Oligopoly

A market structure in which there are few firms, each of which is large relative to the total industry.

Key idea: Decision of firms are interdependent.

The Simple Mathematics of Oligopoly

Given: P = 200 – 2Q TC = 500 + 40Q + 2Q2

What is the profit maximizing price? $160If this firm’s current price was $150 and it raised its price, how would its competition respond? IF YOU DO NOT KNOW THE COMPETITOR’S RESPONSE, IT IS DIFFICULT TO PREDICT WHAT THE NEW DEMAND CURVE WILL BE!!! THEREFORE OUR SIMPLE PROBLEM HAS BECOME A BIT MORE COMPLICATED!!

Game Theory Game Theory – the study of how

individuals make decisions when they are aware that their actions affect each other and when each individual takes this into account.

History: Introduced in 1944 by John von Neumann and Oskar Morgenstern in “The Theory of Games and Economic Behavior.”

The work of von Neuman and Morgenstern was expanded upon by John Nash.

Introduction to Game Theory

A game is a situation in which a decision-maker must take into account the actions of other decision-makers. Interdependency between decision-makers is the essence of a game.

In games people must make strategic decisions. Strategic decisions are decision that have implications for other people.

Strategy – a decision rule that describes that 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 from alternative strategies.

Cooperative and Non-Cooperative Games

Non-Cooperative Games are games in which players cannot enter binding agreements with each other before the play of the game.

Cooperative Games are games in which players can enter binding agreements with each other before the play of the game.

In class we only review non-cooperative games.

Two Types of Games Simultaneous move game – Game in

which each player makes decisions without knowledge of the other players’ decision. Examples: Pitching in baseball, Calling

plays in football Sequential move game – Game in

which one player makes a move after observing the other player’s move. Example: Chess

Elements of a Game1. Set of Players.2. Order of Play.3. Description of the information available

to any player at any point during the game.4. Set of actions available to each player

when making a decision.5. Outcomes that result from every possible

sequence of actions by the players.6. A payoff from the outcomes.7. Strategic situations with the above

elements is considered to be well defined.

Actions, Strategies, and Payoffs Actions – The set of choices available at each decision in a

game. Pure strategy – a rule that tells the player what action to

take at each of her information sets in the game. Mixed strategy – when players can choose randomly

between the actions available to them at every information set. Example: Play calling in sports is a mixed strategy.

Payoffs, for our purposes, consist of either profits to firms, or income to individuals. Payoffs can also be characterized in terms of utility.

Solving Games: Nash Equilibrium Solution Concept – a methodology

for predicting player behavior. Nash Equilibrium - a collection of

strategies one for each player, such that every player's strategy is optimal given that the other players use their equilibrium strategy.

The Opie Equilibrium [Inside Business 10-1]

Dominant and Dominated Strategies

Payoff matrix – a matrix that displays the payoffs to each player for every possible combination of strategies the players could choose.

Dominant Strategy – a strategy that is always strictly better than every other strategy for that player regardless of the strategies chosen by the other players.

Dominated Strategy – a strategy that is always strictly worse than some other strategy for that player regardless of the strategies chosen by the other players.

Weakly Dominate Strategies Weakly dominant strategy - a strategy that is

always equal to or better than every other strategy for that player regardless of the strategies chosen by the other players.

Weakly Dominated Strategy – a strategy that is always equal to or worse than some other strategy for that player regardless of the strategies chosen by the other players.

Prisoner’s DilemmaPrisoner’s Dilemma

Scenario: Two people are arrested for a Scenario: Two people are arrested for a crimecrime

The elements of the game:The elements of the game: The players:The players: Prisoner One, Prisoner Prisoner One, Prisoner

TwoTwo The strategies:The strategies: Confess, Don’t Confess, Don’t

ConfessConfess The payoffs:The payoffs:

– Are on the following slide Are on the following slide – Payoffs read Prisoner 1, Prisoner 2Payoffs read Prisoner 1, Prisoner 2

Prisoner’s Dilemma, cont.Prisoner’s Dilemma, cont.

Prisoner 2Prisoner 2

ConfessConfess Don’t Don’t ConfessConfess

ConfessConfess 6 years, 6 years6 years, 6 years 1 1 year, 10 yearsyear, 10 years

Prisoner 1Prisoner 1

Don’t ConfessDon’t Confess 10 year, 1 year10 year, 1 year 3 years, 3 years3 years, 3 years Dominant strategy equilibrium: In this game, the Dominant strategy equilibrium: In this game, the

dominant strategy for each prisoner is to confess. So dominant strategy for each prisoner is to confess. So the outcome of the game is that they each get six the outcome of the game is that they each get six years.years.

This illustrates the prisoner’s dilemma: games in This illustrates the prisoner’s dilemma: games in which the equilibrium of the game is not the outcome which the equilibrium of the game is not the outcome the players would choose if they could perfectly the players would choose if they could perfectly cooperate.cooperate.

The Advertising GameThe Advertising Game

Scenario: Two firms are determining Scenario: Two firms are determining how much to advertise.how much to advertise.

The elements of the game:The elements of the game: The players:The players: Firm 1, Firm 2Firm 1, Firm 2 The strategies:The strategies:

– High advertising, low advertisingHigh advertising, low advertising

Advertising Game, Cont.Advertising Game, Cont.

The payoffs are as follows: (payoffs read 1,2)The payoffs are as follows: (payoffs read 1,2)

Firm 2Firm 2HighHigh LowLow

HighHigh 40,4040,40 100, 10100, 10Firm 1Firm 1

LowLow 10, 10010, 100 60,6060,60

Dominant strategy equilibrium: In this game, the Dominant strategy equilibrium: In this game, the dominant strategy for firm 1 and firm 2 is high. So dominant strategy for firm 1 and firm 2 is high. So the outcome of the game is 40,40.the outcome of the game is 40,40.

Again, this is an example of the prisoner’s Again, this is an example of the prisoner’s dilemma. The equilibrium of the game is not the dilemma. The equilibrium of the game is not the outcome the players would choose if they could outcome the players would choose if they could cooperate.cooperate.

More Prisoner DilemmasMore Prisoner Dilemmas Industrial Organization ExamplesIndustrial Organization Examples

– Cruise Ship Lines and the move towards ‘glorious Cruise Ship Lines and the move towards ‘glorious excess’. Royal Caribbean offers a cruise with an 18 hole excess’. Royal Caribbean offers a cruise with an 18 hole miniature golf course. Princess Cruises has a ship with miniature golf course. Princess Cruises has a ship with three lounges, a wedding chapel, and a virtual reality three lounges, a wedding chapel, and a virtual reality theater.theater.

– Owners of professional sports teams and the bidding on Owners of professional sports teams and the bidding on professional athletes.professional athletes.

Non-IO ExamplesNon-IO Examples– Politicians and spending on campaigns. Politicians and spending on campaigns. – Worker effort in teams. The incentive exists to shirk, a Worker effort in teams. The incentive exists to shirk, a

strategy that if followed by all workers, reduces the strategy that if followed by all workers, reduces the productivity of the team. More on shirking later.productivity of the team. More on shirking later.

Iterated Dominant Iterated Dominant StrategiesStrategies

What if a dominant strategy What if a dominant strategy does not exist?does not exist?

We can still solve the game by We can still solve the game by iterating towards a solution.iterating towards a solution.

The solution is reached by The solution is reached by eliminating all strategies that eliminating all strategies that are strictly dominated.are strictly dominated.

Example of Iterated DominanceExample of Iterated Dominance

Down is Firm 1, Across is Firm 2Down is Firm 1, Across is Firm 2

F1,F2F1,F2 HighHigh MediuMediumm

LowLow

HighHigh 100,80100,80 95,8595,85 80,10080,100

MediuMediumm

85,9585,95 110,105110,105 110,100110,100

LowLow 80,10080,100 130,110130,110 120,115120,115

Alternative Solution Alternative Solution StrategiesStrategies

Nash Equilibrium - a strategy combination in which Nash Equilibrium - a strategy combination in which no player has an incentive to change his strategy, no player has an incentive to change his strategy, holding constant the strategies of the other players.holding constant the strategies of the other players.

Joint Profit Maximization: This is the objective of a Joint Profit Maximization: This is the objective of a cartel.cartel.

Cut-Throat: A strategy where one seeks to minimize Cut-Throat: A strategy where one seeks to minimize the return to her/his opponent. the return to her/his opponent.

Secure Strategy: A strategy that guarantees the Secure Strategy: A strategy that guarantees the highest payoff given the worst possible scenario.highest payoff given the worst possible scenario.

How does the previous game change when we How does the previous game change when we change the objectives of the players?change the objectives of the players?

This is one of the advantages of game theory. We do This is one of the advantages of game theory. We do not have to assume profit maximization. We still not have to assume profit maximization. We still need to be able to identify the objectives of the need to be able to identify the objectives of the players.players.

Infinitely Repeated Games

A game that is played over and over again forever in which players receive payoffs during each play of the game.

Present Value Across an Infinite Horizon If the profits earned by a firm are the same

in each period and the horizon is infinite, the present value of a firm simplifies to the following formula:

PVFIRM = PROFIT * (1+i)/(i)

Trigger Strategy

A strategy that is contingent on the past play of a game and in which some particular past action triggers a different action by a player.

Example: Two firms charge high prices. Cheating is a trigger which forces the non-cheating firm to cut prices.

Should a firm cheat?

A firm should cheat if the one-time payoff from cheating exceeds the present value of future profits earned from not cheating.

Payoff from cheating vs.

non-cheating profits * (1+i)/i Key issue:

Size of the payoff from cheating Interest rate earned

Pricing Game

The payoffs are as follows (payoffs read 1,2)

Firm 2Low High

Low 0,0 200, 10Firm 1

High 10, 200 20,20

Dominant strategy equilibrium: In this game, the dominant strategy for firm 1 and firm 2 is low. So the outcome of the game is 20,20.

There is an incentive to cheat an earn an one-time payoff of 100.

Solving the Pricing Game

Present value from cheating = $200 Present value from not cheating =

20 * (1+i)/i At what interest rate is cheating not a good idea?

200 = 20*(1+i)/i 200i = 20 + 20i 180i = 20 i = 1/9 = 11.1% If the interest rate is less 11.1%, the payoff from cheating

is too low.

Factors impacting collusion

Knowing identity of rivals Knowing the customers of rivals Knowing when rivals cheat Be able to punish rivals who cheat

Firm and Industry characteristics that impact collusion Number of firms

More firms increase monitoring costs Size of firms

Smaller firms cannot afford monitoring History of the markets

Tacit collusion cannot work if punishing is ineffective.

Punishment mechanisms Can the punishing firm price discriminate? Price discrimination lowers the cost of punishing.

Mixed Strategy

Pure Strategy is a rule that tells the player what action to take at each information set in the game.

Mixed strategy allows players to choose randomly between the actions available to the player at every information set. Thus a player consists of a probability distribution over the set of pure strategies.

Examples of mixed strategy games: Play calling in sports To shirk or not to shirk

The Shirking Game Scenario: A worker is hired but does not wish to work. The

firm will not pay the worker if there is no work, but the firm cannot directly observe the workers effort level or output.

Players: The worker, the firm Strategy: Work or not work, monitor or not monitor Payoffs: Work pays $100, but the worker’s reservation wage

is $40. Worker can produce $200 in revenue, but it costs $80 to monitor.

The Shirking Game, Cont.

There is no dominant strategy, or iterated dominant strategy.

There is also no clear Nash Equilibrium. In other words, no combination of actions makes both sides happy given what the other side has chosen.

Monitor Don’t

Monitor

Work 60, 20 60, 100

Shirk 0, -80 100, -100

The Shirking Game, cont. There are many mixed strategies. The worker could work with

probability (p) of 0.7, 0.6. 0.25, etc... The same is true for the firm. Which mixed strategy should they choose?

If the worker is most likely to shirk, the firm should monitor. Likewise, if the firm is more likely to monitor, the worker should work. In any scenario, no Nash equilibrium will be found. The key is to find a strategy that makes the opponent indifferent to his/her potential choices.

A person is indifferent when the expected return from action A equals the expected return form action B.

The Firm’s Solution

How much should the firm monitor? E(work) = 60p + 60(1-p) = 60 E(shirk) = 0p + 100(1-p) = 100 - 100p 100 - 100p = 60 40 = 100p p = .40 The worker is indifferent when the probability of

monitoring is 40% and the probability of not monitoring is 60%.

The Worker’s Solution How much should the worker work? E(monitor) = 20p + -80(1-p) = 100p - 80 E(Not monitor) = 100p + -100(1-p) = 200p - 100 100p -80 = 200p - 100 20 = 100p p = .2 The firm is indifferent when the probability of working is

20% and the probability of not working is 80%. How does the cost of monitoring and the worker’s

reservation wage impact behavior?