Our Pollution Problem: Insights from Game Theory

Game theory can provide fascinating insights into many of our problems. One such popular instance is the ‘Pollution Game’. The problem is as follows : There are ‘n' countries. Each country incurs a cost of 3 units if they decide to invest in pollution control measures. However, if they choose to remain ignorant of the problem, then it is detrimental for all countries and everyone incurs a cost of 1 unit. So, what should be the Nash equilibrium of such a game?

The following assumptions need to be considered:

• All players are rational, i.e., they do what is best for themselves.
• They do not cooperate among themselves, every player takes a decision unilaterally.
• It is a complete information game, which means that all the above assumptions as well as the cost structure is common knowledge.

The last assumption of common knowledge applies to nested beliefs as well. If there is a piece of information ‘I' that is common knowledge, players A and B have access to it. Further, A knows that B knows ‘I’. Similarly, B knows that A knows that B knows ‘I’. And ad infinitum. Nested beliefs greatly influence decision making. Let’s get back to the problem at hand.

Observe that, if all countries choose to pollute, everyone incurs a cost of ‘n' units. However, if one country chooses to deviate, its cost goes up to (n+2) units while other countries incur (n-1) units. So, no country has an incentive to unilaterally deviate. Hence, the strategy n-tuple where all countries choose to pollute, is a Nash equilibrium. But, is this the only Nash equilibrium? A game can have multiple. Let’s find out.

What about the case where all countries invest in pollution control? Here, all countries incur a cost of 3 units. Does anyone have an incentive to deviate unilaterally? Yes. If country ‘C' chooses to deviate, its cost reduces to 1 unit while all other countries incur an increased cost of 4. Hence ‘C' will deviate the first chance it gets. So clearly this isn’t a Nash equilibrium.

Let us finally consider the general case where ‘k' countries invest in pollution control and the remaining don’t. Those who do, incur a cost of (n-k+3) while the rest incur a cost of (n-k). So any country with a pollution control strategy clearly has an incentive to deviate and incur a lower cost. Hence none of these strategy tuples are stable.

Turns out there is only one Nash equilibrium for the pollution game. And sadly, it means that the living world will slowly degrade and die.