Secure (M+1) st-Price Auction with Automatic Tie-Break

In auction theory, little attention has been paid to a situation where the tie-break occurs because most of auction properties are not affected by the way the tie-break is processed. Meanwhile, in secure auctions where private information should remain hidden, the information of the tie can unnecessarily reveal something that should remain hidden. Nevertheless, in most of existing secure auctions, ties are handled outside the auctions, and all the winning candidates or only the non-tied partial bidders are identified in the case of ties, assuming that a subsequent additional selection or auction to finalize the winners is held publicly. However, for instance, in the case of the $$M+1$$ st-price auction, the tied bidders in the $$M+1$$ st-price need to be identified for such a selection, which implies that their bids unnecessary private information are revealed. Hence it is desirable that secure auctions reveal neither the existence of ties nor the losing tied bidders. To overcome these shortcomings, we propose a secure $$M+1$$ st-price auction protocol with automatic tie-breaks and no leakage of the tie information by improving the bit-slice auction circuit without increasing much overhead.