Algorithmic Game Theory Due: 12 April 2015

Homework 1

Lecturer: Michal Feldman Assistant: Ophir Friedler

1 Routing Games

Problem 1 : (Non-atomic selfish routing).In this problem we consider non-atomic selfish routing networks with one source, one sink,one unit of selfish traffic, and affine cost functions (of the form ce(x) = aex+be for ae, be ≥ 0).In parts (a)-(c), we consider the objective of the maximum cost incurred by a flow f :

maxP :fP>0

∑e∈P

ce(fe)

The price of anarchy is then defined in the usual way, as the ratio between the maximumcost of an equilibrium flow and that of a flow with minimum-possible maximum cost. (Ofcourse, in an equilibrium flow, all traffic incurs exactly the same cost; this is not generallytrue in a non-equilibrium flow.)

(a) Prove that in a network of parallel links (each directly connecting the source to thesink), the price of anarchy with respect to the maximum cost objective is 1.

(b) Prove that the price of anarchy with respect to the maximum cost objective can be aslarge as 4/3 in general networks (with affine cost functions, one source and one sink).

(c) Prove that the price of anarchy with respect to the maximum cost objective is neverlarger than 4/3 (in networks with affine cost functions, one source and one sink). [Hint:try to reduce this to facts you already know.]

(d) A flow that minimizes the average cost of traffic generally routes some traffic on costlierpaths than others. Prove that the ratio between the cost of the longest used path andthat of the shortest used path in a minimum-cost flow is at most 2 (in networks withaffine cost functions, one source and one sink). Prove that this bound can be achieved.

Problem 2: (Atomic selfish routing)

(a) Consider an atomic selfish routing game in which all players have the same source vertexand sink vertex (and each controls one unit of flow). Assume that edge cost functions

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are nondecreasing, but do not assume that they are affine. Prove that a (pure-strategy)Nash equilibrium (i.e., an equilibrium flow) can be computed in polynomial time. [Hint:Remember the potential function. You can assume without proof that the minimum-cost flow problem can be solved in polynomial time. If you havent seen the min-costflow problem before, you can read about it in any book on combinatorial optimization.Be sure to discuss the issue of fractional vs. integral flows, and explain how (or if) youuse the hypothesis that edge cost functions are nondecreasing.]

(b) Prove that in an atomic selfish routing network of parallel links, every equilibrium flowminimizes the potential function.

(c) Show by example that (b) does not hold in general networks, even when all players havea common source and sink vertex.

2 Potential Games

Problem 3: (Better Replies in Potential Games)An generalized ordinal potential function Φ(·) is a function that satisfies for all i:

ui(σ′i, σ−i)− ui(σ) > 0⇒ Φ(σ′i, σ−i)− Φ(σ) > 0

Φ(·) is an ordinal potential function if for all i:

ui(σ′i, σ−i)− ui(σ) > 0⇔ Φ(σ′i, σ−i)− Φ(σ) > 0

A better reply sequence in an n-player game G is a sequence (finite or infinite) of strategyprofiles s0, s1, . . . , st, . . . where s0 is arbitrary, and for each t > 0, st is obtained from st−1

by a single player i switching his strategy to a better one. I.e. for each t > 0 there exists isuch that st−i = st−1−i and ui(s

t) > ui(st−1). We say that all better reply dynamics converge

on G if there are no infinite better reply sequences.

(a) Show that there exists an ordinal potential game with n players, where each player hasonly two strategies, that has better-reply sequences of length 2n.

(b) Prove that G has a generalized ordinal potential game if and only if all better replydynamics on G converge.

(c) Prove or show counter example: G has an ordinal potential game if and only if all betterreply dynamics on G converge.

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3 Network Formation Games

Problem 4: (Network formation with path-length objectives)In this question, we address network formation games with length objectives (this modelcorresponds to real-life settings such as communication protocols that assume that a messagepasses a pre-defined length before reaching its destination; examples are onion routing, andproof-of-work protocols.)

Let G = (V,E) be a directed graph, s ∈ V a source node, and let the cost of every edge be1. Consider a network formation game with n agents, where the objective of player i is toform a path of length li originating at s. Each agent pays for every edge in her path a costproportional to her usage of the edge. Observe that every agent may use an edge multipletimes, i.e., the paths are multi-sets. Formally, given the selected paths P = (P 1, . . . , Pn),let P i(e) be number of times e appears in P i, and let P (e) =

∑ni=1 P

i(e), then the cost ofplayer i is:

costi(P ) =∑e∈Pi

P i(e)

P (e)

(a) Does this game admit a potential function? If it does, specify it, and if it doesn’t proveso. [Hint: What does the existence of a potential imply?]

(b) Establish the best lower and upper bounds you can on the PoA.

(c) Does the social optimum always correspond to a Nash equilibrium of the game?

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