There are several different degrees of incentive-compatibility:
- The stronger degree is Dominant-strategy incentive-compatibility (DSIC).:415 It means that truth-telling is a weakly-dominant strategy. I.e, you fare best by being truthful, regardless of what the others do (even if they are irrational or even collude against you). In a DSIC mechanism, strategic considerations cannot help any agent achieve better outcomes than the truth; hence, such mechanisms are also called strategyproof:244,752 or truthful.:415 See the page about strategyproofness for more information about this property.
- A weaker degree is Bayesian-Nash incentive-compatibility (BNIC).:416 It means that there is a Bayesian Nash equilibrium in which all participants reveal their true preferences. I.e, if all the others act truthfully, then it is also best for you to be truthful.:234
Every DSIC mechanism is also BNIC, but a BNIC mechanism may exist even if no DSIC mechanism exists.
Incentive-compatibility in randomized mechanisms
A randomized mechanism is a probability-distribution on deterministic mechanisms. There are two ways to define incentive-compatibility of randomized mechanisms::231-232
- The stronger definition is: a randomized mechanism is universally-incentive-compatible if every mechanism selected with positive probability is incentive-compatible (e.g. 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 (e.g. if truth-telling gives the agent an optimal expected value).
The famous Revelation principle comes in two flavors 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, can be implemented by a BNIC mechanism.
- Lindahl tax
- Preference revelation
- Monotonicity (mechanism design)