1. (Public Goods Game) Suppose that there are two people, Agent 1 and Agent 2 in...

1. (Public Goods Game) Suppose that there are two people, Agent 1 and Agent 2 in a town.
Assume that there is no street light in the town. To build a street light, someone should pay
costs and once it is built, everyone can enjoy the beneﬁt of street light as there is no way to
force not to use it. Once the street light is build, while Agent 1 has 10 payoﬀ, Agent 2 has 5
payoﬀ. If only one person paid for it, the cost of building a street light is 6. If both agents
paid, each person needs to pay only half of it, thus the cost in this case is 3. As a result, the
payoﬀ matrix of this public goods game is as follows.

	pay	not pay
pay	7,2	4,5
not pay	10,1	0,0

(a) What are BR1(Pay) and BR1(Not Pay)? And what are BR2(Pay) and BR2(Not Pay)?
(b) What is the Nash Equilibrium (NE)?
(c) Suppose that there are many agents who have the same preference of Agent 2. In that
case, what would be the NE? Please explain this with Free Rider problem.

Solutions

Expert Solution

a) BR1(Pay) ie the best response function of agent 1 when agent 2 is paying

BR1(Pay)= not pay

BR1(not pay ) best response function of agent 1 when agent 2 is not paying

BR1(not pay )= pay

Br2(pay ) = not pay

BR2( not pay) = pay

b) Agent 1 would prefer to pay when agent 2 is not paying because in that case his utility is more. Then agent 1 prefers to not pay when agent 2 is paying because his payoff would be more in that case.

Similarly for agent 2

Agent 2 would prefer to pay when agent 1 is not paying because in that case his utility is more. Then agent 2 prefers to not pay when agent 1 is paying because his payoff would be more in that case.

So the nash equilibrium in this case is (pay, not pay ) and (not pay , pay )

c) If many agents have a preference similar to agent 2, then it implies that they have lesser payoff than agent 1. Also it implies that there would be a free rider problem in this case. Since street light is a public good , it would be provided when even 1 of the residents pay for it . So there is an incentive for the other agents to hide their true value and free ride when the good is provided.

This free rider problem sometines leads to market failure when all of the agents try to free ride and at the end no one ends up paying for the provision of public good which would make the payoffs (0,0) if no one pays which would be the nash equilibrium.

(you can comment for doubts )

Rahul Sunny answered 1 year ago

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