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We study linear Fisher markets with satiation. In these markets, sellers have earning limits, and buyers have utility limits. Beyond applications in economics, they arise in the context of maximizing Nash social welfare when allocating indivisible items to agents. In contrast to markets with either earning or utility limits, markets with both limits have not been studied before. They turn out to have fundamentally different properties. In general, the existence of competitive equilibria is not guaranteed. We identify a natural property of markets (termed money clearing) that implies existence. We show that the set of equilibria is not always convex, answering a question posed in the literature. We design an FPTAS to compute an approximate equilibrium and prove that the problem of computing an exact equilibrium lies in the complexity class continuous local search ([Formula: see text]; i.e., the intersection of polynomial local search ([Formula: see text]) and polynomial parity arguments on directed graphs ([Formula: see text])). For a constant number of buyers or goods, we give a polynomial-time algorithm to compute an exact equilibrium. We show how (approximate) equilibria can be rounded and provide the first constant-factor approximation algorithm (with a factor of 2.404) for maximizing Nash social welfare when agents have capped linear (also known as budget-additive) valuations. Finally, we significantly improve the approximation hardness for additive valuations to [Formula: see text]. Funding: J. Garg was supported by the National Science Foundation [Grant CCF-1942321 (CAREER)]. M. Hoefer was supported by Deutsche Forschungsgemeinschaft [Grants Ho 3831/5-1, Ho 3831/6-1, and Ho 3831/7-1].
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An Algorithm for the Assignment Game Beyond Additive Valuations
The assignment game, introduced by Shapley and Shubik [1971], is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. There has since been substantial work studying various extensions of the assignment game. The first main area of extension is to imperfectly transferable utility, which is when frictions, taxes, or fees impede the transfer of money between agents. The second is with more complex valuation functions, in particular gross substitutes valuations, which describe substitutable goods. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in each of these two settings. However, these lines of work have been mostly independent, with no algorithmic results combining the two. Our main result is an efficient algorithm for computing competitive equilibria in a setting encompassing both those generalizations. We assume that sellers have multiple copies of each items. A buyer $$i$$'s quasi-linear utility is given by their gross substitute valuation for the bundle $$S$$ of items they are assigned to, minus the sum of the payments $$q_{ij}(p_j)$$ for each item $$j \in S$$, where $$p_j$$ is the price of item $$j$$ and $$q_{ij}$$ is piecewise linear, strictly increasing. Our algorithm combines procedures for matroid intersection problems with augmenting forest techniques from matching theory. We also show that in a mild generalization of our model without quasilinear utilities, computing a competitive equilibrium is NP-hard.
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- PAR ID:
- 10526932
- Publisher / Repository:
- 25th ACM Conference on Economics and Computation
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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