There are n ≥ 3 doctors who have created a partnership. In each period, each doctor decides how hard to work. Let eti denote the effort chosen by doctor i in period t, and assume that eti can take 1 of 10 levels: 1, 2, ..., 10. The partnership's profit is higher when the doctors work harder. More specifically, total profit for the partnership equals twice the amount of total effort:

Profit=2×(et1 +et2 +...+etn)

A doctors payoff is an equal share of the profits, less the personal cost of effort,
which is assumed to equal the amount of effort; thus,

Doctor i's payoff = (1/n) × 2 × (et1 + et2 + ... + etn) - eti

This stage game is infinitely repeated, where each doctors payoff is the present value of the payoff stream and doctor is discount factor is δi.

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a. Assume that the history of the game is common knowledge. That is, in period t, the past choices of effort for all doctors over periods 1, ..., t - 1 are observed. Derive a subgame perfect Nash equilibrium in which each player chooses an effort e∗ > 1. (Take e∗ > 1 as a constant.)

b. Assume that the history of the game is not common knowledge. In period t, only the total effort (eτ1 +eτ2 +...+eτn), in period τ is observed by all players (for all τ ≤ t-1). (By the assumption of perfect recall, a player knows his own past effort, but you can ignore that information.) Find a subgame perfect Nash equilibrium in which each player chooses an effort e∗ > 1. (Take e∗ > 1 as a constant.)