1 Game Theory. 2 Definitions Game theory -- formal way to analyze interactions among a group of rational agents behaving strategically Agents – players.

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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