Second price auction, page 82-84 of Osborne's An Introduction to Game Theory

Question

Consider the second price auction defined and discussed on pages 82-84 of Osborne's An Introduction to Game Theory $($pages 80-82 here in this online draft version of the textbook: Martin J. Osborne, An Introduction to Game Theory$)$. Since a bid equal to his valuation weakly dominates all his other bids, the second price auction game form is straightforward. I am confused as to why this can't be regarded as 1. dictorial, and 2. a counterexample to Gibbard's Theoren.

Answer 1

The second price auction is not dictatorial, because the outcome is not dependent on a single bidder's report. Although single price auction is strategy-proof $($because the game form is straightforward$)$, has more than three alternatives, and is not dictatorial as we have shown above, it is not a counterexample to Gibbard-Satterthwaite. In Gibbard-Satterthwaite, it is $($perhaps tacitly$)$ assumed that individuals are allowed to have any preference over the set of alternatives. This is not the case in second price auction.

To see this, let us explicitly write down what the possible set of alternatives are in second price auction. With $n$ bidders, the outcome is expressed in terms of $(x_i,\, p_i)_{i=1}^n$ where $x_i = 1$ denotes that the object is allocated to $i$ and $p_i$ denotes the payment $i$ makes. We have $\sum_i x_i \le 1$, $x_i = \{0,\,1\}$, and $p_i \ge 0$. In second price auction, bidder $i$ can only express her preference by submitting report $r_i \in \mathfrak{R}_+$. The report confines the preference of bidder $i$ to something that says $``$I like winning, given that payment $p_i$ is less than $r_i$. I also want to pay less given the allocation of the good.$"$ This is a strict subset of all the possible preferences over alternatives.