Incentive compatibility

A mechanism is called incentive-compatible (IC)^[1]^: 415 if every participant can achieve their own best outcome by reporting 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.

Typical examples of DSIC mechanisms are second-price auctions and a simple majority vote between two choices. Typical examples of non-DSIC mechanisms are ranked voting with three or more alternatives (by the Gibbard–Satterthwaite theorem) or first-price auctions.

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 (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).

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.

References

^ ^a ^b ^c ^d ^e ^f ^g Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
^ "Incentive compatibility | game theory". Encyclopedia Britannica. Retrieved 2020-05-25.
^ James Jr, Harvey S. (2014). "Incentive compatibility". Britannica.
^ Jackson, Matthew (December 8, 2003). "Mechanism Theory" (PDF). Optimization and Operations Research.

[agt07-1] ^ ^a ^b ^c ^d ^e ^f ^g Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.

[2] "Incentive compatibility | game theory". Encyclopedia Britannica. Retrieved 2020-05-25.

[3] James Jr, Harvey S. (2014). "Incentive compatibility". Britannica.

[4] Jackson, Matthew (December 8, 2003). "Mechanism Theory" (PDF). Optimization and Operations Research.

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In randomized mechanisms

Revelation principles

See also

References