An existence theorem for a Pareto equilibrium (Q1175148)

The author considers the following multicriteria game problem: Given are \(N\) players playing noncooperatively with their respective strategy sets \(X_ i\), \(i=1,\dots,N\). Minimize the payoff functions \((f^ i_ 1(x),\dots,f^ i_{k_ i}(x))\) of player \(i\) over the set of feasible outcomes \(x\in X\), where \(X=X_ 1\times X_ 2\times\dots\times X_ N\). He introduces the concept of a Pareto equilibrium for such game problems and proves a theorem on the existence of a Pareto equilibrium under some convexity and compactness assumptions.