Coordination Games and Continuous Strategy Spaces More Complicated Simultaneous Move Games Other Coordination Games Suppose you and a partner are asked to choose one element from the following sets of choices choices If you both make the same choice you earn 1 otherwise nothing Red Green Blue Heads Tails 7 100 13 261 555 Write down an answer to the following questions questions If your partner writes the same answer you win 1 otherwise nothing A positive number A month of the year A woman s name Anti Coordination Anti Coordination Games Not the same dress Fashion Not the same words ideas Writing LUPI a Swedish S di h llottery game d designed i d in i 2007 Choose a positive integer from 1 to 99 inclusive The winner is the one person who chooses the lowest unique positive integer LUPI If no unique integer choice no winner Another Example of Multiple Equilibria in Pure Strategies The Alpha Beta game Strategies for Alpha Row are Up U Middle M or Down D Strategies for Beta Column are Left L or Right R Finding Equilibria by Eliminating Weakly Dominated Strategies Strategies U and M are weakly dominated for Player Alpha by t t D strategy D Suppose we eliminate strategy U first The resulting game is If eliminate L and then M the solution is D R Can Can Lead to the Wrong Conclusion Conclusion Suppose instead we eliminated Alpha s M strategy first The resulting game is Now if we eliminate R and then U the solution is D L Lesson If there are weakly dominated strategies consider all possible orders for removing these strategies when searching for the Nash equilibria of the game game Finding Equilibria via Best Response Analysis Always Works There are two mutual best responses D L and D R These are the two Nash equilibria q of the game Cournot Competition p A game where two firms duopoly compete in terms of the quantity sold market share of a homogeneous good is referred to as a Cournot game after the French economist who first studied it Let q1 and q2 be the number of units of the good that are brought to market by firm 1 and firm 2 Assume the market price P is determined by market demand P a b q1 q2 if a b q1 q2 P 0 otherwise P a Slope b q1 q2 Firm 1 s profits are P c q1 and firm 2 s profits are P c q2 where c is the marginal cost of producing each unit of the good Assume both firms seek to maximize profits Numerical Example Discrete Choices Suppose P 130 q1 q2 so a 130 b 1 The marginal cost per unit c 10 for both firms Suppose there are just three possible quantities that each firm i 1 2 can choose qi 30 40 or 60 There are 3x3 9 possible profit outcomes for the t firms two fi For example if firm 1 chooses q1 30 and firm 2 chooses q2 60 60 then P 130 30 60 40 P 130 30 60 40 Firm 1 s profit is then P c q1 40 10 30 900 Firm Fi 2 s 2 profit fit is i then th P c q P 2 40 10 60 1800 40 10 60 1800 Cournot Game Payoff y Matrix Firm 2 q2 30 q1 30 Firm 1 q1 40 q1 60 q2 40 q2 60 1800 1800 1500 2000 900 1800 2000 1500 1600 1600 800 1200 1800 900 1200 800 0 0 Depicts all 9 possible profit outcomes for each firm Find the Nash Equilibrium Firm 2 Nash Equilibrium q1 30 Firm 1 q1 40 q1 60 q2 30 q2 40 q2 60 1800 1800 1500 2000 900 1800 2000 1500 1600 1600 800 1200 1800 900 1200 800 0 0 q 60 is strictly dominated for both firms use cell by cell inspection to complete the search for the equilibrium Continuous Pure Strategies In many instances the pure strategies available to players do not consist of just 2 or 3 choices but instead consist of infinitely many possibilities We handle these situations by finding each player s reaction function a continuous function revealing the action the player will choose as a function of the action chosen by the other p y player For illustration purposes let us consider again the two firm Cournot quantity competition game Duopoloy two firms only competition leads to an outcome in between monopoly 1 firm maximum possible profit and perfect competition p p many y firms each earningg 0 profits p p c p Profit Maximization with Continuous Strategies Firm 1 s profit 1 P c q1 a b q1 q2 c q1 a bq2 c q1 b q1 2 Firm 2 s profit 2 P c q2 a b q1 q2 c q2 a bq1 c q2 b q2 2 Both firms seek to maximize profits We find the profit maximizing amount using calculus Firm 1 d 1 dq1 a bq2 c 2bq1 At a maximum maximum d 1 dq1 0 0 q1 a bq2 c 2b This is firm 1 s best response function Firm 2 d 2 dq q2 a bq q1 c 2bq q2 At a maximum d 2 dq q2 0 q2 a bq1 c 2b This is firm 2 s best response function In our numerical example firm 1 s best response function is q1 a bq2 c 2b 130 q2 10 2 60 q2 2 Similarly firm 2 s best response function in our example is q2 a bq a bq1 c 2b c 2b 130 q 130 q1 10 2 60 q 10 2 60 q1 2 2 Equilibrium with Continuous Strategies Equilibrium can be found algebraically or graphically Algebraically q1 60 q 60 q2 2 and q2 60 q 60 q1 2 so substitute out using one of these equations q1 60 60 q1 2 2 60 30 q1 4 so q1 1 1 4 30 q1 30 75 40 Similarly you can show that q2 40 as well the problem is perfectly symmetric Graphically q1 120 60 q 60 q1 2 40 60 q q2 2 40 120 q2 Multiple Equilibria F off Life Fact Lif or Problem P bl to b be R Resolved l d Suppose we have multiple Nash equilibria the Alpha Beta game is an example What can we say about the behavior of players in such games One answer is we can say nothing both equilibria are mutual best responses so what we have is a coordination problem as to which equilibrium players will select Such coordination problems seem endemic to lots of interesting strategic environments For instance there are two ways to drive on the right and on the left and no amount of theorizing has led to the conclusion that one way to drive is better than another The U S and France drive on the right the U K and Japan drive on the left A second answer is that if economics strives to be a predictive science then multiplicity of equilibria is a problem that has to be …