<p>As @brunosalcedo suggests in the comment, the agent's payoff, <span class="math-container">$\pi(a,s)$</span>, should depend on i) the realized state <span class="math-container">$s$</span> and ii) the announced state <span class="math-container">$a$</span>. Note that the announcement <span class="math-container">$a$</span> should in general be a function of the realized state <span class="math-container">$s$</span>, i.e. <span class="math-container">$a:S\to S$</span>. Incentive compatibility should thus be 
<span class="math-container">\begin{equation}
\pi(s,s)\ge \pi(a',s),\quad\forall a'\ne s. 
\end{equation}</span>
In words, incentive compatibility requires that the agent's payoff when his announcement agrees with the state (i.e. the state is <span class="math-container">$s$</span> and his announcement is also <span class="math-container">$s$</span>) be no lower than his payoff when his announcement disagrees with the state (i.e. the state is <span class="math-container">$s$</span> but his announcement is <span class="math-container">$a'\ne s$</span>). </p>

<p>Check out the following book chapters for more detail: </p>

<ul>
<li><a href="https://global.oup.com/ushe/product/microeconomic-theory-9780195073409" rel="nofollow noreferrer">Mas-Colell, Whinston, and Green (1995)</a> (aka MWG) chapter 14.C discusses IC in the context of moral hazard </li>
<li>MWG chapter 23.B introduces IC for truthful implementation </li>
<li><a href="https://www.elsevier.com/books/auction-theory/krishna/978-0-12-374507-1" rel="nofollow noreferrer">Krishna (2010)</a> chapter 5 is similar to the previous MWG chapter, but draws out more implications of IC.</li>
</ul>

CC BY-SA 4.0
2019-08-13T11:52:56.277Z
(+1) could you write out MWG once? Just to make the reference clear...
Maarten Punt
CC BY-SA 4.0
2019-08-13T17:47:23.670Z
@MaartenPunt: Thanks. The reference has been added.
Herr K.
CC BY-SA 4.0
2019-08-12T18:08:58.787Z
2019-08-13T17:46:33.983Z
2019-08-13T17:46:33.983Z
Herr K.
<p>Suppose we are interested in constructing an incentive compatibility constraint for an agent to announce his state truthfully.</p>

<p>Consider a simple case of the state space being <span class="math-container">$S=\{H,L\}$</span> and his payoffs to be <span class="math-container">$\pi_L,\pi_H$</span>. </p>

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

<p><span class="math-container">$$\pi_H\leq\pi_L.$$</span></p>

<p>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.</p>

CC BY-SA 4.0
2019-08-12T15:01:54.657Z
I am not clear about your notation. I believe that the payoffs should depend both on the true type and the announced type.
brunosalcedo
<game-theory>
Incentive Compatibility Construction Rule
Frank Swanton
# Incentive Compatibility Construction Rule

(Asked by: username_0 on 2019-08-12)

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.

**Tags:** game-theory
> **Comments:**
> **username_1**: I am not clear about your notation. I believe that the payoffs should depend both on the true type and the announced type.  

---

## 1 answers

### Answer by username_2 on 2019-08-12. Score: 1 | Accepted answer

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)](https://global.oup.com/ushe/product/microeconomic-theory-9780195073409) (aka MWG) chapter 14.C discusses IC in the context of moral hazard
* MWG chapter 23.B introduces IC for truthful implementation
* [Krishna (2010)](https://www.elsevier.com/books/auction-theory/krishna/978-0-12-374507-1) chapter 5 is similar to the previous MWG chapter, but draws out more implications of IC.

> **Comments:**
> **username_3**: (+1) could you write out MWG once? Just to make the reference clear...  
> **username_2**: @MaartenPunt: Thanks. The reference has been added.  

---

