On existence of stable and efficient outcomes in games with public and private objectives (Q1185766)

Let \(X_ i\) be the compact strategy set of player \(i\in N\), \(\rho_ i(x_ i)\) with \(x_ i\in X_ i\) the private payoff to \(i\), and \(\psi(x)=\psi(x_ 1,\dots,x_ i,\dots)\) a social payoff. The outcome- functions \(\rho_ i\), \(\psi\) are assumed to be continuous. Individuals derive utility from private and social payoffs according to \(u_ i(x)=F(\psi(x),\rho_ i(x_ i))\) where \(F\) is assumed to be continuous and monotonic. The question is which specifications of \(F\) ensure the existence of Pareto-optimal Nash equilibria. It is demonstrated that \(F\) admits an efficient Nash equilibrium if and only if it is of the form \(F(\psi,\rho)=\min\{\max\{\lambda_ 1(\psi),\lambda_ 2(\rho)\},\lambda_ 3(\rho)\}\). It is largely a matter of taste whether such utility specifications can be considered to be realistic. In as far as they seem to be (im-)plausible the theorem is an (im-)possibility result on the efficiency of Nash equilibria in economies with private and public goods.