History-Independent Distributed Multi-agent Learning

  • Amos Fiat
  • Yishay Mansour
  • Mariano Schain
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)


How should we evaluate a rumor? We address this question in a setting where multiple agents seek an estimate of the probability, b, of some future binary event. A common uniform prior on b is assumed. A rumor about b meanders through the network, evolving over time. The rumor evolves, not because of ill will or noise, but because agents incorporate private signals about b before passing on the (modified) rumor. The loss to an agent is the (realized) square error of her opinion.

Our setting introduces strategic behavior based on evidence regarding an exogenous event to current models of rumor/influence propagation in social networks.

We study a simple Exponential Moving Average (EMA) for combining experience evidence and trusted advice (rumor), quantifying its resulting performance and comparing it to the optimal achievable using Bayes posterior having access to the agents private signals.

We study the quality of \(p_T\), the prediction of the last agent along a chain of T rumor-mongering agents. The prediction \(p_T\) can be viewed as an aggregate estimator of b that depends on the private signals of T agents. We show that
  • When agents know their position in the rumor-mongering sequence, the expected mean square error of the aggregate estimator is \(\varTheta (\frac{1}{T})\). Moreover, with probability \(1-\delta \), the aggregate estimator’s deviation from b is \(\varTheta \left( \sqrt{\frac{\ln (1/\delta )}{T}}\right) \).

  • If the position information is not available, and agents act strategically, the aggregate estimator has a mean square error of \(O(\frac{1}{\sqrt{T}})\). Furthermore, with probability \(1~-~\delta \), the aggregate estimator’s deviation from b is \(\widetilde{O}\left( \sqrt{\frac{\ln (1/\delta )}{\sqrt{T}}}\right) \).


Mean Square Error Strategic Behavior Symmetric Equilibrium Quadratic Loss Private Signal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abernethy, J., Frongillo,R.M.: A collaborative mechanism for crowdsourcing prediction problems (2011). CoRR, abs/1111.2664Google Scholar
  2. 2.
    Aumann, R.J.: Agreeing to disagree. Ann. Stat. 4(6), 1236–1239 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bala, V., Goyal, S.: Learning from neighbours. Rev. Econ. Stud. 65(3), 595–621 (1998)zbMATHCrossRefGoogle Scholar
  4. 4.
    Banerjee, A., Fudenberg, D.: Word-of-mouth learning. Games Econ. Behav. 46(1), 1–22 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Banerjee, A., Guo, X., Wang, H.: On the optimality of conditional expectation as a Bregman predictor. IEEE Trans. Inf. Theory 51(7), 2664–2669 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    David, E., Jon, K.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York (2010)zbMATHGoogle Scholar
  7. 7.
    Ellison, G., Fudenberg, D.: Word-of-mouth communication and social learning. Q. J. Econ. 110(1), 93–125 (1995)zbMATHCrossRefGoogle Scholar
  8. 8.
    Geanakoplos, J., Polemarchakis, H.M.: We can’t disagree forever. Cowles Foundation Discussion Papers 639, Cowles Foundation for Research in Economics, Yale University, July 1982Google Scholar
  9. 9.
    McKelvey, R.D., Page, T.: Common knowledge, consensus, and aggregate information. Econometrica 54(1), 109–127 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ostrovsky, M.: Information aggregation in dynamic markets with strategic traders. Research Papers 2053, Stanford University, Graduate School of Business, March 2009Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Tel Aviv UniversityTel AvivIsrael
  2. 2.GoogleMountain ViewUSA

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