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Economics S4415 Columbia University Summer 2001 Professor Dutta Solutions to Problem Set 3 Chapter 9 9.2 Consider the payo¤ matrix of any 2 £ 2 game, i.e., any game with two players and two pure strategies: P layer 1nP layer 2 t c t a; a d; e c e; d b; b i) Write down parameter restrictions so that (t; t) is a symmetric Nash equilibrium. ii) Under what restrictions, can (c; c) be a symmetric Nash equilibrium? Are the restrictions in parts i) and ii) compatible with each other, i.e., can such a game have multiple pure strategy symmetric Nash equilibria? Answer: i) a ¸ e ii) b ¸ d. Yes. 9.5 Consider the de…nition of a two-player symmetric game. Using the de…nition, prove the following statement in a semi-rigorous fashion: if the two players play identical actions, they get exactly the same payo¤. Answer: Suppose the action is a pure strategy choice. By de…nition of a symmetric game, if each player chooses thew same pure strategy then their two payo¤s must be equal. Suppose instead that the two players choose identical mixed strategies, say each player chooses strategies s1 , s2 , ::: sM with probabilities p1 , p2 , ::: pM . With probability p2k each player chooses the same pure strategy sk (k = 1, ::: M ) and in that case their two payo¤s are identical. Consider two di¤erent pure strategies instead say s1 and s2 . With probability p1 £ p2 , player 1 picks s1 and 2 picks s2 . In this case their payo¤s need not be the same - say they are ¼(s1 ; s2 ) and r(s1 ; s2 ) respectively. However there is an identical probability p1 £ p2 - that these choices will be reversed, i.e., that player 1 picks s2 and 2 picks s1 . And if that happens the two payo¤s will be reversed as well, i.e., player 1 will get r(s1 ; s2 ) and 2 will get ¼(s1 ; s2 ). Hence the expected payo¤s arising out of this pair of pure strategies is exactly the same for each player, i.e., is p1 £ p2 £ [¼(s1 ; s2 ) + r(s1 ; s2 )]. The same argument applies to any two strategies sk and sk0 - hence the expected payo¤s are identical for the two players. 1
9.8 i) Redo parts i) and ii) of the previous question if the costs of staying increase from 5 dollars to 10 dollars so that the payo¤ matrix becomes: F irm 1 n F irm 2 date 0 date 1 date 2 date 0 0; 0 0; 15 0; 30 date 1 15; 0 ¡10; ¡10 ¡10; 5 date 2 30; 0 5; ¡10 ¡20; ¡20
ii) How does this increase a¤ect the expected pro…ts in the symmetric equilibrium? Explain. Answer: i) Note that a mixed strategy - 0.5 on 0 and 0.5 on 2 dominates the pure strategy date 1. Hence we can concentrate on …nding a mixed strategy equilibrium in which only dates 0 and 2 are played (why?). Suppose the probability of the former is p. The expected payo¤ to playing 2 is therefore 30p ¡ 20(1 ¡ p). This equals the expected payo¤ to playing 0 - 0 - if p = 52 . Expected payo¤ must be 0 (why?). ii) The expected pro…ts are the same as in 2.1 - the probabilities are di¤erent and there is a greater likelihood of early exit by each …rm. 9.16 Consider the following bankruptcy model (with collection costs of two dollars): # of other creditors grabbing 0 1 2 grab 822 ref rain 514 i) Compute the symmetric mixed strategy equilibrium. ii) Compare the probability that each creditor ref rains with the probability that was derived in the text. Explain your answer. iii) How successful is voidable preference law as a deterrent in this case? Answer: Suppose that the probability of grab is p. Then the expected payo¤ from grabbing is 8(1 ¡ p)2 + 4p(1 ¡ p) + 2p2 while the expected payo¤ from refraining is 5(1 ¡ p)2 + p(1 ¡ p) + 4p2
p
The two payo¤s are equal if 3(1 ¡ p) = 2p2 ; i.e., p = ¡3+4 33 , i.e., approximately 0:7 ii) There is a greater probability of ref rain in this case because collection costs are higher. 2
3 3 iii) Everybody refraining has probability ( 10 ) , i.e., 2.7% - not very high even in this case! 9.17 i) Redo the previous question when collection costs are four dollars. ii) How high would the costs need to be for all three creditors to ref rain from stripping the insolvent company’s assets? Chapter 11 11.2 Redo the theatre game of F igure 11.1 to allow for the possibility that a cab can get caught in a tra¢c jam (and in that case is slower than the subway although it is faster than the bus). Suppose that road conditions are known to both players before they make their transportation choices. Be sure to carefully explain the payo¤s that you assign. Answer: Exercise 11.2
b Player 2
c s
π1(Τ,b), π2(N,b) π1(Ν,b), π2(T,c) π1(Ν,b), π2(T,s)
b b Player 1
Player 2
c
c s
s Player 2
b c s
π1(Τ,c), π2(N,b) π1(Τ,c), π2(N,c) 0.75π1(Τ,c)+0.25π1(N,c), 0.75π2(N,s)+0.25π2(T,s) π1(Τ,s), π2(N,b) 0.75π1(Ν,σ)+0.25π1(T,s), 0.75π2(T,c)+0.25π2(N,c) π1(Τ,s), π2(N,s)
11.5 Use backward induction to solve the game from a con…guration of (1; 0)? What about (2; 0) and (3; 0)? Answer: Use the convention that the …rst player to move is player 1. >From (1; 0) clearly player 2 wins. From (2; 0) player 1 can force a win by moving to (1; 0) but loses otherwise. From (3; 0) again player 1 can force a win by moving to (1; 0). 11.11 Consider Example 2 now. Show that there are no (fully) mixed strategy Nash equilibria in that game of entry, i.e., the only Nash equilibria are the pure strategy Nash equilibria that I discussed in the text. (Hint: Show that if Pepsi mixes between T and A, then Coke will never play any of the strategies ET T; EAT; and EAA - since these are all dominated by ET A. Can you show that if Coke plays a mixed strategy in which she puts positive probability on ET A, Pepsi will never want to mix between T and A?) 3
Answer: Consider the strategic form given on p. 16. ET A dominates ET T; EAT; and EA 6 A and hence if Pepsi mixes between T and A, then Coke will never play any of the strategies ET T; EAT; and EAA. On the other hand if Coke plays ET A then Pepsi has a strict best response, i.e., A. Indeed if if Coke plays any mixed strategy with positive probability on ET A and the four strategies involving O, then Pepsi has a strict best response - to play A. So there is no mixed strategy Nash equilibrium in which Pepsi plays T and A with positive probabilities. 11.17 Show that if we attach a backwards induction solution to the n ¡ 1 step problem to the best choices at the …nal decision nodes, we get a backwards induction solution to the n step problem. Answer: This is true virtually by de…nition. Suppose we are at a penultimate decision node of the original game - this corresponds to a …nal decision node of the n¡1 step problem. At this node the backwards induction solution of the n¡1 step problem involves the best choice after incorporating the continuation best choices of the last decision nodes. But that of course is what we would do if we folded the tree two steps.... Chapter 12 12.4 Suppose that the patent is worth $25. Everything else is unchanged. Solve the R&D game by backwards induction. Answer: Exercise 12.4
End R Satety Zone I for R
Trigger Zone I
(3,3)
Safety Zone II for R
Trigger Zone II
(6,6) Trigger Zone III
Safety Zone II sor S
(8,8)
Safety Zone I for S
End S
Trigger Zone IV (10,10) (11,11) (12,12)
12.9 Show then that Safety Zone I, for A, is made up of all locations in which Firm A is within 3 steps - and Firm B is more than 2 steps from …nishing, i.e., is made up of all locations (a; b) such that a 3 and b > 2. Similarly, show that Safety Zone I, for B, is made up of all 4
locations in which Firm B is within 2 steps - and Firm A is more than 3 steps from …nishing, i.e., is made up of all locations (a; b) such that a > 3 and b 2. Answer: Consider (3; 3). Then Firm B …nds it unpro…table to move 3 steps and 2 steps or less is wasted money (given the Trigger Zone). Similarly for (2; 3) and (1; 3). But that means from (3; 4) Firm B …nds it unpro…table to move 1 , 2 or 3 periods, etc. .. A symmetric argument establishes the safety zone of Firm B. 12.13 Find the backwards induction solution to this game. Answer: Exercise 12.13 End B Trigger Zone I Safety Zone I for B (3,2) Trigger Zone II (5,4)
Safety Zone I for A
(7,5) (8,6)
12.17 Redo the analysis of the book when public policy subsidizes Firm A’s R&D by 50%, i.e., it costs Firm A, 1, 3.5 and 7.5 dollars to make. respectively, 1, 2 and 3 steps of progress.
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End A
Answer: Exercise 12.17 End B Trigger Zone I
Safety Zone I for B
(3,3) Trigger Zone II (6,5) (9,7) (12,8) (15,9) (17,10)
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Safety Zone I for A
End A
