(Pollute the lake) Each of n factories draws water from 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 that do not treat their waste before discharging it into the lake; it costs any factory b to treat its waste. Assume that c≤b≤nc.

a. Model this situation as a coalitional game under the assumption that the worth v(S) of any coalition S is the highest payoff that S can guarantee (that is, v(S) is the highest payoff of S under the assumption that none of the other factories treats its waste).

b. Find the conditions under which the game has a nonempty core and the conditions under which the core is a singleton.

c. Discuss the interpretation of the core of this game, taking into account that the definition of v(S) makes assumptions about the behavior of the players outside S.