What is the minimum discount rate for which (Grim, Grim) is an equilibrium?

Management by Geoffrey

5 Basic Game Theory Questions

  1. Suppose the first player plays Tit for Tat nicely – starting with C. The second player plays nasty Tit for Tat – that is, starts at D. What would the first seven pairs of moves be? Could there be any reason that the first player would start with C knowing the other is going to defect?
  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.
  • (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   1,8

D   8,1   2,2

  • (b) What is the minimum discount rate for which (Grim, Grim) is an equilibrium?

 

  1. 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

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%) 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.)

 

  1. There are exactly four candidates, A, B, C and D in an election, and there are exactly five citizens voting on them.

Citizen 1.   C, A, D, B

Citizen 2.   B, D, C, A

Citizen 3.   B, D, C, A

Citizen 4.   C, D, A, B

Citizen 5.   C, D, B, A

  • (a) Which candidates are Pareto efficient for these four citizens?  If a candidate is not Pareto efficient say why not.
  • (b) Is there a Condorcet Winner?  If there is, who is it?
  • (c)  Who would win by the Borda count?
  • (d) Who would win by approval voting, assuming Citizens 1 and 2 voted for only one candidate, but citizens 3, 4, and 5 voted for two.

 

  1. In class I had four movies D, C, S and R and a preference ordering for three people who were voting on which one to go see. Use these orderings instead

person 1:  D R C S

person 2:  S R D C

person 3:  C S D R

  • (a) Suppose the voting method was to choose between the pairs D, C versus S, R, and then choose between whichever pair wins.  Who would win?  (I showed the general method in Thursday’s class.)
  • (b) Suppose instead they started with a choice between either D, S or C, R, then went on to choose one of that pair.  Who would win?
  • (c) Suppose they started with a choice between D, R versus C, S.  Who would win?
  • (d) When we make these predictions about who would win, are we assuming that the players know each other’s preferences, or are we assuming only that they know their own?