Sunday, October 25, 2009

Sornette, Why Stock Markets Crash, part 2--Herding and Minority Games

Game theory, an analytical tool in support of Adam Smith’s “invisible hand,” maintains that all the decisions we make are optimization problems. Didier Sornette disputes this claim. People, he argues, “have natural intuitive mechanisms—mind modules that serve them well in daily interchanges—enabling them to ‘read’ situations and the intentions and likely reactions of others without deep, tutored, cognitive analysis.” (Why Stock Markets Crash, p. 84) Even the ill informed and error prone may converge over time to Smith’s general equilibrium, where everything works out for the best. Then again, equilibrium may be upset.

One way to upset equilibrium is through herding. Sornette distinguishes multiple types of herding, but the most interesting is the informational cascade. An informational cascade occurs when “the existing aggregate information becomes so overwhelming that an individual’s single piece of private information is not strong enough to reverse the decision of the crowd. Therefore, the individual chooses to mimic the action of the crowd.” Sornette continues: “The two crucial ingredients for an informational cascade to develop are: (i) sequential decisions with subsequent actors observing decisions (not information) of previous actors; and (ii) a limited action space.” (p. 95) In the Internet bubble it was irrelevant to know that a particular business model was doomed (I still use my mousepad); the crowd was buying the stock, so the only sensible thing to do was to go along for the ride.

Since herding is such a prevalent phenomenon in markets (from analysts’ recommendations to momentum trading), if a trader lacks information it is optimal for her to imitate. She should look to the crowd, in essence “polling the members” to analyze how likely they are to behave in the future. Mathematically, she is taking part in an infinitely iterative loop. “The opinion si at time t of an agent i is a function of all the opinions of the other ‘neighboring’ agents at the previous time t – 1, which themselves depend on the opinion of the agent i at time t – 2, and so on.” (p. 103)

The problem with following the herd is that investors cannot all win at the same time, so here and there they have to take a minority view. In the simplest terms the investor would want to be in the minority when buying but in the majority during the holding period of the investment. Therefore, the relative impact of contrarian behavior on majority behavior is the ratio of the entry time to the holding time. Sornette speculates about the impact of this ratio on the intraday trader. “The large amount of works on minority games . . . suggests that changing one’s strategy often may be profitable in that situation. It also suggests that only when the information complexifies or when the number of traders decreases will the traders be able to make consistent profits. In contrast, the buy-and-hold strategies profit as long as the information remains simple, such as when a trend remains strong. The problem then boils down to exit/reverse before or at the reversal of the trend.” (p. 119) Sornette oversimplifies the profile of the intraday trader and overlooks the existence of intraday trends, but you get the idea.

Okay, intraday traders, you now have a new project—to learn about minority games, abstractions of the famous El-Farol Bar problem. Although I don’t understand why the total number of traders is relevant to the profitability of individual traders, my projects right now are out of control!

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