1 game theory. game theory definition the study of strategic decision making. more formally, it is...

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Game Theory

Game Theory Definition

The study of strategic decision making. More formally, it is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.

Game Theory is used to analyze how firms interact but has many other applications.

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Other Applications of Game Theory National Defense – Terrorism and Cold War Movie Release Dates and Program

Scheduling Auctions http://en.wikipedia.org/wiki/Spectrum_auction

http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction

Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent

on campaigning Nanny Monitoring Group of Birds Feeding Mating Habits

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Grey’s Anatomy vs. The DonaldNBC delays 'Apprentice' premiereBy Nellie Andreeva

NBC is taking the premiere of "Celebrity Apprentice" out of the cross-hairs of the last original episode of ABC's "Grey's Anatomy"... or so it seems.NBC on Wednesday said that it will push the launch of "Apprentice" from Jan. 3 to Jan. 10, expanding "Deal or No Deal" to two hours on Thursday, Jan. 3.The move follows ABC's midseason schedule announcement Friday that included the last original episode of "Grey's" airing Jan. 3,…

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Grey’s Anatomy vs. The Donald'Grey' move has NBC red Peacock shifts 'Apprentice' backBy Nellie Andreeva

The Thursday night scheduling tango between NBC and ABC continued Thursday morning when ABC officially announced that it will move the last original episode of "Grey's Anatomy" from Jan. 3 to Jan. 10.That led to a reversal in NBC's Wednesday decision to push the premiere of "Celebrity Apprentice" from Jan. 3 toJan. 10 to avoid the first-run "Grey's."NBC said Thursday afternoon that "Apprentice," hosted by Donald Trump, will now launch Jan. 3 as originally planned.

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Game Theory and Movie Release DatesThe Imperfect Science of Release DatesNew York Times

On Dec. 25, which this year happens to be a Thursday, five new movies will be released in theaters -- six, if you count a new Disney IMAX film called ''Young Black Stallion.'' As with the Fourth of July and Thanksgiving, there is a special cachet to opening a film on Christmas Day…. The casual moviegoer rarely ponders why a particular bubbly romantic comedy, serial-killer thriller, literary costume drama or animated talking-farm-animals movie opens on the day it does. Movies come; movies go; movies wind up on video. To those responsible for putting those films on the screen, however, nothing about the timing of their releases is arbitrary.

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Game Theory and Movie Release Dates (cont.)

Last December featured one of the most dramatic games of chicken in recent memory, when two films starring Leonardo DiCaprio were both slated to open on Christmas weekend. Ultimately, Miramax blinked first, moving the release of Martin Scorsese's ''Gangs of New York'' five days earlier and ceding the holiday to the other DiCaprio film, DreamWorks' ''Catch Me if You Can.'' ''We didn't think about moving,'' says Terry Press, the head of marketing for DreamWorks. ''We had been there first, and 'Catch Me if You Can' was perfect for that date.'' This year, DreamWorks chose to schedule a somber psychological drama, ''House of Sand and Fog,'' for the day after Christmas, deferring a bit to Miramax. ''I don't want our reviews to run on the same day as 'Cold Mountain,''' Press says.

Ever wonder why a movie theater shows a preview of an upcoming movie that is to be released in 2 years?

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Other Applications of Game Theory National Defense – Terrorism and Cold War Movie Release Dates and Program

Scheduling Auctions http://en.wikipedia.org/wiki/Spectrum_auction

http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction

Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent on

campaigning Nanny Monitoring Group of Birds Feeding Mating Habits Parenting

Game Theory Secrets for ParentsWall Street Journal July, 2014

The party is over, and you're down to the last bit of cake. All three of your children want it. If you're familiar with game theory, you might think of the classic strategy in which one person cuts the cake and the other chooses the slice. But how do you divide it three ways without anyone throwing a fit?

Game theory is, in essence, the science of strategic thinking—a way of making the best decision possible based on the way you expect other people to act. It was once the domain of Nobel Prize-winning economists and big thinkers on geopolitics, but now parents are getting in on the act. Though game theory assumes, as a technical matter, that its players are rational, it applies just as well to not-always-rational children.

A key lesson in game theory, says Barry Nalebuff, a professor at the Yale School of Management, is to understand the perspective of the other players. It isn't about what you would do in another person's shoes, he says; it's about what they would do in their shoes. "Good game theory," he says, "appreciates the quirks and features that make us unique and takes us as we are." The same could be said of good parenting.

So how to deal with the problem of dividing a piece of cake into three equal shares? Try this: After the first child cuts and the second one chooses, each child further cuts his or her own slice into thirds. The third child then chooses a third of a slice from each plate. It might get messy, but all three should feel fairly treated.

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Game Theory Secrets for ParentsWall Street Journal July, 2014

Credible Punishments: In game theory as in parenting, you have to deliver on your threats, like actually turning off the TV if you said you were going to, even if it punishes you too. Joshua Gans, an economist at the University of Toronto and the author of "Parentonomics," offers advice for gaining a credible reputation at home. When his children were young and would disobey, he would say, "I'm thinking of a punishment." It's much easier to pretend to think of a punishment than to come up with a new one every time, he notes—or, worse, to issue a noncredible threat in the heat of the moment. ("That's it, I'm canceling Christmas!") Once he earned his credibility, he found that he had only to close his eyes and count to 10, and his children would spring into action.

The Bedtime Ultimatum: For shortening the bedtime routine with several children to tuck in, one parent advises using an ultimatum game of take-it-or-leave-it. Before bed, just have the children play rocks, paper, scissors and allow the winning child to choose the book. If the others don't agree with the choice, no one gets a story.

Sleep Training 101: Game theory can work from the earliest days of parenthood. Prof. Nalebuff applies the concept of backward induction to help new mothers get some sleep. If a mother repeatedly gets up in the middle of the night with the child, he explains, eventually the child will only respond to the mother comforting him. Instead, mothers should look forward and reason backward: If you ever want your husband to get up in the middle of the night, then you have to get him involved at the very start. Everyone, it seems, needs to be sleep trained.

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Game Theory Terminology Simultaneous Move Game – Game in

which each player makes decisions without knowledge of the other players’ decisions (ex. Cournot or Bertrand Oligopoly).

Sequential Move Game – Game in which one player makes a move after observing the other player’s move (ex. Stackelberg Oligopoly).

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Game Theory Terminology

Strategy – In game theory, a decision rule that describes the actions a player will take at each decision point.

Normal Form Game – A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.

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Example 1: Prisoner’s Dilemma(Normal Form of Simultaneous Move Game)

Martha’s options

Don’t Confess Confess

Peter’s Options

Don’t Confess M: 2 years P: 2 years

M: 1 yearP: 10 years

Confess M: 10 yearsP: 1 year

M: 6 years P: 6 years

What is Peter’s best option if Martha doesn’t confess?

What is Peter’s best option if Martha confess?

Confess (1<2)Confess (6<10)

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Example 1: Prisoner’s Dilemma

Martha’s options


Peter’s Options

Don’t Confess M: 2 years P: 2 years

M: 1 yearP: 10 years

Confess M: 10 yearsP: 1 year

M: 6 years P: 6 years

What is Martha’s best option if Peter doesn’t confess?

What is Martha’s best option if Peter Confesses?

Confess (1<2)Confess (6<10)

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Example 1: Prisoner’s Dilemma

Martha’s options


Peter’s Options

Don’t Confess 2 years , 2 years 10 years , 1 year

Confess 1 year , 10 years 6 years , 6 years

Dominant Strategy – A strategy that results in the highest payoff to a player regardless of the opponent’s action.

First Payoff in each “Box” is Row Player’s Payoff .

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Example 2: Price Setting Game

Firm B’s options

Low Price High Price

Firm A’s Options

Low Price 0 , 0 50 , -10

High Price -10 , 50 10 , 10

Is there a dominant strategy for Firm B? Is there a dominant strategy for Firm A?

Low Price Low Price

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Nash Equilibrium

A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other player’s strategy. (Every player is doing the best they possibly can given the other player’s strategy.)

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Example 1: Nash?

Martha’s options


Peter’s Options

Don’t Confess 2 years , 2 years 10 years , 1 year

Confess 1 year , 10 years 6 years , 6 years

Nash Equilibrium: (Confess, Confess)

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Example 2: Nash?

Firm B’s options

Low Price High Price

Firm A’s Options

Low Price 0 , 0 50 , -10

High Price -10 , 50 10 , 10

Nash Equilibrium: (Low Price, Low Price)

Chump, Chump, Chump

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http://videosift.com/video/Game-Theory-in-British-Game-Show-is-Tense?loadcomm=1

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EXAMPLE 3: Entry into a fast food market:

Burger King’s options

Enter Skaneateles

Don’t Enter Skaneateles

McDonalds’ Options

Enter Skaneateles BK= -40

M= -30

BK= 0

M= 50

Don’t Enter Skaneateles

BK= 40

M= 0

BK= 0

M= 0

Is there a dominant strategy for BK? NO Is there a dominant strategy for McD? NO

Is there a Nash Equilibrium(ia)?

Yes, there are 2 – (Enter, Don’t Enter) and (Don’t Enter, Enter). Implies, no need for a dominant strategy to have NE.

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EXAMPLE 4: Monitoring Workers

Worker’s options

Work Shirk

Manager’s Options

Monitor W: 1M: -1

W: -1M: 1

Don’t Monitor W: -1M: 1

W: 1M: -1

Is there a dominant strategy for the worker? Is there a dominant strategy for the manager?

NO NO

Is there a Nash Equilibrium(ia)? Not a pure strategy Nash Equilibrium– player chooses to take one action with probability 1

Randomize the actions yields a Nash = mixed strategy

John Nash proved an equilibrium always exists

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Mixed (randomized) Strategy

Definition:

A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her actions.

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Calculating Mixed Strategy EXAMPLE 4: Monitoring Workers Manager randomizes (i.e. monitors with

probability PM) in such a way to make the worker indifferent between working and shirking.

Worker randomizes (i.e. works with probability Pw) in such a way as to make the manager indifferent between monitoring and not monitoring.

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Example 4: Mixed Strategy

Worker’s options

Work Shirk

Manager’s Options

Monitor W: 1M: -1

W: -1M: 1


W: 1M: -1

PM

1-PM

PW1-PW

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Manager selects PM to make Worker indifferent between working and shirking (i.e., same expected payoff)

Worker’s expected payoff from working

PM*(1)+(1- PM)*(-1) = -1+2*PM

Worker’s expected payoff from shirking

PM*(-1)+(1- PM)*(1) = 1-2*PM

Worker’s expected payoff the same from working and shirking if PM=.5. This expected payoff is 0 (-1+2*.5=0 and 1-2*.5=0). Therefore, worker’s best response is to either work or shirk or randomize between working and shirking.

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Worker selects PW to make Manager indifferent between monitoring and not monitoring.

Manager’s expected payoff from monitoring

PW*(-1)+(1- PW)*(1) = 1-2*PW

Manager’s expected payoff from not monitoring

PW*(1)+(1- PW)*(-1) = -1+2*PW

Manager’s expected payoff the same from monitoring and not monitoring if PW=.5. Therefore, the manager’s best response is to either monitor or not monitor or randomize between monitoring or not monitoring .

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Nash Equilibrium of Example 4

Worker works with probability .5 and shirks with probability .5 (i.e., PW=.5)

Manager monitors with probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5)Neither the Worker nor the Manager can increase their expected payoff by playing some other strategy (expected payoff for both is zero). They are both playing a best response to the other player’s strategy.

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Example 4A: What if costs of Monitoring decreases and Changes the Payoffs for Manager

Worker’s options

Work Shirk

Manager’s Options

Monitor W: 1M: -1

W: -1M: 1


W: 1M: -1

-.5 1.5

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Nash Equilibrium of Example 4A where cost of monitoring decreased Worker works with probability .625 and

shirks with probability .375 (i.e., PW=.625) Same as in Ex. 5, Manager monitors with

probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5)

The decrease in monitoring costs does not change the probability that the manager monitors. However, it increases the probability that the worker works.

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Example 5: Mixed Strategy and Tennis http://www.fuzzyyellowballs.com/introducing-the-fyb-strategy-quiz/

Game:Server’s Possible Strategies:

(Serve Left , Serve Right)

Receiver’s Possible Strategies: (Defend Left , Defend Right)

Receiver has a high probability of winning the point if she defends the side the server serves to.

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Example 5: Mixed Strategy and Tennis(Payoffs are probability of winning point)

Receiver

Defend Left Defend Right

Server Serve Left .6 , .4 .75 , .25

Serve Right .7 , .3 .55 , .45

SL

1-SL

DL 1-DL

Mixed Strategy Equilibrium (SL =.5 , DL =.67)This results in the probability of the server winning the point to be .65 irrespective of whether he serves to the left or right.

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Example 5: Mixed Strategy and TennisWhat about the Real World?

Minimax Play at WimbletonWalker and Wooders (AER 2001)

http://www.finance.uts.edu.au/staff/johnwooders/WimbledonAER.pdf

“We use data from classic professional tennis matches to provide an empirical test of the theory of mixed strategy equilibrium. We find that the serve-and-return play of John McEnroe, Bjorn Borg, Boris Becker, Pete Sampras and others is consistent with equilibrium play.”

Results: Probability Server wins is the same whether serve right or left. Which side server serves is not “serially independent”.

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Example 6

A Beautiful Mind

http://www.youtube.com/watch?v=CemLiSI5ox8

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Example 6: A Beautiful Mind

Nash Equilibria: (Pursue Blond, Pursue Brunnette 1) (Pursue Blond, Pursue Brunnette 2) (Pursue Brunnette 1, Pursue Blond) (Pursue Brunnette 2, Pursue Blond)

Other Student’s Options

Pursue

Blond

Pursue

Brunnette 1

Pursue

Brunnette 2

John Nash’s

Pursue

Blond0 , 0 100 , 50 100 , 50

Options Pursue

Brunnette 150 , 100 0 , 0 50 , 50

Pursue

Brunnette 250 , 100 50 , 50 0 , 0

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Sequential/Multi-Stage Games

Extensive form game: A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoffs resulting from alternative strategies.

(Often used to depict games with sequential play.)

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Potential Entrant

Don’t Enter Enter

Incumbent Firm

Price War Share Market (Hard) (Soft)

Potential Entrant: -1 +5Incumbent: +1 +5

Potential Entrant: 0 Incumbent: +10

Example 7

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Potential Entrant

Don’t Enter Enter

Incumbent Firm Potential Entrant

and so on….

Example 7: With “Downstream” Actions

PIM1

PIM2 PID1

PE1

Incumbent Firm Incumbent Firm

Potential Entrant

PE2

Suppose each period the incumbent sets the optimal price as a monopolist and maximizes the present discounted value of profits which is +10.

and so on…. The present discounted value of profits for the incumbent and potential entrant depends on their strategies.

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Potential Entrant

Don’t Enter Enter

Incumbent Firm




Example 7

What are the Nash Equilibria?

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Nash Equilibria

1. (Potential Entrant Enter,

Incumbent Firm Shares Market)

2. (Potential Entrant Don’t Enter, Incumbent Firm Price War)

Is one of the Nash Equilibrium more likely to occur? Why?

Perhaps (Enter, Share Market) because it doesn’t rely on a non-credible threat.

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Subgame Perfect Equilibrium

A condition describing a set of strategies that constitutes a Nash Equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies.

(Basically eliminates all Nash Equilibria that rely on a non-credible threat – like Don’t Enter, Price War in Prior Game)

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Potential Entrant

Don’t Enter Enter

Incumbent Firm




Example 7

What is the Subgame Perfect Equilibrium?

(Enter, Share Market)

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Big Ten Burrito

Enter Don’t Enter

Chipotle Chipotle

Enter Don’t Enter Don’tEnter Enter

BTB: -25 +40 0 0Chip: -50 0 +70 0

Example 8

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Big Ten Burrito

Enter Don’t Enter

Chipotle Chipotle

Enter Don’t Enter Don’tEnter Enter

BTB: -25 +40 0 0Chip: -50 0 +70 0

Use Backward Induction to Determine Subgame Perfect Equilibrium.

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Subgame Perfect Equilibrium

Chipotle should choose Don’t Enter if BTB chooses Enter and Chipotle should choose Enter if BTB chooses Don’t Enter.

BTB should choose Enter given Chipotle’s strategy above.

Subgame Perfect Equilibrium:

(BTB chooses Enter, Chipotle chooses Don’t Enter if BTB chooses Enter and Enter if BTB chooses Don’t Enter.)

Example 9: Limit Pricing

When a firm sets it price and output so that there is not enough

demand left for another firm to enter the market profitably.

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Lower Price, PL Monopoly Price, PM

Potential Potential Entrant Entrant Don’t Don’t

Enter Enter Enter Enter

PE: -1 +5 0 0 -1 +5 0 0

Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10

Example 9:Incumbent (suppose monopolist)

Hard Soft PL PM Hard Soft PL PM Ball Ball Ball Ball

Incumbent Incumbent

Note: Incumbent’s profits are $10 per period if set monopoly price and $8 per period if set lower price. What price the incumbent sets initially does not influence second period profits for incumbent or potential entrant. For simplicity, second period payoffs are not discounted.

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PE: -1 +5 0 0 -1 +5 0 0

Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10

Example 9:Incumbent (suppose monopolist)


Incumbent Incumbent


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PE: -1 -.5 0 0 -1 -.5 0 0

Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10

Example 9a:Incumbent (suppose monopolist)


Incumbent Incumbent


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PE: -1 -.5 0 0 -1 -.5 0 0

Inc: 8+1 8+5 8+8 8+10 10+1 10+5 10+8 10+10

Example 9a:Incumbent (suppose monopolist)


Incumbent Incumbent


Questions:

1. Can you think of examples where the price the incumbent sets the first period could influence second period profits of the incumbent and perhaps the entrant?

2. Are there other actions the incumbent can take prior to the potential entrant’s entry decision that could influence this decision? (R&D, Capital Investment, Lobbying, etc.)

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Predatory Pricing

Definition: When a firm first lowers its price in order to drive rivals out of business (and scare off potential entrants), and then raises its price when its rivals exit the market.

What insights does the analysis on limit pricing provide for the logic of predatory pricing?

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Dan Conlin

Invest in Firm Don’t InvestSpecific Knowledge

Dan Conlin: wI-CI wDI

Marsh&McClennan: 200-wI 150-wDI

Example 10: The Hold-Up Problem

Dan Conlin and M&M negotiate salary


Let wI and wDI denote Dan’s wage if he invests and doesn’t invest in the firm specific knowledge, respectively. Let the cost of investing for Dan be CI and let CI=30. Dan Conlin is worth 200 to M&M if he invests and is worth 150 if he doesn’t.

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Dan Conlin


Dan Conlin: wI-CI wDI

Marsh&McClennan: 200-wI 150-wDI




Assume that Dan’s best “outside option” is a wage of 100 whether or not he invests in the firm specific knowledge and that the outcome of the negotiations are such that Dan and M&M split the surplus. This means that wI=150 and wDI=125.

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Dan Conlin


Dan Conlin: wI-CI=150-30 wDI=125Marsh&McClennan: 200-wI=200-150 150-wDI=150-125




Subgame Perfect Equilibrium outcome has Dan Conlin not investing in the firm specific knowledge and receiving a wage of 125 even though the cost of the knowledge is 30 and it increases his value to the firm by 50.

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Dan Conlin


Dan Conlin: wI-CI=150-30 wDI=125Marsh&McClennan: 200-wI=200-150 150-wDI=150-125




What would you expect to happen in this case?

Dan Conlin and M&M would divide cost of obtaining the knowledge.

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Dan Conlin

Invest in Don’t InvestGeneral Knowledge

Dan Conlin: wI-CI=160-30 wDI =125Marsh&McClennan: 200-wI=200-160 150-wDI=150-125

Example 11: General Knowledge Investment



Assume the game is as in the “hold-up” problem but that Dan’s best “outside option” is a wage of 120 if he invests in general knowledge and 100 if he does not. This means that wI=160 and wDI=125 (assuming split surplus when negotiate).

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Dan Conlin

Invest in Don’t InvestGeneral Knowledge

Dan Conlin: wI-CI=160-30 wDI =125Marsh&McClennan: 200-wI=200-160 150-wDI=150-125

Example 11: General Knowledge Investment



Subgame Perfect Equilibrium outcome has Dan Conlin investing in the general knowledge and receiving a wage of 160.

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Example 12: Hold-up Problem (same idea as the Fisher Auto-body / GM situation)

Suppose there are two players: a computer chip maker (MIPS) and a computer manufacturer (Silicon Graphics). Initially, MIPS decides whether or not to customize its chip (the quantity of which is normalized to one) for a specific manufacturing purpose of Silicon Graphics. The customization costs $75 to MIPS, but adds value of $100 to the chip only when it is used by Silicon Graphics . The value of customization is partially lost when the chip is sold to an alternative buyer, who is willing to pay $60. If MIPS decides not to customize the chip, it can sell a standardized chip to Silicon Graphics at a price of zero and Silicon Graphics earns a payoff of zero from using the chip. If MIPS customizes the chip, the two players enter into a bargaining game where Silicon Graphics makes a take-it-or-leave-it price offer to MIPS. In response to this, MIPS can either accept the offer (in which case the game ends) or reject it (in which case MIPS approaches an alternative buyer who pays $60).

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MIPS

Customize Don’t Customize

MIPS: p-75 60-75= -15Silicon Graphics: 100-p 0

Example 12: Hold-Up Problem

Subgame Perfect Equilibrium – MIPS accepts price p if p>60. Silicone Graphics offers a price p=60. MIPS does not customize. The outcome of this game is that MIPS does not customize even though there is a surplus of $25 to be gained.

MIPS

Silicone Graphics

Accept Reject

Offer Price p

0 : MIPS

0 : Silicon Graphics

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Is the Hold-Up Problem Applicable to other Situations?1. Upstream Firm Investing in Specific Capital to produce

input for Downstream Firm.

Coal Mines located next to Power Plants.

2. An academic buying a house before getting tenure or a big promotion.

3. Taxing of Oil and Gas Lines by local jurisdictions.

4. Multinational firms operating in foreign countries (Foreign Direct Investment)

5. East Lansing Public Schools allocating a certain amount of money for capital expenditures and a certain amount for operating expenditures

YES

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Applying Game Theory to an Oligopoly Market

Definition of an Oligopoly-

A market structure in which there are only a few firms each of which is large relative to the total industry (results in strategic interaction).

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Warning Due to the complexity involved in

analyzing oligopolies and the differences across industries/markets, there is no single model that is relevant to all oligopolies.

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Cournot Oligopoly Example

1. Few firms in market serving many customers.

2. Firms produce either differentiated or homogeneous products.

3. Each firm believes rivals will hold their output constant if it changes its output.

4. Barriers to entry exist.

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Numerical Example of Cournot Oligopoly

Two Firms: Firm 1 and Firm 2 Firms produce a homogenous

product Market Demand is P=100-Q Q=Q1+Q2 where Q1 is Firm 1’s

output and Q2 is Firm 2’s output Each firm has constant marginal

cost of 20 and zero fixed costs.

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What if the firms perfectly collude? What total output should they produce?

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

Q=40. Can’t have more profits than what a monopolist would.

MR

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Suppose firms collude where both firms produce an output of 20 (i.e., Q1=Q2=20)

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

=AVC=ATC

Firm 1’s Profits = 60*20-20*20=800

Firm 2’s Profits = 60*20-20*20=800

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Why might you expect that the firms will not be able to collude in this manner?

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

=AVC=ATC

Firm 1’s Profits = 50*30-20*30=900

Firm 2’s Profits = 50*20-20*20=600

If Firm 1 thinks Firm 2 will produce 20, then Firm 1 can increase his profits to 900 if produce 30.

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Nash Equilibrium A situation in which neither firm has

an incentive to change its output given the other firm’s output. (Also called Cournot Equilibrium.)

In this numerical example, it is for each firm to produce an output of 26.67. If Firm 2 produces an output of 26.67, Firm 1 maximizes profits by producing an output of 26.67 (and vice versa).

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Profits from Cournot Equilibrium: Q1=26.67 and Q2=26.67 so Q=Q1+Q2=53.3

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

53.33

46.66

=AVC=ATC

Firm 1 Profits=46.66*26.67-20*26.67= 713

Firm 2 Profits=46.66*26.67-20*26.67= 713

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Cournot Equilibrium compared to Perfect Collusion Cournot Equilibrium

Q1=26.67 , Firm 1 Profits = 713

Q2=26.67 , Firm 2 Profits = 713

Perfect Collusion

Q1=20 , Firm 1 Profits = 800

Q2=20 , Firm 2 Profits = 800

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What if Firms Interact Repeatedly:Infinitely Repeated Interaction

Suppose Firm 1 thinks Firm 2 won’t deviate from Q2=20 if Firm 1 doesn’t deviate from collusive agreement of Q1=20 and Q2=20. In addition, Firm 1 thinks Firm 2 will produce at an output of 80 in all future periods if Firm 1 deviates from collusive agreement of Q1=20 and Q2=20.

Firm 1’s profits from not cheating

Firm 1’s profits from cheating (by producing Q1=30 Today)

Today In 1 Year In 2 Years In 3 Years In 4 Years

800 800 800 800 800 …

Today In 1 Year In 2 Years In 3 Years In 4 Years

900 0 0 0 0 …

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Industry Characteristics that Facilitate Collusion

2. Stable Industry

3. Few Number of Firms

4. If a firm cheats on a collusive agreement, the probability the firm is “caught” is high.

5. Ability to Credibly Punish in a Severe Manner.

6. Industry demand is growing.

7. Expectation of firms’ behavior is clear.

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My All Time Favorite example of how expectations are formed

Coca-Cola, PepsiCo Set To Call Off Bitter Soft-Drink Price War Staff Reporter of The Wall Street Journal

ATLANTA -- A brief but bitter pricing war within the soft-drink industry might be drawing to a close -- all because no one wants to be blamed for having fired the first shot.

Coca-Cola Enterprises Inc., Coca-Cola Co.'s biggest bottler, said in a recent memorandum to executives that it will "attempt to increase prices" after July 4 amid concern that heavy price discounting in most of the industry is squeezing profit margins.

The memo is a response to statements made to analysts last week by top PepsiCo Inc. executives. Pepsi, of Purchase, N.Y., said "irrational" pricing in much of the soft-drink industry might temporarily squeeze domestic profits, and it laid the blame for the price cuts at Coke's door.

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My All Time Favorite example of how expectations are formed

In the June 5 memo, Summerfield K. Johnston Jr. and Henry A. Schimberg, the chief executive and the president of Coca-Cola Enterprises, respectively, said the bottler's plan is to "succeed based on superior marketing programs and execution rather than the short-term approach of buying share through price discounting."

"This is a first step to disengagement," said Andrew Conway, an analyst in New York for Morgan Stanley & Co. "Coke and Pepsi are out to improve profitability for the category, not destroy it, so this would bode for a stabilization."

For all the signals of a truce, though, Coca-Cola Enterprises' memo could just as easily be seen as throwing down the gauntlet. Messrs. Johnston and Schimberg said in the memo that should "the competition" view the attempt to raise prices "as an opportunity to gain share through predatory pricing, we will, as we have in the past, respond immediately."

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Bertrand Oligopoly

1. Few firms in market serving many customers.

2. Firms produce a homogeneous product at a constant marginal cost (need not actually be the case).

3. Firms engage in price competition and react optimally to prices charged by competitors.

4. Consumers have perfect information and there are no transaction costs.

5. Barriers to entry exist.

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What if Firm 1 and Firm 2 choose price and react optimally to price charged by other firm?Will firms be able to collude on a price of $60?

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D

MCQ

Firms could not collude on a price of $60 because each firm would have incentive to undercut other firm. In the end, you would expect both firms to set a price of $20 (equal to MC) and have zero profits.

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Using Game Theory to Devise Strategies in Oligopolies that Increase ProfitsExamples:1. Price Matching- advertise a price and promise to

match any lower price offered by a competitor.

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Bertrand Oligopoly

In the end, you would expect both firms to set a price of $20 (equal to MC) and have zero profits.

79

Using Game Theory to Devise Strategies in Oligopolies that Increase ProfitsExamples:1. Price Matching- advertise a price and promise

to match an lower price offered by a competitor. In Bertrand example, perhaps each firm would set a price of $60 and say will match.

2. Induce Brand Loyalty – frequent flyer program3. Randomized pricing – inhibits consumers

learning as to who offers lower price and reduces ability of competitors to undercut price.

1 game theory. game theory definition the study of strategic decision making. more formally, it is...

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