# What is the definition of “First Best”, “Second Best”, etc. in contract theory?

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?

• 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 Feb 23 '15 at 20:04
• Actually both. My confusion between "first best" and "ex-post efficiency" comes from the ambiguity of the definition of "first best". – Joe Li Feb 24 '15 at 22:56

In contract theory

The first-best refers to the best you could do if you knew agents' preferences over labor an 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 is not much connexion 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 be 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 (beware this is where it 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).

• 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 Feb 23 '15 at 18:07
• 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 Feb 23 '15 at 19:52

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.

• 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 Feb 23 '15 at 17:55
• 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. – TMB Feb 23 '15 at 20:03

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...