top priorities is at least as high. Lexicographic truthfulness (lex-truthfulness): The vector of probabilities that an agent receives by being truthful has lexicographic dominance over the vector of probabilities he gets by misreporting. That is: the probability of getting the top priority is higher OR (the probability of getting the top priority is equal and the probability of getting one of the two top priorities is higher) OR ... (the probability of getting the first m-1 priorities priority is equal and the probability of getting one of the m top priorities is higher) OR (all probabilities are equal). Weak stochastic-dominance truthfulness (weak-SD-truthfulness): The vector of probabilities that an agent receives by being truthful is not first-order-stochastically-dominated by the vector of probabilities he gets by misreporting. Universal implies strong-SD implies Lex implies weak-SD, and all implications are strict. Truthfulness with high probability For every constant , a randomized mechanism is called truthful with probability if for every agent and for every vector of bids, the probability that the agent benefits by bidding non-truthfully is at most , where the probability is taken over the randomness of the mechanism. If the constant goes to 0 when the number of bidders grows, then the mechanism is called truthful with high probability. This notion is weaker than full truthfulness, but it is still useful in some cases; see e.g. consensus estimate. False-name-proofness A new type of fraud that has become common with the abundance of internet-based auctions is false-name bids – bids submitted by a single bidder using multiple identifiers such as multiple e-mail addresses. False-name-proofness means that there is no incentive for any of the players to issue false-name-bids. This is a stronger notion than strategyproofness. In particular, the Vickrey–Clarke–Groves (VCG) auction is not false-name-proof. False-name-proofness is importantly different from group strategyproofness because it assumes that an individual alone can simulate certain behaviours that would normally require the collusive coordination of multiple individuals. See also Incentive compatibility Individual rationality - means that a player cannot lose by playing the game (i.e. a player has no incentive to avoid playing the game). Further reading Parkes, David C. (2004), On Learnable Mechanism Design, in: Tumer, Kagan and David Wolpert (Eds.): Collectives and the Design of Complex Systems, New York u.a.O., pp. 107–133. On Asymptotic Strategy-Proofness of Classical Social Choice Rules An article by Arkadii Slinko about strategy-proofness in voting systems. References Game theory Mechanism design