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Directed Gossip Algorithms, Consensus Problems, and Stability Effects of Noise Trading

Directed Gossip Algorithms, Consensus Problems, and Stability Effects of Noise Trading. Martin Schmalz (Univ. Stuttgart / Tokyo Institute of Technology) Prof. Masayuki Fujita, PhD (Tokyo Institute of Technology) Prof. Dr.-Ing. Oliver Sawodny (Univ. Stuttgart).

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Directed Gossip Algorithms, Consensus Problems, and Stability Effects of Noise Trading

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  1. Directed Gossip Algorithms, Consensus Problems, and Stability Effects of Noise Trading Martin Schmalz (Univ. Stuttgart / Tokyo Institute of Technology) Prof. Masayuki Fujita, PhD (Tokyo Institute of Technology) Prof. Dr.-Ing. Oliver Sawodny (Univ. Stuttgart)

  2. http://www.robertyin.com/fishschools/5820%20Fish%20Pork%20fish%20Anisotremus%20taeniatus%20School%20Cortez%209-2-.jpghttp://www.robertyin.com/fishschools/5820%20Fish%20Pork%20fish%20Anisotremus%20taeniatus%20School%20Cortez%209-2-.jpg Source: Olfati-Saber 2007 [C1] Overview Herding in Social and Economic Systems Consensus Problems Directed GossipAlgorithms Novel Interpretations of Herd Behavior and Noise Trading in Financial Markets Conclusions

  3. Source: Olfati-Saber 2007 [C1] Consensus Problems in Networked Multi-Agent Systems

  4. Consensus Problems – Introduction Cooperative Control Theory and Consensus Problems • [1] “Consensus means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents. A consensus algorithm (or protocol) is an interaction rule that specifies the information exchange between an agent and all of its neighbors on the network.” • More specific, a consensus algorithm is a decision rule that results in the convergence of the states of all network nodes to a common value. [01]: Olfati-Saber 2007

  5. „Rendez-vous in space“ Consensus of (scalar) space states Source: Olfati-Saber 2007 [C1] Consensus: Convergence of the states of all agents to a common value xi = xj = …= xconsensus Other technical applications:- sensor fusion in sensor networks - robot synchronization - distributed computation over networks

  6. Solution: Gossip Only one agent communicates with one other agent at a time But eventually, everybody knows the „news“ j i Consensus Problems – Information Consensus (Randomized) Gossip Algorithms • Computer Science (Boyd et al., 2006) – distributed computing • Computational Limitations Note: Same restrictions for human networks! http://static.howstuffworks.com/gif/brain-intro.gif • Communicational Limitations Sources: http://www.nos.org/htm/f9.gif

  7. Herding in Social and Economic Systems http://www.robertyin.com/fishschools/5820%20Fish%20Pork%20fish%20Anisotremus%20taeniatus%20School%20Cortez%209-2-.jpg http://www.pbs.org/wgbh/nova/stockmarket/images/traders.jpg

  8. Herding in Social and Economic Systems Relevance of Correlated Decision Making (CDM) in Markets • CDM leads to… • …deviations of prices from fundamental value •  misallocation of resources  prosperity of economy suboptimal • …bubbles and crashes • Synchronization(Abreu/Brunnermeier) • WHY? • Rational: • All make same decision if exposed to same information (Feng et al, 2006) • Boundedly Rational: • Social dynamics and fashions (Shiller), speculation… • Try to profit from others‘ information, uncertainty • Explanation: „contagion“ (Kirman) Common feature: Consensus of Decisions

  9. j i Herding in Social and Economic Systems Ants, Rationality, and Recruitment (Kirman 1993) • Ant colony – homogeneous agents • Symmetrical food sources • Asymmetric exploitation and switching • Reason: Distributed Computation, Local Communication • Graph-theoretic formalization of Kirman‘s ant model • Directed random network • Full adaptation of i‘s state to one neighbor j‘s state • Directed Gossip Algorithm Assymetricbehavioralthough central controller would split the groupsymmetrically Stochasticaggregate behavior although individuals actdeterministic

  10. j i Directed Gossip Algorithms

  11. Directed Gossip Algorithms Directed Gossip Algorithms • Set-up • discrete-time, discrete-value(xi∈ {0,1}) • x[k] = P[k] * x[k-1]where with probability 1/n * pij the random matrix P[k] isP = I – ei*(ej – ei)T • P…stochastic transition matrix: transfers x[k-1] to x[k] – equivalent to “Graph Laplacian” L in continuous-time case

  12. j=3 j=3 i=2 i=2 Directed Gossip Algorithms Directed Gossip Algorithms • P = I – ei * (ej – ei)T x[k] = P[k] * x[k-1] • Example: i=2, j=3 (random)  ei = [0 1 0 0]T, ej = [0 0 1 0]T 0 1 0 0 0 0 0 0 0 1 0 1 0 = 0 0 1 0 * 0 0 0 0 0 1 0 x[k] = P[k] * x[k-1] • i adapts to j’s state • All other nodes do not change their state • Corresponds to noise trading(Shleifer et al. -- only motivation is different)

  13. Probability of consensus formation # of agents n # of interactions Directed Gossip Algorithms Directed Gossip Algorithms • Consensus is guaranteed (!) •  The process necessarily gets stuck in one of the extremes {0,1}  DGA is a consensus protocol • Waiting time until consensus is • gamma-distributed • increases progressively with n • (Subject to ongoing research) • [ Technical Applications • Distributed Computing • Distributed Control of Multi-Vehicle Formations / Fish Schools / Bird Flocks • Wireless applications (communication is costly) ]

  14. Source: Kirman (1993) H-3 Remember the ants? Markets „The stock market is a manic depressive beast – it swings from expansive mania to fearful depression.“1 1: Alexander Elder: „Trading for a living“ http://www.progressiveart.com/bryan/stock_market.jpg

  15. Novel Interpretation of Investor Behavior & Markets Social Groups and Markets – Application of the ants model • 0 is a pessimistic, 1 is an optimistic attitude - „contagion“ through communication, social interaction • Substantiation for social influence (Shiller) and (randomly) following others: • Time pressure  can‘t calculate beliefs fast enough • „Stupid“  Investors unable to calculate beliefs • Uncertainty  Try to profit from other‘s information (and belief)

  16. Novel Interpretation of Investor Behavior & Markets Social Groups and Markets • More substantiation from crowd psychology • „The euphoria of London flows to New York, and the gloom of Tokyo infects Hong Kong.“1 • „Most people feel a strong urge to join the crowd and to act like everybody else.“1 • „They search for a leader, and react to emotions instead of using their intellect.“1 • „The greater the uncertainty, the stronger our wish to join and to follow.“1  e.g. escape panic (Helbing, Viczek) 1: Alexander Elder: „Trading for a living“, personal communication with traders of large investment banking firms

  17. Novel Interpretation of Investor Behavior & Markets Directed Gossip Algorithms Applied to Financial Markets • Representation of „following/mimicking others“ by Directed Gossip Algorithm • Interpretation of Consensus Proof: • Markets necessarily get stuck in pure optimism or pure pessimism • (Eigen-Dynamics of the market is instable!) • But: Markets do not get stuck in pure optimism / pessimism • So what is reality like? How can we drive the process to, e.g., x* = 0.5?  Introduce „leaders“ who never change their states  „Virtual Nodes“

  18. „Virtual World“ - VN „ Real World“ - RN Novel Interpretation of Investor Behavior & Markets Directed Gossip with External Influence • Introduction of „Virtual Nodes“ (VN) • VN never update their states  act as leaders • RN can also randomly access VN‘s states now • xavg,VN = 0.5 = const. m n

  19. p 0 1 0 1 0 1 m < β m = β m > β Novel Interpretation of Investor Behavior & Markets Directed Gossip and Markets • Characteristics of Stationary Markov Process: • Given by (in-)equality • No more equilibria, but equilibrium distribution, or PDF > < = mβ(n) β = 2 * (n-1) / (n-3)

  20. Novel Interpretation of Investor Behavior & Markets Real-World Interpretation • What are „Virtual Nodes“? • fundamental influnce (fundamental information / potential events) • Switching (0  1 / 1  0) corresponds to trading / volatility • Surprising result: • The higher the number of „stupid“ investors n in the market, the more „efficient“ will the market be! • Why? „Bigger swarms need fewer leaders.“

  21. Novel Interpretation of Investor Behavior & Markets Interpretation of „Bigger swarms need fewer leaders“ • #edges/agent = const. (directed gossip, humans) • „Diameter“ of the system increases with increasing number of nodes n • Convergence time is longer • Consensus is harder to reach in large systems • It is not the presence of destabilizing behavior (n) that destabilizes markets, but the absence of fundamentalists (m) is •  Prevent „decision monopolies“ •  Keep the number of effective market participants high • Alternative Interpretation: Thin market = few participants  thin markets more volatile than developed markets

  22. j i Source: Olfati-Saber 2007 [C1] Critique and Conclusions http://www.robertyin.com/fishschools/5820%20Fish%20Pork%20fish%20Anisotremus%20taeniatus%20School%20Cortez%209-2-.jpg http://www.progressiveart.com/bryan/stock_market.jpg

  23. j i Conclusion Conclusions: Directed Gossip • Generalization of „Gossip Algorithms“ to directed graphs • Suitable, e.g., for sensing applications, technical and non-technical • Found proof for consensus property of Directed Gossip • New as a purely graph theoretic consideration • Interpreation: Markets without sufficient fundamental information / influence necessarily converge to extreme attitudes • Interpreted model applied to financial markets • there must necessarily be volatility and trade volume •  the Eigen-Dynamics of the market is instable! •  stability comes from external information • Surprising result: Many noise traders  high efficiency

  24. Conclusion Critique • Humans can only listen to one particular source, but speak to many (no change of qualitative system behavior expected) • Communication Graph is not random • In particular: „hubs“, e.g. corporate financial reports (will make consensus faster – no change of qualitative system behavior) • Agents are not homogenous in real-world economic systems

  25. Outlook Further Research • Quantitative evaluation of Trade Volume, Volatility… Final Conclusion • A system (market) is NOT the pure sume of its subsystems (market participants).

  26. End Thank You!

  27. More Simulation Results

  28. Consensus Problems – Introduction Cooperative Control Theory and Consensus Problems • [01] “Cooperation [means] (…) giving consent to providing one’s state and following a common protocol that serves the group objective.” • Consensus problem one example of a cooperative control problem • global objective: reach consensus of the states of all agents • agents not necessarily conscious about group objective (!)  Nash Equilibria [01]: Olfati-Saber 2007

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