1. Distributed Decision Making Jason L. Speyer UCLA MURI Kick Off Meeting May 14, 2000

2. Old Results and Current Thoughts  Distributed Estimation  Example: Target Association  Static Teams  Decentralized Control with Relaxed Communication  Static Games and Mixed Strategies  Static Team on Team

3. Distributed Estimation A historical viewpoint Spatially distributed measurements, but same dynamic system  Global Kalman filter estimates may be algebraically assembled from local Kalman filter estimates and a local correction term  These local transmissions can occur anytime  Speyer (1979) Distributed measurements and dynamic system  Generalized above results for Gauss-Markov systems  Willsky, Bello, Castanon, Levy, and Verghese (1982)

4. Distributed Estimation (cont.) Formulated the distributed Gauss-Markov estimation problem using the information state  Levy, Castanon, Verghese, and Willsky (1983) Distributed estimation algorithm extended to apply to the track association problem  Overlap of error variances used to associate track  New approach using fault detection ideas  Applies to passive radar (only bearings to target measured at each station)

5. Example of Distributed Estimation: Target Association

6. Target Association Problem Each Station has a “track” of the angle-only measurement history of each target Associate each track at one station to the track corresponding to the same target at another station Standard technique (Pao, et. Al):  Each station estimates the position and corresponding error envelope of each target  For each target at station 1, find the track at another station whose error envelope comes closet to that of station 1’s target  Each target’s position can be estimated using the Modified Gain Extended Kalman Filter (MGEKF)  Problem: the estimated position using only 1 station’s angular measurements may not be precise

7. Target Association Using Detection Methods Detection approach to track association  To associate targets between two stations  For each track at station 1, construct a bank of MGEKFs, each using data from that track and a track from station 2  Mismatched tracks bias the residual of the MGEKF  By contrast, matching tracks generate a good estimate with a small residual  To associate targets from an additional station  Use estimated target position from 1 st 2 stations  Parity test associates their tracks to those at additional stations

8. Stochastic Static Team Strategies Stochastic Static Teams  Minimize L is a convex function, x is the state of the world, z i (x) are local measurements, u i (z i ) are the team strategies  Radner (1962) showed that person by person optimality implied stationary  Optimal strategies to the static LQG problem are affine in the measurements  Hard to verify condition (local finiteness) was circumvented by Krainak, Speyer, and Markus (1982)  Optimal strategies to the static LEG problem are affine in the measurements

9. Simulation Results Detection filter residuals of two tracks: the diagonal plots are correctly associated Parity test of two tracks: the diagonal plots are correctly associated

10. Stochastic Dynamic Team Problems Decentralized stochastic control requires information patterns that allow a dynamic programming recursion  In the dynamic programming formalism, a stochastic static team problem is solved to determine the optimal strategies  Previous LQG and LEG solutions, using only a one-step delayed information sharing pattern, produced affine strategies  New results for the LQG problem uses a control only sharing information pattern  Local controller affine only in the local measurement history  Requires increasing the state of the world to include the process noise sequence  Appears to be the minimal information pattern which retains affine strategies

11. Static Game Problems Results of current work in resource allocation in an air campaign shows that mixed strategies are seemingly generic  General approach is to construct primitives so that a descretization of strategies can be determined  In the following simple games, a decomposition into primitives naturally occurs Static Game (Bryson and Ho)  Find the saddle point strategies of Where u minimizes and v  maximizes L  Note that L uu > 0 implies a pure saddle point strategy, u=0, L vv  > 0 implies a mixed strategy,

12. Static Games (cont.) Find the saddle point strategies of  Reduces to a matrix game of discrete primitives  Probabilities for mixed strategies obtained by Linear Programming Generalized: For measurements z 1 =z 1 (x) and z 2 =z 2 (x) (x state of the world) find the saddle point strategies where u(z 1 ) is to minimize and v(z 1 ) is to maximize E[L(x, u, v)]

13. Team on Team Stochastic Static Games  Strategies dependent on local information Generalize stochastic static games to many players on each side  Stochastic static team on team  Zero-sum games  None-zero sum games Solutions basis for approximation methods Consider implications to dynamic team on team