Direct Reduction of PPAD Linear Complementarity Problems to Bimatrix Games

It is well known that the problem of finding a Nash equilibrium for a bimatrix game (2-NASH) can be formulated as a linear complementarity problem (LCP). In addition, 2NASH belongs to the complexity class PPAD (Polynomialtime Parity Argument Directed). Based on the close connection between the graph associated with the Lemke algorithm, a vertex following algorithm for LCP, and the graph used to certify a problem as belonging to PPAD, it is possible to identify most LCPs processable by the Lemke algorithm (that is, problems for which the algorithm either finds a solution or provides a certificate that no solution exists) as belonging to PPAD. The discovery that 2-NASH is PPAD-complete means that every PPAD LCP can be reduced to a 2-NASH. However, the ingeniously constructed reduction (which is designed for any PPAD problem) is very complicated, so while of great theoretical significance, it is not practical for actually solving an LCP via 2-NASH, and it does not provide the potential insight that can be gained from studying the game obtained from a problem formulated as an LCP (e.g. market equilibrium). The main result of this paper is the construction of a simple explicit reduction of PPAD LCPs to symmetric 2-NASH problems. In particular, the cost matrix associated with the resulting game is constructed from the coefficient matrix of the LCP with one extra row and column (for LCPs with guaranteed solutions) or with two extra rows and columns (for the other PPAD LCPs). In addition, we show that the reduction is a bijection and discuss its implications for solving LCPs via 2-NASH and the potential for getting a deeper insight into these LCPs.