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## Oligopoly

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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? $160

If this firm’s current price was $170 and it lowered 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 Game

- Set of Players.
- Order of Play.
- Description of the information available to any player at any point during the game.
- Set of actions available to each player when making a decision.
- Outcomes that result from every possible sequence of actions by the players.
- A payoff from the outcomes.
- 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.

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 Dilemma

- Scenario: Two people are arrested for a crime
- The elements of the game:
- The players: Prisoner One, Prisoner Two
- The strategies: Confess, Don’t Confess
- The payoffs:
- Are on the following slide
- Payoffs read Prisoner 1, Prisoner 2

Prisoner’s Dilemma, cont.

Prisoner 2

Confess Don’t Confess

Confess 6 years, 6 years 1 year, 10 years

Prisoner 1

Don’t Confess 10 year, 1 year 3 years, 3 years

- Dominant strategy equilibrium: In this game, the dominant strategy for each prisoner is to confess. So the outcome of the game is that they each get six years.
- This illustrates the prisoner’s dilemma: games in which the equilibrium of the game is not the outcome the players would choose if they could perfectly cooperate.

The Advertising Game

- Scenario: Two firms are determining how much to advertise.
- The elements of the game:
- The players: Firm 1, Firm 2
- The strategies:
- High advertising, low advertising

Advertising Game, Cont.

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

Firm 2

High Low

High 40,40 100, 10

Firm 1

Low 10, 100 60,60

- Dominant strategy equilibrium: In this game, the dominant strategy for firm 1 and firm 2 is high. So the outcome of the game is 40,40.
- Again, this is an example of the prisoner’s dilemma. The equilibrium of the game is not the outcome the players would choose if they could cooperate.

More Prisoner Dilemmas

- Industrial Organization Examples
- Cruise Ship Lines and the move towards ‘glorious excess’. Royal Caribbean offers a cruise with an 18 hole miniature golf course. Princess Cruises has a ship with three lounges, a wedding chapel, and a virtual reality theater.
- Owners of professional sports teams and the bidding on professional athletes.
- Non-IO Examples
- Politicians and spending on campaigns.
- Worker effort in teams. The incentive exists to shirk, a strategy that if followed by all workers, reduces the productivity of the team.

Iterated Dominant Strategies

- What if a dominant strategy does not exist?
- We can still solve the game by iterating towards a solution.
- The solution is reached by eliminating all strategies that are strictly dominated.

Example of Iterated Dominance

- Down is Firm 1, Across is Firm 2

Alternative Solution Strategies

- Nash Equilibrium - a strategy combination in which no player has an incentive to change his strategy, holding constant the strategies of the other players.
- Joint Profit Maximization: This is the objective of a cartel.
- Cut-Throat: A strategy where one seeks to minimize the return to her/his opponent.
- Secure Strategy: A strategy that guarantees the highest payoff given the worst possible scenario.
- How does the previous game change when we change the objectives of the players?
- This is one of the advantages of game theory. We do not have to assume profit maximization. We still need to be able to identify the objectives of the players.

Sequential Move Game

- Return to the advertising game. What if Firm 1 has first-mover advantage?
- What is the solution?
- Backwards induction: Start at the end of the game and work backwards.
- If Firm 1 chooses High, Firm 2's best strategy is to pick High.
- If Firm 1 chooses Low, Firm 2's best strategy is to pick High.
- Firm 1, knows that Firm 2 will always pick High. Knowing this, Firm 1 picks High also. Thus the solution is High, High.

More sequential move games

- What if firm 1 has first mover advantage? How does this change the game?
- Solution concept: Profit max
- If firm 1 were to raise its price, firm 2 would keep
- If firm 1 were to keep its price, firm 2 would lower
- If firm 1 were to lower its price, firm 2 would keep.
- Given this, firm 1 should lower its price and the Nash equilibrium is lower, keep

More sequential move games

- Solution concept: Cut-throat (firm 1 moves first)
- If firm 1 were to raise its price, firm 2 would keep
- If firm 1 were to keep its price, firm 2 would lower
- If firm 1 were to lower its price, firm 2 would lower.
- Given this, if firm 1 plays cut-throat, the solution strategy is lower, lower.
- if firm 1 plays profit-max, the solution strategy is keep, lower.

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.
- UNDERSTAND THE SIMPLE CARTEL MODEL AND WHY CARTELS ARE UNSTABLE

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.

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?

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