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![Page 1: 1 Game Theory. 2 Definitions Game theory -- formal way to analyze interactions among a group of rational agents behaving strategically Agents – players.](https://reader030.fdocuments.net/reader030/viewer/2022032517/56649c7f5503460f94936085/html5/thumbnails/1.jpg)
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Game Theory
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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…..)
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• Rational – When accounting for interdependence, each player chooses their best action
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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
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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
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Games can vary
• By move– Simultaneous or sequential
• Payoffs– Zero sum or non zero sum
• Players– 2 or many
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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
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Firm B
Option C Option D
Firm A
Option C πac,πbc πac,πbd
Option D πad,πbc πad,πbd
Player on horizontal is1st Payoff in the pair
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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
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P1 $10 $20 $10 20
Q1 30 15 60 0
Profits1 $150 $225 $300 $0
P2 $10 $20 20 $20
Q2 30 25 0 60
Profits2 $150 $225 $0 $300
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Firm Pricing Payoff Matrix
Toys R Us
Price High Price Low
Wal-Mart
Price High 225,225 0,300
Price Low 300,0 150,150
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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
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Player B
Rock Paper Scissor
Rock 0,0 -1, 1 1, -1
Player A Paper 1, -1 0,0 -1, 1
Scissor -1, 1 1, -1 0,0
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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
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Toys R Us
Price High Price Low
Wal-Mart
Price High 225,225 0,300
Price Low 300,0 150,150
Does Wal-Mart have a dominant strategy?
Hold TRU action at “Price low”, what is best decision?
Hold TRU decision at Price High, what is best you can do?
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• Dominant strategy equilibrium – where each player follows a dominant strategy
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Player B
Rock Paper Scissor
Rock 0,0 -1, 1 1, -1
Player A Paper 1, -1 0,0 -1, 1
Scissor -1, 1 1, -1 0,0
Does A have a dominant Strategy?
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Player 2
Left Middle Right
Player 1
Up 2,2 1,1 4,0
Down 1,2 4,1 3,5
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Player 2
Left Middle Right
Player 1
Up 2,2 1,1 4,0
Down 1,2 4,1 3,5
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• Look at player 2. In this case, M is a dominated strategy. If 1plays 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
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Prisoner’s dilemma
• 2 suspects thought to be involved in a crime are 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
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Prisoner’s Dilemma
Prisoner B
Quiet Betrays
Prisoner A
Quiet -3,-3 -10,-1
Betrays -1,-10 -5,-5
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Is there a Dominant Strategy?
Prisoner B
Quiet Betrays
Prisoner A
Quiet -3,-3 -10,-1
Betrays -1,-10 -5,-5
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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
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• Some games do not have dominant strategies
• You will need a different solution concept
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Game of Chicken
Buzz Gunderson
Swerve Don’t
Jim
Stark
Swerve 0,0 -5, 5
Don’t 5, -5 -10,-10
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Game of Chicken
Buzz Gunderson
Swerve Don’t
Jim
Stark
Swerve 0,0 -5, 5
Don’t 5, -5 -10,-10
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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
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Player 2
Left Middle Right
Player 1
Up 2,2 1,1 4,0
Down 1,2 4,1 3,5
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Player 2
Left Middle Right
Player 1
Up 2,2 1,1 4,0
Down 1,2 4,1 3,5
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Prisoner’s Dilemma
Prisoner B
Quiet Betrays
Prisoner A
Quiet -3,-3 -10,-1
Betrays -1,-10 -5,-5
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How is Nash Equilibrium explained in the movie?
Youtube link
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Game of Chicken
Buzz Gunderson
Swerve Don’t
James Dean
Swerve 0,0 -5, 5
Don’t 5, -5 -10,-10
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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?
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Battle of the Sexes
Boyfriend
Die Hard IV
Becoming
Jane
Girlfriend
Die Hard IV
5,10 0,0
Becoming Jane
2,2 10,5
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Battle of the Sexes
Boyfriend
Die Hard IV
Becoming
Jane
Girlfriend
Die Hard IV
5,10 0,0
Becoming Jane
2,2 10,5
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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
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No second strike capability
Russia
First
Strike Not
US
First
Strike
-10,-10 0,-8
Not -8,0 0,0
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Nuclear Deterrence
Russia
First
Strike Not
US
First
Strike
-10,-10 -9,-8
Not -8,-9 0,0
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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
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• 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
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Cartel Pricing
Iraq
Produce
Low
Produce
High
Iran
Produce
Low
46, 42 26, 44
Produce
High
52,22 32, 24
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Cartel Pricing
Iraq
Produce
Low
Produce
High
Iran
Produce
Low
46, 42 26, 44
Produce
High
52,22 32, 24