Incentive compatibility

In game theory and economics, 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 The concept is attributed to the Russian-born American economist Leonid Hurwicz.²

There are several different degrees of incentive-compatibility:⁴

• The stronger degree is dominant-strategy incentive-compatibility (DSIC).^1: 415 This 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 This 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.

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.