Bayesian Decision Making in Groups is Hard

We study the computations that Bayesian agents undertake when exchanging opinions over a network. The agents act repeatedly on their private information and take myopic actions that maximize their expected utility according to a fully rational posterior belief. We show that such computations are NP-hard for two natural utility functions: one with binary actions, and another where agents reveal their posterior beliefs. In fact, we show that distinguishing between posteriors that are concentrated on different states of the world is NP-hard. Therefore, even approximating the Bayesian posterior beliefs is hard. We also describe a natural search algorithm to compute agents' actions, which we call elimination of impossible signals, and show that if the network is transitive, the algorithm can be modified to run in polynomial time.

[1]  R. Radner,et al.  Team Decision Problems , 1962 .

[2]  D. Blackwell,et al.  Merging of Opinions with Increasing Information , 1962 .

[3]  R. Radner,et al.  Economic theory of teams , 1972 .

[4]  M. Degroot.  Reaching a Consensus , 1974 .

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  J. Geanakoplos,et al.  We Can't Disagree Forever , 1982 .

[7]  J. Cave.  Learning to agree , 1983 .

[8]  J. Stiglitz,et al.  The Architecture of Economic Systems: Hierarchies and Polyarchies , 1984 .

[9]  Teofilo F. GONZALEZ,et al.  Clustering to Minimize the Maximum Intercluster Distance , 1985, Theor. Comput. Sci..

[10]  Joseph E. Stiglitz,et al.  Human Fallibility and Economic Organization , 1985 .

[11]  John N. Tsitsiklis,et al.  On the complexity of decentralized decision making and detection problems , 1985 .

[12]  M. Bacharach.  Some extensions of a claim of Aumann in an axiomatic model of knowledge , 1985 .

[13]  Drew Fudenberg,et al.  The Folk Theorem in Repeated Games with Discounting or with Incomplete Information , 1986 .

[14]  Yannis M. Ioannides.  On the architecture of complex organizations , 1987 .

[15]  J. Stiglitz,et al.  Committees, hierarchies and polyarchies , 1987 .

[16]  John N. Tsitsiklis,et al.  The Complexity of Markov Decision Processes , 1987, Math. Oper. Res..

[17]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[18]  Rohit Parikh,et al.  Communication, consensus, and knowledge , 1990 .

[19]  A. Banerjee,et al.  A Simple Model of Herd Behavior , 1992 .

[20]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[21]  Ehud Lehrer,et al.  Merging and learning , 1996 .

[22]  Lones Smith,et al.  Pathological Outcomes of Observational Learning , 2000 .

[23]  Spyros Vassilakis,et al.  Accelerating New Product Development by Overcoming Complexity Constraints , 2002 .

[24]  R. Gibbons.  Incentives in Organizations , 1998 .

[25]  S. Bikhchandani,et al.  Learning from the behavior of others : conformity, fads, and informational cascades , 1998 .

[26]  S. Goyal,et al.  Learning from neighbours , 1998 .

[27]  Michael I. Jordan,et al.  Loopy Belief Propagation for Approximate Inference: An Empirical Study , 1999, UAI.

[28]  Kumaraswamy Velupillai,et al.  Computable Economics: The Arne Ryde Memorial Lectures , 2000 .

[29]  Douglas Gale,et al.  Bayesian learning in social networks , 2003, Games Econ. Behav..

[30]  P. DeMarzo,et al.  Persuasion Bias, Social Influence, and Uni-Dimensional Opinions , 2001 .

[31]  Scott Aaronson,et al.  The complexity of agreement , 2004, STOC '05.

[32]  Luca Trevisan,et al.  Average-Case Complexity , 2006, Found. Trends Theor. Comput. Sci..

[33]  R. Aumann.  Agreeing to disagree. , 1976, Nature cell biology.

[34]  Luca Trevisan.  Average-case Complexity , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[35]  David J. Spiegelhalter,et al.  Probabilistic Networks and Expert Systems - Exact Computational Methods for Bayesian Networks , 1999, Information Science and Statistics.

[36]  Ilan Lobel,et al.  BAYESIAN LEARNING IN SOCIAL NETWORKS , 2008 .

[37]  Venkat Chandrasekaran,et al.  Complexity of Inference in Graphical Models , 2008, UAI.

[38]  Nicolas Vieille,et al.  Informational externalities and emergence of consensus , 2009, Games Econ. Behav..

[39]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[40]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

[41]  Matthew O. Jackson,et al.  Naïve Learning in Social Networks and the Wisdom of Crowds , 2010 .

[42]  Causing Human Actions , 2010 .

[43]  Thorbjørn Knudsen,et al.  Design of Decision-Making Organizations , 2010, Manag. Sci..

[44]  Sanjeev Arora,et al.  Computational complexity and information asymmetry in financial products , 2011, Commun. ACM.

[45]  Asuman E. Ozdaglar,et al.  Opinion Dynamics and Learning in Social Networks , 2010, Dyn. Games Appl..

[46]  J. Kwisthout.  The Computational Complexity of Probabilistic Inference Technical Report , 2011 .

[47]  Manuel Mueller-Frank,et al.  A general framework for rational learning in social networks , 2011 .

[48]  Ali Jadbabaie,et al.  Non-Bayesian Social Learning , 2011, Games Econ. Behav..

[49]  Felipe A. Csaszar,et al.  Organizational Decision Making: An Information Aggregation View , 2012 .

[50]  Elchanan Mossel,et al.  Strategic Learning and the Topology of Social Networks , 2012, ArXiv.

[51]  Yashodhan Kanoria,et al.  Tractable Bayesian Social Learning on Trees , 2013, IEEE J. Sel. Areas Commun..

[52]  Yashodhan Kanoria,et al.  Tractable Bayesian Social Learning on Trees , 2012, IEEE Journal on Selected Areas in Communications.

[53]  Elchanan Mossel,et al.  Making Consensus Tractable , 2010, TEAC.

[54]  M. Rabin,et al.  Extensive Imitation is Irrational and Harmful , 2014 .

[55]  Elchanan Mossel,et al.  Asymptotic learning on Bayesian social networks , 2012, Probability Theory and Related Fields.

[56]  Vikram Krishnamurthy,et al.  Online Reputation and Polling Systems: Data Incest, Social Learning, and Revealed Preferences , 2014, IEEE Transactions on Computational Social Systems.

[57]  Elchanan Mossel,et al.  On the Speed of Social Learning , 2014, SSRN Electronic Journal.

[58]  Elchanan Mossel,et al.  WHEN MORE INFORMATION REDUCES THE SPEED OF LEARNING , 2015 .

[59]  Elchanan Mossel,et al.  Groupthink and the Failure of Information Aggregation in Large Groups , 2014 .

[60]  Elchanan Mossel,et al.  Opinion Exchange Dynamics , 2014, Probability Surveys.

[61]  Quentin Berthet,et al.  Statistical and computational trade-offs in estimation of sparse principal components , 2014, 1408.5369.

[62]  Alejandro Ribeiro,et al.  Learning to Coordinate in Social Networks , 2014, Oper. Res..

[63]  Elchanan Mossel,et al.  Efficient Bayesian Learning in Social Networks with Gaussian Estimators , 2010, 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[64]  Yudong Chen,et al.  Statistical-Computational Tradeoffs in Planted Problems and Submatrix Localization with a Growing Number of Clusters and Submatrices , 2014, J. Mach. Learn. Res..

[65]  Ali Jadbabaie,et al.  Bayesian Heuristics for Group Decisions , 2016, ArXiv.

[66]  Ali Jadbabaie,et al.  Bayesian Learning Without Recall , 2016, IEEE Transactions on Signal and Information Processing over Networks.

[67]  Ali Jadbabaie,et al.  A Theory of Non‐Bayesian Social Learning , 2018 .

[68]  Krishna Dasaratha,et al.  Bayesian Social Learning in a Dynamic Environment , 2018, EC.

[69]  B. Golub,et al.  SOCIAL LEARNING IN A DYNAMIC ENVIRONMENT , 2018 .

[70]  Subhash Khot,et al.  Pseudorandom Sets in Grassmann Graph Have Near-Perfect Expansion , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[71]  Xu Tan,et al.  Locally Bayesian Learning in Networks , 2019, Theoretical Economics.

[72]  Elchanan Mossel,et al.  Social Learning Equilibria , 2012, EC.

[73]  Wei Li.  Locally Bayesian Learning in Networks ∗ , 2018 .

[74]  S. Nageeb Ali,et al.  Herding with costly information , 2018, J. Econ. Theory.

[75]  Elchanan Mossel,et al.  Reasoning in Bayesian Opinion Exchange Networks Is PSPACE-Hard , 2018, COLT.

[76]  Yakov Babichenko,et al.  Robust Non-Bayesian Social Learning , 2019, EC.

[77]  Yakov Babichenko,et al.  Naive Learning Through Probability Matching , 2019, EC.