Multi-Agent Influence Diagrams

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1 Multi-Agent Influence Diagrams CS 886: Multi-Agent Systems Multi-Agent influence diagrams for representing and solving games [1] Authored by D. Koller and B. Milch Tyler Nowicki Computer Science University of Waterloo

2 Why do we like MAIDs? 2

3 Why do we like MAIDs? Well.. these MAIDs won't make dinner, but they do clean up our games! MAID form games can be more compact and readable than an extensive form game. 3

4 Road Example 2n agents own plots on a road that is being built. When the road reaches a pair of plots the agents put up 1 of 3 buildings. MAID Form # variables ~ 6n Size Extensive Form # nodes ~ 3 2n (leaves) Agents 4

5 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 5

6 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 6

7 Bayesian Networks Probabilistic graphical model of a Set of variables x 1,...,x n where x i is restricted to a finite set dom(x i ) Also called var(x i ) Arcs connect variables Arrows imply parent/descendant relationship Conditional Probability Distribution (CPD) Given node X and its parents Pa(X) Pr(X Pa(X)) gives a distribution over the domain of X for each parent. 7

8 Example P(E) e 0 e Earthquake Burglary P(B) b 0 b P(R E) e 0 e 1 r 0 r Radio P(C A) a 0 a 1 c 0 c Alarm Call P(A B,E) b 0,e 0 b 0,e 1 b 1,e 0 b 1,e 1 a 0 a Reproduce of [1], Figure 1. 8

9 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 9

10 Flow of Influence Influence flows through the BN Flow is activated or blocked by observed variables When flow between two variables is blocked, they are independent (aka d-separated) Let E be evidence or observed variables Only 3 Cases to Consider Let x, y, z be variables 10

11 Case 1 Head-Tail Variable x z y Active if z E, block otherwise. Given z variables x and y become independent. x z y z E p(x,y,z) = p(x)p(z x)p(y z) z E p(x,y z) = p(x z)p(y z) 11

12 Case 2 Tail-Tail Variable x z y Active if z E, block otherwise. Given z variables x and y become independent. z E p(x,y,z) = p(x z)p(y z)p(z) z E p(x,y z) = p(x z)p(y z) z x y 12

13 Case 3 Head-Head Variable x z y Active if z or a descendant is in E, block otherwise. Given z variables x and y become dependent. Without z, x and y are independent. x y z z E p(x,y,z) = p(x)p(y) z E p(x,y z) = p(x)p(y)p(z x,y) 13

14 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 14

15 Influence Diagrams Variables: Decision (rectangle) Chance (oval) Utility (diamond) Decision variables are a choice (bet) Filled in with a CPD and Becomes a chance node. Chance variables are defined by the game Cards, dice, lady luck. Utility variables do not have children Payoff is defined by the game. 15

16 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 16

17 Multi-Agent Influence Diagrams Each players decisions and utilities are specified in the same game. Example Alice reasons about her building plans. 17

18 Tree Killer Example 18

19 MAID Terminology Given the MAID. A strategy profile is a set of CPD for a set of decision variables. Applying to = [ ] results in chance variables for the strategy profile. A strategy is fully mixed if the strategy defines a CPD for each decision variable. 19

20 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 20

21 Calculating Utility Where Ua be the set of utility variables for agent a. EU a = U U a u dom U P M [ ] U =u u Goal is to select an optimal to maximize utility for agent a. 21

22 Nash Equilibrium Let ε be a Let be a partial strategy profile over ε is optimal if for all partial strategies ' EU a, EU a, ' 22

23 Computing NE? Exponential blow-up prevents us from simply turn the MAID into a extensive form game. Lets look at that flow again. 23

24 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 24

25 Finding a Nash Equilibrium Basic Idea Transform the MAID into a relevance graph. Pick an arbitrary fully mixed strategy. Compute the optimal strategy for each decision rule in the relevance graph according to the topological ordering. Strategy has been optimized! 25

26 Relevance Graph What is a topological order? How do we construct the relevance graph? 26

27 Constructing a Relevance Graph Consists of decision nodes of MAID. Edges are formed only when two nodes are strategically reachable in. Is Acyclic! Topological ordering (or ancestral ordering) is derived from the relevance graph. 27

28 Constructing a Relevance Graph For each variable For all other variables ' Determine if ' is s-reachable from Then add edge ' to the graph Or Shachter's Bayes-Ball runs in linear time. 28

29 s-reachable Let U be any descendant utility variable of. Let * be a virtual parent of ' ' is s-reachable from if there exists an active path from * to U given and Pa( ). 29

30 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 30

31 Advantages & Disadvantages How does the requirement for a topological ordering affect the usefulness of this method? 31

32 Advantages & Disadvantages Influence diagram must be acyclic Single-player perfect recall games Multi-player perfect information games But only some imperfect information games! 32

33 Advantages & Disadvantages Games with cycles must be separated into a set of strongly connected components. Compact, readable, intuitive graphs. Requires care when designing the graph. Finds one Nash Equilibrium Left up to reader to find multiple. 33

34 Topics Bayesian Networks Flow of influence Influence Diagrams Multi-Agent Influence Diagrams Utility & Nash Equilibrium Finding a Nash Equilibrium Advantages & Disadvantages References & Questions 34

35 References 1. D. Koller and B. Milch, Multiagent influence diagrams for representing and solving games, Games and Economic Behavior, 45(1), p , Mudgal, C., Vassileva, J. An Influence Diagram Model for Multi- Agent Negotiation, Proceedings of the Fourth International Conference on MultiAgent Systems (ICMAS-2000), p , D. Koller, Structured models of complex decision problems, Invited talk at the First International Congress of the Game Theory Society, ( 4. Sargur Srihari, Lecture Notes on Graphical Models, CSE 574, University at Buffalo, Fall 2007 & ( 35

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