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Suppose that we have a set G of gin merchants and a set W of tonic water merchants. For Є G, the ith gin merchant has a litres of gin, and for j Є W, the jth tonic water merchant has ẞi litres of tonic water. Assume that α; and ẞj are real positive numbers for each i and j (also mind that, in general, a₁'s are different for different i and Bi's are different for different j). The merchants can form any coalition (i.e., any subset of GUW), and when producing the gin and tonic cocktail they should 1) use only gin and tonic water which are available to them, 2) use twice bigger volume of tonic water than (that of) gin. In other words, the ratio of tonic water to gin in the cocktail has to be 2 1. The value of any coalition S ≤ GUW is then defined as the maximal volume of gin and tonic cocktail which S can produce.1 (i) Let G = {G₁, G₂} and W = {W₁} with α ₁ = 7, α₂ = 8 and ẞ₁ = 20. Find the values of all coalitions of these three merchants using the rule described above, and write out the characteristic function v using the values of coalitions thus found. (ii) For the resulting game in characteristic function form, find its 0 - 1 reduced form u. Next, write out the system of constraints defining the core of u, draw this core as a part of the triangle S3 = {x E R³: ': X1 + X2 + X3 = 1, x₁ ≥ 0, X2 ≥ 0, ׳ ≥ 0} and describe it as the convex hull of a finite set of vertices (i.e., find these vertices using certain intersections of lines in your drawing). (iii) Let G and W be arbitrary sets. Write out the formula for the value v(S) of arbitrary coalition S≤ GUW (using the rule explained above), and prove that (G U W, v) is a game in characteristic function form. (iv) Is (GU W, v) essential? Is it constant-sum? Give your arguments.

Suppose that we have a set G of gin merchants and a set W of tonic water merchants. For Є G, the ith gin merchant has a litres of gin, and for j Є W, the jth tonic water merchant has ẞi litres of tonic water. Assume that α; and ẞj are real positive numbers for each i and j (also mind that, in general, a₁'s are different for different i and Bi's are different for different j). The merchants can form any coalition (i.e., any subset of GUW), and when producing the gin and tonic cocktail they should 1) use only gin and tonic water which are available to them, 2) use twice bigger volume of tonic water than (that of) gin. In other words, the ratio of tonic water to gin in the cocktail has to be 2 1. The value of any coalition S ≤ GUW is then defined as the maximal volume of gin and tonic cocktail which S can produce.1 (i) Let G = {G₁, G₂} and W = {W₁} with α ₁ = 7, α₂ = 8 and ẞ₁ = 20. Find the values of all coalitions of these three merchants using the rule described above, and write out the characteristic function v using the values of coalitions thus found. (ii) For the resulting game in characteristic function form, find its 0 - 1 reduced form u. Next, write out the system of constraints defining the core of u, draw this core as a part of the triangle S3 = {x E R³: ': X1 + X2 + X3 = 1, x₁ ≥ 0, X2 ≥ 0, ׳ ≥ 0} and describe it as the convex hull of a finite set of vertices (i.e., find these vertices using certain intersections of lines in your drawing). (iii) Let G and W be arbitrary sets. Write out the formula for the value v(S) of arbitrary coalition S≤ GUW (using the rule explained above), and prove that (G U W, v) is a game in characteristic function form. (iv) Is (GU W, v) essential? Is it constant-sum? Give your arguments.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.2: Norms And Distance Functions
Problem 10EQ
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Suppose that we have a set G of gin merchants and a set W of tonic water merchants. For Є G, the ith gin merchant has a litres of gin, and for j Є W, the jth tonic water merchant has ẞi litres of tonic water. Assume that α; and ẞj are real positive numbers for each i and j (also mind that, in general, a₁'s are different for different i and Bi's are different for different j). The merchants can form any coalition (i.e., any subset of GUW), and when producing the gin and tonic cocktail they should 1) use only gin and tonic water which are available to them, 2) use twice bigger volume of tonic water than (that of) gin. In other words, the ratio of tonic water to gin in the cocktail has to be 2 1. The value of any coalition S ≤ GUW is then defined as the maximal volume of gin and tonic cocktail which S can produce.1 (i) Let G = {G₁, G₂} and W = {W₁} with α ₁ = 7, α₂ = 8 and ẞ₁ = 20. Find the values of all coalitions of these three merchants using the rule described above, and write out the characteristic function v using the values of coalitions thus found. (ii) For the resulting game in characteristic function form, find its 0 - 1 reduced form u. Next, write out the system of constraints defining the core of u, draw this core as a part of the triangle S3 = {x E R³: ': X1 + X2 + X3 = 1, x₁ ≥ 0, X2 ≥ 0, ׳ ≥ 0} and describe it as the convex hull of a finite set of vertices (i.e., find these vertices using certain intersections of lines in your drawing). (iii) Let G and W be arbitrary sets. Write out the formula for the value v(S) of arbitrary coalition S≤ GUW (using the rule explained above), and prove that (G U W, v) is a game in characteristic function form. (iv) Is (GU W, v) essential? Is it constant-sum? Give your arguments.
Transcribed Image Text:Suppose that we have a set G of gin merchants and a set W of tonic water merchants. For Є G, the ith gin merchant has a litres of gin, and for j Є W, the jth tonic water merchant has ẞi litres of tonic water. Assume that α; and ẞj are real positive numbers for each i and j (also mind that, in general, a₁'s are different for different i and Bi's are different for different j). The merchants can form any coalition (i.e., any subset of GUW), and when producing the gin and tonic cocktail they should 1) use only gin and tonic water which are available to them, 2) use twice bigger volume of tonic water than (that of) gin. In other words, the ratio of tonic water to gin in the cocktail has to be 2 1. The value of any coalition S ≤ GUW is then defined as the maximal volume of gin and tonic cocktail which S can produce.1 (i) Let G = {G₁, G₂} and W = {W₁} with α ₁ = 7, α₂ = 8 and ẞ₁ = 20. Find the values of all coalitions of these three merchants using the rule described above, and write out the characteristic function v using the values of coalitions thus found. (ii) For the resulting game in characteristic function form, find its 0 - 1 reduced form u. Next, write out the system of constraints defining the core of u, draw this core as a part of the triangle S3 = {x E R³: ': X1 + X2 + X3 = 1, x₁ ≥ 0, X2 ≥ 0, ׳ ≥ 0} and describe it as the convex hull of a finite set of vertices (i.e., find these vertices using certain intersections of lines in your drawing). (iii) Let G and W be arbitrary sets. Write out the formula for the value v(S) of arbitrary coalition S≤ GUW (using the rule explained above), and prove that (G U W, v) is a game in characteristic function form. (iv) Is (GU W, v) essential? Is it constant-sum? Give your arguments.
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