Arrow’s voting theorem of Economics

2011/03/08 · by · in Economics, Fun, Ideas, People

I was in the dining hall, and the TV there had some talking heads babbling about the stock market. I couldn’t really make out what they were saying, but it set the stage in my mind for some other thoughts. I was informed recently that the High Frequency Traders, are really just a natural response to the exchange incentives (HTF’ers get paid for volume created, so all they have to do is make sure they trade alot every day, and break even on any price differential). This really quite perturbed be, as I want a market that is more easily accessible by someone with my income level (slave-wage grad student). I also want a market that is more fair (whatever that might mean).

Deep in the recesses of my mind, I knew something about fairness. It is not always to be had. For example, there is Arrow’s Impossibility Theorem:

In short, the theorem proves that no voting system can be designed that satisfies these three “fairness” criteria:

  • If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  • If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs like X and Z, Y and Z, or Z and W change).
  • There is no “dictator”: no single voter possesses the power to always determine the group’s preference.

So, sometimes fairness isn’t achievable. In particular, Ken Arrow has proven it unattainable in a voting system. But, isn’t the stock market, really just a giant online voting system? where people casting their bets as dollars can be seen as people casting their votes as ballots? Doesn’t the market clearing algorithm have to ensure some fairness criteria akin to that listed above? I don’t really have time to look into the issue too deeply, but since Ken Arrow is a very esteemed economist, he may have already published something on the topic. At least, I would expect results concerning market behavior, though my idea concerning a proof of impossibility for a fair market clearing algorithm might be a bit too specific.

Then, if we accept the hypothesis that no fair market clearing algorithm exists, it is simply a natural state of affairs that some companies with inevitably ‘game’ the algorithms which are implemented. Perhaps the only ‘fair’ algorithm is to rotate among a collection of different clearing algorithms, so that the unfairness is amortized across each round of clearing. (This strategy might be problematic though, as I remember it was once possible to make a guaranteed winning at poker if the house rotated among different rules (5 card stud, Texan hold-em, etc) and you changed your betting strategy appropriately. This is actually a specific instance of a more general game theory result that it is sometimes possible, to make a guaranteed win out of two games of chance which, when either is played alone, are a guaranteed loss. I cannot at the moment remember what such a paring of games is called.)