COMP30024 Artificial Intelligence
Week 9 Problem Sheet
1. Auctions [T] Your class will conduct a set of in-class auctions to gain experience with each
type of action. The basic process is the following:
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1. The auctioneer reveals the good to be auctioned.
2. You are given a card that indicates your redemption value (V ) of the good. You can think of this as your private value for the good.
3. Think of a rational bidding strategy for each type of auction. Record the bids made.
4. The auctioneer conducts the auction.
5. If you win the auction, you ’pay’ the auctioneer the appropriate amount based on the type of auction. Your profit/loss is P = V − W .
Follow the above process for (1) an English auction, (2) a Dutch auction, (3) a first-price, sealed-bid auction and (4) a second-price, sealed-bid auction. In the case of a sealed-bid auction, you should write your name and bid on a piece of paper, which is discreetly handed to the auctioneer.
(a) What strategy did you use in each auction?
(b) How did the amount paid to the auctioneer vary between auctions?
(c) How did the profit of the winning bidder vary between auctions?
2. Efficient Auctions [T] Argue that the optimal strategy for a participant in a sealed-bid, second-price auction is to bid at their private valuation.
3. Successive Wins [T] To encourage Emma’s promising tennis career, her father offers her a prize if she wins (at least) two tennis sets in a row in a three-set series to be played with her father and the club champion alternately: father-champion-father or champion-father- champion, according to Emma’s choice. The champion is a better player than Emma’s father. Which series should Emma choose?
4. An Inconvenient Truth [T] The students in your tutorial tell the truth one third of the time. They lie with probability two-thirds. On an occasion, after the student to your right made a statement, you ask the student to your left ‘was that statement true?’ and they respond ‘yes’. What is the probability that the statement was indeed true?
5. The Truel [T] A, B, and C are to fight a three-cornered pistol duel. All duelists are aware that A’s chance of hitting their target is 0.3, C’s is 0.5, and B never misses. A hit duelist loses further turns and is no longer shot at. They must fire at their choice of target in succession in the order A, B, C, cyclically until only one is left standing.
(a) What is the probability of A winning a duel with C, assuming C goes first?
(b) Suppose A shoots at C first, what is the probability A wins?1
(c) Now suppose A shoots at B first, what is the probability A wins? Hence what is the optimal strategy for A?
(d) Suppose A can deliberately miss on the first shot, should they2?
1Note here that any player in a sequential truel maximizes their survival probability by firing at the more dangerous opponent, as the problem reduces to a duel if the shot connects, and it is always preferable to duel a weaker opponent.
2Follow-up: is this an honourable course of action?
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