Power Points Chapter Ten (Game Theory)Presentation Transcript
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 + 2Q 2
What is the profit maximizing price? $160
If 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 – 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.
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.
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.
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
Are on the following slide
Payoffs read Prisoner 1, Prisoner 2
Prisoner’s Dilemma, cont.
Confess Don’t Confess
Confess 6 years, 6 years 1 year, 10 years
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
High advertising, low advertising
Advertising Game, Cont.
The payoffs are as follows: (payoffs read 1,2)
High 40,40 100, 10
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.
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. More on shirking later.
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
120,115 130,110 80,100 Low 110,100 110,105 85,95 Medium 80,100 95,85 100,80 High Low Medium High F1,F2
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.
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:
PV FIRM = PROFIT * (1+i)/(i)
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
Size of the payoff from cheating
Interest rate earned
The payoffs are as follows (payoffs read 1,2)
Low 0,0 200, 10
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.
Can the punishing firm price discriminate?
Price discrimination lowers the cost of punishing.
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.
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?