Generalized equilibrium results for games with incomplete information.

*(English)*Zbl 0658.90104A noncooperative game with incomplete information is given by the following data: The game has n players. Each player i, \(i=1,...,n\), has a topological action space \(A_ i\) and obtains his information from a set of types \(T_ i\), which is equipped with a \(\sigma\)-algebra \({\mathcal T}_ i\). Write \(T=\prod^{n}_{i=1}T_ i\) with generic element \(t=(t_ 1,...,t_ n)\) etc. Let \(\eta\) be a probability on the product space (T,\({\mathcal T})\) which governs the random behavior of the information, with marginals \(\eta\). After learning his own type \(t_ i\in T_ i\) each player has to choose an action \(a_ i\in A_ i\). A behavioral strategy \(\delta_ i\) for player i is a transition probability from \((T_ i,{\mathcal T}_ i)\) to \((A_ i,{\mathcal B}(A_ i))\); denote by \(\Delta_ i\) the set of all behavioral strategies for i. The expected payoff \(E_ i:\) \(\Delta\) \(\to R\) for player i is
\[
E_ i(\delta_ 1,...,\delta_ n)=\int_{T}[\int_{A_ 1}...\int_{A_ n}U_ i(t,a)\delta_ 1(t_ 1;da_ 1)...\delta_ n(t_ n;da_ n)]\eta (dt),
\]
where \(U_ i:\) \(T\times A\to R\) is i’s utility function. A strategy n-tuple \(\delta^*\in \Delta\) is a Nash equilibrium if for all \(i=1,...,n:\) \(E_ i(\delta^*)\geq E_ i(\delta^*_ 1,...\delta_ i...,\delta^*_ n)\) for all \(\delta_ i\in \Delta_ i.\)

Under some mild measurability, continuity and boundedness assumptions on \(U_ i\) and \(\eta\) it is shown that a Nash equilibrium exists if \(A_ i\) is compact metric. Under an additional restriction for the admissible strategies, the result is also extended to certain noncompact action spaces.

The paper generalizes earlier existence results of P. R. Milgram and R. J. Weber [ibid. 10, 619-632 (1985; Zbl 0582.90106)] and J. W. Mamer and K. E. Schilling [ibid. 11, 627-631 (1986; Zbl 0625.90094)] by using the theory of weak convergence for transition probabilities, which is also recapitulated briefly.

Under some mild measurability, continuity and boundedness assumptions on \(U_ i\) and \(\eta\) it is shown that a Nash equilibrium exists if \(A_ i\) is compact metric. Under an additional restriction for the admissible strategies, the result is also extended to certain noncompact action spaces.

The paper generalizes earlier existence results of P. R. Milgram and R. J. Weber [ibid. 10, 619-632 (1985; Zbl 0582.90106)] and J. W. Mamer and K. E. Schilling [ibid. 11, 627-631 (1986; Zbl 0625.90094)] by using the theory of weak convergence for transition probabilities, which is also recapitulated briefly.

Reviewer: M.Nermuth