Managerial Economics. Game Theory

Transcription

1 Managerial Economics Koç University Graduate School of Business MGEC 1 Levent Koçkesen Game Theory Game theory studies strategic interactions within a group of individuals actions of each individual have an effect on the outcome that is of interest to all individuals are aware of that fact Individuals are rational have well-defined objectives over the set of possible outcomes implement the best available strategy to pursue them Rules of the game and rationality are common knowledge 1

2 Price Competition Toys R Us and Wal-Mart have to decide whether to sell a particular toy at a high or low price They act independently and without knowing the choice of the other store Wal-Mart Toys R Us,2 2, What should Toys R Us play? Does that depend on what it thinks Wal-Mart will do? is an example of a dominant strategy it is optimal independent of what other players do How about Wal-Mart? (, ) is a dominant strategy equilibrium A lesson we learned from oligopoly models Individual rationality does not imply collective rationality Strategic Form Games It is used to model situations in which players choose strategies without knowing the strategy choices of the other players Three components: 1. Players: N = {Toys R Us, Wal-Mart} 2. Strategies: S T = {, }, S W = {, } Outcomes S = {(, ), (, ), (, ), (, )} 3. Payoffs: For each player assigns a number to each outcome u T (,) = Reflects players rankings of outcomes Wal-Mart Toys R Us,2 2, 2

3 Price Guarantee Toys R Us web page has the following advertisement Sounds like a very good deal for consumers How does this change the game? Price Guarantee What happens if we add price matching as a strategy for both stores? Match: post a high price and match the other store s price Wal-Mart Match 2, Toys R Us,2 Match Is ever an optimal strategy? is weakly dominated by Match Is Match a dominant strategy? A rational player should not use a dominated strategy What happens to the game once you eliminate the dominated strategies? Is there a dominated or dominant strategy in the new game? becomes weakly dominated Match becomes weakly dominant Unique solution is (Match, Match) The above procedure is known as Iterated Elimination of Dominated Strategies (IEDS) 3

4 Iterated Elimination of Dominated Strategies (IEDS) is weakly dominated and Toys R Us is rational it should not use is weakly dominated and Wal-Mart is rational it should not use If Toys R Us knows that Wal-Mart is rational, it knows that Wal-Mart will not use This is where we use common knowledge of rationality To be a good strategist try to see the world from the perspective of your rivals and understand that they will most likely do the same This makes a weakly dominated strategy for both Wal-Mart Match 2, Toys R Us,2 Match Dominant Strategy Equilibrium A strategy X strictly dominates another strategy Y, if X always gives a strictly higher payoff than Y no matter what other players do strictly dominates A strategy X weakly dominates another strategy Y, if X never gives less payoff than Y and sometimes gives a strictly higher payoff Right weakly dominates Left dominant strategy: it dominates every other strategy it is optimal independent of what other players do strictly dominant: strictly dominates every other strategy is strictly dominant weakly dominant: weakly dominates every other strategy Right is weakly dominant If every player has a dominant strategy, then the corresponding outcome is a dominant strategy equilibrium (, ) is a strictly dominant strategy equilibrium (Right, Right) is a weakly dominant strategy equilibrium Left Right 2, Up 5,,2 Down,5 4

5 Iterated Elimination of Dominated Strategies dominated strategy: never optimal no matter what other players do strictly dominated: there is a strategy that strictly dominates it is strictly dominated weakly dominated: there is a strategy that weakly dominates it Up is weakly dominated Iterated elimination of strictly dominated strategies: every strategy eliminated is a strictly dominated strategy (U, M) is the unique outcome that survives IESDS U D L 1,0 0,3 3 M 1,2 0,1 Iterated elimination of weakly dominated strategies: at least one strategy eliminated is a weakly dominated strategy R 0,1 2,0 (Match, Match) is the unique outcome that survives IEWDS 1 2 Price Guarantee: It Takes Two to Tango What if only Toys R US is aware of this smart strategy? Wal-Mart 2, Toys R Us,2 Match becomes weakly dominant for Toys R US If Wal-Mart believes that Toys R US is rational, it will play as well 5

6 Two-Tiered Tender Offer Robert Campeau made a tender offer for Federated Department Stores in 18 The following is a simplified version of the actual offer Pre-takeover price of a Federated share is $0 Campeau offers to pay $5 for the first % of the shares tendered and $0 for the remainder All shares, however, are bought at the weighted average price. If s is the percentage share tendered, then the price of each share tendered is given by 5, p = s 5 + 0, s s s < s If the takeover succeeds (s ), the shares that were not tendered is worth $0 each; if it does not succeed they are worth $0 How much does each share cost Campeau if everybody tenders? There are 0 Federated shareholders (including you) each of whom owns one share What would you do: Tender or Not? Let s* be the number of shares tendered not including you s* < 4 s* = 4 s* Tender (/(s*+1)) Not Game of Chicken There are two providers of satellite radio: XM and Sirius XM is the industry leader with 5 million subscribers; Sirius has 2.2 million In the long-run the market can sustain only one provider Sirius Stay Exit XM Stay Exit 0, 0 0,0 0,0 0,0 Is there a dominated strategy? What are the likely outcomes? Could (Stay, Stay) be an outcome? If XM expects Sirius to exit, what is its best strategy (best response)? If Sirius expects XM to stay what is its best response? (Stay, Exit) is an outcome such that 1. Each player best responds, given what they believe the other will do 2. Their beliefs are correct It is a Nash Equilibrium 6

7 Nash Equilibrium Nash equilibrium is a strategy profile (a collection of strategies, one for each player) such that each strategy is a best response (maximizes payoff) to all the other strategies Nash equilibrium is self-enforcing: no player has an incentive to deviate unilaterally One way to find Nash equilibrium is to first find the best response correspondence for each player Best response correspondence gives the set of payoff maximizing strategies for each strategy profile of the other players and then find where they intersect XM s best response to Stay is Exit Its best response to Exit is Stay Sirius best response to Stay is Exit and to Exit is Stay Best response correspondences intersect at (Stay, Exit) and (Exit, Stay) These two strategy profiles are the two Nash equilibria of this game We would expect one of the companies to exit in the long-run Sirius Stay Exit XM Stay Exit 0, 0 0,0 0,0 0,0 Sequential Move Games Strategic form games are used to model situations in which players choose strategies without knowing the strategy choices of the other players In some situations players observe other players moves before they move Consider the following entry game: Kodak is contemplating entering the instant photography market and Polaroid can either fight the entry or accommodate K Stay out Enter P 0, Fight Accommodate 5, 0, Sequential moves games can be represented using a game tree What should Polaroid do if Kodak enters? Given what it knows about Polaroid s response to enter, what should Kodak do? This is an example of a backward induction equilibrium 7

8 Backward Induction Equilibrium At a backward induction equilibrium each player plays optimally at every decision node in the game tree (i.e., plays a sequentially rational strategy) (Enter, Accommodate) is the unique backward induction equilibrium of the entry game (Stay out, Fight) is a Nash equilibrium of this game Polaroid Fight Accommodate Kodak Stay Out Enter 0,0 5,0 0,0 (Stay out, Fight) benefits Polaroid It can only be sustained by issuing a threat If Polaroid could commit itself to fight it would benefit Commitments (tying your own hands) could be beneficial in strategic interactions Commitments to what seem to be ex post irrational strategies could be beneficial We have already seen such a beneficial commitment in Stackelberg game Hernan Cortes ordered his men to burn all but one ship in his conquest of Mexico Credibility of the commitment is key Kodak entered instant photography in late 170s, but in a gentle fashion Designed cameras and film that were incompatible with Polaroid s cameras reassuring Polaroid that it would be able to continue to make money, selling to its customer base Yet Polaroid chose war: launched patent infringement lawsuits, lowered prices, introduced new products Edwin Land (founder of Polaroid) This is our very soul our whole life. For them, it s just another field. (NYT: April 28, 176) Is the threat credible? Polaroid limited itself to only one industry. Loosing that market would mean the end of the company, whereas Kodak was diversified. Polaroid won the war Commitments 8

9 Price Protection General Electric and Westinghouse were the only manufacturers of large electric turbine generators during 60s and 70s 163: GE offered its customers price protection for 6 months This is known as most-favored-customer-clause (MFCC) Westinghouse responded by offering the same deal This looks like a policy that benefits the customers Antitrust Division of the U.S. Department of Justice demanded that GE and Westinghouse refrain from this policy claiming that it harmed the customers What is going on here? Price Protection Suppose it costs GE $,000 per unit to build a turbine There are potential customers with reservation prices as in the table below Potential Buyer Reservation Price TR MR TC MC Profit What is the profit maximizing price for GE? GE sets its price at $60,000 and sells a turbine to the first 5 customers There are still 5 more customers who are willing to do business with GE

10 Price Protection After GE sells to the first 5 customers the remaining market is as follows What is the profit maximizing price for GE now? GE sets its price at $40,000 and sells a turbine to customers 6 and 7 Potential Buyer Reservation Price 40 TR MR - TC MC Profit The market becomes GE sets its price at $,000 and sells a turbine to customer 8 Potential Buyer 8 Reservation Price TR 40 MR TC MC Profit GE sets its price at $,000 and sells a turbine to customer Potential Buyer Reservation Price TR MR TC MC Profit No more profitable trade for GE Price Protection So, if the customers are not willing to wait, i.e., they are impatient, GE can implement price discrimination over time Period Price Customers What happens if the customers are patient? Anticipating a price cut by GE, customer 8 will wait until the next period In general, as long as the current price is greater than the MC the monopolist will have an incentive to reduce the price next period. Customers anticipate this and do not buy unless P = MC The unique backward induction equilibrium of the game is P = MC Coase conjecture: a monopolist selling a durable good to very patient customers has no monopoly power GE would like to commit itself to not lower the price in the future. If it can do that, then it would set price at the monopoly price of $60,000 Most-favored-customer-clause provides the commitment mechanism: GE cannot price discriminate over time. ering the price next period means lowering the price for all the units that had been sold before. MFCC reduces incentives to cut prices It also makes it easier to collude on monopoly price since price cutting is costly