TY - GEN

T1 - On computability of equilibria in markets with production

AU - Garg, Jugal

AU - Vazirani, Vijay V.

PY - 2014

Y1 - 2014

N2 - Even though 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. For markets with separable, piecewise-linear concave (SPLC) utilities and SPLC production, we obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria, and we further give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 [14]. 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 odd- ness 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 is practical. Also, 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.

AB - Even though 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. For markets with separable, piecewise-linear concave (SPLC) utilities and SPLC production, we obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria, and we further give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 [14]. 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 odd- ness 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 is practical. Also, 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.

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U2 - 10.1137/1.9781611973402.98

DO - 10.1137/1.9781611973402.98

M3 - Conference contribution

AN - SCOPUS:84902094574

SN - 9781611973389

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1329

EP - 1340

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery

T2 - 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

Y2 - 5 January 2014 through 7 January 2014

ER -