Lottery over joint action profiles (${\rho }_{ij}\ge 0,{\sum }_{ij}{\rho }_{ij}=1$)

A probability distribution over joint action profiles implements a convex combination of their two-player payoff vectors. The attainable expected payoffs form the convex hull of the pure joint payoffs. The distribution need not factor into independent mixed strategies, and a freely enforceable lottery need not satisfy the incentive constraints of a Nash equilibrium.