CM30174 - CM50206 Introduction to Intelligent Agents Semester 1, - PDF document

What is Game Theory Strategic Games Repeated Games Extensive Games with Perfect Information CM30174 - CM50206 Introduction to Intelligent Agents Semester 1, 2010-11 Marina De Vos, Julian Padget Game Theory / 20101025 / version 0.6 October 25, 2010 De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 1 / 45 What is Game Theory Strategic Games Repeated Games Extensive Games with Perfect Information Authors/Credits for this lecture Primary author: Marina De Vos. Material sourced from Martin J. Osborne and Ariel Rubinstein, A course in game theory (MIT Press, 1994) [1]. De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 2 / 45 What is Game Theory Strategic Games Repeated Games Extensive Games with Perfect Information Game Theory Game Theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact Two basic assumptions: rationality strategy Application Domains: multi-agent systems, political competitions, theoretical computer science, distributions of tongue length in bees and tube length in flowers. Game theory uses mathematics to express its ideas formally First publications in this area are from Von Neumann and Morgenstern in 1944 De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 4 / 45

Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Strategic Games (I) A strategic game models a situation where several agents (called players) independently make a decision about which action to take, out of a limited set of possibilities The result of the actions is determined by the combined effect of the choices separately made by each player. A collection of actions, one for each player, is called a profile. A profile determines the outcome of the game. Each agent has a preference relation, � , over the set of outcomes. De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 6 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Strategic Games (II) Definition A strategic game is a tuple � N , ( A i ) i ∈ N , ( � i ) i ∈ N � where N is a finite set of players; for each player i ∈ N , A i is a nonempty set of actions that are available to her; and for each player i ∈ N , � i is a preference relation on A = × j ∈ N A j . An element a ∈ A is called a profile. For a profile a we use a i to denote the action of agents i , while a − i denotes the actions taken by the other agents. De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 7 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies An Example Assume we just have two agents N = { 1 , 2 } with each two actions A i = { C , D } . Then we have four outcomes or profiles A = { CC , DD , CD , DC } CC ≻ 1 DD ≻ 1 CD � 1 DC DD ≻ 2 CC ≻ 2 CD � 2 DC De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 8 / 45

Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Payoff/Utility and Preference The preference relation is often too abstract and therefore replaced by a payoff or utility function u i for each agent. u i : A → R such that u i ( a ) ≥ u i ( b ) iff a � i b , for each a , b ∈ A . For two players this gives a nice representation, called the payoff matrix: C D s C 3,1 0,0 D 0,0 1,3 De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 9 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies The Prisoner’s Dilemma Two men are collectively charged with a crime and held in separate cells, with 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, then each will be jailed for two years Both prisoners know that if neither confesses, then they will each be jailed for one year. De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 10 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies The Prisoner’s Dilemma (II) Payoff matrix for this game could be: Silent Confess Silent 2,2 0,3 Confess 3,0 1,1 Top Left: If both remain silent, the both are rewarded for mutual cooperation. Top Right: If the first cooperates by remaining silent and the second confesses, the first gets the sucker’s payoff of zero, while the second gets the maximum payoff. Bottom Left: Reverse of Top Right. Bottom Right: If both confess, they both get punished for mutual defection. De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 11 / 45

Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Nash Equilibrium Playing a game � N , ( A i ) i ∈ N , ( � i ) i ∈ N � consists of each player i ∈ N selecting a single action from the set of actions A i available to her. Since players are thought to be rational, it is assumed that a player will select an action that leads to a “preferred” profile. Definition A Nash equilibrium of a strategic game � N , ( A i ) i ∈ N , ( � i ) i ∈ N � is a profile a ∗ satisfying ∀ a i ∈ A i · ( a ∗ − i , a ∗ i ) ≥ i ( a ∗ − i , a i ) . De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 12 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Find the Nash Equilibria (1) C D C D A 4,4 1,1 A 1,0 2,1 B 3,4 2,4 B 2,1 3,1 De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 13 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Find the Nash Equilibria (2) C D A B A 3,3 2,4 A -1,-1 1,2 B 2,1 -1,-1 B 1,2 4,1 De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 14 / 45

Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Find the Nash Equilibria (3) A B C H T A 1,3 2,4 1,3 H 1,-1 -1,1 B 1,2 2,1 4,2 T -1,1 1,-1 C 3,1 1,2 1,4 De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 15 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Nash Equilibria: Problems Rational agents have no incentive to deviate from a Nash equilibrium. Unfortunately: Not every interaction scenario has a Nash equilibrium Some interaction scenarios have more than one Nash equilibrium, possibly resulting in an outcome which is not a Nash equilibrium while every agent plays an action belonging to a Nash equilibrium. De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 16 / 45 Definition What is Game Theory The Prisoner’s Dilemma Strategic Games Nash Equilibria Repeated Games Best Response Function Extensive Games with Perfect Information Dominated Strategies Best Response Function Definition Let G = � N , ( A i ) i ∈ N , ( � i ) i ∈ N � be a strategic game. For any a − i ∈ A − 1 , we define the best response function of player i as the set-valued function B i ( a − i ) such that: B i ( a − i ) = { a i ∈ A i | ( a − i , a i ) � i ( a − i , a ′ i ) , ∀ a ′ i ∈ A i } . Definition A Nash equilibrium of a strategic game � N , ( A i ) i ∈ N , ( � i ) i ∈ N � is a profile a ∗ satisfying: a ∗ i ∈ B i ( a ∗ − i ) , ∀ i ∈ N De Vos/Padget (Bath/CS) CM30174/Game Theory October 25, 2010 17 / 45