# The Price of Anarchy in Selfish Multicast Routing

## Abstract

We study the price of anarchy for selfish multicast routing games in directed multigraphs with latency functions on the edges, extending the known theory for the unicast situation, and exhibiting new phenomena not present in the unicast model. In the multicast model we have *N* commodities (or player classes), where for each \( i=1,\hdots,N\), a flow from a source *s* _{ i } to a finite number of terminals \(t_i^1,\hdots,t_i^{k_i}\) has to be routed such that every terminal *t* _{ i } ^{ j } receives flow *n* _{ i }∈ℝ_{ ≥ 0}.

One of the significant results of this paper are upper and lower bounds on the price of anarchy for edge latencies being polynomials of degree at most *p* with non-negative coefficients. We show an upper bound of \((p+1)\cdot\frac{\nu^{p+1}}{\nu^*}\) in some variants of multicast routing. We also prove a lower bound of *ν* ^{ p }, so we have upper and lower bounds that are tight up to a factor of (*p*+1)*ν*. Here, *ν* and *ν* ^{*} are network and strategy dependent parameters reflecting the maximum/minimum consumption of the network. Both are 1 in the unicast case. Our lower bound of *ν* ^{ p }, where in the general situation we have *ν*>1, shows an exponential increase compared to the Roughgarden bound of *O*(*p*/ln*p*) for the unicast model. This exhibits the contrast to the unicast case, where we have Roughgarden’s (2002) result that the price of anarchy is independent of the network topology. To our knowledge this paper is the first thorough study of the price of anarchy in the multicast scenario. The approach may lead to further research extending game-theoretic network analysis to models used in applications.

## Keywords

Nash Equilibrium Variational Inequality Latency Function Strategy Latency Congestion Game## Preview

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