Q1. Two opponents A and B are to play a game which consists of two inde¬pendent rounds. To win the game a player must win both rounds otherwise the game is declared a draw. Each player must decide in advance, without consulting the other, whether to play aggressively (strategy 1) or defensively (strategy 2) and must play the same way in both rounds. If both play¬ers play aggressively or if both players play defensively then each player is equally likely to win a particular round. When playing aggressively against a defensive opponent, player A wins (and B loses) a round with probability 0.7 but when playing defensively against an aggressive opponent, A wins (and B loses) a round with probability 0.6
(i) Assuming that each player wishes to maximise their probability of winning, construct the pay-off table.
(ii) Find a pair of strategies in equilibrium.
(iii) Without further calculation sketch the pay-off set and identify the pay-offs corresponding to jointly admissible strategies.
(iv) Suppose now that A and B cooperate. Sketch the new pay-off set and identify the negotiation set.
Q2. Two builders, A and B, are competing for a contract to construct a new office block. Each can bid either £3.5 million or £4 million for the job. The builder with the lower bid will win the contract and will be paid the value of their bid once the building is finished. If both bids are equal, a fair coin is tossed to decide who should win the contract. Builder A reckons that his real cost of completing the job is £2 million, whereas builder B reckons it will cost him 2.5 million to complete it.They decide to discuss their bids.
(i) Construct the table of pay-offs (i.e the expected profits) to the two builders and find the maximum profit each can reasonably expect to get.
(ii) Deter-mine the Shapley solution.