The bubble game: an experimental study of speculation (Q2864825)

The authors introduce a bubble game in which an asset, commonly known to be valueless, is traded sequentially but the traders do not always know where they stand in the sequence. The game has a no bubble Nash equilibrium when there is a finite price cap, and an additional bubble equilibrium when there is no price cap. Experiments are run both with and without a price cap and the obtained empirical data is used for the estimation of several behavioral game theory models. The authors conclude that in their setting, quantal responses and analogy-based expectations are important drivers of speculation, while cognitive hierarchies are not.NEWLINENEWLINEThe authors experiment with the following bubble game. There are three traders in the market who trade sequentially. Each trader is assigned a position in the market sequence and can be first, second or third with the same probability 1/3, and traders are not told their position. The first trader is offered to buy at a random price \(P_1 = 10^n\) where \(\mathbb P (n = j) = 1/ 2^{j+1}\), \(j\in\{0,1,\dots\}\). If a trader decides to buy the asset at price \(P_t\), he automatically proposes the asset to the next trader at a price \(P_{t+1} = 10P_t\). The traders are informed about a price cap \(K=10^k\) on the first price where \(k \in \{0,1,\dots,+\infty\}\).NEWLINENEWLINEEach trader is endowed with 1 unit of capital. The additional capital needed to buy the asset if \(P_t > 1\) is provided by an outside financier, the experimenter in practice, who is also the original owner of the asset. Payoffs are divided between the trader and the financier in proportion of the capital initially invested: \(1/P_t\) for the trader and \((P_t - 1)/P_t\) for the financier; in particular, the terminal wealth of a trader is either 0 or 10. The authors note that if \(K = +\infty\), there exist increasing and convex utility functions such that all traders trade, hence a common knowledge rational bubble is formed without an infinite trading horizon or infinite trading speed. It is also mentioned that if each trader has a different outside financier, these financiers may rationally fuel the bubble.NEWLINENEWLINEThe authors study two types of behavioral game models: the Subjective Quantal Response Equilibrium (SQRE) of \textit{B. W. Rogers} et al. [J. Econ. Theory 144, No. 4, 1440--1467 (2009; Zbl 1166.91310)], and the Analogy-Based Expectation Equilibrium (ABEE) of \textit{P. Jehiel} [J. Econ. Theory 123, No. 2, 81--104 (2005; Zbl 1102.91014)]. Special cases of the SQRE model, such as the Cognitive Hierarchy (CH) model of \textit{C. F. Camerer} et al. [Q. J. Econ. 119, No. 3, 861--898 (2004; Zbl 1074.91503)] and the Quantal Response Equilibrium (QRE) model of \textit{R. D. McKelvey} and \textit{T. R. Palfrey} [Games Econ. Behav. 10, No. 1, 6--38 (1995; Zbl 0832.90126)], are also analyzed. For each of the models considered, the probabilities that agents enter the trade when observing a given price are computed for various cap values \(K\). These results are used to fit these models to experimental data obtained by playing the bubble game 78 times with 234 subjects, each participating in one run only.NEWLINENEWLINEBased on their experiment, the authors observe that {\parindent=6mm\begin{itemize}\item[--] the QRE model fits the data better than the CH model; \item[--] the data supports the presence of a ``snowball effect'', i.e.\ that the propensity to enter bubbles increases with the required number of reasoning steps in the backward induction ruling out rational bubble formation, and with the probability not to be last; this feature supports the superiority of QRE over CH since the latter model does not capture the snowball effect while the former does account for it; \item[--] dominance of particular extensions of the QRE model over QRE also indicate that cognitive hierarchies are not necessary to explain speculative behavior in the bubble game; \item[--] ABEE models produce fits comparable to those of the most appropriate QRE models and in addition feature more stability in the estimated parameters.NEWLINENEWLINE\end{itemize}}