CS6998-3: Problem Set # 1
Due in class, on Monday October 13
Problem 1
Recall in class we proved a bound of 4
3on the price of anarchy in networks with linear latency functions. We
also stated (without proof) that the two node, two link networks form the worst case for price of anarchy
calculations.
(a) For a positive integer d≥1, let Mddenote the set {axi:a≥0, i ∈ {0,1, . . . , d}} of monomials with
non negative coefficients with degree at most d. What is the largest price of anarchy possible with cost
functions in Md? Can you give an asymptotic bound in terms of d.
(b) For a positive integer d≥1, let Pddenote the set {Pd
i=0 aixi:a0, a1, . . . , ad≥0}of polynomials with
non negative coefficients and degree at most d. What is the largest price of anarchy possible with cost
functions in Pd?
Problem 2
When talking about selfish routing in class, we focused on what are known as “single-commodity” networks.
That is all traffic has the same source, and same destination. In this problem we extend the definitions to a
“multicommodity” networks. We are given a graph G= (V, E) and linear latency functions `e: [0,1] →R+
on all of the edges. We are given ksource-sink pairs (s1, t1),(s2, t2),...,(sk, tk) with si, ti∈V. We would
like to route 1 unit of flow from each source sito each sink ti.
(a) Extend the definitions of flows at Nash equilibrium and optimal flows to the multicommodity case.
(b) Prove a generalization of the bicriteria theorem we showed in class. That is, if fis a Nash flow, then
f/2 is the optimal flow for routing 1/2 unit from each sito each ti.
Problem 3
In class we showed that the price of anarchy for the network design game can be as bad as k. Prove that
the price of anarchy can never exceed k.
Problem 4
Recall the network design game as we talked about in class. We are given a graph G= (V, E), and kplayers,
where player iwants to go from si∈Vto ti∈V. Assume the edge costs are integral, and bounded above by
C; that is ce∈ {0,1,2, . . . , C −1, C }. As we saw, iterated best response will converge to a Nash equilibrium,
but may do so slowly. One way to deal with this problem is to consider only “large” changes in the potential.
That is, fix a constant > 0, and call a change in the potential large if it decreases Φ by at least an /k
fraction of its value.
(a) Show that if we consider only best responses that cause large changes in the potential, the process
converges in time that is polynomial in |E|,k, log Cand 1
.
(b) Show that the resulting outcome may be far from an equilibrium. That is, even though there are no
deviations that cause large changes in the potential, individual players may still reduce their costs by
an arbitrarily large factor.
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