Market Games

Let us nov consider an economic application of the concepts we have developed. The situation we will describe is that of a "market game". In a market game, there is one consumption good, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/inline18.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> production goods and n players. Each player i has a production function uⁱ(x_l,x₂,...,xhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/inline19.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>), defined for all X_j > 0 and with values in IR. The quantity uⁱ(x_1,x₂,... .xhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/inline20.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>) represents the amount of the single consumption good that i can produce from inputs x₁,X₂,....xhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/inline21.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>. Each player i also has an endowment (aⁱ ₁,aⁱ ₂,...,aⁱ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/inline22.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>) of production goods. Each coalition produces as much of the consumption good as possible so that https://www.w3.org/1998/Math/MathML"> (1)       v ( S ) ⁢     = ⁢     max ⁡ { Σ i ∈ S     u i ( x i ) :   Σ i ∈ S     x i   = Σ i ∈ S         a i ⁢   a n d ⁢         x i ≥ 0 ⁢     f o r ⁢       a l l ⁢       i   } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/math66.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where xⁱ = https://www.w3.org/1998/Math/MathML"> ( x 1 i , x 2 i , ... , x l i ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/inline-math37.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and aⁱ is similarly defined.