On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201

Competitive Allocation of a Mixed Manna

We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads that everyone dislikes, as well as items that some like and others dislike. The seminal work of Bogomolnaia et al. [Econometrica'17] argue why allocating a mixed manna is genuinely more complicated than a good or a bad manna, and why competitive equilibrium is the best mechanism. They also provide the existence of equilibrium and establish its peculiar properties (e.g., non-convex and disconnected set of equilibria even under linear utilities), but leave the problem of computing an equilibrium open. This problem remained unresolved even for only bad manna under linear utilities. Our main result is a simplex-like algorithm based on Lemke's scheme for computing a competitive allocation of a mixed manna under SPLC utilities, a strict generalization of linear. Experimental results on randomly generated instances suggest that our algorithm will be fast in practice. The problem is known to be PPAD-hard for the case of good manna, and we also show a similar result for the case of bad manna. Given these PPAD-hardness results, designing such an algorithm is the only non-brute-force (non-enumerative) option known, e.g., the classic Lemke-Howson algorithm (1964) for computing a Nash equilibrium in a 2-player game is still one of the most widely used algorithms in practice. Our algorithm also yields several new structural properties as simple corollaries. We obtain a (constructive) proof of existence for a far more general setting, membership of the problem in PPAD, rational-valued solution, and odd number of solutions property. The last property also settles the conjecture of Bogomolnaia et al. in the affirmative

Solving equilibrium problems in economies with financial markets, home production, and retention

We propose a new methodology to compute equilibria for general equilibrium problems on exchange economies with real financial markets, home-production, and retention. We demonstrate that equilibrium prices can be determined by solving a related maxinf-optimization problem. We incorporate the non-arbitrage condition for financial markets into the equilibrium formulation and establish the equivalence between solutions to both problems. This reduces the complexity of the original by eliminating the need to directly compute financial contract prices, allowing us to calculate equilibria even in cases of incomplete financial markets. We also introduce a Walrasian bifunction that captures the imbalances and show that maxinf-points of this function correspond to equilibrium points. Moreover, we demonstrate that every equilibrium point can be approximated by a limit of maxinf points for a family of perturbed problems, by relying on the notion of lopsided convergence. Finally, we propose an augmented Walrasian algorithm and present numerical examples to illustrate the effectiveness of this approach. Our methodology allows for efficient calculation of equilibria in a variety of exchange economies and has potential applications in finance and economics

An Improved Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

We present an improved combinatorial algorithm for the computation of equilibrium prices in the linear Arrow-Debreu model. For a market with $n$ agents and integral utilities bounded by $U$, the algorithm runs in $O(n^7 \log^3 (nU))$ time. This improves upon the previously best algorithm of Ye by a factor of \tOmega(n). The algorithm refines the algorithm described by Duan and Mehlhorn and improves it by a factor of \tOmega(n^3). The improvement comes from a better understanding of the iterative price adjustment process, the improved balanced flow computation for nondegenerate instances, and a novel perturbation technique for achieving nondegeneracy.Comment: to appear in SODA 201

Proportional Dynamics in Exchange Economies

We study the Proportional Response dynamic in exchange economies, where each player starts with some amount of money and a good. Every day, the players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates the bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: - For linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. - We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. - For CES utilities in the substitute range $[0,1)$, the dynamic converges for all parameters. This answers an open question about exchange economies with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known. We also note that proportional response is a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.Comment: 25 pages, 6 figure