Class # 5 Highlights

In this class, we studied general solutions to the information problem. We can think about the "game" designed by the principal (seller, employer, etc.) as a message game with an arbitrary message space. The vector of messages is then translated into allocations. For example, in a first-price auction, the bids are messages, and the allocation gives the good to the person sending the highest "message" and obliges them to pay the amount of their message. Viewed in this light, all such situations are indirect mechanisms.

A direct mechanism is a message game with the restriction that the message space is equal to the type space.

The Revelation Principle states that any BNE of any indirect mechanism is a BNE of the direct mechanism under the additional constraint that all agents report their types truthfully.

The intuition for the revelation principle is that reporting untruthfully corresponds to deviating from the equilibrium message in the indirect mechanism.But since we are implementing a BNE of the indirect mechanism then such a deviation must not be profitable, otherwise we didn't have a BNE in the first place.

We then specialized preferences to the simple specification where

u_i = type_i x_i - T_i

In that setting, we derived an equilibrium condition of a direct mechanism. This is simply the FOC evaluated at truth-telling. Then, using the envelope theorem, we know that the change in equilibrium utility with respect to type is simply x_i.. 

A lemma shows that any incentive feasible allocation must have the property that x_i is non-decreasing for all i. Hence, using the fundamental theorem of calculus, we can show that the equilibrium utility under a direct mechanism is simply

u[type_i] = u[type_0] + Integrate[x(s), {s, type_0, type_i}]

(using Mathematica notation) and where type_0 is the lowest possible type. 

Equilibrium utility can also be expressed as

u[type_i] = type_i x_bar[type_i] - T_bar[type_i]

or, equivalently

 T_bar[type_i] = type_i x_bar[type_i] - u[type_0] - Integrate[x(s), {s, type_0, type_i}]

Thus, the payment to the principal depends only on the allocation schedule x and the utility of the lowest type. 

This yields the Revenue Equivalence Theorem: Any mechanism resulting in the same payoff to the lowest type and the same allocation schedule also yields the same expected revenue to the principal. 

In terms of our two auctions, notice that the x schedule was the same for both--in either case, the object was given to the person with the highest value. Moreover, the payment to the lowest type was the same: In each case, that type earned zero. Thus, the two mechanisms are revenue equivalent. But now we know that any auction that produces an efficient outcome and has free-opt out, so the lowest type earns the same amount, will produce the same expected revenue as either of these auctions.

Key things should be able to do after this class:

1. Explain how the revelation principle works--especially why it is without loss of generality to restrict attention to truth-telling strategies.
2. Understand the revenue equivalence theorem and know when it does or does not apply.