# Convergence within binary market scoring rules

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## Abstract

Prediction markets are run to extract information from its participants through financial incentive. The market scoring rule mechanism represents a way of organizing markets in order to foster agents to make sincere predictions. Market scoring rules are usually presented in a context of asset trading, but they also boil down to a sequential probability report process analyzed here. If the future state space is binary (i.e., composed of only two possible states) and only two agents participate alternatively in the market, it is proven that for strictly proper market scoring rules, the report sequences of each agent converge toward limit probability reports which are closer to each other than the subjective probabilities of the agents.

## Keywords

Prediction market Risk aversion Fixed point Favorite-longshot bias Equilibrium## JEL Classification

D47 D79 D82 D83## References

- Abernethy, J., Chen, Y., Vaughan, J.W.: Efficient market making via convex optimization, and a connection to online learning. ACM Trans. Econ. Comput.
**1**(2), 12 (2013)CrossRefGoogle Scholar - Agrawal, S., Delage, E., Peters, M., Wang, Z., Ye, Y.: A unified framework for dynamic prediction market design. Oper. Res.
**59**(3), 550–568 (2011)CrossRefGoogle Scholar - Armantier, O., Treich, N.: Eliciting beliefs: proper scoring rules, incentives, stakes and hedging. Eur. Econ. Rev.
**62**, 17–40 (2013)CrossRefGoogle Scholar - Berg, J., Forsythe, R., Nelson, F., Rietz, T.: Results from a dozen years of elections futures markets research. Handb. Exp. Econ. Res.
**1**, 742–751 (2008)CrossRefGoogle Scholar - Budish, E., Cramton, P., Shim, J.: The high-frequency trading arms race: frequent batch auctions as a market design response. Q. J. Econ.
**130**(4), 1547–1621 (2015)CrossRefGoogle Scholar - Chen, Y., Pennock, D. M.: A utility framework for bounded-loss market makers. (2012). arXiv preprint arXiv:1206.5252
- Chen, Y., Dimitrov, S., Sami, R., Reeves, D.M., Pennock, D.M., Hanson, R.D., Fortnow, L., Gonen, R.: Gaming prediction markets: equilibrium strategies with a market maker. Algorithmica
**58**(4), 930–969 (2010)CrossRefGoogle Scholar - Dimitrov, S., Sami, R.: Non-myopic strategies in prediction markets. In: Proceedings of the 9th ACM Conference on Electronic Commerce, pp. 200–209. ACM (2008)Google Scholar
- Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solutions Mappings. Springer, Berlin (2009)CrossRefGoogle Scholar
- Friend, I., Blume, M.E.: The demand for risky assets. Am. Econ. Rev.
**65**(5), 920–922 (1975)Google Scholar - Gneiting, T., Raftery, A.: Strictly proper scoring rules, prediction and estimation. J. Am. Stat. Assoc.
**102**, 359–378 (2007)CrossRefGoogle Scholar - Gruca, T., Berg, J.E., Cipriano, M.: Consensus and differences of opinion in electronic prediction markets. Electron. Mark.
**15**(1), 13–22 (2005)CrossRefGoogle Scholar - Hanson, R.: Combinatorial information market design. Inf. Syst. Front.
**5**(1), 107–119 (2003)CrossRefGoogle Scholar - He, X.-Z., Treich, N.: Prediction market prices under risk aversion and heterogeneous beliefs. J. Math. Econ.
**70**, 105–114 (2017)CrossRefGoogle Scholar - Iyer, K., Johari, R., Moallemi, C. C.: Information aggregation in smooth markets. In: Proceedings of the 11th ACM Conference on Electronic Commerce, pp. 199–206. ACM (2010)Google Scholar
- Kadane, J.B., Winkler, R.L.: Separating probability elicitation from utilities. J. Am. Stat. Assoc.
**83**, 357–363 (1988)CrossRefGoogle Scholar - Lange, J., Economides, N.: A parimutuel market microstructure for contingent claims. Eur. Financ. Manag.
**11**(1), 25–49 (2005)CrossRefGoogle Scholar - Manski, C.F.: Interpreting the predictions of prediction markets. Econ. Lett.
**91**(3), 425–429 (2006)CrossRefGoogle Scholar - Offerman, T., Sonnemans, J., Van de Kuilen, G., Wakker, P.P.: A truth serum for non-bayesians: correcting proper scoring rules for risk attitudes. Rev. Econ. Stud.
**76**(4), 1461–1489 (2009)CrossRefGoogle Scholar - Ostrovsky, M.: Information aggregation in dynamic markets with strategic traders. Econometrica
**80**(6), 2595–2647 (2012)CrossRefGoogle Scholar - Ottaviani, M.: Price reaction to information with heterogeneous beliefs and wealth effects: underreaction, momentum, and reversal. Am. Econ. Rev.
**105**(1), 1–34 (2014)CrossRefGoogle Scholar - Ottaviani, M., Sørensen, P. N.: The favorite-longshot bias: an overview of the main explanations. In: Hausch, D.B., Ziemba, W.T. (eds.) Handbook of Sports and Lottery Markets, pp. 83–101. North Holland, Amsterdam (2008)CrossRefGoogle Scholar
- Weitzman, M.: Utility analysis and group behavior: an empirical study. J. Polit. Econ.
**73**(1), 18–26 (1965)CrossRefGoogle Scholar