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Game Theory Quad.Key PLAN • Classical games • introduction to basic notations and concepts • Repeated games GAME THEORY • games that are played repeatedly CSE 311A: Introduction to Intelligent Agents Using Science Fiction • programming assignment 1 is on repeated games Spring 2020 • Stackelberg and Bayesian games • games where one player moves before the other • games with uncertainty • Applications • application of game theory for social good 1 2 GAMES • Players 1 and 2 are given 4 chocolates • Player 1 proposes how to divide the chocolates. CLASSICAL GAME THEORY • Player 2 can either accept or reject the proposal. • If proposal is accepted, both players get the chocolates according to the proposal. • If proposal is rejected, both players get no chocolates. 3 4 GAMES GAMES • Players 1 and 2 have a choice: Be nice or be smart. • Players 1 and 2 are given 2 chocolates each. • If both players are nice, then they both get 2 chocolates. • Both players have a choice: heads or tails • If one player is nice and the other is smart, then the smart player • If both choose heads or both choose tails, then player 1 gets 1 gets 3 chocolates, the nice player gets no chocolate. chocolate from player 2 • If both players are smart, then they both get 1 chocolate. • If one chooses heads and the other chooses tails, then player 2 gets 1 chocolate from player 1. 5 6 INTRODUCTION INTRODUCTION • Definition of Game Theory • Definition of Game Theory • The analysis of competitive situations using mathematical models. • The analysis of competitive situations using mathematical models. • How can I maximize the number of chocolates I will get assuming my • How can I maximize the number of chocolates I will get assuming my opponent will do the same? opponent will do the same? • What is it about? • What is it about? A very important assumption: • Fundamentally about the study of decision-making • FundamentallyThat about all the players study are of decision-makingbehaving rationally! • Seeks to answer: • Seeks to answer: • What possible choices are there for each player? • What possible choices are there for each player? • What set of choices lead to good outcomes? • What set of choices lead to good outcomes? 7 8 INTRODUCTION HISTORY • Definition of Game Theory • John von Neumann (1903-1957) • The analysis of competitive situations using mathematical models. • Father of Game Theory • How can I maximize the number of chocolates I will get assuming my • Photographic memory opponent will do the same? • Inspired by poker and the concept of bluffing :) • Published Theory of Games and Economic Behavior • What is it about? with Oskar Morgenstern in 1944 • Fundamentally about the study of decision-making • Application of mathematical models to • Seeks to answer: broadly analyze games • What possible choices are there for each player? • What set of choices lead to good outcomes? • Worked in the Manhattan Project during WWII • Awarded the Presidential Medal of Freedom by Eisenhower in 1956 9 10 GAMES DEFINITIONS Player 2 • Players: Participants in the game • Rules: Specifi cation of the game nice smart • Players 1 and 2 have a choice: Be nice or be smart. • If both players are nice, then they both get 2 chocolates. nice 2,2 0,3 • If one player is nice and the other is smart, then the smart player gets 3 chocolates, the nice player gets no chocolate. Player 1 Player smart 3,0 1,1 • If both players are smart, then they both get 1 chocolate. • Normal form game: Common representation for simultaneous games. • Simultaneous games: Players make their moves at the same time. 11 12 DEFINITIONS DEFINITIONS Player 2 • Players: Participants in the game Player 2 • Players: Participants in the game • Rules: Specifi cation of the game • Rules: Specifi cation of the game nice smart • Moves: Possible actions for each nice smart • Moves: Possible actions for each player player nice 2,2 0,3 nice 2,2 0,3 • Payoffs: Amount received by each player at the end of the game Player 1 Player smart 3,0 1,1 1 Player smart 3,0 1,1 • Normal form game: Common • Normal form game: Common representation for simultaneous games. representation for simultaneous games. • Simultaneous games: Players make their • Simultaneous games: Players make their moves at the same time. moves at the same time. 13 14 DEFINITIONS DEFINITIONS Player 2 • Players: Participants in the game Player 2 • Players: Participants in the game • Rules: Specifi cation of the game • Rules: Specifi cation of the game nice smart • Moves: Possible actions for each nice smart • Moves: Possible actions for each player player nice 2,2 0,3 • Payoffs: Amount received by nice 2,2 0,3 • Payoffs: Amount received by each player at the end of the each player at the end of the game game Player 1 Player smart 3,0 1,1 1 Player smart 3,0 1,1 • Strategies: Set of move for • Strategies: Set of move for each possible situation in the each possible situation in the • Normal form game: Common game • Normal form game: Common game representation for simultaneous games. • In this simple game, strategy is a representation for simultaneous games. • Pure strategies: a move is • Simultaneous games: Players make their single move. • Simultaneous games: Players make their always chosen with certainty moves at the same time. moves at the same time. • Mixed strategies: moves are chosen according to some probability distribution 15 16 BEST RESPONSE BEST RESPONSE Player 2 • Players: Participants in the game Player 2 • Players: Participants in the game • Rules: Specifi cation of the game • Rules: Specifi cation of the game nice smart • Moves: Possible actions for each nice smart • Moves: Possible actions for each player player nice 2,2 0,3 • Payoffs: Amount received by nice 2,2 0,3 • Payoffs: Amount received by each player at the end of the each player at the end of the game game Player 1 Player smart 3,0 1,1 1 Player smart 3,0 1,1 • Strategies: Set of move for • Strategies: Set of move for each possible situation in the each possible situation in the • Normal form game: Common game • Normal form game: Common game representation for simultaneous games. • Pure strategies: a move is representation for simultaneous games. • Pure strategies: a move is • Simultaneous games: Players make their always chosen with certainty • Simultaneous games: Players make their always chosen with certainty moves at the same time. • Mixed strategies: moves moves at the same time. • Mixed strategies: moves are chosen according to some are chosen according to some probability distribution probability distribution 17 18 BEST RESPONSE BEST RESPONSE Player 2 • Players: Participants in the game Player 2 • Players: Participants in the game • Rules: Specifi cation of the game • Rules: Specifi cation of the game nice smart • Moves: Possible actions for each nice smart • Moves: Possible actions for each player player nice 2,2 0,3 • Payoffs: Amount received by nice 2,2 0,3 • Payoffs: Amount received by each player at the end of the each player at the end of the game game Player 1 Player smart 3,0 1,1 1 Player smart 3,0 1,1 • Strategies: Set of move for • Strategies: Set of move for each possible situation in the each possible situation in the • Normal form game: Common game • Normal form game: Common game representation for simultaneous games. • Pure strategies: a move is representation for simultaneous games. • Pure strategies: a move is • Simultaneous games: Players make their always chosen with certainty • Simultaneous games: Players make their always chosen with certainty moves at the same time. • Mixed strategies: moves moves at the same time. • Mixed strategies: moves are chosen according to some are chosen according to some probability distribution probability distribution 19 20 BEST RESPONSE DOMINANCE Player 2 • Players: Participants in the game Player 2 • Players: Participants in the game • Rules: Specifi cation of the game • Rules: Specifi cation of the game nice smart • Moves: Possible actions for each nice smart • Moves: Possible actions for each player player nice 2,2 0,3 • Payoffs: Amount received by nice 2,2 0,3 • Payoffs: Amount received by each player at the end of the each player at the end of the game game Player 1 Player smart 3,0 1,1 1 Player smart 3,0 1,1 • Strategies: Set of move for • Strategies: Set of move for each possible situation in the each possible situation in the • Normal form game: Common game Dominating• Normal formaction game: Best: Common response for all possiblegame actions of the opponent representation for simultaneous games. • Pure strategies: a move is representation for simultaneous games. • Pure strategies: a move is • Simultaneous games: Players make their always chosen with certainty • Simultaneous games: Players make their always chosen with certainty moves at the same time. • Mixed strategies: moves moves at the same time. • Mixed strategies: moves are chosen according to some are chosen according to some probability distribution probability distribution 21 22 NASH EQUILIBRIUM PRISONER’S DILEMMA Player 2 • Players: Participants in the game Player 2 • First formalized by Merill Flood • Rules: Specifi cation of the game and Melvin Dresher at RAND in nice smart co- defect 1950 • Moves: Possible actions for each operate player • Represents cold war scenarios: nice 2,2 0,3 • Payoffs: Amount received by cooperate R,R S,T • Nuclear arms rivalry each player at the end of the • Build hydrogen bomb (defect) or game to not build (cooperate) Player 1 Player smart 3,0 1,1 1 Player defect T,S P, P • Strategies: Set of move for • If both build, then both waste each possible situation in the money since neither will use Nash• Normal equilibrium form game: Both: Common players do not have gamean incentivize to deviate.
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