Amechanism is called incentive-compatible (IC) or truthful[1]: 415 if every participant can achieve their own best outcome by acting according to their true preferences.[1]: 225 [2] For example, there is incentive compatibility if high-risk clients are better off in identifying themselves as high-risk to insurance firms, who only sell discounted insurance to high-risk clients. Likewise, they would be worse off if they pretend to be low-risk. Low-risk clients who pretend to be high-risk would also be worse off.[3]
There are several different degrees of incentive-compatibility:[4]
The stronger degree is dominant-strategy incentive-compatibility (DSIC).[1]: 415 It means that truth-telling is a weakly-dominant strategy, i.e. you fare best or at least not worse by being truthful, regardless of what the others do. In a DSIC mechanism, strategic considerations cannot help any agent achieve better outcomes than the truth; such mechanisms are called strategyproof,[1]: 244, 752 truthful or straightforward.
A weaker degree is Bayesian-Nash incentive-compatibility (BNIC).[1]: 416 It means there is a Bayesian Nash equilibrium in which all participants reveal their true preferences. In other words, if all other players act truthfully, then it is best to be truthful.[1]: 234
Every DSIC mechanism is also BNIC, but a BNIC mechanism may exist even if no DSIC mechanism exists.
A randomized mechanism is a probability-distribution on deterministic mechanisms. There are two ways to define incentive-compatibility of randomized mechanisms:[1]: 231–232
The stronger definition is: a randomized mechanism is universally-incentive-compatible if every mechanism selected with positive probability is incentive-compatible (i.e. if truth-telling gives the agent an optimal value regardless of the coin-tosses of the mechanism).
The weaker definition is: a randomized mechanism is incentive-compatible-in-expectation if the game induced by expectation is incentive-compatible (i.e. if truth-telling gives the agent an optimal expected value).
The revelation principle comes in two variants corresponding to the two flavors of incentive-compatibility:
The dominant-strategy revelation-principle says that every social-choice function that can be implemented in dominant-strategies can be implemented by a DSIC mechanism.
The Bayesian–Nash revelation-principle says that every social-choice function that can be implemented in Bayesian–Nash equilibrium (Bayesian game, i.e. game of incomplete information) can be implemented by a BNIC mechanism.
