Abstract
This chapter builds an evolutionary model of one-sided buyer auction market toexamine if the informational efficiency would still occur as a long run outcome. Here, each trader’s behavior is preprogramed with its own inherent and fixed probabilities of overpredicting, predicting correctly, and underpredicting the fundamental value of the asset. This chapter shows that, if each buyer’s initial wealth is sufficiently small relative to the market supply and if the variation in the asset’s random shock is sufficiently small, then as time gets sufficiently large, the proportion of time, that the asset price is arbitrarily close to the fundamental value, converges to one with probability one.
This chapter is based on my article published in the Journal of Financial Markets, 6, 163–197, 2003.
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Notes
- 1.
Sometimes, secondary markets do exist for commercial paper and private placements. But usually the transaction costs of setting up these markets have proven to be prohibitive.
- 2.
Other examples of auctions, where there is a fixed supply of an asset and no short sales, can be found in both the theoretical and experimental economics literature (e.g., Wilson (1979) and Forsythe et al. (1982)).
- 3.
Allen and Yannelis (2001) present a comprehensive discussions on various rational equilibrium outcomes in various differential information economies.
- 4.
Their results are partly due to the difference in the utility functions of both types of traders, the absence of wealth accumulation and the absence of wealth flows between the two types of traders.
- 5.
This one-sided buyer market with one unit of an asset being supplied each time period is also used in Blume and Easley (1992).
- 6.
This specification of beliefs or predictions is consistent with Grossman (1976, 1978), Figlewski (1978), and Hellwig (1980).
- 7.
For negative predictions not to occur, the lower bound for predictions is constrained to zero. That is b s t = maxz s + u s t, 0.
- 8.
Alternatively, one can assume different initial endowments of wealth for all traders. However, the same results with respect to convergence of the asset price will hold, provided that the initial endowment of wealth for all entering traders is bounded from above.
- 9.
For simplicity it is assumed that each trader potentially invests all of his or her wealth in each period. The results of this chapter would still hold if each trader spends only a fraction of his or her total wealth on trading activity. A smaller fraction could reflect more risk aversion on the part of the trader.
- 10.
It is apparent that, given the demand function, any trader’s wealth can never be negative. This is due to the fact that each trader begins with a positive level of initial wealth and short sales are not allowed.
- 11.
A related paper which examines the relationship between accuracy of agents’ predictions, wealth accumulation and the convergence of prices to the true value, is Sandroni (2000).
- 12.
That is, for trader t where b s t = p s , \({q}_{s}^{t} = \frac{{V }_{s-1}^{t}} {{\sum }_{{t}^{{\prime}}\in {M}_{s}}{V }_{s-1}^{{t}^{{\prime}}}}\left [1 -{\sum }_{{t}^{{\prime}}\in {A}_{s}}{q}_{s}^{{t}^{{\prime}} }\right ]\) where \({A}_{s} =\{ {t}^{{\prime}}\mid {b}_{s}^{{t}^{{\prime}} } > {p}_{s}\}\) and \({M}_{s} =\{ {t}^{{\prime}}\mid {b}_{s}^{{t}^{{\prime}} } = {p}_{s}\}\). It would not alter the results if any alternative method of allocating shares of the asset is used in this situation.
- 13.
In fact, there can be a positive probability of traders with other types of predictive behavior. For example, such trader types could enter with prediction behavior that imitates past successful traders (e.g., Lettau (1997)). Nevertheless, as long as the θ − Assumption holds, the presence of these types of traders would not disrupt convergence.
- 14.
The variation in random shock here refers to E(exp(ω t ) − 1)ω t > 0.
- 15.
For the simulations, as T grows, the computer (c.p.u.) time to complete the simulations grows in proportion to T 2. To conduct 100 simulations in a reasonable length of c.p.u. time, T was constrained to be 3000. However, for five simulations the time was extended to 5000. By time period 5000, on average across five simulations, \(\frac{\#\left \{s\leq 5000:\left \vert {p}_{s}-{z}_{s}\right \vert <0.10\right \}} {5000} \ = 0.64.\) This shows further convergence.
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Luo, G.Y. (2012). Evolution, Noise Traders, and Market Efficiency in a One-Sided Auction Market. In: Evolutionary Foundations of Equilibria in Irrational Markets. Studies in Economic Theory, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0712-6_6
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