Far from the madding crowd: collective wisdom in prediction markets

Abstract

We investigate market selection and bet pricing in a simple Arrow security economy which we show is equivalent to the repeated prediction market models studied in the literature. We derive the condition for long run survival of more than one agent (the crowd) and quantify the information content of prevailing prices in the case of two 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 believes of all surviving agents. Moreover, we show that by adapting their beliefs to past prices, agents further decrease the agreement between market prices and true probabilities.

Betting, Selection, and Luck: A Long-Run Analysis of Repeated Betting Markets

“…In a market populated by utility maximizers, the agent who trades knowing the correct probabilities always realizes a non-negative expected profit [ 7 ]. In the case of bettors using the fractional Kelly rule, a generalization of the Kelly rule that includes a risk-aversion parameter, sufficient and, apart from hairline cases, necessary conditions for strategy dominance or survival has been derived [ 8 , 9 ]. These conditions generalize and correct previous tentative results based on numerical simulations [ 10 ].…”

Wealth and price distribution by diffusive approximation in a repeated prediction market

“…Notice that (7) is always satisfied for c sufficiently small, as long as π 1 and π 2 are one larger and one smaller than π * . Remarkably, the condition for the persistent heterogeneity derived by imposing the existence of the invariant distribution of the diffusive approximation are identical to those derived for the original process by Bottazzi and Giachini (2016) applying the results based on martingale convergence theorem in Bottazzi and Dindo (2015). In this respect (4) perfectly replicates the qualitative behavior of the original, discrete time, model.…”

“…In fact, as discussed in Bottazzi and Giachini (2016), the system has two possible long-run outcomes: or one agent ends up owning the entire wealth, that is lim t→∞ w t = 0, 1, and in this case the market price converges to the belief of that agent, lim t→∞ p t = π 2 , π 1 , or, alternatively, both agents stay in the market in the long run, their wealth shares persistently fluctuate and p t keeps moving in the interval (π 1 , π 2 ). The second outcome constitutes a situation of persistent heterogeneity, in which agents with different beliefs can indefinitely coexist in the market, and (6) is a sufficient and necessary condition for its occurrence.…”

“…Bottazzi and Giachini (2016) show that the model discussed here is equivalent, through a simple change of variables, to the repeated prediction market model discussed inKets et al (2014).…”

Momentum and reversal in financial markets with persistent heterogeneity

“…See Section 2 for an account of the related literature.2 As we shall make clear in Section 4, these portfolio and saving rules correspond to those of an agent who maximizes the expected discounted stream of consumption when her beliefs get influenced by market prices, see, or the related work on the Fractional Kelly rule in heterogeneous agent economiesKets et al (2014),Bottazzi and Giachini (2016),…”