Far from the madding crowd: collective wisdom in prediction markets

Abstract

We investigate market selection and bet pricing in a repeated prediction market model. We derive the conditions for long-run survival of more than one agent (the crowd) and quantify the information content of prevailing prices in the case of fractional Kelly traders with heterogeneous beliefs. It turns out that, apart some non-generic situations, prices do not converge, neither almost surely nor on average, to true probabilities, nor are they always nearer to the truth than the beliefs of all surviving agents. This implies that, in general, prediction market prices are not maximum likelihood estimators of the true probabilities. However, when more than one agent survives, the average price emerging from a prediction market approximates the true probability with lower information loss than any individual belief.

Keywords:

• Prediction markets
• Wisdom of crowds
• Collective intelligence
• Market selection
• Machine learning
• Model aggregation

JEL Classification:

• C60
• D53
• G11
• G12

Acknowledgments

If there is any good idea in this paper, it probably emerged in some discussion with Pietro Dindo. In any case, we retain the exclusive paternity of all the bad ones. We thank Filippo Massari and Remco Zwinkels for useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

G. Bottazzi http://orcid.org/0000-0003-2565-4732

D. Giachini http://orcid.org/0000-0002-0001-5240

Notes

1 See Wolfers and Zitzewitz (2004) for a discussion of prediction markets. More generally, trading in some financial instruments is basically equivalent to betting on the outcome of an uncertain event. For instance, the price of Credit Default Swaps is generally recognized as a prediction about the probability of default of the underlying bond issuer.

2 The case of one-shot prediction markets has also been addressed by the literature, see e.g. Gjerstad (2005), Wolfers and Zitzewitz (2006), Manski (2006), He and Treich (2017).

3 The fractional Kelly rule is a generalization of Kelly betting and consists in investing in each asset proportionally to a linear combination of the individual belief and the market price, see MacLean et al. (1992, 2004, 2005). It can be derived from intertemporal expected log-utility maximization under the assumption that agents naively learn from prices (Dindo and Massari 2017).

4 This change of variables clarifies that the analytic results in Kets et al. (2014) are in fact identical to those derived in Blume and Easley (1992). For related contributions in the same setting see also Plott and Chen (2002), Einbinder (2006), Ottaviani and Sorensen (2009), Blume and Easley (2009).

5 Under these assumptions, the model matches exactly the example provided in Bottazzi and Dindo (2014), Section 2.

6 We use (7(7) $\underset{t\to \mathrm{\infty }}{lim}{w}_t=1\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{y}(\mathrm{a}.\mathrm{s}.),$(7) ) and (9(9) $\underset{t\to \mathrm{\infty }}{lim sup}{w}_t>0\mathrm{a}.\mathrm{s}.\wedge \underset{t\to \mathrm{\infty }}{lim inf}{w}_t<1\mathrm{a}.\mathrm{s}..$(9) ) for consistency with Bottazzi and Dindo (2014). In the present simplified framework one can equivalently define dominance and survival requiring that $\underset{t\to \mathrm{\infty }}{lim}\mathrm{E}\left[w_t\right]=1$ and $\underset{t\to \mathrm{\infty }}{lim}\mathrm{E}\left[w_t\right]>0$, respectively, as in Kets et al. (2014).

7 The wealth of the strategy with zero relative entropy, the one of a Kelly trader (c=1) with correct beliefs, never decreases in expectation. This is the optimal strategy for an agent, irrespective of what the other agents do, but it unrealistically requires the precise knowledge of the probability ${\pi }_{\ast }$.

8 This is not a general property of this kind of market models, though. It is the case here because agents adopt fractional Kelly rules. What might happen with myopic CRRA strategies, for instance, is discussed in Bottazzi and Dindo (2013) and Bottazzi and Giachini (2018).

9 Conversely, the agent's mixing parameter does not affect the ability of a trader to dominate the market. However, as MacLean et al. (1992) argue, there is a trade-off between risk and expected growth. Nonetheless here we are interested in asymptotic outcomes, thus, assuming that we are in a situation in which an agent dominates, the fact that her c is small only implies that her wealth will converge to 1 slower than a case in which her c is large.

10 Following the same reasoning, it is easy to show that this also applies to the more general case when $c_1\ne c_2$.

11 As a specific example, take ${\pi }_{\ast }=0.2$, ${\pi }_1=0.1$, ${\pi }_2=0.32$ and c=0.96. Numerical simulations show that in this case it is $\mathrm{E}\left[p\right]=0.3191\pm 0.00003$.

12 See proof of theorem 5 in Appendix 1 of Barbu and Lay (2012).

13 While the investigation of this trade-off is per se interesting, neither the subject nor the length of this paper are sufficiently wide to accommodate it.

Additional information

Funding

We gratefully acknowledge the funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 640772—DOLFINS.