Convex vNM–stable sets for a semi-orthogonal game. Part III: A small economy - uniqueness and multiple solutions

This paper constitutes the third part in a series dealing with vNM-Stable Sets, see J. Rosenmüller, "Convex vNM-Stable Sets for a Semi Orthogonal Game. Part I" [IMW Working Paper no. 483 (2013)], "Convex vNM-Stable Sets for a Semi Orthogonal Game. Part II" [IMW Working Paper no. 498 (2014)]. We consider (cooperative) linear production games with a continuum of players. The coalitional function is generated by r + 1 "production factors" (non atomic measures). r factors are given by orthogonal probabilities ("cornered" production factors) while factor r + 1^th is provided "across the corners" of the market. We consider convex vNM-Stable Sets of this game. Within this third part we exhaustively discuss the situation in a small but very significant economy or game. In tis situation, there are two corners of the market (factors represented by orthogonal probabilities), each of which being divided into two sectors of constant density of the non cornered commodity (a measure exhibiting mass across all corners of the market). For short, this is the 2 x 2-case, the foundations of which have been laid in Example 2.1 of Part I [IMW Working Paper no. 483 (2013)]. It turns out that, depending on the boundary conditions, we obtain two different scenarios. The first one reflects a situation that exhibits a unique vNM-Stable Set. The second scenario allows for a variety of vNM-Stable Sets including but not equal to the core of the game.