SECOND PART ONLY PLEASE Global warming or the \'\'tragedy of the commons\". Ten

Question

SECOND PART ONLY PLEASE

Global warming or the ''tragedy of the commons". Ten countries are considering fighting global warming. Country i must decide to spend an amount x_1 with 0 lessthanorequalto x_i lessthanorequalto 1 to reduce its carbon emissions. The total benefits produced by theses expenditures is 6(x_1 +... x_10) and each country receives 1/10 of the benefits. This is a game with ten players 1, 2, 3,... 10. Write down the payoff for country I P_i(x_1...,x_10) = Benefits - Expenditures = ? Solve the game by showing that for each country x_i = 0 (i.e. spend nothing) is a dominating strategy.

Explanation / Answer

2. Let us first consider one country and try to show that spend nothing will be a dominant strategy. It will be that same for all other countries.

Let x1 > 0 be a different strategy for country 1.

Let x2, . . . , xn be any strategies for the other countries.

Then 1(0, x2, . . . , x10) 1(x1, x2, . . . , x10)

= ( 3/5 (0 + x2 + · · ·+ x10) 0) ( 3/5 (x1 + x2 + · · ·+ x10) x1)

= 0 – 3/5x1+x1

= 2/5x1 >0

Hence we can expect each country to spend nothing in order to fight Global warming where in each country might get a payoff of 0.

However if we assume that all countries somehow agree to spend $1 each to fight Global Warming, then each country’s payoff would be

3/5 (1+1+1+1+1+1+1+1+1+ 1) 1 = 30/5 1 =6-1= 5,

This scenario explains that each country would be better off. The fact being that each country would actually receive higher benefits for an expense of $1. But we all know that each country will only try to cheat each other.

The total benefits to all countries gets reduced by $5yi, when there is a reduction in country I’s expenditures by $yi. But it is also evident that it will reduce the benefits of the country i by $3/5yi.

The result makes it evident that the problem of Global Warming is a game of Prisoner’s Dilemma. This is also an example of a Public Goods Game where the individual share of payoffs is less than the cost of cooperating i.e when there is cooperation among the players, the value of payoffs of all players increases than the individual's cost of cooperating.

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