Leontief Economies Encode Nonzero Sum Two-Player Games Bruno Codenotti * Amin Saberi † Kasturi Varadarajan ‡ Yinyu Ye § Abstract We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Un- like bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria. We give a reduction from two-player games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash equilibria of any game are in one-to-one correspondence with the equilibria of the corresponding economy. Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies (where an equilibrium is guaranteed to exist) is at least as hard as finding a Nash equilibrium for two-player nonzero sum games. As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [17]. In particular, among other results, we show that it is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium. Perhaps more importantly, we also prove that it is NP- hard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known PPAD-completeness result of [30], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer’s Fixed Point Theorem. On the algorithmic side, we present an algorithm for finding an approximate equilibrium for some special Leontief economies, which achieves quasi-polynomial time whenever each trader does not demand too much more of any good than some other good. * IIT-CNR, Pisa, Italy. This work was done while visit- ing the Toyota Technological Institute at Chicago. Email: bruno.codenotti@iit.cnr.it. † Department of Management Science and Engineering, Stan- ford University, Stanford CA 94305. Email: saberi@stanford.edu. ‡ Department of Computer Science, The University of Iowa, Iowa City IA 52242. Email: kvaradar@cs.uiowa.edu. Partially supported by NSF CAREER grant CCR-0237431. § Department of Management Science and Engineering, Stanford University, Stanford CA 94305. Email: yinyu- ye@stanford.edu. 1 Introduction In the last few years, there has been a lot of interest in the computation of market equilibrium prices in an economy. In a very short time, polynomial-time algo- rithms have been developed for computing the prices for different special cases of this problem using tech- niques such as primal-dual [9, 21], auction algorithms [15, 16], and convex programming [29, 20, 32, 5, 4, 3]. However, it seems that all the markets for which these polynomial-time algorithms have been derived share a common property: their equilibrium set is convex. Roughly speaking, these results take advantage, ex- plicitly or implicitly, of settings where the market’s re- action to price changes is well-behaved either because the market demand retains some properties of the indi- vidual demands or thanks to the special structure of the individual utility functions (e.g., linear, Cobb-Douglas, CES in a certain range of its defining parameter, the elasticity of substitution). In this paper, we study economies in which the players have Leontief utility functions. A Leontief utility function describes the behavior of an extreme CES consumer, who desires goods in fixed proportions. These utility functions have a very nice combinatorial description and they come up in different contexts such as modeling congestion control mechanisms like TCP [22]. An economy with Leontief consumers can lead to very “expressive” market demand functions. 1 . The set of equilibria in these markets can be disconnected [18, 6]. Furthermore, no efficient algorithm is known for computing the equilibrium prices in these markets, except in the case of proportional endowments, where the set of equilibria is convex [5]. Our result shows that polynomial time algorithms handling the equilib- rium problem in such a scenario where multiple discon- nected equilibria can readily appear, would have an ex- tremely important computational consequence for bi- matrix games. In particular, we can show that any al- gorithm which computes an equilibrium price for a (spe- cial case of a) market with Leontief utility functions can 1 For instance, it is known that an economy with Leontief con- sumers can generate the Jacobian of any market excess demand at a given price (see [25], p.119). 1