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Concepts of Game Theory I

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

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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.

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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

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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

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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

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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).

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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

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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

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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.

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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

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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…

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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

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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.

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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

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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

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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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