Question

Consider the public good game studied in class where there are just 2 players, i and j, but the players have different endowments, ωi and ωj. The payoff to player i is written as: πi = ωi - xi + β(xi + xj) where xi (xj) is player i's (player j's) contribution to the public good, and 1/2 < β < 1.

a. Suppose that player i's endowment is greater than player j's endowment, i.e., ωi > ωj. What is the Nash equilibrium contribution of players i, j, in this case? Does it differ from the case studied in class where both players have the same endowment, ωi = ωj = ω? Explain/justify your answer.

b. Now consider a rather different version of the public good game where β = 1, and player i's payoff is given by:
πi = ωi - xi + (xi + xj) + (ωi - xi)(xi + xj) = ωi + xj + (ωi - xi)(xi + xj). The last term in the payoff function is an interaction effect, wherein player i's payoff from the group total is also weighted by how much player i kept in his private account. Player j's payoff is similar, but with j's replacing the i's and vice versa.

What is the Nash equilibrium contribution of players i, j, in this case? Suppose again that player i's endowment, ωi > ωj. Does one player contribute more than the other? Explain/justify your answer. You may further assume that ωj > 2ωi.

Answer 1

In the public good game with different endowments, the Nash equilibrium contribution of players i, j will depend on their relative endowments. When player i's endowment is greater than player j's endowment, player i will contribute less compared to the case where both players have the same endowment. In a modified version of the game with an interaction effect, player i will contribute more than player j due to their higher endowment.

Step-by-step explanation:

In the public good game with different endowments, the Nash equilibrium contribution of players i, j can be determined by finding the best response of each player given the other player's contribution. In the case where player i's endowment is greater than player j's endowment (ωi > ωj), player i will contribute less than player j. This is because player i values the public good less than player j, as player i has a higher endowment and therefore has less to gain from the public good.

In the modified version of the public good game with β = 1, the Nash equilibrium contribution of players i, j can also be found by analyzing their best responses. Given player i's higher endowment (ωi > ωj), player i will contribute more than player j. This is because the last term in player i's payoff function creates an interaction effect that rewards player i for keeping more in their private account. Player j's payoff is similar, but with the roles reversed. Therefore, player i will contribute more in this case.