Summary: CSCI 6963 Fall 2011 Algorithmic Game Theory
Problem Set 3
Due October 20
Problem 1. Do Problem 19.16 from the textbook. To clarify, you must show that this is a monotone
valid-utility game, and thus (by Theorem 19.19), the price of anarchy is at most 2. You must also give an
example (or a series of examples) to show that the price of anarchy can be arbitrarily close to 2.
Problem 2. Consider the following cut game. We are given an undirected graph G = (V, E). The players
are nodes in this graph, and each player can choose between 2 strategies: belonging to set A or set B. Every
edge (u, v) E has a weight wuv, which represents how much u and v don't like each other. Thus, a node
v's cost if v chooses set A is uA wuv, and if v chooses set B, its cost is uB wuv. As usual, the social
cost is the sum of all player costs, i.e., 2[ uA,vA wuv + uB,vB wuv].
(a) Show that every centrally optimal solution is a Strong Equilibrium.
(b) Show that this is a potential game.