Core of landowner-worker game:-

Check that no coalition can improve upon any action of the grand coalition in which the output received by every worker is nonnegative and at most f(n) - f(n - 1). (Use the fact that the form of f implies that f(n) - f(k) ≥ (n - k)(f(n) - f(n - 1)) for every k ≤ n.)

We conclude that the core of the game is the set of all actions of the grand coalition in which the output xi obtained by each worker i satisfies 0 ≤ xi ≤ f(n) - f(n - 1) and the output obtained by the landowner is the difference between f(n) and the sum of the workers' shares. In economic jargon, f(n) - f(n - 1) is a worker's "marginal product".

Thus in any action in the core, each worker obtains at most her marginal product. The workers' shares of output are driven down to at most f(n) - f(n - 1) by competition between coalitions consisting of the landowner and workers.

If the output received by any worker exceeds f(n) - f(n - 1) then the other workers, in cahoots with the landowner, can deviate and increase their share of output. That is, each worker's share of output is limited by her comrades' attempts to obtain more output.

The fact that each worker's share of output is held down by inter-worker competition suggests that if the workers were to agree not to join deviating coalitions except as a group then they might be better off. You are asked to check this idea in the following exercise.