On the Complexity of Nash Equilibria and Other Fixed Points

We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, $\Gamma$, with 3 or more players, and given $\epsilon>0$, compute an approximation within $\epsilon$ of some (actual) Nash equilibrium. We show that approximation of an actual Nash equilibrium, even to within any nontrivial constant additive factor $\epsilon<1/2$ in just one desired coordinate, is at least as hard as the long-standing square-root sum problem, as well as a more general arithmetic circuit decision problem that characterizes P-time in a unit-cost model of computation with arbitrary precision rational arithmetic; thus, placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis $\{+,*,-,/,\max,\min\}$ with rational constants. We show that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for FIXP. The price equilibrium problem for exchange economies with algebraic demand functions is another FIXP-complete problem. We show that the piecewise linear fragment of FIXP equals PPAD. Many other problems in game theory, economics, and probability theory can be cast as fixed point problems for such algebraic functions. We discuss several important such problems: computing the value of Shapley's stochastic games and the simpler games of Condon, extinction probabilities of branching processes, probabilities of stochastic context-free grammars, and termination probabilities of recursive Markov chains. We show that for some of them, the approximation, or even exact computation, problem can be placed in PPAD, while for others, they are at least as hard as the square-root sum and arithmetic circuit decision problems.