Question

Question 1 [40 pts] – Free Rider Problem –Game Theoretic Modeling

Consider two individuals who are deciding to pay for a public good or not. The value of the public good is 10 for each individual and the cost of the public good is 12 TL.

• If they both vote yes and agree to pay, they will share the cost equally and the public good is provided.
• If only one vote yes and agrees to pay, full cost will be borne by this individual and the public good will be provided.
• If they both opt out from paying by voting no, no funds will be collected and the public good will not be provided.

Given the valuation for the public good, cost and the cost sharing mechanism above and our knowledge about public goods this decision process is depicted as a simultaneous form game below.  [Note on notation: the first payoff in each cell corresponds to the payoff of the row player, Individual 1. The second payoff in each cell corresponds to the payoff of the column player, Individual 2.]

			Ind 2
				Vote Y	Vote N
Ind 1		Vote Y		4 , 4	-2 , 10
		Vote N		10 , -2	0 , 0

Modeling: – Using Game Theory

1. (5 pts) Public goods are non-rival and non-excludable. Check the payoffs (net benefit; benefit after costs) from each outcome and note that while writing the payoffs of the game these two characteristics of pure public goods is used. Refer to these characteristics first and explain how these characteristics are incorporated into the game.

2. (4 pts) Explain why this problem about the provision of a public good can be represented using game theory; say as opposed to the problem of an individual deciding to buy a t-shirt for himself or not.

One-Shot Game: Impossibility of reaching the cooperative outcome.

3. (2 pts) What is Individual 1’s best response to other player choosing to pay for the public good? Briefly explain.
4. (2 pts) Explain what a dominant action is and then indicate if there is a dominant action here for any player.

5. (2 pts) Solve for the Nash equilibrium of the game. Explain your work.

6. (5 pts) Evaluate the Nash equilibrium of the game. Is this equilibrium socially efficient? Explain by referring to the market failure we encounter with the public goods.

7. (5 pts) Can the two individuals just promise each other to vote yes and then sustain the promised cooperative outcome in this game that is played once? Show that each individual has an incentive to cheat and vote no instead when the other is sticking to its promise.

Answer 1

1.a. Non rival means that the use of a public good by one individual does not reduce the utility from the public good for other individuals. This is incorporated in the game model. See that when both say yes, they share the cost (12/2 = 6 each) and receive 10 benefit each and hence the pay off is 10-6 = 4 for both the players. The use of the public by Ind 1 does not reduce value for Ind 2.

Non excludability means a public good is available to every individual and no individual can be exempted from its use. In the model, see that when Ind 1 votes yes and Ind 2 votes no, Ind 1 pays the total cost of 12 and receives pay off 12-10 = -2 while even though Ind 2 who is not paying still gets to use the public good and thus receives pay off of 10.

a. This problem of provision of public good can be represented by a game because the decision of the availability of public good depends on what the public (here Individual 1 and 2) are choosing. Hence the availability of the public good and the utility every individual receives from it depends on the actions of other individuals. Say both individuals say no, the public good wont be provided. In contrast when you go to buy a t-shirt (a private good) there are no other buyers who influence your decision of buying the t-shirt by their actions and nor does your action of not buying the t-shirt affects the market for t-shirts. There is no game involved. The one buying the t-shirt would be the only player if it was a game model.

a. When Individual 2 chooses to pay for the public good, then individual 1 can vote yes and reveive 4 pay off or say no and earn 10 pay off. A rational player would surely choose no and earn a higher pay off.

b. Dominant action is that action which is the best action for a player given any action his opponent may take.

Now if Ind 1 votes Y, then best option for Ind 2 is to vote N (10>4) and if Ind 1 votes N, then best option for Ind 2 is to vote N (0>-2). Hence voting No is dominant action for Ind 2.

Similarly if Ind 2 votes Y, then best option for Ind 1 is to vote N (10>4) and if Ind 2 votes N, then best option for Ind 1 is to vote N (0>-2). Hence voting No is dominant action for Ind 1.

Voting No is dominant strategy for both players.

a. Nash Equilibrium is that outcome from which the individuals do not have incentive to deviate. From the previous answer on dominant strategies, it is clear that both players will choose their dominant strategies of voting no and nash equilibrium pay off will be 0,0. This is the Nash equilibrium as deviating from this strategy of voting no is risky and loss making for both players. If any individual chooses to vote yes, he receives lesser pay off than what he could have received by playing his dominant strategy.

a. The Nash equilibrium of the game shows that both individuals due to the risk of having to pay the full price by himself, opts to vote no and finally not receive the public good as a whole. This occurs because of the free rider problem faced in the provision of public goods. As both player want to receive higher pay off by free riding, the final result is that the market for the public good does not exist because of the free riding tendancy. This equilibrium is not socially efficient as there is no public good available in the market and the pay off for both individuals is 0,0.

a. Well, when both players promise and cooperate on voting yes, they both receive 4,4 pay off which is higher than the previously obtained Nash equilibrium. This will be socially optimum result as public good is available and the market exists and there are positive benefits to individuals.

It is true that there is high chance of the other player cheating (free riding) when one player is sticking to his promise. When one is sticking to vote yes, then the other can vote no and get a higher pay off of 10>4. This means the cheater avoids paying and avails himself the benefit of the public good as it is non excludable. The incentive to cheat is there as deviating from the promise of voting yes given higher pay off of 10 to the cheater (free rider).