Consider 2 identical players (i.e. i = 1, 2) with utility function:

Question

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 public good and is 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. Assume that b = 1/2 and c = 1. The players have two possible actions: to cooperate to the public good (C), or not to cooperate to the public good (NC). The players choose actions simultaneously and only one time. a) Write down the game in strategic form. b) What is the Nash equilibrium in pure strategies?

Answer 1

Player 1’s utility: π1=1/2(q1 + q2)-q1 Player 2’s utility: π2=1/2(q2 + q1)-q2 • When player 1 and player 2 both play (C) : π1C=1/2(1 + 1)-1 => π1C=1/2*2-1 => π1C=1-1=> π1C=0 By symmetry, π1C= π2C => π2C=0 • When player 1 plays (C), and player 2 plays (NC), or vice versa : πC=1/2(1 + 0)-1=> πC=1/2*1-1=> πC= -1/2

πNC=1/2(0+1)-0=> πNC=1/2*1-0=> πNC=1/2 • When player 1 and player 2 both play (NC): π1 NC=1/2(0+0)-0 => π1 NC= 0 By symmetry, π2 NC= 0

Answer 2

So, then if we are trying to answer the point b. and find what is the nash equilibrium in pure strategies to this exercise..how do we answer? because the way I see it, there is no nash equilibrium because:

• when Player 1 plays, he will prefer player 2 to play as well because 0>-1/2
• when Player 1 plays, player 2 will prefer to not play because 1/2>0

this shows that player 1 and 2 cannot meet, and cannot have the same preferences at the same time and this is the same for all 4 situations so there is no nash equilibrium. Is this right?