What is the optimal solution to this common resource pool problem?

Question

I have come across the following problem: There are 6 people taking part in an experiment. Each person may belong to either a 'blue' or 'green' group (no further detail on what group membership entails). Each person is given £2. They may choose to either keep all the money, or donate an amount of money (in increments of £0.1) to a common pool. After everyone has made their choice, the common pool is doubled and divided equally among each participant, regardless of the amount they put in.

What is the decision each person should make that results in the strongest Nash equilibrium?

To me the obvious decision would be that each person donates £2, as then then everyone walks away with £4. This is the maximum amount a person can walk away with. Every other choice will result in everyone walking away with less money.

Do you assume that each person thinks this through logically? Because if not, it might be the case that everyone thinks they will keep their £2 because that's guaranteed earnings, whereas if they put in their whole £2 but nobody else does, they walk away with only £0.66.

This is bugging me, because unlike in the prisoner's dilemma, where there is no solution that results in both people getting the best possible outcome (walking free), here there does seem to be an obvious best outcome.

Is my reasoning flawed in some way?

Answer 1

If each person donates $x_i$, each one gets $y_i=(2-x_i) + 2(\sum_{i=1}^N x_i)/N$. The total is $2 N + \sum_{i=1}^N x_i$ .A particular person gets

$$y_1= (2-x_1) + 2\frac{x_1+\sum_{i=2}^N}{N} = 2 - x_1 \frac{N-2}{N}+ 2\frac{\sum_{i=2}^Nx_i}{N}$$

Then, for each person, the optimum decision is $x_1=0$ (donate zero), and this does not depend on the decisions of the others. Hence this is the Nash equilibrium (which does no lead to the global optimum, of course; but the same happens in the prisoner dilemma).

Answer 2

The situation where everybody donates £2 to the common pool is not an equilibrium.

If one person decides to keep £0.3 for themself (and everyone else's actions are unchanged), they will get £0.1 less from the pool at the end, and still come out with a net advantage of £0.2.

This holds whenever anyone is giving anything to the common pool: Assuming everyone else's actions don't change, giving less will lead to a net win.

The only Nash equilibrium is where nobody gives anything to the common pool. (On the other hand, this is not a Pareto-optimal collective behavior, whereas any situation where at least 4 persons give all they have to the common pool would be, no matter how miserly the last 2 are).

This situation is a formalized instance of the tragedy of the commons.