CSCI 6963 Fall 2011 Algorithmic Game Theory Problem Set 4 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 vi wins? (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 Collections: Computer Technologies and Information Sciences