Unionized workers in landowner-worker game:-

Formulate as a coalitional game the variant of the landowner-worker game in which any group of fewer than n - 1 workers refuses to work with the landowner, and find its core. The core of the original game is closely related to the outcomes predicted by the economic notion of "competitive equilibrium".

Suppose that the landowner believes she can hire any number of workers at the fixed wage w (given as an amount of output), and every worker believes that she can obtain employment at this wage. If w ≥ 0 then every worker wishes to work, and if w ≤ f(n) - f(n - 1) the landowner wishes to employ all n - 1 workers. (Reducing the number of workers by one reduces the output by f(n) - f(n - 1); further reducing the number of workers reduces the output by successively larger amounts, given the shape of f .) If w > f(n) - f(n - 1) then the landowner wishes to employ fewer than n - 1 workers, because the wage exceeds the increase in the total output that results when the (n - 1)th worker is employed.

Thus the demand for workers is equal to the supply if and only if 0 ≤ w ≤ f(n) - f(n - 1); every such wage w is a "competitive equilibrium". A different assumption about the form of f yields a different conclusion about the core. Suppose that each additional worker produces more additional output than the previous one. An example of a function f with this form is shown in Figure 1. Under this assumption the economy has no competitive equilibrium: for any wage, the landowner wishes to employ an indefinitely large number of workers. The next exercise asks you to study the core of the induced coalitional game.