Question

•Suppose two consumers can choose to contribute or free-ride on a public good that can clean up the air.

•To monetarize the benefit, if a consumer contributes to the public good, it generates 50% of the amount the consumer pay.

•For example, if $100 is contributed, the consumer can get back a benefit of $150.

•As the public good is non-exclusive, the benefit is evenly share between the two consumers.

•Form a game matrix and determine the Nash.

Answer 1

Let the two players be P1 and P2. Let the two strategies be 'Contribute' and 'Free Ride'

Let 'Contribute' mean a player gives $X to the public good.

Let 'Free Ride' mean a player gives nothing to the public good.

Hence for a total payment of Y, benefits are 1.5Y. and since they are evenly distributed, each player gets 0.75Y.

Let us break down this into cases:

Case 1: Both contribute X

In this case, each gets a benefit of 1.5X so net benefit/payoff = 1.5X - X = 0.5X

Case 2: One player contributes X and other free rides

So in the case that one player who contributes gets 0.75X benefits and hence net benefits = 0.75X - X = -0.25X

The other player who free rides has net payoff = 0.75X

Case 3: Both free ride

In this case as contribution = 0, net benefit/payoff = 0

The payoff matrix looks as follows:

*	*	Player	2
*	Payoff	Contribute	Free Ride
Player	Contribute	(0.5X, 0.5X)	(-0.25X, 0.75X)
1	Free Ride	(0.75X, -0.25X)	(0,0)

Let X = 1

*	*	Player	2
*	Payoff	Contribute	Free Ride
Player	Contribute	(0.5, 0.5)	(-0.25, 0.75)
1	Free Ride	(0.75, -0.25)	(0,0)

We can see that at (Free Ride, Free Ride); no player benefits from unilaterally changing their strategy to Contribute (0 > -0.25).

Hence (Free Ride, Free Ride) i.e (0,0) is the Nash equilibrium.