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Despite attractive theoretical properties, Vickrey auctions are seldom used due to the risk of information revelation and fear of cheating. CVAs (Cryptographic Vickrey Auctions) have been proposed to protect bidders' privacy or to prevent cheating by the bid taker. This paper focuses on incentive issues for certain CVAs. First, it defines the CVAs of interest and identifies ideal goals for this class of CVAs. One of the criteria identifies an incentive problem that is new to the literature on CVAs, the disincentive of bidders to complete the protocol once they have learned that they lost the auction. Any auction protocol that requires losing bidders to do additional work after learning they have lost the auction must provide the losers with proper incentives to follow the protocol. Second, this paper shows that for a class of CVAs, some losers must continue to participate even though they know they have lost. Finally, it describes two new CVA protocols that solve the protocol-completion incentive problem. Both protocols use bidder-bidder comparisons based on a modified Yao's Millionaires' protocol. The first protocol performs O(n 2) bidder-bidder comparisons, while the second protocol performs O(n) comparisons.