Queen Elizabeth has decided to auction off the crown jewels, and there are two bidders: Sultan Hassanal Bolkiah of Brunei and Sheikh Zayed Bin Sultan Al Nahyan of Abu Dhabi. The auction format is as follows: The Sultan and the Sheikh simultaneously submit a written bid. Exhibiting her well-known quirkiness, the Queen specifies that the Sultan’s bid must be an odd number (in hundreds of millions of English pounds) between 1 and 9 (that is, it must be 1, 3, 5, 7, or 9) and that the Sultan’s bid must be an even number between 2 and 10. The bidder who submits the highest bid wins the jewels and pays a price equal to his bid. The winning bidder’s payoff equals his valuation of the item less the price he pays, whereas the losing bidder’s payoff is 0. Assume that the Sultan has a valuation of 8 (hundred million pounds) and that the Sheikh has a valuation of 7.a. In matrix form, write down the strategic form of this game.b. Derive all Nash equilibria.
Queen Elizabeth has decided to auction off the crown jewels, and there are two bidders: Sultan Hassanal Bolkiah of Brunei and Sheikh Zayed Bin Sultan Al Nahyan of Abu Dhabi. The auction format is as follows: The Sultan and the Sheikh simultaneously submit a written bid. Exhibiting her well-known quirkiness, the Queen specifies that the Sultan’s bid must be an odd number (in hundreds of millions of English pounds) between 1 and 9 (that is, it must be 1, 3, 5, 7, or 9) and that the Sultan’s bid must be an even number between 2 and 10. The bidder who submits the highest bid wins the jewels and pays a price equal to his bid. The winning bidder’s payoff equals his valuation of the item less the price he pays, whereas the losing bidder’s payoff is 0. Assume that the Sultan has a valuation of 8 (hundred million pounds) and that the Sheikh has a valuation of 7.
a. In matrix form, write down the strategic form of this game.
b. Derive all Nash equilibria.
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