103.] as a measure of fairness. Kurokawa et al. [Kurokawa D, Procaccia AD, Wang J (2018) Fair enough: Guaranteeing approximate maximin shares. J. ACM 65(2):8.] were the first to investigate this fundamental problem in the additive setting. They showed that in delicately constructed examples, not everyone can obtain a utility of at least her maximin value. They mitigated this impossibility result with a beautiful observation: no matter how the utility functions are made, we always can allocate the items to the agents to guarantee each agent’s utility is at least 2/3 of her maximin value. They left open whether this bound can be improved. Our main contribution answers this question in the affirmative. We improve their approximation result to a 3/4 factor guarantee.

The Review of Methods for Assignment of Elective Courses at Universities

Abstract In this paper we present a review of matching algorithms that are used for matching students with elective courses at universities. This is an example of a market where price mechanism cannot be used to determine the equilibrium allocation. In the Random Serial Dictatorship students select courses based on their position in a random queue. This mechanism is not ex post Pareto-efficient and its drawback is overcome in the Probabilistic Serial Assignment, although this mechanism is not strategy-proof. In the auction mechanism, students’ bids for courses do not represent their true preferences, since bids depend on their beliefs about the popularity of courses. The efficient allocation is obtained when Deferred Acceptance Algorithm is used - where the priority of students is determined based on their bids. Harvard draft mechanism is based on changing the order of students in the random queue from one round to another and this mechanism is not strategy-proof; however, it is, by all means, better for students than Random Serial Dictatorship. The Wharton Business School mechanism is based on the calculation of approximate competitive equilibrium.

Inapproximability of Nash Equilibrium

We prove that finding an $\epsilon$-approximate Nash equilibrium is $\mathsf{PPAD}$--complete for constant $\epsilon$ and a particularly simple class of games: polymatrix, degree 3 graphical games, in which each player has only two actions. As corollaries, we also prove similar inapproximability results for Bayesian Nash equilibrium in a two-player incomplete information game with a constant number of actions, for relative $\epsilon$-well supported Nash equilibrium in a two-player game, for market equilibrium in a nonmonotone market, for the generalized circuit problem defined by Chen, Deng, and Teng [J. ACM, 56 (2009)], and for approximate competitive equilibrium from equal incomes with indivisible goods.

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 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.

Changing the Course Allocation Mechanism at Wharton

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.

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.

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 computation of approximate competitive equilibrium is PPAD-hard

Arrow and Debreu showed in 1954 that, under mild conditions, a competitive economy always has an equilibrium. In this paper, we show that, given a competitive economy that fully respects all the conditions of Arrow–Debreu's existence theorem, for any positive constant h > 0 , it is PPAD-hard to compute a 1 n h -approximate competitive equilibrium.

Asymptotical behavior of finite and possible discontinuous economies

We consider a finite economy in which the data are depending on an exogenous parameter and the utilities satisfy a condition, previously introduced by the authors and called sequential pseudocontinuity, weaker than sequential lower semicontinuity and than sequential upper semicontinuity. We show that the economy has a nice asymptotical behavior, that is: for any sequence (xn)n of exogenous parameters converging to xo, any convergent subsequence of a sequence of approximate competitive equilibria of \({(\mathcal{E}(x_n ))_n}\) converges to an exact equilibrium of \({\mathcal{E}(x_o )}\) .

Dissertation Abstract: Decision-Theoretic Models of Market Equilibration

There has been a surprising dearth of decision-theoretic approaches to market equilibration within the extended Arrow-Debreu-McKenzie framework. While game theory is replete with learning and evolutionary models in which Nash equilibria are not necessarily the only stable states of a game, general equilibrium theorists have generally been content to accept the attainment of competitive equilibrium as a matter of faith. The first two chapters of this dissertation advance the study of behaviorally plausible models of market equilibration. First, we develop the epsilon-intelligent competitive equilibrium algorithm. According to this model, the actions of minimally sophisticated agents based on local information will lead an exchange economy to approximate competitive equilibrium in a larger set of economics than Walras’ tatonnement. The algorithm also supports a behavioral interpretation of Negishi’s existence proof of competitive equilibrium. The second chapter is an empirical analysis of laboratory markets designed to test the extent to which human behavior is consistent with the algorithm’s behavioral restrictions, and suggest alternative hypotheses. The chief finding is that while subjects sufficiently sophisticated to consistently secure competitive utility for themselves exist, the majority are satisficers who follow small modifications of a simple utility-improvement rule. While general equilibrium theory has lacked a behaviorally plausible foundation of price equilibration, behavioral economic models typically ignore general equilibrium implications. In the third chapter of this dissertation, I present the surprising result that loss aversion and diminishing sensitivity for gains imply a greater redistribution of endowments than when otherwise identical preferences are reference-independent. The result is surprising because loss aversion was developed in part to account for the status quo bias, whereby people tend to value a good more when it is in their possession than when it is not. One might reasonably suspect trade to be inhibited by this bias. The counter-intuitive result is driven by the fact that, given the axiomatization of loss aversion and diminishing sensitivity for gains, the acquisition of some quantity of a good increases one’s taste for that good, thus perpetuating a taste for more trade.

Combination Exchange Mechanisms for Efficient Bandwidth Allocation

The size, scale and multiple ownership of communication network resources makes it important to consider an economic framework wherein we can investigate the efficiency of network operation taking agents' incentives into account. Such a framework has been considered in the design and analysis of pricing mechanisms to regulate congestion and share bandwidth over short time scales. We consider time scales of a few months over which owners of communication links lease bandwidth to network service providers. As is well-known, economic efficiency is related to how close an allocation is to a competitive equilibrium. We first show that achieving economic efficiency through a market mechanism depends on network topology. We then show that in finite networks a competitive equilibrium may not exist. But a competitive equilibrium does exist in an idealized continuum model, in which all agents are infinitesimal compared with the size of the network. This suggests that approximate competitive equilibria with good performance may be attainable in real networks. We finally introduce a market mechanism called the combinatorial seller's bid double auction whose outcome, in the continuum model, is a competitive equilibrium. 1. Introduction. Communication networks have increased in scale and hetero- geneity. There are multiple network owners and operators, each with their own hetero- geneous endowments and privately known cost and revenue models. So, an allocation of network resources efficient from an engineering perspective need not be realized in the market. Hence, it is useful to find economic mechanisms that result in efficient resource allocations among selfish agents. The efficiency of bandwidth allocation among competing users in a communica- tion network has been studied within an economic framework before. The previous research literature focuses attention on congestion that results when aggregate user demand exceeds capacity, and proposes usage-dependent pricing as a method for con- trolling congestion (23, 24). However, several difficulties must be faced in coming up with a good price-adjustment mechanism. First, congestion occurs over a very short time scale: Aggregate traffic is bursty and exceeds nominal capacity for periods of a few seconds (10). So the pricing mechanism must react very quickly to changes in aggregate demand. Second, the 'correct' congestion price depends on how much users are willing to pay for the service, which is not known. This has prompted invention of distributed price iteration algorithms that converge to the 'correct' price (16, 22). These algorithms are not practical but permit a re-interpretation of TCP-like proto-

A Simple Decentralized Institution for Learning Competitive Equilibrium

The epsilon-intelligent competitive equilibrium algorithm is a decentralized alternative to Walrus' tatonnement procedure for markets to arrive at competitive equilibrium. We build on the Gode-Spear-Sunder zero-intelligent algorithm in which random generation of bids and offers from agents' welfare-enhancing opportunity sets generates Pareto optimal allocations in a pure exchange economy. We permit agents to know if they are subsidizing others at such allocations, and to veto such allocations, restricting the subsequent iterations of the algorithm only to those trades that are both Pareto-improving and provide strictly greater wealth, and ultimately utility, for such agents. In this simple institution actions of minimally intelligent agents based on local information can lead the market to approximate competitive equilibrium in a larger set of economies than the tatonnement process would allow. This helps address one of the major shortcomings of the Arrow-Debreu-McKenzie model with respect to the instability of tatonnement in an open set of economies. It also addresses the behavioral critique of mathematically derived equilibria for the inability of cognitively-limited humans to maximize. The proof of convergence of the algorithm presented here also provides a way of showing the existence of competitive equilibrium for monotonic, convex exchange economies with heterogeneous agents and many goods without application of a fixed-point theorem.

Equal yield tax alternatives: General equillibrium computational techniques

In an earlier paper [Shoven and Whalley (1973)] we have described a technique for computing an approximate competitive equilibrium for an economy with ad valorem consumer and producer taxes. A general equilibrium model was used in which producers maximize profits and consumer demands satisfy conventional budget constraints. Each individual consumer or producer is faced with a given set of tax rates on the various commodities. These rates may be different for each commodity and each producer or consumer. Thus, discriminatory taxes including those which introduce extreme distortions may be considered. Equilibrium in these circumstances requires not only that the supply and demand for commodities be equated, but also that no producer do better than break even after payment of taxes. It is a further property of such an equilibrium that the government tax collections match its expenditure. In a subsequent paper [Shoven and Whalley (1974)] the approach was extended to a many country model with tariffs applying to trade flows. In this paper we explicitly recognize that in comparing alternative tax structures, policy makers consider proposals which will generate a particular amount of revenue. Often, the relevant evaluation is between equal yield tax alternatives. Thus, we have modified our earlier computational approach so