Two firms (firm I and firm II) are considering entering a market. If only firm I enters, this firm obtains a value of 3 units. If only firm II enters, firm II obtains a value of 4. No entry means a value of 0. However, if both firms enter the common value will be 9. To see what each firm receives when they both enter the market, the firms agree to characterise the entry game as a cooperative game with two players where N={I,II}. v(I) is then the profit generated when only firm I enters and v(II) is the profit generated when only firm II is in the market. Suppose the firms agree that the profit of v(N)=9 will be divided according to the Shapley rule.
a) Calculate the Shapley value.
Please help
Since this is a simple example I will not use a shapley value formula explicitly.
Shapley value is an average marginal contribution of a player over all the possible different permutations (scenarios) in which coalition can be constructed. In case of two players, coalition can be formed as:
Scenario 1: Firm I first, Firm II second or
Scenario 2: Firm Firm II first, Firm I second.
Scenario 1: marginal value added of firm 1 is 3 units, since it enters an empty market. Marginal value added of firm 2 is 6 units, since the profit when only firm 1 is on the market is 3 units and when both firms are on the market joint profit amounts to 9 units.
Scenario 2: marginal value added of firm 2 is 4 units, since it enters an empty market. Marginal value added of firm 1 is 5 units, since the profit when only firm 2 is on the market is 4 units and when both firms are on the market joint profit amounts to 9 units.
Shapley value of firm 1 is therefore $(3+5)/2 =4$.
Shapley value of firm 2 is $(4+6)/2 =5$.