Funding Dilema (game)

[Jared Russell]

The first question when playing a game like this is to establish what your objective function should be. Normally in the prisoner's dilemma, the objective is to maximize your payoff without considering your opponent's. However, one could certainly make the argument that in the case of FIRST the objective should be to maximize the total amount of sponsorship for both teams without worrying about who gets what. Interestingly, examining the "worst case" payoff matrix for the game as posed yields what I believe to be the first mathematical proof of GP...

Here is the payoff matrix of IKE's game:

A shares, B doesn't - $1000 total
B shares, A doesn't - $1000 total
A shares, B shares - $800 total
A doesn't, B doesn't - $2 total

The first thing we need to mention is that the game has to be double-blind in order for there truly to be a dilemma. If one team can contact the other, then arranging a $1000 donation to one team that then gets shared 50/50 would be the optimal fair solution. But let's imagine we can't do this...

Clearly, one team sharing when the other does not yields the maximum total payoff. However, if you don't know what the other team is going to say, you can't be sure that you are arranging this scenario (i.e. you need to know what the other guy is going to say in order to say the opposite). So let's examine what happens if team A chooses either option...

If A won't share, the best possible outcome is that B decides to share ($1000 payoff). The worst possible outcome is that B doesn't share ($2 payoff). Assuming that B has a 50/50 chance of choosing either, the "expected value" (EV) of A's "don't share" decision is (1000+2)/2 = $501.

If A does share, the best possible outcome is that B does not ($1000 payoff). The worst possible outcome is that B shares as well ($800 payoff). With the same 50/50 assumption, the EV is now (1000+800)/2 = $900.

Clearly, in both the worst case ($2 vs. $800) and "average" case ($501 vs. $900), A is better off sharing (the best case is identical, $1000 either way). B would have the same payoff metrics for their own decision.

Thus, when maximizing the payoff to the whole, sharing is always the better decision. If both A and B are rational players, they would both share, and $800 is the result.

Now let's look at what would happen if our objective was to maximize our OWN payoff instead of the total.

The payoff matrix (for team A) is:
A shares, B doesn't - $0
B shares, A doesn't - $1000
A shares, B shares - $400
Neither shares - $1

If A shares, the best case is $400, the worst case is $0, and the EV is $200.

If A doesn't share, the best case is $1000, the worst case is $1, and the EV is $500.50.

In this scenario, not sharing is the best choice. So A and B, both being rational, choose not to share and each gets $1.

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Notice how when the objective is to maximize the "greater good", each team walks away with $400. But when the objective is to maximize your OWN payoff, each team only walks away with $1.

GP isn't just a good idea, it's mathematically optimal.