Computational Economics

, Volume 48, Issue 1, pp 155–178 | Cite as

Belief Aggregation with Automated Market Makers

  • Rajiv SethiEmail author
  • Jennifer Wortman Vaughan


We consider the properties of a cost function based automated market maker aggregating the beliefs of risk-averse traders with finite budgets. Individuals can interact with the market maker an arbitrary number of times before the state of the world is revealed. We show that the resulting sequence of prices is convergent under general conditions, and explore the properties of the limiting price and trader portfolios. The limiting price cannot be expressed as a function of trader beliefs, since it is sensitive to the market maker’s cost function as well as the order in which traders interact with the market. For a range of trader preferences, however, we show numerically that the limiting price provides a good approximation to a weighted average of beliefs, inclusive of the market designer’s prior belief as reflected in the initial contract price. This average is computed by weighting trader beliefs by their respective budgets, and weighting the initial contract price by the market maker’s worst-case loss, implicit in the cost function. Since cost function parameters are chosen by the market designer, this allows for an inference regarding the budget-weighted average of trader beliefs.


Prediction markets Automated market makers Belief aggregation 


  1. Abernethy, J., Chen, Y., & Vaughan, J. W. (2013). Efficient market making via convex optimization, and a connection to online learning. ACM Transactions on Economics and Computation, 1(2), 12:1–12:39.CrossRefGoogle Scholar
  2. Abernethy, J., Frongillo, R., Li, X., & Vaughan, J. W. (2014). A general volume-parameterized market making framework. In Proceedings of the fifteenth ACM conference on economics and computation (pp. 413–480).Google Scholar
  3. Aumann, R. J. (1976). Agreeing to disagree. Annals of Statistics, 4(6), 1236–1239.CrossRefGoogle Scholar
  4. Berg, J., Forsythe, R., Nelson, F., & Rietz, T. (2008). Results from a dozen years of election futures markets research. Handbook of Experimental Economics Results, 1, 742–751.CrossRefGoogle Scholar
  5. Broughton, P. D. (2013, April 24). Prediction markets: Value among the crowd. Financial Times.Google Scholar
  6. Charette, R. (2007, February 28). An internal futures market. Information Management.Google Scholar
  7. Chen, K. Y., & Plott, C. (2002). Information aggregation mechanisms: Concept, design and field implementation. Social Science Working Paper no. 1131. Pasadena: CalTech.Google Scholar
  8. Chen, Y., & Pennock, D. M. (2007). A utility framework for bounded-loss market makers. In Proceedings of the 23rd conference on uncertainty in artificial intelligence (UAI).Google Scholar
  9. Chen, Y., Ruberry, M., & Vaughan, J. W. (2012). Designing informative securities. In Proceedings of the 28th conference on uncertainty in artificial intelligence (UAI).Google Scholar
  10. Cowgill, B., Wolfers, J., & Zitzewitz, E. (2009). Using prediction markets to track information flows: Evidence from Google. In Auctions, market mechanisms and their applications (Vol. 14).Google Scholar
  11. Foley, D. K. (1994). A statistical equilibrium theory of markets. Journal of Economic Theory, 62(2), 321–345.CrossRefGoogle Scholar
  12. Geanakopolos, J. D., & Polemarchakis, H. M. (1982). We can’t disagree forever. Journal of Economic Theory, 28(1), 192–200.CrossRefGoogle Scholar
  13. Gjerstad, S. (2004). Risk aversion, beliefs, and prediction market equilibrium. Working Paper, Economic Science Laboratory, University of Arizona.Google Scholar
  14. Hahn, F. H., & Negishi, T. (1962). A theorem on non-tâtonnement stability. Econometrica, 30(3), 463–469.CrossRefGoogle Scholar
  15. Hanson, R. D. (2003). Combinatorial information market design. Information Systems Frontiers, 5(1), 107–119.CrossRefGoogle Scholar
  16. Hanson, R. D. (2007). Logarithmic market scoring rules for modular combinatorial information aggregation. Journal of Prediction Markets, 1(1), 1–15.Google Scholar
  17. Hayek, F. A. (1945). The use of knowledge in society. American Economic Review, 35(4), 519–530.Google Scholar
  18. Iyer, K., Johari, R., Moallemi, C. C. (2010). Information aggregation and allocative efficiency in smooth markets. In Proceedings of the 11th ACM conference on electronic commerce (EC).Google Scholar
  19. Leigh, A., & Wolfers, J. (2006). Competing approaches to forecasting elections: Economic models, opinion polling and prediction markets. Economic Record, 82(258), 325–340.CrossRefGoogle Scholar
  20. Li, X., & Vaughan, J. W. (2013). An axiomatic characterization of adaptive-liquidity market makers. In Proceedings of the fourteenth ACM conference on electronic commerce (EC).Google Scholar
  21. Manski, C. F. (2006). Interpreting the predictions of prediction markets. Economics Letters, 91(3), 425–429.CrossRefGoogle Scholar
  22. Ostrovsky, M. (2012). Information aggregation in dynamic markets with strategic traders. Econometrica, 80(6), 2595–2648.CrossRefGoogle Scholar
  23. Othman, A., & Sandholm, T. (2010). When do markets with simple agents fail? In Proceedings of the 9th international conference on autonomous agents and multiagent systems (AAMAS).Google Scholar
  24. Othman, A., Pennock, D. M., Reeves, D. M., & Sandholm, T. (2013). A practical liquidity-sensitive automated market maker. ACM Transactions on Economics and Computation, 1(3), 14.CrossRefGoogle Scholar
  25. Pennock, D. M. (1999). Aggregating probabilistic beliefs: Market mechanisms and graphical representations. PhD thesis, The University of Michigan.Google Scholar
  26. Rothschild, D. M. (2009). Forecasting elections comparing prediction markets, polls, and their biases. Public Opinion Quarterly, 73(5), 895–916.CrossRefGoogle Scholar
  27. Rothschild, D. M. (2015). Combing forecasts: Accurate, relevant, and timely. International Journal of Forecasting, 31(3), 952–964.CrossRefGoogle Scholar
  28. Rothschild, D. M., & Sethi, R. (2015). Wishful thinking, manipulation, and the wisdom of crowds: Evidence from a prediction market, available at SSRN:
  29. Sethi, R., & Yildiz, M. (2012). Public disagreement. American Economic Journal: Microeconomics, 4(3), 57–95.Google Scholar
  30. Sethi, R., & Yildiz, M. (2015). Perspectives, opinions, and information flows, available at SSRN:
  31. Ungar, L., Mellors, B., Satopää, V., Baron, J., Tetlock, P., Ramos, J., & Swift, S. (2012). The good judgment project: A large scale test of different methods of combining expert predictions, AAAI Technical Report FS-12-06.Google Scholar
  32. Wolfers, J., & Leigh, A. (2002). Three tools for forecasting federal elections: Lessons from 2001. Australian Journal of Political Science, 37(2), 223–240.CrossRefGoogle Scholar
  33. Wolfers, J., & Zitzewitz, E. (2006). Interpreting prediction market prices as probabilities. NBER Working Paper No. 12200.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Economics, Barnard CollegeColumbia UniversityNew YorkUSA
  2. 2.Microsoft ResearchNew YorkUSA

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