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Concepts of Game Theory I. 2 What are Multi-Agent Systems? Organisational relationship Interaction Agent Environment Spheres of influence.

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Presentation on theme: "Concepts of Game Theory I. 2 What are Multi-Agent Systems? Organisational relationship Interaction Agent Environment Spheres of influence."— Presentation transcript:

1 Concepts of Game Theory I

2 2 What are Multi-Agent Systems? Organisational relationship Interaction Agent Environment Spheres of influence

3 3 A Multi-Agent System Contains: A number of agents that interact through communication are able to act in an environment have different spheres of influence (which may coincide) will be linked by other (organisational) relationships.

4 4 Utilities of agents (1) Assume that we have just two agents: AG = {i, j } Agents are assumed to be self-interested: –They have preferences over environmental states

5 5 Utilities of agents (2) Assume that there is a set of outcomes that agents have preferences over: = { 1, 2, } –Example: odd-or-even game (alternative to head-or-tail) = {(0,0),…,(0,5),(1,0),…,(1,5),…(5,0),…,(5,5)} These preferences are captured by utility functions: u i : u j : –Example: odd-or-even game (alternative to head-or-tail) u even ((0,0)) = 1 u even ((0,1)) = 0 u even ((0,2)) = 1 … u odd ((0,0)) = 0 u odd ((0,1)) = 1 u odd ((0,2)) = 0 … Or, more simply, u even ((m,n)) = 1, if m +n is an even number; otherwise 0 u odd ((m,n)) = 0, if m +n is an even number; otherwise 1

6 6 Utilities of agents (2) Utility functions lead to preference orderings over outcomes: i means u i ( ) u i ( ) j means u j ( ) u j ( ) But, what is utility? In some domains, utility is analogous to money; e.g. we could have a relationship like this: Money Utility

7 7 Agent Encounters To investigate agent encounters we need a model of the environment in which agents act: –agents simultaneously choose an action to perform, –the actions they select will result in an outcome ; –the actual outcome depends on the combination of actions; Assume each agent has just two possible actions it can perform: – C (cooperate) – D (defect).

8 8 The State Transformer Function Lets formalise environment behaviour as: : Ac i Ac j Some possibilities: –Environment is sensitive to the actions of both agents: (D,D) 1 (D,C ) 2 (C,D) 3 (C,C ) 4 –Neither agent has influence on the environment: (D,D) (D,C ) (C,D) (C,C ) 1 –The environment is controlled by agent j. (D,D) 1 (D,C ) 2 (C,D) 1 (C,C ) 2

9 9 Rational Action (1) Suppose an environment in which both agents can influence the outcome, with these utility functions: u i ( 1 ) 1u i ( 2 ) 1 u i ( 3 ) 4 u i ( 4 ) 4 u j ( 1 ) 1u j ( 2 ) 4 u j ( 3 ) 1 u j ( 4 ) 4 Including choices made by the agents: u i ( (D,D)) 1 u i ( (D,C )) 1 u i ( (C,D)) 4 u i ( (C,C )) 4 u j ( (D,D)) 1 u j ( (D,C )) 4 u j ( (C,D)) 1 u j ( (C,C )) 4

10 10 Rational Action (2) Then, the preferences of agent i are: (C,C ) i (C,D) i (D,C ) i (D,D) C is the rational choice for i : –Agent i prefers outcomes that arise through C over all outcomes that arise through D.

11 11 Pay-off Matrices We can charaterise this scenario (& similar scenarios) as a pay-off matrix : Agent i is the column player Agent j is the row player i j DefectCoop Defect Coop

12 12 Dominant Strategies Given any particular strategy s (either C or D) for agent i, there will be a number of possible outcomes s 1 dominates s 2 if every outcome possible by i playing s 1 is preferred over every outcome possible by i playing s 2 A rational agent will never play a strategy that is dominated by another strategy –However, there isnt always a unique strategy that dominates all other strategies…

13 13 Nash Equilibrium Two strategies s 1 and s 2 are in Nash Equilibrium if: –under the assumption that agent i plays s 1, agent j can do no better than play s 2 ; and –under the assumption that agent j plays s 2, agent i can do no better than play s 1. Neither agent has any incentive to deviate from a Nash equilibrium!! Unfortunately: –Not every interaction has a Nash equilibrium –Some interactions have more than one Nash equilibrium… John Forbes Nash, Jr

14 14 Competitive and Zero-Sum Interactions When preferences of agents are diametrically opposed we have strictly competitive scenarios Zero-sum encounters have utilities which sum to zero:, u i ( ) u j ( ) 0 –Zero sum implies strictly competitive Zero sum encounters in real life are very rare –However, people tend to act in many scenarios as if they were zero sum.

15 15 The Prisoners Dilemma Two people are collectively charged with a crime –Held in separate cells –No way of meeting or communicating They are told that: –if one confesses and the other does not, the confessor will be freed, and the other will be jailed for three years; –if both confess, both will be jailed for two years –if neither confess, both will be jailed for one year Albert W. Tucker

16 16 Defect = confess; Cooperate = not confess Numbers in pay-off matrix are not years in jail They capture how good an outcome is for the agents –The shorter the jail term, the better The utilities thus are: u i (D,D) 2 u i (D,C ) 5 u i (C,D ) 0 u i (C,C ) 3 u j (D,D) 2 u j (D,C ) 0 u j (C,D ) 5 u j (C,C ) 3 The preferences are: (D,C ) i (C,C ) i (D,D) i (C,D ) (C,D ) j (C,C ) j (D,D) j (D,C ) The Prisoners Dilemma Pay-Off Matrix

17 17 The Prisoners Dilemma Pay-Off Matrix Top left: both defect, both get 2 years. Top right: i cooperates and j defects, i gets suckers pay-off, while j gets 5. –Bottom left is the opposite Bottom right: reward for mutual cooperation. i j DefectCoop Defect Coop Defect = confess Coop = not confess


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