Question

1. (Level B) Each of n people chooses whether or not to contribute a fixed amount toward the provision of a public good. The good is provided if and only if at least k people contribute where 2 ≤ k ≤ n; if it is not provided, then the contributions are not refunded. Each person ranks the outcomes from best to worst as follows:

(i) Any outcome in which the good is provided and he/she does not contribute.

(ii) Any outcome in which the good is provided and he/she contributes.

(iii) Any outcome in which the good is not provided and he/she does not contribute.

(iv) Any outcome in which the good is not provided and he/she contributes. Answer the following questions.

(a) Formulate this situation as a strategic (normal-from) game.

(b) Find ALL pure strategy Nash equilibria of this game. Is there a Nash equilibrium in which more than k people contribute? One in which k people contribute? One in which fewer than k people contribute?

(c) Provide a definition of a Pareto optimal strategy profile. Which pure strategy equilibria of this game are Pareto optimal? Briefly explain.

Answer 1

It is given that the public good is provided only if at least k people contribute where 2 ≤ k ≤ n. And there is no refund for the amount contributed, even if the good is not provided.

Consider there are 2 players (k=2). Let C denotes the contribution provided by each person, where 0 ≤ C ≤ 1. For each of the 4 situations, we can rank them based on the payoffs. Let C denotes contribute and D denotes don't contribute.

(i) Any outcome in which the good is provided and he/she does not contribute (Payoff  =1 )

(ii) Any outcome in which the good is provided and he/she contributes.(Payoff = 1-C)

(iii) Any outcome in which the good is not provided and he/she does not contribute.(Payoff = 0)

(iv) Any outcome in which the good is not provided and he/she contributes.(Payoff = -C)

A) Therefore the game will look like,

	Player 2
C	D
Player 1	C	(1-C,1-C)	(-C,0)
D	(0,-C)	(0,0)

When both players contribute it will give both of them payoff 1-C. Good is provided and contribution C is also contributed. If both the players decide not to contribute the payoff will be 0 and the good won't be provided. If one of them contributes and the other not, then the good won’t be provided. However the one who contributed will face -C payoff, whereas the other will get a 0 payoff.

B) Nash equilibrium involves strategic choice that once made, provide no incentive for players to change their behaviour further. From game found in part A, the game has 2 pure strategy Nash equilibrium, (C, C) and (D,D). And there is no dominant strategy in this game. However if it is not a player game, to answer this question we have to consider various possibilities that can occur.

(i) Is there a Nash equilibrium in which more than k people contribute?

No. Consider n> x> k. Since for player i, there is an incentive to deviate from the position contribute. That is, for him the good will be provided even if one person doesn't contribute. So there will not be any Nash equilibrium.

(ii) Is there a NE where exactly k people contribute?

Yes. In this case, if at least one player decides not to Contribute, the public good won't be provided and simultaneously their payoff will fall (consider payoffs in part AA). So there exists Nash Equilibrium.

(iii) Is there NE where strictly less than k people contribute?

Yes, there is one Nash Equilibrium where no one contributes. Each of the player has an incentive to move to strategy D. Here the number of people is less than K. So anyway the public good will not be provided. So no one will be willing to Contribute in this case. NE exits at (D,D)

C) An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off. That is moving from Pareto optimal strategy profile will make at least one player worse off. So, if we consider the normal form of the game, which is given in part A, it achieves Pareto optimal strategy profile at (Contribute, Contribute). If one player moves from the strategy, other gets worse off. Though (D,D) is also a Pareto optimal strategy, deviating from ones strategy without a change in other player's strategy will reduce his/her payoff itself.