 
Summary: CSCI 6963 Fall 2011 Algorithmic Game Theory
Problem Set 1
Due September 15
Problem 1. Chapter 1, Problem 4, from the textbook.
Problem 2. Consider the following game. There are M servers, each containing a copy of a large file
available for download. There are N players, with each player desiring to download this file. The strategy
of each player p is to choose a server to download from, i.e., the strategy set is Sp = {1, . . . , M}. Thus,
an outcome of this game is as assignment of players to servers. (A player is not allowed to download from
multiple servers at the same time.)
For a solution s = (s1, . . . , sN ), let ni(s) be the number of players assigned to server i in this solution.
We will also refer to ni(s) as the load of server i in solution s. If more players choose the same server, the
time it takes them to download the file becomes slower. To make this precise, assume that every server i
has a (possibly different) nondecreasing delay function ri(x), which denotes how long it takes players to
download the file from server i if x players choose server i to download from (we will assume that ri(0) = 0
for all i). The cost function of a player p is simply the time it takes p to download the file, i.e., if sp = i,
then the cost of player p is cp(s) = ri(ni(s)).
(a) Give an algorithm to find a pure Nash equilibrium in the server assignment game. This algorithm should
run in time polynomial in N and M. (Note: A pure Nash equilibrium always exists in this game; you
do not need to prove this, although it should be a trivial consequence of your algorithm.)
(b) Consider assignments of players to servers that minimize the maximum download time, i.e., solutions s
