Consider the following payoff matrix of a simultaneous move game, where x is the last digit of your student number. (LAST DIGIT = 3)
\ 2 Left Middle Right
Top 5, 1 5, 4 5, 3
High 4, 5 3, x 5, 4
Low 2, 2 4, 4 5, 6
Find all pure strategy Nash equilibria and Pareto efficient outcomes in this game. Discuss pros and cons of using Nash equilibrium to solve a simultaneous move game.
[14 marks]
Can you solve this game by iterated dominance? Discuss.
[6 marks]
What is the best response of the column player when the row player plays the mixed strategy σ1 = (2/3,0,1/3).
[7 marks]
Compute the expected payoff to the row player if the mixed strategy profile (σ1, σ2) = ((0, 2/3,1/3), (3/4,0,1/4)) is played.
[6 marks]
Word count: 250 (not including mathematical expressions and graphs)
You and your sister have just inherited a house worth 2+x millions, where x is the last digit of your student number. You both negotiate your shares of the inheritance according to the following protocol: You make an initial offer. Your sister either accepts the initial offer or makes a counteroffer. Then, you either accept her counteroffer or the house is assigned randomly to one of you with equal probabilities. Each time an offer is not accepted, 1 million is due in lawyers’ fees, which are paid by the house owner. Further, assume that both of you maximize expected money earnings and value future payments just as much as current payments.
Draw the complete game tree
[10 marks]
Describe carefully the relevant solution concept and calculate the equilibrium shares
[10 marks]
Who benefits, you or your sister, when lawyers’ fees increase or when you both become risk averse? Discuss.
[6 marks]
Carefully explain the process of backward induction in sequential games.
[7 marks]
Word count: 250 (not including mathematical expressions and graphs)
Alpha and Beta don’t observe the content of a box but they know that, with equal probabilities, it is either empty or it contains £4,000. They bid simultaneously either £0 or £1,000 for the box. The highest bidder wins the box (with probability ½ if both bidders submit the same bid) and pays her bid.
Represent the game in normal and extensive form.
[10 marks]
Solve the game with the relevant solution concept(s).
[9 marks]
Represent the game in normal and extensive form when Alpha observes the content of the box before bidding.
[6 marks]
Solve the game in c) with the relevant solution concept(s). Does Alpha benefit from observing the content?
[8 marks]
Word count: 250 (not including mathematical expressions and graphs)
Demonstrate the nature of the chain store paradox by constructing, solving and discussing a suitable game theoretical example with at least three players.
[33 marks]
Word count: 350 (not including mathematical expressions and graphs)
