Summary: CSCI 6963 Fall 2011 Algorithmic Game Theory
Problem Set 4
Due November 10
Problem 1. In class we proved that the Vickrey single-item auction (also known as the second-price
auction) is truthful, i.e, it is a dominant strategy for every player to tell the truth. In this problem, assume
that we have a set of players participating in a Vickrey auction.
(a) Let vi be player i's true valuation, and vi be another possible bid. Prove that there exist some values
v-i of bids by other players such that player i is strictly better off bidding his true value instead of vi.
(b) Usually, when discussing auctions, we consider only deviations by single players. Suppose instead that
some subset S of our n players decided to collude, i.e., coordinate their bids in such a way that the
total utility of S will improve. Give a precise description of what types of player sets S (in terms of
the true player values vi) are able to strictly improve their utility in a Vickrey auction by not bidding
their true valuations.
When designing auctions, we usually care about dominant strategies in the resulting game, such as whether
telling the truth is a dominant strategy. Below we instead consider Nash equilibria of single-item auctions.
(c) Does there exist a Nash equilibrium in a Vickrey auction, such that the player with the lowest valuation
(d) Consider the third-price auction (highest-bid player wins, and pays the third-highest bid). Is this auction
truthful? Does there always exist a pure Nash equilibrium?
Problem 2. Suppose we are given a tree T. Every player is located at some node of this tree, but only