Abstract
We investigate the behavior of experts who seek to make predictions with maximum impact on an audience. At a known future time, a certain continuous random variable will be realized. A public prediction gradually converges to the outcome, and an expert has access to a more accurate prediction. We study when the expert should reveal his information, when his reward is based on a proper scoring rule (e.g., is proportional to the change in log-likelihood of the outcome).
In Azar et al. (2016), we analyzed the case where the expert may make a single prediction. In this paper, we analyze the case where the expert is allowed to revise previous predictions. This leads to a rather different set of dilemmas for the strategic expert. We find that it is optimal for the expert to always tell the truth, and to make a new prediction whenever he has a new signal. We characterize the expert’s expectation for his total reward, and show asymptotic limits.
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Notes
- 1.
Taking logs transforms a lognormal random walk into a Gaussian one.
- 2.
When a normal variable with prior distribution \(N(0,\sigma _0^2)\) is sampled with known variance t at value \(x_t\), its Bayesian posterior distribution is normal with mean \(\frac{x_t/t}{1/\sigma _0^2+1/t}\) and variance \(\frac{1}{1/\sigma _0^2+1/t}\). Assuming \(\sigma _0^2 \gg T_{max} \ge t\), this simplifies to \(N(x_t,t)\).
- 3.
Since the expert is better informed than the market, his prediction depends on his signal alone. This is formally proved in Proposition 1.
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Ban, A., Azar, Y., Mansour, Y. (2017). The Strategy of Experts for Repeated Predictions. In: R. Devanur, N., Lu, P. (eds) Web and Internet Economics. WINE 2017. Lecture Notes in Computer Science(), vol 10660. Springer, Cham. https://doi.org/10.1007/978-3-319-71924-5_4
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