Formulate the above problem as a matrix game. (Use: Only use the following payoffs: 10,5,-5,-10)
Find all pure Nash Equilibria in the game you described in the previous part.
Compute the average payoff of the strategy pair (1/2,1/2) for both Bob and Alice. Is this a NE?
(OPTIONAL 5 Points Extra Credit)
Meanwhile Bob and
Alice luckily both ended up at the cinema. In the movie, five pirates have
stolen 100 gold coins. Now they are faced with the problem of how to divide
their treasure. As all five pirates have different heights, they agreed to the
following plan: The shortest pirate should suggest a way of how to split the
coins (and he can suggest whatever he wants e.g. even giving everything to
himself) and then ALL pirates should have a vote (including the one who just
made the proposal). If he gets a MAJORITY voting for his split (half of the
votes is not enough), then whatever he suggested will be done. If he doesn't
get a majority, he will be thrown over board and the exact same procedure will
be repeated with the remaining pirates.
Calculate the outcome of the treasure split. [There is only one correct
solution.]
(A full answer should include for each pirate - shortest to
biggest - how much they get or if they will have to go over board and some
reasoning why. Don't forget to clearly state the outcome of each vote that you
mention.)
(25 Points)
After the five pirates finished dividing up
their treasure, the tallest pirate gets upset and challenges the shortest
pirate to a duel. The shortest pirate makes the first move. The shortest
pirate can either PUNCH or KICK. In return, the big pirate can DUCK or JUMP.
Ducking makes him avoid a punch, and by jumping he can avoid a kick. The small
pirate knows that whether he hits the big pirate or not, he has two options
after that: He can STAY or FLEE. If he manages to hit, it would be very
satisfying to stay in order to "teach the big pirate a lesson". Otherwise,
however, it would be much healthier to flee.
Write the above fight (only up to stay/flee) as a game in extended form (state your assumptions and use appropriate payoffs)
Now translate the game you wrote above into a matrix game with the same payoffs.
Write down a best strategies for each player.
(30 Points)
Write the four conditions which, if satisfied, qualify a 2-player, 2-action game as a prisoner's dilemma game.
Consider this prisoner's dilemma game:
C | D | |
C | -1,-1 | -10,0 |
D | 0,-10 | -8,-8 |
Use the inequality aversion model with alpha=.5, beta =.2 (the same alpha and beta for both players.) Give the resulting matrix of *utilities* (not payoffs) for both players.
What are all the Nash equilibria of this modified, behavioral game?
Given the values alpha=.2 and beta=.1 for both players, can you write a game that is a prisoner's dilemma game (as defined in the Schelling book) but has an IA Nash equilibrium of (defect, defect)? If so, write one such game. If not, write "IMPOSSIBLE" and briefly explain why it is impossible.
Keeping the payoff as they are originally defined above, can you find values for alpha_row, beta_row, alpha_column, and beta_column that give a (cooperate, defect) equilibrium? (Note that the two players are now allowed to have different alpha and beta).