Game Theory 1 Definitions Game theory formal way

Game Theory 1

Definitions • Game theory -- formal way to analyze interactions among a group of rational agents behaving strategically • Agents – players in the game • Strategic behavior: account for interdependence in behavior (I know that you know that I know…. . ) 2

• Rational – When accounting for interdependence, each player chooses their best action 3

Some simple examples • Go into restaurant with a group of people. – Situation 1: everyone pays for their own food – Situation 2: divide the bill evenly • Ultimatum game – $100 is on the table, Player 1 splits the pot between players 1 and 2, player 2 accepts or rejects the offer • You can bid on a jar of silver dollars. Unknown amount of dollars in the jar, highest bid wins the jar 4

Strategy • Complete set of contingent plans for playing the game • Detailed book on chess rather than just one move • In simple games we consider, strategy is one move – so one decision or action is the strategy 5

Games can vary • By move – Simultaneous or sequential • Payoffs – Zero sum or non zero sum • Players – 2 or many 6

Simple game • • 2 players 2 potential moves 4 potential outcomes Great for pedagogical reasons, but, also many real world examples • Can easily outline the players, strategies and payoffs in a simple 2 x 2 matrix 7

Player on horizontal is 1 st Payoff in the pair Firm B Option C Option D Option C πac, πbc πac, πbd Option D πad, πbc πad, πbd Firm A 8

New toy • Cost to firm is $5 • Two firms will sell • Can sell the toy at a high ($20/toy) or low ($10/toy) price • If prices are the same, firms split the market. • If you are the low price firm, you get the lions share of the market and hence profits 9

P 1 $10 $20 $10 20 Q 1 30 15 60 0 $150 $225 $300 $0 P 2 $10 $20 20 $20 Q 2 30 25 0 60 $150 $225 $0 $300 Profits 1 Profits 2 10

Firm Pricing Payoff Matrix Toys R Us Price High Price Low Price High 225, 225 0, 300 Price Low 300, 0 150, 150 Wal-Mart 11

Zero Sum Game • • Game of conflict – what you win, I lose Payoffs in any situation sum to zero Rock/paper/scissors You win, you get a $1, I win, I get a $1, tie no money is exchanges 12

Player B Rock Player A Paper Scissor 0, 0 -1, 1 1, -1 0, 0 13

Dominance • Dominant strategy – a strategy that work at least as well as any other one, no matter what the other player does – If one exists, will play the dominant strategy… – But, none may exist • To solve the game, eliminate dominated strategies • Key assumption – player is acting in own self interest – doing the same thing you are doing 14

Does Wal-Mart have a dominant strategy? Hold TRU action at “Price low”, what is best decision? Toys R Us Price High Price Low Price High 225, 225 0, 300 Price Low 300, 0 150, 150 Wal-Mart Hold TRU decision at Price High, what is best you can do? 15

• Dominant strategy equilibrium – where each player follows a dominant strategy 16

Does A have a dominant Strategy? Player B Rock Player A Paper Scissor 0, 0 -1, 1 1, -1 0, 0 17

Player 2 Up Player 1 Down Left Middle Right 2, 2 1, 1 4, 0 1, 2 4, 1 3, 5 18

Player 2 Up Player 1 Down Left Middle Right 2, 2 1, 1 4, 0 1, 2 4, 1 3, 5 19

• Look at player 2. In this case, M is a dominated strategy. If 1 plays Up, Left is optimal. If 1 plays down, Right is optimal • Now consider player 1. He knows player 2 will never take Middle so this is irrelevant • For player 1, up is a dominant strategy • Player 2 knows 1 will play up. Therefore 2’s only option is 2 20

Prisoner’s dilemma • 2 suspects thought to be involved in a crime arrested • Have enough evidence for a minor conviction, but police know they were involved in more major crimes • Put in separate rooms • Each suspect is offered a deal. – If they turn states evidence, get reduced sentence – If the other person turns states evidence, you get hung out to dry 21

Prisoner’s Dilemma Prisoner B Quiet Betrays Quiet -3, -3 -10, -1 Betrays -1, -10 -5, -5 Prisoner A 22

Is there a Dominant Strategy? Prisoner B Quiet Betrays Quiet -3, -3 -10, -1 Betrays -1, -10 -5, -5 Prisoner A 23

Notice 2 things • Although to betray is a dominant strategy, both parties could do better by keeping quiet • Therefore, the dominant strategy does not necessarily generate the optimal outcome 24

• Some games do not have dominant strategies • You will need a different solution concept 25

Game of Chicken Buzz Gunderson Jim Stark Swerve Don’t Swerve 0, 0 -5, 5 Don’t 5, -5 -10, -10 26

Game of Chicken Buzz Gunderson Jim Stark Swerve Don’t Swerve 0, 0 -5, 5 Don’t 5, -5 -10, -10 27

Nash Equilibrium • No player has an incentive to deviate from equilibrium play because their strategy is best response to her belief about the other player’s strategy • Way to evaluate equilibrium or outcome of game • All dominant strategies are Nash Equilibriums but not vice versa 28

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Player 2 Up Player 1 Down Left Middle Right 2, 2 1, 1 4, 0 1, 2 4, 1 3, 5 30

Player 2 Up Player 1 Down Left Middle Right 2, 2 1, 1 4, 0 1, 2 4, 1 3, 5 31

Prisoner’s Dilemma Prisoner B Quiet Betrays Quiet -3, -3 -10, -1 Betrays -1, -10 -5, -5 Prisoner A 32

How is Nash Equilibrium explained in the movie? Youtube link 33

Game of Chicken Buzz Gunderson James Dean Swerve Don’t Swerve 0, 0 -5, 5 Don’t 5, -5 -10, -10 34

Battle of the Sexes • Male/female want to go to the movies – He prefers Die Hard IV – She prefers Becoming Jane – They prefer to go together • How should they decide what move to go to? 35

Battle of the Sexes Boyfriend Die Hard IV Becoming Jane Die Hard IV 5, 10 0, 0 Becoming Jane 2, 2 10, 5 Girlfriend 36

Battle of the Sexes Boyfriend Die Hard IV Becoming Jane Die Hard IV 5, 10 0, 0 Becoming Jane 2, 2 10, 5 Girlfriend 37

Nuclear Strategy • Two super powers with incredible nuclear arsenal • At any point in time, you can strike first or not • Suppose that in a first strike, the nuclear arsenal of the attacked will be destroyed 38

No second strike capability Russia US First Strike Not First Strike -10, -10 0, -8 Not -8, 0 0, 0 39

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Nuclear Deterrence Russia US First Strike Not First Strike -10, -10 -9, -8 Not -8, -9 0, 0 41

Cartel Pricing • Two oil produces • Each can produce either 2 or 4 million barrels per day • Iran can produce output at $2/barrel • Iraq can produce at $4/barrell • World output will be 6, 6 or 8 million barrels/day • Price/barrel in these three scenarios would be $25, $15, and $10 42

• Iran’s profits with low production • π =revenues – costs = $25(2) – 2(2) = $46 million • Iraq’s profits w/ low production • π = $25(2) – 4(2) = $42 million 43

Cartel Pricing Iraq Iran Produce Low Produce High Produce Low 46, 42 26, 44 Produce High 52, 22 32, 24 44

Cartel Pricing Iraq Iran Produce Low Produce High Produce Low 46, 42 26, 44 Produce High 52, 22 32, 24 45