Pareto dominant

An outcome $s^*$ strongly Pareto dominates $s^\prime$ if:

An outcome $s^*$ weakly Pareto dominates $s^\prime$ if:

$s^*$ is strongly Pareto optimal if whenever $s^\prime$ weakly Pareto dominates $s^*$ , $\forall i\leq n\left[u_i(s^*)=u_i(s^\prime)\right]$ . That is, there is no strategy which provides at least as large a payoff to each player and a larger one to at least one. $s^*$ is weakly Pareto optimal if there is no $s^\prime$ such that $s^\prime$ strongly Pareto dominates $s^*$ .