Eco 40, J. Sandford, fall 0 October 4, 0 Homework #4 answers Problem Consider the following simltaneos-move game: Player Player Y Z W a,b c,d X e,f g,h a. List all ineqalities that mst hold for (W, Y ) to be a dominant strategy eqilibrim. (hint: an example of an ineqality wold be g h) a > e, c > g, b > d, f > h. b. List all ineqalities that mst hold for (W, Y ) to be a Nash eqilibrim. a > e, b > d. Problem This problem demonstrates a seeming pecliarity abot mixed strategy Nash eqilibria. Consider the following game between the Chicago Bears offense and the Detroit Lions defense. Payoffs are the nmber of yards advanced (positive yards for Chicago are negative yards for Detroit). Chicago Detroit rn defense pass defense rn -,,- pass,-, a. Find all pre strategy Nash eqilibria, if any. Then find the mixed-strategy Nash eqilibrim of the game. There are no pre strategy Nash eqilibria. There is a mixed-strategy Nash eqilibrim where Chicago rns fraction 6 of the time, and passes 7 of the time, and Detroit plays a rn defense fraction 4 7 time, and a pass defense 7 of the time. b. Now sppose that the Bears improve their rn game by bringing Mike Ditka ot of retirement: of the Chicago Detroit rn defense pass defense rn -, 0,-0 pass,-, Find the mixed-strategy Nash eqilibrim of the new game. Now Chicago rns fraction 4 7 of the time, and passes 7 of the time, while Detroit plays a rn defense fraction 9 7 6 of the time, and a pass defense fraction 6 of the time. c. When rnning the football becomes a more attractive option for the Bears, do they rn more often, or pass more often? Can yo explain why? While Ditka played tight end, the combination of his blocking and the downfield threat he poses as a receiver, even at 7, wold help their rnning game immeasrably.
Eco 40, J. Sandford, fall 0 October 4, 0 The Bears rn less often when their rnning game improves. The reason is that after Mike Ditka signs with the Bears, the Lions will become relatively more inclined to play a rn defense, lowering the Bear s yardage on rn plays. Problem Consider the seqential move game below: U D A B C 4 Y Z Y Z 7 6 0 0 What is the eqilibrim otcome of this game? plays D, plays A, and plays Z at both nodes (thogh the game ends before player has a chance to make his decisions). Payoffs are 4,,. Problem 4 Consider the following game played between a taxpayer and the IRS:
Eco 40, J. Sandford, fall 0 October 4, 0 Taxpayer IRS Adit Don t adit Cheat on taxes -60,60 -, Don t Cheat -0, -0,0 The game has no pre strategy Nash eqilibria, bt has one mixed strategy Nash eqilibrim. Find it. The taxpayer cheats of the time, the IRS adits of the time. Problem Consider the following interaction between two entrepreners (players and ) who are working on a joint project, and a ventre capitalist (player ) who is a potential investor in the project. First, player decides whether to devote high or low effort to preliminary work on the project. Player observes this choice and then decides whether to devote high or low effort himself. They then make a presentation to the ventre capitalist, who can observe which, if any, of the entrepreners devoted high effort to the project, and decides whether or not to invest. The payoffs are as follows. Each entreprener gets a payment of if the ventre capitalist invests and 0 otherwise. In addition, choosing high effort costs an entreprener, while choosing low effort is free. Investing costs the ventre capitalist, bt if he invests he gains for each entreprener who chose high effort. If the ventre capitalist does not invest, his payoff is 0. Draw the game tree corresponding to this game and find its eqilibrim otcome by solving backwards. See figre at end of answer key. Problem 6 Consider the game below: Thief Man Threaten Don t threaten Steal 0,0 0, Don t steal 40,60 0,0 a. What is the Nash eqilibrim if choices are made simltaneosly? There are Nash eqilibria in pre strategies. Don t steal, Threaten, and Steal, Don t threaten. b. What is the eqilibrim otcome if player chooses first? Yo shold draw the game tree in answering this qestion. If the thief chooses first, he chooses to steal, and the man chooses not to threaten. c. What is the eqilibrim otcome if player chooses first? The man chooses to threaten, and the thief chooses not to steal. Problem 7 Two bills are being considered in Congress (bill A, which wold reinstitte the Volstead Act, and bill B, which wold prohibit anyone of Canadian origin from owning property). Here are the payoffs to Congress and the president depending pon which laws are passed: Otcome Congress President Bill A only 8 - Bill B only - 9 Both bills Neither bill 0 0
Eco 40, J. Sandford, fall 0 October 4, 0 a. Sppose that Congress first decides which of the for options to select. The president can then either sign or veto, in which case no law is passed. Which bills become laws in the eqilibrim of this seqential game? Explain, with aid of a diagram. See the diagram on the last page. In the eqilibrim, Congress passes both bills, and the President signs them. b. Now sppose that the president has a line-item veto, so that if Congress passes both bills, he can choose to sign bill A or bill B only. However, he cannot enact laws that Congress does not pass. Which bills become laws in the eqilibrim of this game? Explain. Now, if both bills are sent to the President together, he will veto Bill A, bt not Bill B. As sch, the payoffs to Congress passing both bills are now (, 9). The game is otherwise the same as in part a. In tehprobab eqilibrim, neither bill becomes law. c. It is often sggested that giving the president a line-item veto wold be a good way to make government work more efficiently, as then he wold not have to veto entire bills jst becase he felt one provision of the bill wold make a bad law. In light of this qestion, what do yo think of this sggestion? A line-item veto wold have the direct effect of allowing the President to eliminate parts of legislation he views as harmfl, bt wold have the indirect effect of hindering compromise; nder a line-item veto, only bills that both Congress and the President agree on can become law, and it may be difficlt to work ot a compromise where each side contribtes his own favored policies to a bill.