I am having trouble understanding this coalitional game problem. It is stated below

Each of n factories draws water from the a lake and discharges waste into the same lake. Each factory requires pure water. It costs any factory kc to purify its water supply, where k is the number of factories which do not purify their water before discharging it into the lake. It costs any factory to treate its waste. assume c<=b<=nc.

How do I model this as a coalitional game with transferrable utility, under the assumption that v(S) of any coalition S is the highest payoff that S can guarantee, i.e. v(S) assumes that no factories treat their waste.