Leontief economies encode nonzero sum two-player games

Abstract

We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike 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 NPhard 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.

On the complexity of market equilibria with maximum social welfare

“…In comparison, a major result of Codenotti, Saberi, Varadarajan and Ye [4] states that finding the equilibrium price is NP-hard. However, the economy they defined may not have an equilibrium price at all.…”

“…In the model, there are equal number of traders and goods and trader i holds one unit of good i initially. It has been proven to be very useful in building the connection between bi-matrix games and market equilibria [4].…”

“…A recent surprising result by Codenotti, Saberi, Varadarajan and Ye [4] proved that computing an market equilibrium of Leontief economy with certain properties is NP-hard, for which economy equilibrium may not exist at all. Informally, in a Leontief economy, each agent demands a bundle of goods proportional to a constant vector a = (a 1 , a 2 , ..., a n )…”

“…Cases led to NPhardness include: (1) the uniqueness of market equilibria; (2) The existence of equilibrium with positive prices on a given set of goods; (3) The existence of equilibrium with at least (or at most) k goods positive priced (c.f. [4]). Moreover, such results can be extended to prove the NP-hardness of deciding the existence of an equilibrium for Leontief economy (which is known not to have an equilibrium for some cases).…”

Pricing for Fairness: Distributed Resource Allocation for Multiple Objectives

“…One example is the function i log x i , which is particularly important for two reasons: it is the objective maximized by TCP Vegas [20] which is an important version of TCP, and this also corresponds to the objective function required to solve Leontief economies [8].…”