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Article Excerpt
Summary. We examine whether a simple agent-based model can generate asset price bubbles and crashes of the type observed in a series of laboratory asset market experiments beginning with the work of Smith, Suchanek and Williams (1988). We follow the methodology of Gode and Sunder (1993, 1997) and examine the outcomes that obtain when populations of zero-intelligence (ZI) budget constrained, artificial agents are placed in the various laboratory market environments that have given rise to price bubbles. We have to put more structure on the behavior of the ZI-agents in order to address features of the laboratory asset bubble environment. We show that our model of "near-zero-intelligence" traders, operating in the same double auction environments used in several different laboratory studies, generates asset price bubbles and crashes comparable to those observed in laboratory experiments and can also match other, more subtle features of the experimental data.
Keywords and Phrases: Bubbles, Zero-intelligence traders, Double auction, Agent-based models, Experimental economics.
JEL Classification Numbers: D83, D84, G12.
1 Introduction
Experimental economists beginning with Smith, Suchanek and Williams (1988) have devised laboratory environments that generate asset price bubbles and crashes as evidenced by the behavior of inexperienced human subjects who are placed in these environments. The Smith et al. (1988) finding of price bubbles and crashes has been replicated in several other experiments and found to be robust to a number of modifications of the laboratory environment specifically aimed at eliminating bubbles. (3)
A difficulty with these laboratory asset markets is that they do not map well into existing theories of price determination in markets with a single common-value good. The laboratory markets that give rise to bubbles have known, finite horizons and are set up so that rational, profit-maximizing agents would never choose to engage in any trade in these environments. By contrast, a theoretical literature on bubble formation demonstrates how bubbles can arise in infinite horizon environments despite the fact that agents are (typically) homogeneous and have rational expectations. (4) These rational bubble theories are therefore of little use in understanding the laboratory asset bubble phenomenon. Surprisingly, the experimentalists themselves have little to say as to why bubbles and crashes regularly occur and appear to be puzzled by their own inability to eliminate asset bubbles in a wide range of laboratory environments. As Smith et al. (2000, p. 568) notes, "controlled laboratory markets price bubbles are something of an enigma."
Our aim in this paper is to take a further step toward understanding the laboratory asset price bubble and crash phenomenon, not by conducting additional experiments with paid human subjects, but by placing a population of artificial adaptive agents in the same laboratory environments that have given rise to price bubbles and determining how the artificial agents must behave so as to generate the asset price bubbles and related features found in the experimental data. (5) Theoretical analyses of individual behavior in the double auction market mechanism has turned out to be quite difficult due to the large multiplicity of equilibria that are possible in this environment (Friedman (1993)). Agent-based techniques provide an alternative means of gaining insight into the features of these environments that may be responsible for generating asset price bubbles and crashes in laboratory studies. (6)
At the same time, agent-based models are subject to a number of arbitrary modeling decisions. We address this difficulty in two ways. First, we attempt to use the simplest model of agent behavior. In particular,we follow Gode and Sunder's (1993, 1997) approach of using "budget-constrained zero-intelligence machine traders" as a means of focusing attention on the institutional features, e.g. the rules of the laboratory market environment. As we show later in the paper, we have to modify the basic ZI approach in several respects in order to capture features of the experimental data we seek to understand. However, the modificiations we make are, again, the simplest possible; indeed we explore the marginal contribution of the two modifications we have to make to the ZI methodology and show how both are critical to our findings.
Second, we impose further discipline on our modeling exercise by requiring that our calibrated artificial agent simulations match several key features of the experimental data as reported in the various laboratory studies that have given rise to bubbles. We then ask how the data from the simulations match other, more subtle features of the experimental data. These features of the data are ones that we did not seek to match in choosing our calibration of the model.
Unlike Gode and Sunder (1993, 1997), we are not interested in the effect of various market procedures on allocative efficiency; instead our aim is to determine whether our calibrated agent-based model can deliver, both qualitatively and quantitatively, results that are similar to those found in a variety of different laboratory bubble experiments, beginning with the orginal study of Smith et al. (1988). We then examine the performance of our model in two alternative market environments that experimentalists have proposed and examined in an effort to eliminate asset price bubbles. We find that our model continues to track experimental results well in the first of these alternative environments, due to Noussair et al. (2001), even though it is not calibrated to match any of the features of that environment. For the second alternative environment proposed by Lei et al. (2001), we have to simplify our model so that it corresponds more closely to the environment those authors study. A calibrated version of this simpler model continues to perform well in tracking the features of the data observed in the Lei et al. (2001) experiment. We conclude that agent-based modeling approaches provide one means of assessing new experimental designs or market mechanisms designed to eliminate or reduce the frequency of asset price bubbles.
2 Laboratory market price bubbles
The original market environment of Smith et al. (1988) involved 9 or 12 inexperienced traders who participated in T = 15 or 30 trading periods of a computerized market. Each subject began the experimental session with an endowment of x units of cash and y units of the single asset. In each trading period, subjects could submit both bid and ask quotes for a unit of the asset (only one unit could be traded at a time) subject to budget/endowment constraints. Bid or ask prices that did not improve on pre-existing bid or ask prices were ranked relative to the current best bid and ask prices and placed in an order book queue. Agents were free to buy or sell a unit at a time at the current best bid or ask prices which were the only prices shown on each subjects' trading screens. When a unit was sold, the inventory and cash balances of the two traders was adjusted accordingly, and the transaction price was revealed to all traders. The next best bid and ask prices from the queue became the new best available bid and ask prices on all traders' screens. Trading was halted at the end of each 240 second trading period.
Following the completion of each trading period, subjects earned a dividend payment per unit of the asset that they owned at the end of the period. The dividend amount was a random variable consisting of a uniform draw from a distribution with support: {[d.sub.1], [d.sub.2], [d.sub.3], [d.sub.4]} where [less than or equal to] [d.sub.1] < [d.sub.2] < [d.sub.3] < [d.sub.4], so the expected dividend payment was [bar.d] = [1/4] [[summation].sub.i=1.sup.4] [d.sub.i]. After dividends were paid out, provided the last trading period T had not been reached, another 240 second trading period would commence.
At the beginning of each experimental session, i.e. before the start of trading period t = 1, player i's initial cash balance, [x.sup.i], and endowment of the single asset, [y.sup.i] satisfy:
[x.sup.i] + [bar.D.sub.1.sup.T][y.sup.i] = c
where [bar.D.sub.1.sup.T] is defined below and c is a constant that is the same for all i. Given that all traders have the same expected value for their initial endowment at the start of the experiment, they should be indifferent between not trading in any period, or trading at the fundamental market price in period t, which earns them zero profits. (7) Since players are told the session will last T periods, the fundamental expected market price of the asset at the beginning of trading period t is:
[bar.D.sub.t.sup.T] = [bar.d](T - t + 1) + [bar.D.sub.T+1.sup.T],
and is decreasing as t [right arrow] T. Here, [bar.D.sub.T+1.sup.T] denotes the expected default value (or buy-out value) of the asset after period T. If there are any trades, the traded prices should track the fundamental expected market price, [bar.D.sub.t.sup.T] over time and should steadily decrease by an increment of [bar.d] per trading period.
Following the start of the experiment, individual agents' cash endowments and inventories become endogenous, reflecting individual trading decisions. Endowments were not reinitialized at the start of each new trading period. Dividend earnings from the previous period become available for making cash purchases in the following period. All trades are allowed provided that the two parties to a trade have the necessary asset and cash endowments to complete the trade; these endowment amounts are updated in real time in the laboratory session using computerized software, and we follow the same practice in the artificial agent simulations. At the end of T trading periods, the standard practice was to pay out the period T dividend realization amounts per share and then pay out the default (buy-out) value of the asset.
The basic finding reported by Smith et al. (1988) is that with inexperienced subjects, there is a considerable volume of trade especially in the early periods of the market, and that the mean traded price exhibits a "hump-shaped" pattern. Initially the mean traded price lies below the fundamental price, but quickly soars above this fundamental price. Substantially higher than fundamental prices are sustained for some number of trading periods near the beginning of the session despite the declining fundamental value of the asset. Such a sustained departure of prices from fundamentals is labeled a "bubble" by the experimenters. (8) Finally, sometime during the last few trading periods, a market crash becomes a likely event, with the mean traded price falling precipitously to values close to or even below the fundamental asset...
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