Game Theory

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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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Player on horizontal is 1st Payoff in the pair
Firm B Firm B
Option C Option D
Firm A Option C pac,pbc pac,pbd
Firm A Option D pad,pbc pad,pbd
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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 Toys R Us
Price High Price Low
Wal-Mart Price High 225,225 0,300
Wal-Mart 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 Player B 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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Does Wal-Mart have a dominant strategy?
Hold TRU action at Price low, what is best
decision?
Toys R Us Toys R Us
Price High Price Low
Wal-Mart Price High 225,225 0,300
Wal-Mart Price Low 300,0 150,150
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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Does A have a dominant Strategy?
Player B Player B 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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Player 2 Player 2 Player 2
Left Middle Right
Player 1 Up 2,2 1,1 4,0
Player 1 Down 1,2 4,1 3,5
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Player 2 Player 2 Player 2
Left Middle Right
Player 1 Up 2,2 1,1 4,0
Player 1 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 2s
  only option is 2

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Prisoners 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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Prisoners Dilemma
Prisoner B Prisoner B
Quiet Betrays
Prisoner A Quiet -3,-3 -10,-1
Prisoner A Betrays -1,-10 -5,-5
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Is there a Dominant Strategy?
Prisoner B Prisoner B
Quiet Betrays
Prisoner A Quiet -3,-3 -10,-1
Prisoner A 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 Buzz Gunderson
Swerve Dont
Jim Stark Swerve 0,0 -5, 5
Jim Stark Dont 5, -5 -10,-10
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Game of Chicken
Buzz Gunderson Buzz Gunderson
Swerve Dont
Jim Stark Swerve 0,0 -5, 5
Jim Stark Dont 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 players
  strategy
• Way to evaluate equilibrium or outcome of game
• All dominant strategies are Nash Equilibriums but
  not vice versa

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(No Transcript)
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Player 2 Player 2 Player 2
Left Middle Right
Player 1 Up 2,2 1,1 4,0
Player 1 Down 1,2 4,1 3,5
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Player 2 Player 2 Player 2
Left Middle Right
Player 1 Up 2,2 1,1 4,0
Player 1 Down 1,2 4,1 3,5
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Prisoners Dilemma
Prisoner B Prisoner B
Quiet Betrays
Prisoner A Quiet -3,-3 -10,-1
Prisoner A 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 Buzz Gunderson
Swerve Dont
James Dean Swerve 0,0 -5, 5
James Dean Dont 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 Boyfriend
Die Hard IV Becoming Jane
Girlfriend Die Hard IV 5,10 0,0
Girlfriend Becoming Jane 2,2 10,5
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Battle of the Sexes
Boyfriend Boyfriend
Die Hard IV Becoming Jane
Girlfriend Die Hard IV 5,10 0,0
Girlfriend 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 Russia
First Strike Not
US First Strike -10,-10 0,-8
US Not -8,0 0,0
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(No Transcript)
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Nuclear Deterrence
Russia Russia
First Strike Not
US First Strike -10,-10 -9,-8
US 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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• Irans profits with low production
• p revenues costs
• 25(2) 2(2) 46 million
• Iraqs profits w/ low production
• p 25(2) 4(2) 42 million

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