Incentive Compatibility – along with mechanism design – is one of the key ideas for which the 2007 Nobel Prize in Economic Sciences was awarded. A mechanism is said to be incentive-compatible if every participant can achieve the best outcome for him/herself just by revealing his or her true preferences, or openly acting on his or her preferences (without “cheating”). https://en.wikipedia.org/wiki/Incentive_compatibility
Eric Maskin, in his lecture, “An Introduction to Mechanism Design” (Warwick Economics Summit 2014) uses the simple example of a parent wanting to divide a cake between two children. The goal is to divide the cake in such a way that both children are happy. The mechanism could be that one child cuts the cake while the other has first choice of the pieces. The goal of making a “fair division” is known in advance. The mechanism is designed specifically to achieve that goal.
This “one cuts, the other chooses” mechanism is incentive-compatible because it allows both children to achieve their goal, defined as getting at least half the cake.