Economics S4415Columbia University Summer 2001 Professor Dutta Solutions to Problem Set 1 Chapter 1 1.4 Consider the purchase of a house. By carefully examining each of the four components of a game situation - group, interaction, rationality, and strategy - discuss whether this quali…es as a game. Answer: Group - seller and potential buyers (bidders); interaction - only one of the bidders is going to eventually buy the house and will do so at his winning bid and this bid will be what the seller gets from selling the house; strategy - for bidders a determination of how much they are willing to go up to and for the seller a determination of how much he is willing to come down to; rationality - a strategy for the bidder that is consistent with the maximization of some objective (such as the expected gain from buying the house) and similarly a strategy for the seller that is consistent with the maximization of some objective (such as the expected gain from selling the house). 3.1 Show in detail that player 2 has a winning strategy in Nim if the two piles of matches are balanced. (Your answer should follow the formalism I set up in the text; in particular, every con…guration of matches should be written as (m; n) and removing matches should be represented as a reduction in either m or n.) Answer: Initially, the con…guration of matches is (m; m) for some number m. Suppose that after player 1’s move the con…guration is (k; m) where k is a number between 0 and m ¡ 1. If k = 0, then player 2 can remove the m matches from the second pile and win the game. If k > 0, then player 2 can move the game to (k; k). Now it is player 1’s turn to move; suppose he moves the game to (l; k). If l = 0, player 2 can remove the k matches from the second pile and win the game. Else. he can move the game to (l; l). In this fashion, after no more than m moves of player 1, the game must arrive at a con…guration such as (0; p) where p > 0. Then player 2 removes the last p matches from the second pile. (In this answer I simpli…ed the exposition by assuming that player 1 always removes from the …rst pile and I can do this without loss of generality since after every move of player 2 there is exactly the same number of matches in each pile. Hence, if player 1 removes from the second pile then player 2 can remove from the …rst so as to keep the piles balanced.) 1 by question 3. 3 Consider the following model of price competition. Chapter 3 1. 0). he is a little happier if he is at the football game by himself (he is happier still if he is with his wife at the opera and is the happiest if they are both at the football game)? (Likewise.A wins against the e¤ective alternative choice of N. 1 0. How would you modify the payo¤s to (f. Answer: Player 1 can balance the piles in his …rst move. ¡1 o ¡1. or b) (m. This strategy involves matching the other player . Truthful voting …rst round . What would be the outcome of truthful voting in this case? What about strategic voting? Answer: Truthful voting second round . In the former case player removes all but one match from the only pile remaining.1 Consider the game of Battle of the Sexes. If …rm 1 is the lower priced …rm. 3. show that if the two piles are unbalanced. the wife is unhappiest when she is at the football game by herself. then it is …rm 1 that meets all of the demand and conversely if …rm 2 is the lower priced out…t. For example. where m ¸ 2. 1).9. if …rms 1 and 2 post prices 2 . Thereafter. In the latter case he removes all m matches from the larger pile. Strategic voting …rst round . Strategic voting second round .f) to re‡ect the following: the husband is unhappiest when he is at the opera by himself. she is a little happier if she is alone at the opera. player 1 has a winning strategy. happier still if she is with her husband at the football game and is the happiest if they are both at the opera.o) and (o.12 Suppose voter 3’s preferences were the following: N ÂBÂA (instead of N Â A Â B as in the text). 0 1.) Answer: Modi…ed Battle of the Sexes HnW f o f 3.10 Finally.A wins against B.in order to kep the two piles balanced unless player 2 has a) moved the game to (m.A wins against N but B loses to N. he has a winning strategy.A wins against N but B loses to N. Two …rms set prices in a market whose demand curve is given by: Q= 6¡p where p is the lower of the two prices.3. then …rm 1. 1.and hence sells 3 units.e.5 Consider voter 1. Finally.9 Show that the strategy of volunteering for 1 hour (weakly) dominates the strategy of volunteering for 2 hours.. 2. after some rearranging. Suppose.equal to 2 and 4 dollars respectively. …rm 1 would have been better o¤ pricing at p ¡ 1 (work the pro…t …gures out for yourself). If the rival’s price is $5. 2 prices can be 0. squaring both sides q q 2+y 2+y+2 1+y >2+y and that last inequality always holds. then they each get half the market. i. then the latter has a zero pro…t but the former has a strictly positive pro…t of 25 . furthermore. Consider any price between p = 2 and p = 6. 5 or 6 dollars. if the rival’s price is $6.or vice-versa? 3 . is equivalent to q 1+y+1> or. as the lower priced …rm. each of the two strategies yields a pro…t of 0. there is no dominant strategy.. then …rm 1 would have been better o¤ pricing at 3 instead. In that case. Clearly a price of 0 is not a dominant strategy either.2 Show that the strategy of posting a price of $5 (weakly) dominates the strategy of posting a price of $6.e.. Suppose that prices can only be quoted in dollar units. 3. i. they each get 6¡p . On the other hand p = 1 is not a dominant strategy because if the rival …rm prices at 6.4 Is there a dominant strategy for player 1? Explain. Does it strongly dominate as well? Answer: If the rival’s price is $4 or less. Answer: No. that costs of production are zero for both …rms. There are two strategies that involve truthvoting in the second stage. i. AAN and BAN . Does AAN dominate BAN .e. meets all of the market demand . Does it strongly dominate as well? Answer: The strategy of volunteering for 1 hour (strongly) dominates the strategy of volunteering for 2 hours if and only if q 1+y¡1> q 2+y¡2 That inequality. then the former strategy yields a pro…t of 5 while the latter yields a pro…t of 0. 4. 3. Chapter 4 1. and suppose that …rm 2 happens to match that price. 3. 3. Hence the former strategy (weakly) dominates the latter. If the two …rms post the same price p. Right . 1 0. then it makes no di¤erence to the overall outcome . 2. say m. Do this for both i) strong as well as ii) weak domination. 0 1. b) The US vetoes B (and Africa plays its dominant strategy). then veto A (and thereby elect H). Answer: The monopoly price. 4. Finally. If the rival prices between m and m + k.Answer: Yes . Answer: The following game has an IEDS solution of U p. If B has already been vetoed. Then A would go on to win the second round of voting and that outcome would be strictly preferred by this voter to the alternative under which B gets elected in the …rst round but then loses the second round election. then elect B by vetoing the other remaining candidate. when only one of the other two voters votes for A). 1 1. There is however no dominant strategy. If the vote switch makes no di¤erence to the …rst round election’s outcome.3 Give an example of a game that has an outcome to IEDS although no player has a dominant strategy.3 a) What is the dominant strategy for Africa? b) What is the IEDS solution? Answer: a) If B has not been vetoed. the …rm has strictly higher pro…ts from pricing at m (why?). 0 4 .. by de…nition.and hence payo¤. 0 0. 2 Down 1.in any price competition model . then both prices yield zero pro…ts. If the rival prices at or above m + k then. say m+k above that level.regardless of whether we use strong or weak dominance to eliminate strategies. dominates any price.switching a …rst-round vote from B to A might get A passed in a circumstance where it might not have passed otherwise (i. 1. H gets elected. if the rival prices below m.e. that prices above the monopoly price.9 Based on your answer to the previous two questions can you give a reason why . 1n2 Lef t Center Right U p 0. again the monopoly price yields a higher pro…t (since the pro…t is 0 for price m + k).a duopoly …rm would never want to price above the monopoly price? (Hint: when can a duopoly …rm. make positive profits? What would happen to those pro…ts if the …rm charged a monopoly price instead?).