Externalities, Nonconvexities, and Fixed Points

By our main approximation result, for all z we haveη(z) ⊆ F ∗ (z) ⊆ N (z),and by the Tychonoff Fixed Point Theorem (see Theorem 17.56 in Aliprantis andBorder, 2006), each w ∗ -w ∗ -continuous approximating function, f n k(·), hasafixedpoint. Thus for k =1, 2,...,wehaveforsomez n k∈ Z, z n k= f n k(z n k). WLOG,assume that z n k→ z∗ .GiventhatF ∗ (·) is an CUSCO, by (72), we have (z ∗ ,z ∗ ) ∈ F ∗ .w ∗Thus,z ∗ ∈ F ∗ (z ∗ ) ⊂ N (z ∗ ).Rather than prove our fixed point result by continuous approximation, we couldinstead just apply Ward’s Fixed Point Theorem (see Theorem 3 in Ward, 1961). Inparticular, By our main approximation result, we have for any minimal Nash USCO ηof N the CUSCO, κ(η(·)), such that, κ(η(z)) ⊆ N (z). By Ward’s Theorem, becauseZ is a dendrite and κ(η(·)) has compact connected values in Z, κ(η(·)) has a fixedpoint. Thus,z ∗ ∈ κ(η(z ∗ )) ⊂ N (z ∗ ).For the sake of the reader we state Ward’s Theorem 3. First recall that a spacehas the fixed point property for a class of self-mappings if all self-mappings from thisclass and defined on the space in question have fixed points.Theorem 18 (Ward, 1961)If Z is a Peano continuum then the following statements are equivalent:(1) Z is a dendrite.(2) Z has the fixed point property for the class of upper semicontinuous, continuumvaluedmappings.(3) Z has the fixed point property for the class of continuous, closed-valued mappings.8 Fulfilled Expectations Nash Equilibria in Network FormationGames8.1 Consensus Beliefs and Strategic Network FormationReferring to the example in Subsection 2.6, example (2), the belief-parameterizedcollection of strategic form games over random networks is given byG := (P, {P i , Φ i (·),u i (·, (·, ·))} i∈N ) .We will assume that player i 0 s payoff function,u i (·, (·, ·)) : P×(P i ×P −i ) → R,42