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Follow the Money

Brian Hayes

Trading with Zeno

The various economic models discussed here differ in many details, but they can be classified in two broad families: those where the economy falls into a black hole, with one trader acquiring nearly everything of value, and those where the distribution of wealth reaches some stable equilibrium. What is the root of the difference?

Figure 4. Taxes and welfare . . .Click to Enlarge Image

Dragulescu and Yakovenko point out that transactions like those in the yard-sale model break time-reversal symmetry. For an example of a transaction rule that is reversible, consider the marriage-and-divorce model. Lumping together two fortunes and then splitting the sum is a process that works the same both forward and backward. If two traders report that they have $5 and $3 at one moment, and $7 and $1 at another moment, with a single transaction between these states, you can't tell which report is earlier and which is later. The lumping-and-splitting rule could apply in either direction. In the yard-sale model, on the other hand, the crucial step is taking the minimum of the two amounts, and reversing this operation cannot always restore the initial configuration. A transaction carried out under the yard-sale rule can go from the $5-and-$3 state to the $7-and-$1 state, but not the other way.

The irreversibility of the yard-sale rule acts as a kind of ratchet: Once the economy wanders into a state with an unbalanced distribution of wealth, it takes a long while to find its way out again. To see more clearly how the ratchet works, consider an even simpler model—an economy pared down to just two participants. Now the changing fortunes of either trader can be represented by a random walk along a line extending from zero to the total wealth available. All activity stops if the trader reaches either end of this line. A random walk that takes steps of uniform length is guaranteed to hit an end point sooner or later (a fate known as gambler's ruin). But this is not what is going on in the yard-sale model. There the steps are not of fixed size; because transactions are limited to the lesser of the trading partners' assets, the steps get smaller as the walk approaches either end point. If there is no smallest unit of currency, the random walk becomes a "Zeno walk," which spends most of its time in the neighborhood of an end point but never actually gets there.

To simplify the model still further, we can take a Zeno walk on the interval from 0 to 1, choosing to go left or right at random but letting the step size always be half the distance to the nearer end point (rather than a random fraction of this distance). If we begin at the point 11/42, the initial step size is 11/44. Suppose the first move is to the right, reaching the point 31/44. Now the step size is 11/48. If we turn back to the left, we do not return to our starting point but instead stop at 51/48. Where will we wind up after n steps? The probability distribution for this process has an intricate fractal structure, so there is no simple answer, but the likeliest landing places get steadily closer to the end points of the interval as n increases. This skewed probability distribution is the ratchetlike mechanism that drives the yard-sale model to states of extreme imbalance.

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