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Home , Backward induction, Game theory, Oligopoly, Rock paper scissors, Simultaneous game

Chapter 16 “ 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 and consider how others might respond to that action.”

Oligopoly Oligopoly

• “Oligopoly is a structure in which only a few • “Figuring out the environment” when there are sellers offer similar or identical products.” rival firms in your market, means guessing (or • As we saw last time, oligopoly differs from the two ‘’ inferring) what the rivals are doing and then cases, perfect and . choosing a “” • In the ‘ideal’ cases, the firm just has to figure out the environment ( for the perfectly competitive firm, • This means that firms in oligopoly markets are for the monopolist) and select to playing a ‘game’ against each other. maximize profits • To understand how they might act, we need to • An oligopolist, on the other hand, also has to figure out the understand how players play . environment before computing the best output. • This is the role of Game Theory. Some Concepts We Will Use

• Strategies • Strategies are the choices that a player is allowed • Payoffs to make. • Sequential Games •Examples: • Simultaneous Games – In game trees (sequential games), the players choose paths or branches from roots or nodes. • Best Responses – In games players choose rows or columns • Equilibrium – In market games, players choose prices, or quantities, • Dominated strategies or R and D levels. • Dominant Strategies. – In Blackjack, players choose whether to stay or draw.

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

• Simultaneous games occur when players have to \Player Left Right make their choices without seeing what Player \ Two their rivals have done. • Note: These do not literally have to be One \ simultaneous. Up •Examples: – Rock, paper scissors, Snap – Pricing games by rivalrous firms Down – R and D games by firms

Payoffs Payoffs

• Payoffs describe what a player gets when she • More often, though, payoffs can vary a lot. plays the game. • To fully describe a game, we need to • In some cases, we do not have to be very precise. explain the consequences of every move by • For example, some games are “zero-sum”, when every player. one person wins, another loses. •Examples: • Then it doesn’t really matter what number we assign. – Profits for firms • A winner could get 1 and a loser, -1. Or a winner – Prizes (first, second third) in could get, say, 15 and the loser -15. – in R and D games. A Sequential Game with Payoffs Sequential Game ice h Pr Hig (5,24) • In the above game, the first number refers to L r ow P the (dollar) payoff for the first mover, the nte rice E (-10, 20) second is the payoff for the second mover. rice B h P Hig (0, 27) No t E L (0, 23) nt ow P er A rice

Consider a Pricing Game Between Two Firms B • Firms simultaneously. LOW(P = 1) HIGH(P = 2) • They may price High or Low LOW( P = 1) 4, 4 6, 3 • Both Pricing High yields the highest profits. • But each wants to cheat. A HIGH(P = 2) 3, 6 5, 5

Simultaneous Game Payoffs Equilibrium

• In the above game, the first number refers to • Once we have described strategies and the Row player’s payoffs (the number of payoffs, we can go ahead and try to predict years in prison) how a player will play a game. • The second number refers to the Column • But where do we start? player’s payoffs (the number of years in • “Equilibrium” is the term game theorists use prison). to describe how the game will (likely?) be played.

Equilibrium

• We will focus on two types of equilibrium: “A dominated strategy is any strategy such • Elimination of Dominated Strategies 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 B • If Strategy A always gives a player a lower payoff LOW(P = 1) HIGH(P = 2) than Strategy B, no matter what the rivals do, then it seems foolish ever to use A. LOW( P = 1) 4, 4 6, 3 • One way to simplify a game, and see if we can predict what a player will do, is eliminate all A HIGH(P = 2) 3, 6 5, 5 dominated strategies and see what we have left.

• Pricing High is a dominated by pricing Low for Firm A.

B LOW(P = 1) HIGH(P = 2)

Pricing High is also dominated by LOW( P = 1) 4, 4 6, 3 pricing Low for Firm B. A HIGH(P = 2) 3, 6 5, 5

B LOW(P = 1) HIGH(P = 2)

The only equilibrium is LOW( P = 1) 4, 4 6, 3 both Firms pricing Low. A HIGH(P = 2) 3, 6 5, 5

Big Games B LEFT MIDDLERIGHT • Eliminating dominated strategies can make solving big games a lot easier: TOP 6, 2 10,2 7,3

CENTER 5, 3 9,1 6, 1 A BOTTOM 4, 2 8, 1 3,7

Equilibrium B LEFT MIDDLE RIGHT • When the elimination of dominated strategies leads to a single pair of strategies, TOP 6, 2 9, 3 5, 1 then we call this an equilibrium of a game. • Some games, however, do not have CENTER 5, 3 10, 2 6, 1 dominated strategies, or after elimination, A do not lead to a single pair of strategies. BOTTOM 4, 2 8, 1 7, 3 • For example, the next game has no dominated strategies.

Nash Equilibrium

• John Nash, the celebrated mathematician (Nobel Prize, ) 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 Some Games May Have No Nash equilibria. Equilibria (in non-random strategies) B B LEFT RIGHT LEFT RIGHT TOP 0, 1 -2, 0 A A TOP 0, 0 0, -1 BOTTOM -3, -1 -1, 0 BOTTOM 1, 0 -1, 3 Sequential Games and A sequential game is a game in which players A sequential game is a game in which players make at least some of their decisions at make at least some of their decisions at different times. 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.

Nim Version 1

• Start with a pile of matches 2First • Take turns. When it is your turn, you can 3 take either 1 or 2 matches from the pile. 4 • If it is your turn and there is just 1 match 5 left in the pile, you lose. 6 7 8 2First 2First 3First 3First 4 4 Second 5 5 6 6 7 7 8 8

2First 2First 3First 3First 4 Second 4 Second 5First 5First 6 6First 7 7 8 8 2First 2First 3First 3First 4 Second 4 Second 5First 5First 6First 6First 7 Second 7 Second 8 8First

Nim Version 2 NIM Version 3

• Start with two piles of matches • Pearls Before Swine • Take turns. When it is your turn, you can • http://www.transience.com.au/pearl.html 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. Game

90 – 99 First Two players, A and B take turns choosing a number between 1 and 10 (inclusive). A goes first. The cumulative total of all the chosen is calculated as the game progresses. The player whose choice takes the total to exactly 100 is the winner.

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

90 – 99 First 90 – 99 First 89 Second 89 Second 79 – 88 First 79 – 88 First 78 Second 78 Second 68 – 77 First 68 – 77 First 67 Second 67 Second ………. 1 Second ice (5, 24) e (-5, -5) h Pr oduc A Hig A Pr L e No r ow uc t te Pric (-10, 20) od Prod (100, 0) En e Pr uce B B N No ot rice (0, 27) t P e (0, 100) En h P ro duc te ig du ro r H ce P A A Low Not Price (0, 23) Produ (0, 0) Note: B’s profits are shown first Note: B’s profits are shown first ce

e (-5, 5) oduc A Pr e No uc t January $120 Firm od Prod Pr uce (100, 0) February $80 Union B No March $50 Firm t P e (0, 110) ro duc du Pro ce April $20 Union A Not Produ (0, 0) Note: B’s profits are shown first ce • Solve this problem by going to the end of the • In February, Union must offer Firm at least $30; if game and working backward. Union offers less, Firm will reject the offer and the • In April, Union will ask for $20, leaving Firm strike will continue. Therefore in March Union with $0. will receive $80 - $30 = $50. • In March, Firm must offer Union at least $20; if • In January, Firm must offer Union at least $50; if Firm offers less, Union will reject the offer and the Firm offers less, Union will reject the offer and the strike will continue. Therefore in March Firm will strike will begin. Therefore in January the Firm receive $50 - $20 = $30. will receive $120 - $50 = $70.

A • Backward induction B A

B A • perfect Nash equilibrium A A A B A Chapter 16 Oligopoly and Game Theory • Simultaneous move games • Sequential games – Rock-scissors-paper – Chess, checkers • Backwards induction • Dominant strategy • Credible threats – Prisoner’s dilemma •Examples: – Oligopoly -- firms choose quantities –Nim • Nash equilibrium – Addition Game – Number of Nash equilibria –Entry – Government subsidies – Labor

A with Payoffs B (The Prisoner’s Dilemma Game) Confess Not Confess Confess Not Confess

Confess 8, 8 0, 20 Confess 8, 8 0, 20 A A Not confess 20, 0 1, 1 Not confess 20, 0 1, 1

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

Confess 8, 8 0, 20 Confess 8, 8 0, 20

A Not confess 20, 0 1, 1 A Not confess 20, 0 1, 1

The Only Outcome that Remains is Confess, Confess. B Confess Not Confess

Confess 8, 8 0, 20

A Not confess 20, 01 1, 1

NIKE

ASICS H M L LL H 60,60 36,70 36,35 0,20

M 70,36 50,50 30,35 16,19

L 35,36 35,30 25,25 15,18

LL 20,0 19,16 18,15 14,14

Pricing Game

Server Receiver Left Right Backhand -1,1 1,-1 Forehand 1,-1 -1,1 Tennis Serve Game