Chapter 16 “Game theory is the study of how people Oligopoly 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.” Game Theory Oligopoly Oligopoly • “Oligopoly is a market 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 ‘ideal’ inferring) what the rivals are doing and then cases, perfect competition and monopoly. choosing a “best response” • In the ‘ideal’ cases, the firm just has to figure out the environment (prices for the perfectly competitive firm, • This means that firms in oligopoly markets are demand curve for the monopolist) and select output 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 games. environment before computing the best output. • This is the role of Game Theory. Some Concepts We Will Use Strategies • 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 matrix 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 Sequential Game 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 – Chess 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 strategy 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 competitions could get, say, 15 and the loser -15. – Inventions 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 price 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 • Nash Equilibrium 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 outcome 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, 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 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 Backward Induction 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. Addition 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 numbers 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 (-5, -5) e c u d ro P (100, 0) A oduce Not Pr (0, 100) e c u d e o c r u P d ro (0, 0) P duce t Pro (5, 24) B No Not Produce A ce ri P h g (-10, 20) Hi A rice Low P Note: B’s profits are shown first (0, 27) e r c e i t r n P E h ig (0, 23) H rice ow P B L Not Enter A Note: B’s profits are shown first Bargaining January $120 Firm February $80 Union (-5, 5) March $50 Firm e c u April $20 Union d ro P (100, 0) A oduce Not Pr (0, 110) e c u d e o c r u P d ro (0, 0) P duce t Pro B No Not Produce A Note: B’s profits are shown first • Solve this problem by going to the end of the • In February, Union must offer Firm at least $30; if game and working backward.