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Fundamental Limitations of Networked Decision Systems Munther A. Dahleh Laboratory for Information and Decision Systems MIT AFOSR-MURI Kick-off meeting, Sept, 2009 Smart Grid 2 Drug Prescription: Marketing The drugs your physician prescribes may well depend on the behavior of an opinion leader in his or her social network in addition to your doctor’s own knowledge of or familiarity with those products. 3 Smoking Whether a person quits smoking is largely shaped by social pressures, and people tend to quit smoking in groups. If a spouse quits smoking, the other spouse is 67% less likely to smoke. If a friend quits, a person is 36% less likely to still light up. Siblings who quit made it 25% less likely that their brothers and sisters would still smoke. 4 Social Networks and Politics Network structure of political blogs prior to 2004 presidential elections 5 Outline Human/ Selfish Engineered Networks: Connectivity/capacity Nature of Interaction: Cyclic/Sequential Computation Learning Decisions 6 Learning Over Complex Networks In Collaboration: Daron Acemoglu Ilan Lobel Asuman Ozdaglar The Tipping Point: M. Gladwell The Tipping Point is that magic moment when an idea, trend, or social behavior crosses a threshold, tips, and spreads like wildfire. Just as a single sick person can start an epidemic of the flu, so too can a small but precisely targeted push cause fashion trend, the popularity of a new product, or a drop of crime rate. 8 Objective Develop Models that can capture the impact of a social network on learning and decision making 9 Who's Buying the Newest Phone and Why? I read positive reviews, and Lisa got it. This phone has great functionality. Got it! 1 Got it! 2 Everyone has it, but 3G speeds are rather lacking. 3 5 Got it! Got it! Didn’t. It looks good, but Jane didn’t get it despite all her friends having it. Didn’t. 6 Before I asked around, I thought the phone was perfect. But now I’m getting mixed opinions. 4 7 Should I get it? 10 Formulation: Two States I read positive reviews, and Lisa got it. This phone has great functionality. Got it! 1 Got it! 2 Everyone has it, but 3G speeds are rather lacking. 3 5 Got it! Got it! Didn’t. It looks good, but Jane didn’t get it despite all her friends having it. Didn’t. 6 Before I asked around, I thought the phone was perfect. But now I’m getting mixed opinions. 4 7 Should I get it? 11 General Setup Two possible states of the world µ 2 f 0; 1g both equally likely. A sequence of agents (n = 1; 2; :::) making binary decisions x n . Agent n obtains utility 1 if x n = µ and utility 0 otherwise. Each agent has an iid private signal sn 2 [0; 1]. The signal is sampled from a cumulative density F µ . The neighborhood: - The neighborhoods - n ½ f 1; : : : n ¡ 1g is generated according to arbitrary independent distributions f Q n ; n 2 N g. Information: I n = f sn ; - n ; x k for all k 2 - n g n ½ f 1; : : : n ¡ 1g 12 The World According to Agent 7 13 Rationality Rational Choice: Given information set I n agent n chooses ¾n (I n ) 2 arg max P (µ = yjI n ) : y2 f 0;1g ¾= f ¾n g Strategy profile: Asymptotic Learning: Under what conditions does limn ! 1 P ¾(x n = µ) = 1 14 Equilibrium Decision Rule The belief about the state decomposes into two parts µ ¶¡ dF0 (sn ) p (s ) = P (µ = 1js ) = 1 + n n ¾ n Private Belief, dF1 (sn ) 1 Social belief, P ¾(µ = 1j- n ; x k for all k 2 - n ) Strategy profile ¾ is a perfect Bayesian equilibrium if and only if: P ¾(µ = 1jsn ) + P ¾(µ = 1j- n ; x k for all k 2 - n ) > 1 =) ¾n (I n ) = 1; P ¾(µ = 1jsn ) + P ¾(µ = 1j- n ; x k for all k 2 - n ) < 1 =) ¾n (I n ) = 0: 15 Selfish vs. Engineered Response: Star Topology (Cover) Hypothesis testing: If nodes communicate their observations: L(¢) = Likelihood Ratio 1X Pf L (si ) ¡ EL (S) > dg · en g( d) n i What if nodes communicate only their decisions: x i = f (si ) 1X Pf L (x i ) ¡ EL (x) > dg · en g1 ( d) n i 16 Selfishness and Herding Phenomenon: [Banerjee (92), BHW 92] Setup: Full network: - n = f 1; 2; :::; n ¡ 1g with probability 0.8 Absence of Collective Wisdom 17 Private Beliefs µ dF 0 (sn ) 1+ dF 1 (sn ) ¶¡ 1 Private Beliefs pn (sn ) = P ¾(µ = 1jsn ) = Definition: The private beliefs are called unbounded if sup s2 S dF0 (s) = 1 dF1 (s) and inf s2 S : dF 0 (s) = 0 dF 1 (s) If the private beliefs are unbounded, then there exist some agents with beliefs arbitrarily close to 0 and other agents with beliefs arbitrarily close to 1. Discrete example: with probability 0.8? 18 Expanding Observations Definition: A network topology f Q n gn 2 N is said to have expanding observations if for all ² > 0, and all K 2 R , there exists some N such that for all n ¸ N µ Qn ¶ max b < K b2 - < ² n Conversely Absence of Excessively Influential Agents 19 Influential References Vidyasagar Astrom/Murray New Book New Book Doyle, Francis Tannenbaum Kailath New Book Ljung New Book No “learning”! 20 Summary Expanding Observat ions Ot her Topologies Unbounded Beliefs YES NO Bounded Beliefs USUALLY NO, SOMET IMES YES NO If j- n j · M then learning is impossible for signals with bounded beliefs Line network It is possible to learn with expanding observations and bounded beliefs 21 Deterministic Networks: Examples Full topology: - Line topology: - n n = f 1; 2; :::; n ¡ 1g = f n ¡ 1g 22 Examples: A Random Sample Suppose each agent observes a sample C > 0 of randomly drawn (uniformly) decisions from the past. If the private beliefs are unbounded, then asymptotic learning occurs. 23 Examples: Binomial Sample (Erdős–Rényi) Suppose all links in the network are independent, and for two constants A and B we have Q n (m 2 - n ) = AB ; n If the beliefs are unbounded and B < 1, asymptotic learning occurs If B ¸ 1 , asymptotic learning does not occur 24 Active Research Just scratching the surface…. Presented a simple model of information aggregation Private signal Network topology More complex models for sequential decision making Dependent neighbors Heterogeneous preferences Multi-class agents Cyclic decisions Rationality Topology measures: depth, diameter, conductance Expanding observations Learning Rate Robustness 25