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

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