# Incentive compatibility

A mechanism is called **incentive-compatible** (**IC**) if every participant can achieve the best outcome to themselves just by acting according to their true preferences.^{[1]}^{:225}^{[2]}

There are several different degrees of incentive-compatibility:^{[3]}

- 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; hence, such mechanisms are also called**strategyproof**^{[1]}^{:244,752}or**truthful**.^{[1]}^{:415}(See Strategyproofness) - A weaker degree is
**Bayesian-Nash incentive-compatibility**(**BNIC**).^{[1]}^{: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 or at least not worse for you to be truthful.^{[1]}^{:234}

Every DSIC mechanism is also BNIC, but a BNIC mechanism may exist even if no DSIC mechanism exists.

Typical examples of DSIC mechanisms are majority voting between two alternatives, and second-price auction.

Typical examples of a mechanisms that are not DSIC are plurality voting between three or more alternatives and first-price auction.

## In randomized mechanisms

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 (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).

## Revelation principles

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.

## See also

- Implementability (mechanism design)
- Lindahl tax
- Monotonicity (mechanism design)
- Preference revelation
- Strategyproofness

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007).*Algorithmic Game Theory*. Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0. http://www.cs.cmu.edu/~sandholm/cs15-892F13/algorithmic-game-theory.pdf. - ↑ "Incentive compatibility | game theory" (in en). https://www.britannica.com/topic/incentive-compatibility.
- ↑ Jackson, Matthew (December 8, 2003). "Mechanism Theory".
*Optimization and Operations Research*. https://web.stanford.edu/~jacksonm/mechtheo.pdf.

Original source: https://en.wikipedia.org/wiki/Incentive compatibility.
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