Secure integer comparison has been a popular research topic in cryptography, both for its simplicity to describe and for its applications. The aim is to enable two parties to compare their inputs without revealing the exact value of those inputs. In this paper, we highlight three-party integer comparison (TPIC), where a \emph{judge}, with no private input, wants to know the comparison result, while two \emph{competitors} hold secret integers to do privacy-preserving comparison. The judge actively obtains the result rather than passively waiting for it sent by a competitor. We give two TPIC constructions considering \emph{Mixed adversaries}, who have with different capabilities. One is secure against a semi-honest adversary with low computation and communication cost, while the other is secure against a malicious adversary. Basing on TPIC, we present multi-party comparisons through concrete applications, including a joint bidding scheme and a practical auction. Brief security proofs and analysis for the applications are presented. In comparison, our auction scheme is more efficient with lower cost, making it feasible in practice rather than a theoretical design. All the comparisons and application schemes run on top of blockchain requiring a constant number of rounds.