 
Summary: CSCI 6963 Fall 2011 Algorithmic Game Theory
Problem Set 4
Due November 10
Problem 1. In class we proved that the Vickrey singleitem auction (also known as the secondprice
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
vi 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 singleitem auctions.
(c) Does there exist a Nash equilibrium in a Vickrey auction, such that the player with the lowest valuation
vi wins?
(d) Consider the thirdprice auction (highestbid player wins, and pays the thirdhighest 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
