Market game A set of merchants N1 = {1, 2} sell their wares in the market. Each merchant has an "initial endowment" (0, 1/2 ):
the first number represents the amount of gin the merchant has and the second number the amount of tonic the merchant has.
A second set of merchants N2 = {3, 4, 5} also sells in the same market, and each member of this set has an initial endowment of (1, 0). The total bundle available to a coalition S ⊆ N1 ∪ N2is
(|S ∩ N2|, (1/2)(|S ∩ N1|)),
where |R| denotes the number of members of coalition R.
Consumers will only buy cocktails containing equal parts gin and tonic.
The net profit from selling α units of cocktail is α dollars. Describe this situation as a coalitional game, and write down in detail the coalitional function.