## Wednesday, 2 December 2015

### Lecture 23

In the final lecture I spoke further about auction design, and then showed how the same ideas can be used to design a mechanism for a problem in which agents will share a facility (or a so-called club good). The mechanism receives declarations of agent types, $\theta_1,\dotsc,\theta_n$, and then chooses a size for the facility, $Q$, and extracts payments from the agents to cover its cost, $c(Q)$.

Agents with small $\theta_i$ will not find it worth joining the club and will not be made to make any payments. The analysis harkens back to earlier lectures, in that Lagrangian methods are employed, and a pivoting type algorithm is used to make adjustments to the payments. An agent will find it profitable to join the club iff $\theta_i+\lambda g(\theta_i)\geq 0$, where $\lambda$ is a Lagrange multiplier for the constraint that expected payments cover expected facility cost. The case $\lambda=\infty$ corresponds to maximizing the expected payments taken from the agents.

I gave a talk on this work at London School of Economics: Mechanism Design for a Service Provided in the Cloud. If you view this talk you will see how I managed to work into my talk the playing of the song ‘non, je ne regrette rien’ (‘no, I regret nothing’) (as sung by Edith Piaf).