# Incentive Compatibility Construction Rule

Suppose we are interested in constructing an incentive compatibility constraint for an agent to announce his state truthfully.

Consider a simple case of the state space being $$S=\{H,L\}$$ and his payoffs to be $$\pi_L,\pi_H$$.

For the player to announce the state and payoff truthfully, is it correct to set the IC as:

$$\pi_H\leq\pi_L.$$

If it is correct, why is it so, and if it isn't, what is the correct way of setting up the IC and why? Is there a general rule of thumb in constructing an IC? Any reference would be also helpful.

• I am not clear about your notation. I believe that the payoffs should depend both on the true type and the announced type. – brunosalcedo Aug 12 at 15:01

As @brunosalcedo suggests in the comment, the agent's payoff, $$\pi(a,s)$$, should depend on i) the realized state $$s$$ and ii) the announced state $$a$$. Note that the announcement $$a$$ should in general be a function of the realized state $$s$$, i.e. $$a:S\to S$$. Incentive compatibility should thus be $$$$\pi(s,s)\ge \pi(a',s),\quad\forall a'\ne s.$$$$ In words, incentive compatibility requires that the agent's payoff when his announcement agrees with the state (i.e. the state is $$s$$ and his announcement is also $$s$$) be no lower than his payoff when his announcement disagrees with the state (i.e. the state is $$s$$ but his announcement is $$a'\ne s$$).