Talk:Vickrey–Clarke–Groves mechanism

Notation

Section states "Our goal is to select an outcome that maximizes the sum of values, i.e.:

$x^{opt}(v)=\arg \max _{x\in X}\sum _{i=1}^{n}v_{i}(x)$

In other words, our social-choice function is utilitarian."

But utility is defined:

$u_{i}:=v_{i}(x)+p_{i}$

So the given function should be:

$x^{opt}(u)=\arg \max _{x\in X}\sum _{i=1}^{n}u_{i}(x)$

since $p_{i}$ is not necessarily zero - before tax. — Preceding unsigned comment added by 136.186.248.250 (talk) 01:54, 13 January 2018 (UTC)

Clark pivot rule sign of $p_{i}$ reversed?

The article defines $p_{i}$ as a sum paid to each agent, but the section on the Clark pivot rule seems to interpret it as a sum paid by each agent. As defined above, it is based on $\sum _{j\neq i}v_{j}(x^{*})$, a nonnegative value, but then adjusted by adding $h_{i}(v_{-i})$, which may be negative enough to make the final result negative. Indeed, for

$h_{i}(v_{-i})=-\max _{x\in X}\sum _{j\neq i}v_{j}(x),$

we have

$p_{i}=\sum _{j\neq i}v_{j}(x^{*})-\max _{x\in X}\sum _{j\neq i}v_{j}(x),$

which can be at most 0 if $x$ is chosen as $x^{*}$ (otherwise, it must result in an even greater sum, making the final result negative). This suggests that $p_{i}\leq 0$, not the other way around, and further that $u_{i}=v_{i}(x^{*})+p_{i}\geq 0$, i.e. adding (the negative) $p_{i}$ rather than subtracting it. I propose that we reverse the sign of $p_{i}$ used when describing the Clark pivot rule. Rriegs (talk) 00:59, 26 April 2016 (UTC)

Limitations

There's little discussion of the limitations of this algorithm. I'm no expert but it appears:

• The algorithm assumes that people act rationally based on their own selfish interests, whereas in practice I suspect many people would not understand it well enough to behave 'correctly'. For example, this seems to often be the case on e-bay.
• It can be fooled by a small group of people being untruthful together?
• The outcome may not be socially optimal when the work people are expected to do choosing their valuations is taken into account - imagine doing this with a large population and numerous trivial decisions.

Perhaps someone more familiar with the subject could consider these points?

Geoff55 (talk) 10:44, 15 December 2017 (UTC)