Class #6 Highlights

In this class, we saw the RET in action. The key is to note that the RET implies even stronger conditions about individuals participating in the mechanism. Notably, any mechanism that induces the same allocation x and gives the same utility to the lowest type produces the same utility to each type. This arises from the utility formula offered in the previous class highlights which shows that a type's equilibrium utility only depends on x and on the utility of the lowest type.

Moreover, under the same conditions, a given type makes the same expected payment under any mechanism. In an auction setting, this means that the expected payment under any auction that allocates the good to the same type and has the same utility for the lowest type will be the same as that under the Vickrey auction. Since the Vickrey is quite easy to analyze, this equivalence is very handy to compute things in a given mechanism without having to go through all the calculations to obtain equilibrium.

An example of this is a winner-take-all labor tournament. There are several equivalent interpretations for this mechanism. (1) Suppose that effort is directly measured but that higher types can produce a given level of effort at lower cost than lower types. (2) Output, which is a combination of effort and ability is measured. Effort costs the same for all types, but higher types produce more output per unit of effort than lower types. (3) Effort is measured and all individuals produce effort at the same cost but higher types value the prize more than lower types. Case (3) is obviously the least realistic, but the easiest to map directly into our RET framework. From this, we can deduce the expected payment for each type. Since types "pay" their effort, then the expected payment is the same as the equilibrium effort of each type. Furthermore, since all three settings are isomorphic, this is also the equilibrium effort under either of the other interpretations as well.

The point of the example is twofold: First, it illustrates that settings that are not, recognizably auctions, can be fruitfully modeled in a way that permits us to use the RET to analyze. Second, it illustrates that equilibrium calculations that are otherwise annoying to compute may be calculated quite directly using other implications of the RET.


Key things should be able to do after this class:


1. Explain how and why the RET works.
2. Use the RET to compute equilibria in auction-like games.