Chapter 16 Oligopoly and Game Theory

“Game theory is the study of how people behave in strategic situations. By ‘strategic’ we mean a situation in which each person, when deciding what actions to take, must consider how others might respond to that action.”

Oligopoly “Oligopoly is a market structure in which only a few sellers offer similar or identical products.” As we saw last time, oligopoly differs from the two ‘ideal’ cases, perfect competition and monopoly. In the ‘ideal’ cases, the firm just has to figure out the environment (prices for the perfectly competitive firm, demand curve for the monopolist) and select output to maximize profits An oligopolist, on the other hand, also has to figure out the environment before computing the best output.

Oligopoly “Figuring out the environment” when there are rival firms in your market, means guessing (or inferring) what the rivals are doing and then choosing a “best response” This means that firms in oligopoly markets are playing a ‘game’ against each other. To understand how they might act, we need to understand how players play games. This is the role of Game Theory.

Some Concepts We Will Use Strategies Payoffs Sequential Games Simultaneous Games Best Responses Equilibrium Dominated strategies Dominant Strategies.

Strategies Strategies are the choices that a player is allowed to make. Examples: –In game trees (sequential games), the players choose paths or branches from roots or nodes. –In matrix games players choose rows or columns –In market games, players choose prices, or quantities, or R and D levels. –In Blackjack, players choose whether to stay or draw.

Sequential Games A sequential game is a game that is played in strict order. One person makes a move, then another sees the move and then she moves. Examples: –Chess and checkers, –Nim –Entry games by firms.

A Sequential Game A B B B A A A A A A

Simultaneous Games Simultaneous games occur when players have to make their strategy choices without seeing what their rivals have done. Note: These do not literally have to be simultaneous. Examples: –Rock, paper scissors, Snap –Pricing games by rivalrous firms –R and D games by firms

A Matrix Game \ Player Player \ Two One \ LeftRight Up Down

Payoffs Payoffs describe what a player gets when she plays the game. In some cases, we do not have to be very precise. For example, some games are “zero-sum”, when one person wins, another loses. Then it doesn’t really matter what number we assign. A winner could get 1 and a loser, -1. Or a winner could get, say, 15 and the loser -15.

Payoffs More often, though, payoffs can vary a lot. To fully describe a game, we need to explain the consequences of every move by every player. Examples: –Profits for firms –Prizes (first, second third) in competitions –Inventions in R and D games.

A Sequential Game with Payoffs Enter Not Enter High Price Low Price B A (-10, 20) (0, 27) (0, 23) (5,24)

Sequential Game In the above game, the first number refers to the (dollar) payoff for the first mover, the second is the payoff for the second mover.

A Simultaneous Game with Payoffs (The Prisoner’s Dilemma Game) A

Simultaneous Game Payoffs In the above game, the first number refers to the Row player’s payoffs (the number of years in prison) The second number refers to the Column player’s payoffs (the number of years in prison).

Equilibrium Once we have described strategies and payoffs, we can go ahead and try to predict how a player will play a game. But where do we start? “Equilibrium” is the term game theorists use to describe how the game will (likely?) be played.

Equilibrium We will focus on two types of equilibrium: Elimination of Dominated Strategies Nash Equilibrium

“A dominated strategy is any strategy such that there is some other strategy that always does better for a player in a game regardless of the strategies chosen by other players.”

Elimination of Dominated Strategies If Strategy A always gives a player a lower payoff than Strategy B, no matter what the rivals do, then it seems foolish ever to use A. One way to simplify a game, and see if we can predict what a player will do, is eliminate all dominated strategies and see what we have left.

Not Confess is Dominated by Confess for Player A. B A

Not Confess is Dominated by Confess for Player B. B A

The Only Outcome that Remains is Confess, Confess. B A

Consider a Pricing Game Between Two Firms Firms price simultaneously. They may price High or Low Both Pricing High yields the highest profits. But each wants to cheat.

A B

Big Games Eliminating dominated strategies can make solving big games a lot easier:

A B

Equilibrium When the elimination of dominated strategies leads to a single pair of strategies, then we call this an equilibrium of a game. When only a single strategy remains for a given player, we say that strategy is a dominant strategy. Some games, however, do not have dominated strategies, or after elimination, do not lead to a single pair of strategies. For example, the next game has no dominated strategies.

A B

Nash Equilibrium John Nash, the celebrated mathematician (Nobel Prize, A Beautiful Mind) became famous for suggesting how to solve this problem. A Nash equilibrium is a situation in which economic actors interacting with one another each choose their best strategy given the strategies that all the other actors have chosen

Nash Equilibrium A Best Response to a rival player’s strategy s, is a strategy choice that, assuming the rival plays s is the payoff maximizing choice. A Nash equilibrium is a pair of strategies that are best responses to each other.

Some Games may have many Nash equilibria. A B

Some Games May Have No Nash Equilibria (in non-random strategies) A B

A Employer Employee Game Suppose an employee earns $20 a day but wants to spend a lot of time surfing the web at work. Web- surfing yields him $10 of pleasure. The boss wants him to work, that gives her $20 of profit. It costs her $5 to check up on him. And $8 if she does not check but he surfs. If she checks she fines him $13. What is the equilibrium of this game?

B A

Sequential Games and Backward Induction A sequential game is a game in which players make at least some of their decisions at different times. Sequential games can almost always be solved by ‘Backward Induction’ This is like doing sequential elimination of dominated strategies starting from the end of the game going to the beginning.

The Centipede Game A B AB ,000 1M 9000 out in out in out in out in

Nim Version 1 Start with a pile of matches Take turns. When it is your turn, you can take either 1 or 2 matches from the pile. If it is your turn and there is just 1 match left in the pile, you lose.

Nim Version 2 Start with two piles of matches Take turns. When it is your turn, you can take 1 or more matches from either pile. You are not allowed to take matches from both piles. If it is your turn and there is just 1 match left in one pile and no matches left in the other pile, you lose.

NIM Version 3 Pearls Before Swine

Two players, A and B take turns choosing a number between 1 and 10 (inclusive). A goes first. The cumulative total of all the numbers chosen is calculated as the game progresses. The player whose choice takes the total to exactly 100 is the winner. Addition Game

90 – 99First

90 – 99First 89Second

90 – 99First 89Second 79 – 88First

90 – 99First 89Second 79 – 88First 78Second

90 – 99First 89Second 79 – 88First 78Second 68 – 77First

90 – 99First 89Second 79 – 88First 78Second 68 – 77First 67Second

90 – 99First 89Second 79 – 88First 78Second 68 – 77First 67Second ………. 1Second

Sequential Games and Credibility One point that sequential games and backward induction make clear is that, if threats (or promises) are not credible, they should not be believed. A threat or promise is only credible if it is in your own best interest to follow through on it. As an example, suppose an incumbent monopolist is threatened with entry by a competitor. The monopolist could say, “If you come into the market, I will start a price war and destroy you!” Should the entrant believe her?

Enter Not Enter High Price Low Price B A A (5, 24) (-10, 20) (0, 27) (0, 23) Note: B’s profits are shown first

Order of Moves Is it best to move first or last? The general answer is that it depends. Consider the divide the cake game. Consider the following game where B can choose first whether or not to produce and A must choose second.

Produce Not Produce Produce Not Produce B A A (-5, -5) (100, 0) (0, 100) (0, 0) Note: B’s profits are shown first

Thinking Strategically Barry Nalebuff in Co-opetition (with Adam Brandenburger) suggests that the best way to deal with strategic weakness is to change the game. Suppose that, before the production game is played, A goes to a trusted colleague and makes the following bet: –I bet $10 that I will produce (and compete against B if necessary). –Assuming they take the bet seriously, how has the game changed?

Produce Not Produce Produce Not Produce B A A (-5, -5) (bet won) (100, -10) (bet lost) (0, 100) (bet won) (0, -10) (bet lost) Note: B’s profits are shown first

Order of Moves and Bargaining In bargaining games, though, it is generally best to move last. In the next game, a union and a firm bargain sequentially over how to share the available profits of operating. The firm offers first, then the union responds. The longer the game goes, the smaller the available profits become.

Bargaining

Solve this problem by going to the end of the game and working backward. In April, Union will ask for $20, leaving Firm with $0. In March, Firm must offer Union at least $20; if Firm offers less, Union will reject the offer and the strike will continue. Therefore in March Firm will receive $50 - $20 = $30.

In February, Union must offer Firm at least $30; if Union offers less, Firm will reject the offer and the strike will continue. Therefore in March Union will receive $80 - $30 = $50. In January, Firm must offer Union at least $50; if Firm offers less, Union will reject the offer and the strike will begin. Therefore in January the Firm will receive $120 - $50 = $70.

A B B B A A A A A A

Backward induction Subgame perfect Nash equilibrium

Chapter 16 Oligopoly and Game Theory Simultaneous move games –Rock-scissors-paper Dominant strategy –Prisoner’s dilemma –Oligopoly -- firms choose quantities Nash equilibrium –Number of Nash equilibria

Sequential games –Chess, checkers Backwards induction Credible threats Examples: –Nim –Addition Game –Entry –Changing the game –Labor Negotiations