Abstract
The greater fools explanation of financial bubbles says that traders are willing to pay more for an asset than they deem it worth, because they anticipate they might be able to sell it to someone else for an even higher price. As agents’ beliefs about other agents’ beliefs are at the heart of the greater fools theory, this paper comes to formal terms with the theory by translating the phenomenon into the language and models of dynamic epistemic logic. By presenting a formalization of greater fools reasoning, structural insights are obtained pertaining to the structure of its higher-order content and the role of common knowledge.
My research is financed by the Carlsberg Foundation. I would like to thank Thomas Bolander, Vincent F. Hendricks, Rasmus K. Rensdvig and two anonymous referees for their valuable comments on this paper.
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Notes
- 1.
See [7] for an extensive overview of studies of financial bubbles.
- 2.
The greater fools explanation of a bubble must not be confused with herding phenomena as the two are fundamentally different: where in a herding bubble investors act the same because of an incentive to follow the crowd, investors in a greater fools bubble simply act the same as a result of similar reasoning, as will be elaborately discussed in this paper.
- 3.
See [19] for a more eleborate definition of the fundamentals that influence the fundamental value.
- 4.
This eliminates the situation where Arthur, who owns an apple tree, and Barbara, who owns an orange tree, trade apples for oranges and mutually benefit from the trade.
- 5.
A rational model of greater fools bubble should however not rule out the important role played by irrational behavior and mass psychology.
- 6.
Although the models are a representation of the epistemic states of a few individuals, these individuals can be interpreted to represent a group of homogeneous traders.
- 7.
The relation \(\preceq _i\) being locally connected means that for all \(w, u\in \mathcal {W}\) whenever they are related by the symmetric closure of \(\preceq _i\), then \(w\preceq _i u\) or \(u\preceq _i w\).
- 8.
The second condition has the realistic consequence that when for example Arthur believes he can sell the asset for 10, after which he learns no other agent in fact wants to buy the asset for 10, he will lower the price for which he offers to sell. The price will continue to drop until he either has an agreement to trade, or Arthur realizes he cannot sell the asset for more than the value he deems it worth.
- 9.
Following the protocol, communication should precede the trade such that the trade does not envoke any epistemic change. Moreover, the framework does not formally keep track of who owns the tree, thus a trade should not envoke any atomic change. The chosen encoding of trading fulfills both desiderata.
- 10.
After the update, worlds \(u_1', x_1', y_1'\) and \(z_1'\) become redundant by their bisimulation to worlds \(u_1, x_1, y_1\) and \(z_1\) respectively. Such bisimulation contraction is harmless in epistemic logic.
- 11.
This is not evident, as e.g. [1] merely shows – by simulating a growing bubble with a price that runs up automatically in every period – that if a bubble exists, people are willing to ride the bubble.
- 12.
Note that while a bubble is typically described by referring to an extreme overpricing, this definition allows for a minimal overpricing of \(v_{\mathsf {max}}+1\). It is here chosen to refrain from an ad hoc specification of “extreme”, but if preferred one can easily adjust the definition and examples accordingly.
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van Lee, H.S. (2017). A Formalization of the Greater Fools Theory with Dynamic Epistemic Logic. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_40
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