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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.

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    $\begingroup$ I am not clear about your notation. I believe that the payoffs should depend both on the true type and the announced type. $\endgroup$ – brunosalcedo Aug 12 at 15:01
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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 \begin{equation} \pi(s,s)\ge \pi(a',s),\quad\forall a'\ne s. \end{equation} 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 $a'\ne s$).

Check out the following book chapters for more detail:

  • Mas-Colell, Whinston, and Green (1995) (aka MWG) chapter 14.C discusses IC in the context of moral hazard
  • MWG chapter 23.B introduces IC for truthful implementation
  • Krishna (2010) chapter 5 is similar to the previous MWG chapter, but draws out more implications of IC.
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  • $\begingroup$ (+1) could you write out MWG once? Just to make the reference clear... $\endgroup$ – Maarten Punt Aug 13 at 11:52
  • $\begingroup$ @MaartenPunt: Thanks. The reference has been added. $\endgroup$ – Herr K. Aug 13 at 17:47

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