Finish all the questions please
1.(a) In a repeated Prisoner’s Dilemma player 1 plays the Grim strategy and player 2 plays Nice Tit-for-Tat. What would the first eight pairs of moves look like? Write them out like this, except instead of X put C or D, of course. Player 1: X X X X X X X X Player 2: X X X X X X X X (b) Suppose the first player plays an alternating strategy, starting with C and doing the exact opposite of what he himself did last time. The second player plays Nasty Tit for Tat – that is, starts at D, and then chooses what the first player did on the last move. Fill in the X’s again. [NOTE: The original of this last sentence was “The second player plays Nasty Tit for Tat – that is, starts at D, and then chooses the opposite of what the first player did on the last move. Fill in the X’s again.” The revised version is the correct definition of Nasty Tit-for-Tat, but if you’ve already done written up the answer, no need to change it. The TA’s will take either one as correct.] (c) Draw the Moore graph for (b). 2. Suppose your discount rate is .8 per year. Would you rather have four yearly payments, starting now of $10, $10, $10, and $10, or would you rather have $4 yearly forever? Show your work of course. 3. (a) For the following Prisoner’s Dilemma game, infinitely repeated, is the strategy pair (Grim, Grim) and equilibrium for discount rate δ = .7? Show your work, as always. C D C 3,3 1,6 D 6,1 2,2 (b) What is the minimum discount rate for which (Grim, Grim) is an equilibrium? 4. The 2008 Presidential nomination saw 21 Republican primaries and caucuses happening all on one day, so-called Super Tuesday, Feb 8. There were only four real candidates left afterwards: John McCain, Mitt Romney, Mike Huckabee, and Ron Paul. In the West Virginia caucus, there were seven groups of voters, we’ll assume. Here are their sizes and their preferences. Group 1 (16%) JM MR MH RP Group 2 (28%) MR JM MH RP Group 3 (13%) MR MH JM RP Group 4 (21%) MH MR JM RP Group 5 (12%) MH JM MR RP Group 6 (6%) RP MR MH JM Group 7 (4%) RP MH MR JM (a) Let’s assume the voting procedure was the following. Candidates are eliminated in rounds until only one is left. Assume the voters vote sincerely each time – that is, they vote for their top choice. On the first round Mitt Romney won a plurality of the votes in the first round with 41%. (Plurality means he won more votes than anyone else.) But that doesn’t make him the winner. After each round of the caucus if no one wins a majority, (majority means over 50%0 then the candidate with the fewest votes is dropped. Then his or her supporters vote for one of the remaining candidates. Again they vote sincerely. Who would win? (It probably helps to copy the above table, and cross people in it, as you determine the fewest-vote-getter each time.) (c) Is there a Condorcet winner? As soon as you find someone who beats a certain candidate you can move on to the next. 5. We talked about how to solve the simple equations to find a mixed strategy equilibrium in a 2x2 game. If it’s a bigger game than 2x2, things get complicated and you need some other method, for example, linear programming. Still, if I gave a many-row, many-column game, and I gave you the column-chooser’s probabilities, you could check if that mixed strategy could be part of an equilibrium. How about this game, where Column-chooser uses the probabilities assigned in the top row. Why might Column’s strategy be or not be a part of a Nash equilibrium? .4 .2 .2 .2 W X Y Z A 1,0 0,1 0,1 0,1 B 0,1 0,1 1,0 1,0 C 0,1 1,0 0,1 1,0 D 0,1 1,0 1,0 0,1 6. In class I objected to the voting procedure used in recalling Governor Newsom. The procedure was that first people vote on whether to recall the governor, then they vote on who should replace the governor if the first vote turns out to get over 50% Yes’s. Given my objection, which of the following changes would I like best? Say why in a short sentence. (i) Include the current governor on the list of replacement candidates. (ii) require a 2/3 majority vote to replace the governor (nor just an over-50% majority.) (iii) eliminate recall elections in California (most states don’t have them.) (iv) limit the time someone can be the governor to a single three-year term.