Finding Any Nontrivial Coarse Correlated Equilibrium Is Hard

One of the most appealing aspects of the (coarse) correlated equilibrium concept is that natural dynamics quickly arrive at approximations of such equilibria, even in games with many players. In addition, there exist polynomial-time algorithms that compute exact (coarse) correlated equilibria. In light of these results, a natural question is how good are the (coarse) correlated equilibria that can arise from any efficient algorithm or dynamics. In this paper we address this question, and establish strong negative results. In particular, we show that in multiplayer games that have a succinct representation, it is NP-hard to compute any coarse correlated equilibrium (or approximate coarse correlated equilibrium) with welfare strictly better than the worst possible. The focus on succinct games ensures that the underlying complexity question is interesting; many multiplayer games of interest are in fact succinct. Our results imply that, while one can efficiently compute a coarse correlated equilibrium, one cannot provide any nontrivial welfare guarantee for the resulting equilibrium, unless P=NP. We show that analogous hardness results hold for correlated equilibria, and persist under the egalitarian objective or Pareto optimality. To complement the hardness results, we develop an algorithmic framework that identifies settings in which we can efficiently compute an approximate correlated equilibrium with near-optimal welfare. We use this framework to develop an efficient algorithm for computing an approximate correlated equilibrium with near-optimal welfare in aggregative games.Comment: 21 page

Elastic constants from microscopic strain fluctuations

Fluctuations of the instantaneous local Lagrangian strain $\epsilon_{ij}(\bf{r},t)$, measured with respect to a static ``reference'' lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse- grained by averaging them within subsystems (of size $L_b$) of a system (of total size $L$) in the canonical ensemble. Using a simple finite size scaling theory we predict the behaviour of the fluctuations $$ as a function of $L_b/L$ and extract elastic constants of the system {\em in the thermodynamic limit} at nonzero temperature. Our method is simple to implement, efficient and general enough to be able to handle a wide class of model systems including those with singular potentials without any essential modification. We illustrate the technique by computing isothermal elastic constants of the ``soft'' and the hard disk triangular solids in two dimensions from molecular dynamics and Monte Carlo simulations. We compare our results with those from earlier simulations and density functional theory.Comment: 24 pages REVTEX, 10 .ps figures, version accepted for publication in Physical Review