<p>The solution to an optimal contract problem is called "first best" if it maximizes the principal's objective function subject to all constraints except the incentive constraints. The solution to an optimal contract problem is called "second best" if it maximizes the principal's objective function subject to all constraints, including the incentive constraints. Usually, one calls a contract "second best" only if it differs from the "first best" contract.</p>

CC BY-SA 3.0
2015-02-23T17:55:07.293Z
This seems to be a very clear definition, given that the optimal contract problem is clearly defined. For the example of bilateral trade, it's not clear to me why the seller should be the principal, rather than the buyer. What about the individual rationality constraints? Should it be ex-post, ex-interim or ex-ante for the agent? Does this depend on the timing of the contract even in the discussion of First best?
Joe Li
CC BY-SA 3.0
2015-02-23T20:03:07.583Z
The "principal" will be whoever designs the contract, a seller, a buyer, a social planner, etc. The participation constraints, in particular whether they are ex post, interim, or ex ante, will differ case by case. The phrases "first best" and "second best" don't by themselves imply anything about the precise nature of participation constraints. As a general remark: There is no completely precise, universally agreed upon definition of these phrases. It is best when using these phrases to provide their precise definitions in the context that one is working in.
Tilman Borgers
CC BY-SA 3.0
2015-02-23T16:26:56.353Z
2015-02-23T16:26:56.353Z
Tilman Borgers
<p><em>First Best</em> is the ideal optimal solution of a given problem, i.e. the mathematical solution of the model with "no imperfections".</p>

<p>If that solution is <strong>not attainable</strong> then the solution must be binding to some constraint, in which case we call it a <em>Second Best</em> solution.</p>

<p>'<strong>Not attainable</strong>' here means that there's a discrepancy between the theoretical predictions of the model and what really happens in the economy. Different authors use other terms...</p>

CC BY-SA 3.0
2015-02-23T14:12:53.090Z
2015-02-23T14:12:53.090Z
Ayar ʕʘ̅͜ʘ̅ʔ
<p><strong>In contract theory</strong></p>
<p>The <em>first-best</em> refers to the best you could do if you knew agents' preferences over labor and income (i.e., if you did not have to impose the incentive compatibility constraint), and the <em>second-best</em> is the best you can do if agents have to reveal their preferences themselves.</p>
<p><strong>In mechanism design</strong></p>
<p>A useful reference is Galichon, Alfred, Ex-Ante vs. Ex-Post Efficiency in Matching (May 21, 2011). Available at SSRN: <a href="http://ssrn.com/abstract=1837321" rel="nofollow noreferrer">http://ssrn.com/abstract=1837321</a> or <a href="http://dx.doi.org/10.2139/ssrn.1837321" rel="nofollow noreferrer">http://dx.doi.org/10.2139/ssrn.1837321</a>  :</p>
<blockquote>
<p>&quot;an assignment is called ex-post efficient if no other <em>deterministic assignment</em> is improving on it in a Pareto sense; and ex-ante efficient if no <em>lottery over deterministic assignments</em> is.&quot; (my emphasis)</p>
</blockquote>
<p><strong>Difference between the two</strong></p>
<p>There isn't many connexions between the two notions as defined above. Every combination of the two notions is <em>a priori</em> possible. Both a mechanism and a contract can be</p>
<ol>
<li><p>First-best ex-post efficient (i.e., efficient when incentive compatibility constraint is <em>not</em> imposed and the outcome of the mechanism/contract must be <strong>deterministic</strong>)</p>
</li>
<li><p>First-best ex-ante efficient (i.e., efficient when incentive compatibility constraint is <em>not</em> imposed and the outcome of the mechanism/contract can be <strong>random</strong>)</p>
</li>
<li><p>Second-best ex-post efficient (i.e., efficient when incentive compatibility constraint is imposed and the outcome of the mechanism/contract must be <strong>deterministic</strong>)</p>
</li>
<li><p>Second-best ex-ante efficient (i.e., efficient when incentive compatibility constraint is imposed and the outcome of the mechanism/contract can be <strong>random</strong>)</p>
</li>
</ol>
<p><strong>In the literature</strong></p>
<p>Yet, in general, you are more likely to find the first-best/second-best terminology in contract theory (random contracts are not <em>that common</em> in contract theory), and the ex-post/ex-ante efficiency in mechanism design (knowledge of the preferences is rarely assumed in a mechanism: the fact that we do not know agents preferences is the <em>raison d'être</em> of mechanism design).</p>
<p>Thus you can expect to see 3. and 4. being discussed in the mechanism design literature, and 1. and 3. being discussed in contract theory.</p>
<p><strong>Beware</strong></p>
<p>This being said, the notions of second-best and first-best are often used outside of contract theory in a rather permissive way, which may be confusing.</p>
<p>For some people, first-best simply means &quot;if we do <em>not</em> impose some constraint <span class="math-container">$X$</span>&quot; and second-best means &quot;if we <em>do</em> impose constraint <span class="math-container">$X$</span>&quot;.</p>
<p>Thus you may hear people talk about (this is where things can gets confusing):</p>
<ul>
<li>an ex-post efficient mechanism as &quot;second-best&quot; efficient, because it is &quot;efficient under the constraint that the assignment be deterministic&quot;.</li>
<li>an ex-ante efficient mechanism as &quot;first-best&quot; efficient, that is &quot;efficient if we do <em>not</em> impose constraint <span class="math-container">$X$</span>&quot;.</li>
</ul>
<p>Hope that clarifies it (or at least does not confuse you more).</p>

CC BY-SA 3.0
2015-02-23T18:07:12.997Z
Thanks @Martin. Although I didn't fully understand, but it seems that ex-post efficiency is clearly defined, but first best depend on the context. Is the quote the only definition of first best in the mechanism design theory?
Joe Li
CC BY-SA 3.0
2015-02-23T19:52:53.897Z
I did not read your question correctly, so my answer was confusing. I'll try to rewrite it, hopefully it will make things clearer.
Martin Van der Linden
CC BY-SA 4.0
2015-02-23T14:57:11.153Z
2021-12-01T14:05:43.987Z
2021-12-01T14:05:43.987Z
Martin Van der Linden
<p>What is the definition of "First Best", "Second Best", etc. in contract theory?</p>

<p>Especially, what is the difference between "First Best" in contract theory and "ex-post efficient" in mechanism design?</p>

CC BY-SA 3.0
2015-02-23T20:04:03.137Z
May I suggest that you edit your title or the body of your post? The question in the title is different from the one in your post. Which one is your actual question?
Martin Van der Linden
CC BY-SA 3.0
2015-02-24T22:56:40.713Z
Actually both. My confusion between "first best" and "ex-post efficiency" comes from the ambiguity of the definition of "first best".
Joe Li
<mechanism-design><contract-theory>
What is the definition of "First Best", "Second Best", etc. in contract theory?
Joe Li
# What is the definition of "First Best", "Second Best", etc. in contract theory?

(Asked by: username_0 on 2015-02-23)

What is the definition of "First Best", "Second Best", etc. in contract theory?

Especially, what is the difference between "First Best" in contract theory and "ex-post efficient" in mechanism design?

**Tags:** mechanism-design, contract-theory

> **Comments:**
> **username_1**: May I suggest that you edit your title or the body of your post? The question in the title is different from the one in your post. Which one is your actual question?  
> **username_0**: Actually both. My confusion between "first best" and "ex-post efficiency" comes from the ambiguity of the definition of "first best".  

---

## 3 answers

### Answer by username_1 on 2015-02-23. Score: 12 | Accepted answer

**In contract theory**

The *first-best* refers to the best you could do if you knew agents' preferences over labor and income (i.e., if you did not have to impose the incentive compatibility constraint), and the *second-best* is the best you can do if agents have to reveal their preferences themselves.

**In mechanism design**

A useful reference is Galichon, Alfred, Ex-Ante vs. Ex-Post Efficiency in Matching (May 21, 2011). Available at SSRN: <http://ssrn.com/abstract=1837321> or <http://dx.doi.org/10.2139/ssrn.1837321> :

> "an assignment is called ex-post efficient if no other *deterministic assignment* is improving on it in a Pareto sense; and ex-ante efficient if no *lottery over deterministic assignments* is." (my emphasis)

**Difference between the two**

There isn't many connexions between the two notions as defined above. Every combination of the two notions is *a priori* possible. Both a mechanism and a contract can be

1. First-best ex-post efficient (i.e., efficient when incentive compatibility constraint is *not* imposed and the outcome of the mechanism/contract must be **deterministic**)
2. First-best ex-ante efficient (i.e., efficient when incentive compatibility constraint is *not* imposed and the outcome of the mechanism/contract can be **random**)
3. Second-best ex-post efficient (i.e., efficient when incentive compatibility constraint is imposed and the outcome of the mechanism/contract must be **deterministic**)
4. Second-best ex-ante efficient (i.e., efficient when incentive compatibility constraint is imposed and the outcome of the mechanism/contract can be **random**)

**In the literature**

Yet, in general, you are more likely to find the first-best/second-best terminology in contract theory (random contracts are not *that common* in contract theory), and the ex-post/ex-ante efficiency in mechanism design (knowledge of the preferences is rarely assumed in a mechanism: the fact that we do not know agents preferences is the *raison d'être* of mechanism design).

Thus you can expect to see 3. and 4. being discussed in the mechanism design literature, and 1. and 3. being discussed in contract theory.

**Beware**

This being said, the notions of second-best and first-best are often used outside of contract theory in a rather permissive way, which may be confusing.

For some people, first-best simply means "if we do *not* impose some constraint $X$" and second-best means "if we *do* impose constraint $X$".

Thus you may hear people talk about (this is where things can gets confusing):

* an ex-post efficient mechanism as "second-best" efficient, because it is "efficient under the constraint that the assignment be deterministic".
* an ex-ante efficient mechanism as "first-best" efficient, that is "efficient if we do *not* impose constraint $X$".

Hope that clarifies it (or at least does not confuse you more).

> **Comments:**
> **username_0**: Thanks @Martin. Although I didn't fully understand, but it seems that ex-post efficiency is clearly defined, but first best depend on the context. Is the quote the only definition of first best in the mechanism design theory?  
> **username_1**: I did not read your question correctly, so my answer was confusing. I'll try to rewrite it, hopefully it will make things clearer.  

---

### Answer by username_2 on 2015-02-23. Score: 6

The solution to an optimal contract problem is called "first best" if it maximizes the principal's objective function subject to all constraints except the incentive constraints. The solution to an optimal contract problem is called "second best" if it maximizes the principal's objective function subject to all constraints, including the incentive constraints. Usually, one calls a contract "second best" only if it differs from the "first best" contract.

> **Comments:**
> **username_0**: This seems to be a very clear definition, given that the optimal contract problem is clearly defined. For the example of bilateral trade, it's not clear to me why the seller should be the principal, rather than the buyer. What about the individual rationality constraints? Should it be ex-post, ex-interim or ex-ante for the agent? Does this depend on the timing of the contract even in the discussion of First best?  
> **username_2**: The "principal" will be whoever designs the contract, a seller, a buyer, a social planner, etc. The participation constraints, in particular whether they are ex post, interim, or ex ante, will differ case by case. The phrases "first best" and "second best" don't by themselves imply anything about the precise nature of participation constraints. As a general remark: There is no completely precise, universally agreed upon definition of these phrases. It is best when using these phrases to provide their precise definitions in the context that one is working in.  

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### Answer by username_3 on 2015-02-23. Score: 4

*First Best* is the ideal optimal solution of a given problem, i.e. the mathematical solution of the model with "no imperfections".

If that solution is **not attainable** then the solution must be binding to some constraint, in which case we call it a *Second Best* solution.

'**Not attainable**' here means that there's a discrepancy between the theoretical predictions of the model and what really happens in the economy. Different authors use other terms...


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