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# a - ﬂﬁﬁl sameness-1| The optimal strategy in a...

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Unformatted text preview: ﬂﬁﬁl. sameness-1| The optimal strategy in a Viekrey auction with a single, indiyisible item is for each bidder to bid theirtrue yalue o Let vi be bidder i's yalue forthe item. Let 32;- be bidder i's bid forthe item. 1:- —max .5. if b. 3:. max 15- The payof‘ffor bidderi is 1 Hal 3 I 3%“ 3' 0 otherwise .-aaw§aisswssaa The strategy of oyerbidding is dominated by bidding truthfully. Assume that bidderi bids by 3' vi. Ifmaxyﬁbj 4:: “D,- then the bidder would win the item with a truthful bid as well as an oyerbid. The bid's amount does net change the payoff so the two strategies haye equal payoﬁs in this case. lfIﬂaKjﬁij 33‘ [)1- then the bidderwould lose the item eitherway so the strategies haye equal payoffs in this case. Ifﬂ; {i Iﬂﬂixjgibj (I b:- then only the strategy of oyerbidding would win the auction. The payoffwould be negatiye for the strategy of oyerbidding because they paid more than theiryalue ofthe item, while the payofffor a truthful bid would be zero. Thus the strategy ofbidding higherthan one's true yaluation is dominated by the strategy oftruthfully bidding. The strategy ofunderbidding is dominated by bidding truthfully. Assume that bidder i bids :51“: v1: lfmaxjgéibj 3‘: 1,1:- then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies haye equal payoffs forthis ease. Ifmaxjﬁbj {I [1:- then the bidderwould win the item eitherway so the strategies haye equal payoffs in this case. If fl; *5: IDES-[jagg- bj 4:. oi then only the strategy oftruthfully bidding would win the auction. The payoffforthe truthful strategy would be payoﬁ as they paid less than theiryalue ofthe item, while the payoﬁfor an underbid bid would be zero. Thus the strategy ofunderbidding is dominated by the strategy oftruthfully bidding. Truthful bidding dominates the other possible strategies [underbidding and oyerbidding] so it is an optimal strategy. ...
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