Managerial Economics Koç University Graduate School of Business MGEC 501 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
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 High Low High 10,10 2,15 Toys“R”Us Low 15,2 5,5 � What should Toys“R”Us play? � Does that depend on what it thinks Wal-Mart will do? � Low is an example of a dominant strategy � it is optimal independent of what other players do � How about Wal-Mart? � (Low, Low) 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 = {High, Low}, S W = {High, Low} � Outcomes S = {(High, High), (High, Low), (Low, High), (Low, Low)} 3. Payoffs: For each player assigns a number to each outcome � u T (High ,High) = 10 � Reflects players’ rankings of outcomes Wal-Mart High Low High 10,10 2,15 Toys“R”Us Low 15,2 5,5 2
Low 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? Low 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 High Low Match High 10,10 2,15 10,10 Toys“R”Us Low 15,2 5,5 5,5 Match 10,10 5,5 10,10 � Is High ever an optimal strategy? � High 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? � Low 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
Iterated Elimination of Dominated Strategies (IEDS) � High is weakly dominated and Toys“R”Us is rational → it should not use High � High is weakly dominated and Wal-Mart is rational → it should not use High � If Toys“R”Us knows that Wal-Mart is rational, it knows that Wal-Mart will not use High � 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 Low a weakly dominated strategy for both Wal-Mart High Low Match High 10,10 2,15 10,10 Toys“R”Us Low 15,2 5,5 5,5 Match 10,10 5,5 10,10 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 � Low strictly dominates High � 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 � Low 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 � (Low, Low) is a strictly dominant strategy equilibrium � (Right, Right) is a weakly dominant strategy equilibrium Left Right High Low Up 10,10 5,15 High 10,10 2,15 Low 15,2 5,5 Down 15,5 5,5 4
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 � High 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 L M R U 1,0 1,2 0,1 D 0,3 0,1 2,0 2 3 1 � Iterated elimination of weakly dominated strategies: at least one strategy eliminated is a weakly dominated strategy � (Match, Match) is the unique outcome that survives IEWDS Low Price Guarantee: It Takes Two to Tango � What if only Toys“R”US is aware of this smart strategy? Wal-Mart High Low High 10,10 2,15 Toys“R”Us Low 15,2 5,5 Match 10,10 5,5 � Low becomes weakly dominant for Toys“R”US � If Wal-Mart believes that Toys“R”US is rational, it will play Low as well 5
Two-Tiered Tender Offer � Robert Campeau made a tender offer for Federated Department Stores in 1998 � The following is a simplified version of the actual offer � Pre-takeover price of a Federated share is $100 � Campeau offers to pay $105 for the first 50% of the shares tendered and $90 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 105 , s 50 < p 50 s 50 = − 105 90 , 50 × + × s ≥ s s If the takeover succeeds ( s ≥ 50) , the shares that were not tendered is worth $90 each; if it does not � succeed they are worth $100 � How much does each share cost Campeau if everybody tenders? � There are 100 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* ≥ 50 s* < 49 s* = 49 Tender 105 105 90 + 15(50/( s *+ 1)) Not 100 100 90 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 Stay – 200, – 200 300,0 XM Exit 0,300 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
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 Stay – 200, – 200 300,0 XM Exit 0,300 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 Enter Stay out P 0, 20 Fight Accommodate 10, 10 – 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
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