Rational Convex Programs and Efficient Algorithms for 2-Player Nash and Nonsymmetric Bargaining Games

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are “small”, then its solution can be found in strongly polynomial time. We also give non-succinct linear games whose solution can be found in strongly polynomial time.