Multiplicative weights, equalizers, and P=PPAD
This resolves a long-standing open problem in computational complexity and game theory by proving P = PPAD, with implications for equilibrium computation across AI and economics.
The paper tackles the problem of computing equilibria in symmetric bimatrix games by showing that such games either have an interior symmetric equilibrium or a weakly dominated pure strategy, both of which can be detected and computed efficiently in polynomial time using linear programs. This result, combined with symmetrization methods and known PPAD-completeness results, implies P = PPAD, a major complexity theory breakthrough.
We show that, by using multiplicative weights in a game-theoretic thought experiment (and an important convexity result on the composition of multiplicative weights with the relative entropy function), a symmetric bimatrix game (that is, a bimatrix matrix wherein the payoff matrix of each player is the transpose of the payoff matrix of the other) either has an interior symmetric equilibrium or there is a pure strategy that is weakly dominated by some mixed strategy. Weakly dominated pure strategies can be detected and eliminated in polynomial time by solving a linear program. Furthermore, interior symmetric equilibria are a special case of a more general notion, namely, that of an "equalizer," which can also be computed efficiently in polynomial time by solving a linear program. An elegant "symmetrization method" of bimatrix games [Jurg et al., 1992] and the well-known PPAD-completeness results on equilibrium computation in bimatrix games [Daskalakis et al., 2009, Chen et al., 2009] imply then the compelling P = PPAD.