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**Concepts of Game Theory I**

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**What are Multi-Agent Systems?**

Organisational relationship Interaction Agent Spheres of influence Environment

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**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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**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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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: ui : uj : ueven((0,0)) = 1 ueven((0,1)) = 0 ueven((0,2)) = 1 … uodd((0,0)) = 0 uodd((0,1)) = 1 uodd((0,2)) = 0 … Or, more simply, ueven((m,n)) = 1, if m +n is an even number; otherwise 0 uodd((m,n)) = 0, if m +n is an even number; otherwise 1

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

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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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**The State Transformer Function**

Let’s formalise environment behaviour as: : Aci Acj 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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Rational Action (1) Suppose an environment in which both agents can influence the outcome, with these utility functions: ui (1) 1 ui (2) 1 ui (3) 4 ui (4) 4 uj (1) 1 uj (2) 4 uj (3) 1 uj (4) 4 Including choices made by the agents: ui ( (D,D)) 1 ui ( (D,C )) 1 ui ( (C,D)) 4 ui ( (C,C )) 4 uj ( (D,D)) 1 uj ( (D,C )) 4 uj ( (C,D)) 1 uj ( (C,C )) 4

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** (C,C ) i (C,D) i (D,C ) i (D,D)**

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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Pay-off Matrices We can charaterise this scenario (& similar scenarios) as a pay-off matrix : i Defect Coop 1 4 j Agent i is the column player Agent j is the row player

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Dominant Strategies Given any particular strategy s (either C or D) for agent i, there will be a number of possible outcomes s1 dominates s2 if every outcome possible by i playing s1 is preferred over every outcome possible by i playing s2 A rational agent will never play a strategy that is dominated by another strategy However, there isn’t always a unique strategy that dominates all other strategies…

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**Nash Equilibrium Two strategies s1 and s2 are in Nash Equilibrium if:**

under the assumption that agent i plays s1, agent j can do no better than play s2; and under the assumption that agent j plays s2, agent i can do no better than play s1. 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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**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: , ui () uj () 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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**The Prisoner’s 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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**The Prisoner’s Dilemma Pay-Off Matrix**

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: ui (D,D) 2 ui (D,C ) 5 ui (C,D ) 0 ui (C,C ) 3 uj (D,D) 2 uj (D,C ) 0 uj (C,D ) 5 uj (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 )

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**The Prisoner’s Dilemma Pay-Off Matrix**

Defect Coop 2 5 3 Defect = confess j Coop = not confess Top left: both defect, both get 2 years. Top right: i cooperates and j defects, i gets sucker’s pay-off, while j gets 5. Bottom left is the opposite Bottom right: reward for mutual cooperation.

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