"Consider 2 identical players (i.e. i = 1, 2) with utility function: πi = b(qi + q-i) - cqi. Where qi is equal to one if player i contributes to the provision of a public good and zero if she does not, q-i is the sum of the contributions by all other players, b is the constant marginal benefit of contributing to the public good, and c is the cost of contributing to the good.
Suppose that player 2 will always play C (ie, will always cooperate). For what values of “b” will player 1 play C?"
My question is how is it possible to compute the values of "b" for which Player 1 will play C (provided that Player 2 always plays C) when cqi is not given nor is it any other way through which it can be calculated?
The notation is a bit confusing. Is his profit function $\pi_i=b \cdot (q_i+q_{-i})-c \cdot q_i$?
The marginal benefit of contributing is $b$, while the marginal cost of contributing is $c$, therefore contributing is optimal whenever $b>c$. Without further assumptions on $b$ or $c$, this is as far as you get. If I interpreted the notation correctly, this should be the optimal behavior independent of the choice of the other player.