Sunday, February 27, 2005

SODA Outtakes: Market Equilibria

I know I know. By the time I am done with SODA outtakes, it will be time for SODA 2006. This is why we need more algorithms bloggers ! Machine learning blogs appear to be popping up faster than you can say 'expectation-maximization', but there is no corresponding growth spurt in algorithms. Sigh....

This outtake is much easier, because there is a well-written survey I can point to. The topic is market equilibria, or the problem of determining when equilibrium sets in when traders exchange good at various prices, and how fast this can be determined.

More formally, the problem is: given an initial allocation of goods to traders (a "supply" vector, where each "dimension" represents a single commodity), and a "preference" function for each trader (I have a high affinity for gold and cookies, you have an aversion to silver, but really like beer), determine a set of prices for goods, and a "demand" vector for each trader so that if each trader sells what they have and buys what they want (at the given prices), their preference is maximized.

Here, the preference function component for any commodity is a concave function of its amount. Note that you want a concave function in order to model "diminishing returns". In addition, having concave utilities helps ensure unique optimal solutions (because the average of two solutions has better utility).

There are also sanity conditions (you can only buy with whatever money you have from selling, and the total amount of goods is fixed - this does not model producer economies).

This SIGACT News survey by Codenotti, Pemmaraju and Varadarajan lays out the terrain of market equilbria very nicely, outlining the history, the techniques (some beautiful methods) and what is currently known.

It's a good example of an area that has been studied extensively (in economics) but where the computer science angle brings in questions that were not necessarily of concern earlier: issues of computational feasibility, efficiency, approximations that are characteristic of the algorithmic way of approaching a problem.

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