Penny pinching yields pennies

January 26, 2012

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A Greek Professor of Economics has written up a thought-provoking series of articles on complexity fetishism. As part of his argument, Professor Varoufakis discusses the fallacy of using an analytic-synthetic approach to modeling economics. To highlight this problem, he presents an admittedly stylized example:

Imagine a table piled up with ten thousand gold sovereigns. Ann and Ben are given the opportunity to take them all as long as they respect a simple rule. They must approach the table one at a time and, on each approach, collect a single coin (with Ann kicking the process off). Crucially, they are also given the right to collect two coins on the understanding that, the moment either player exercises that right, the game ends with the remaining coins withdrawn. In short, if the person collecting at any visit takes two coins at once, it is ‘game over’ for both (though they can hold on to the coins they have already collected).

The conclusion is that though logical, following an analytic-synthetic approach yields a sub-optimal outcome, that Ann will end up with 2 coins, Ben with none, and 9998 coins unclaimed. This is a startling result, but it arises naturally from the presumed objective function: maximize the difference between Ann and Ben. This can be modeled as a simle linear programming problem.

Using the linprog package in R, we get the following resul (abridged)t:

> solveLP(c(1,-1),bvec,Amat, TRUE)
1 5001
2 4999

A more reasonable result occurs when a different objective function is defined. Rather than maximizing the difference, define the objective as: maximize the total number of coins accumulated.

This yields a more likely result, though not an even split. Why not? One interpretation is that even if the two players do not communicate, they can still only choose between  the two outcomes. Ann obviously has the upper hand, since she goes first. If she chooses to end the game immediately, then there is nothing to discuss. If she doesn’t, then Ben knows that her objective is the second function. Ben’s logic is similar, except that he already knows that Ann is choosing to go until the end. So unless Ben wants to end the game immediately (his objective function is to win outright), he chooses maximizing the accumulation of coins and accepts that he will end up with fewer coins than Ann. Knowing this, Ann can continue all the way up to the last move taking one coin. When there are two coins left, using the same greedy constraint as the original problem, she will clearly take them both as opposed to an even split. So Ann ends up with 5001 and Ben ends up with 4999.

> solveLP(c(1,1),bvec,Amat, TRUE)
1 5001
2 4999

Getting back to the article, I’m not wholly convinced that economic models and logic are flawed because of their reliance on the analytic-synthetic approach. A bigger concern is how do you define rational behavior? It all depends on the objective.

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