Topological Price of Anarchy Bounds for Clustering Games on Networks
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Frequently Asked Questions (11)
Q2. What is the best-response dynamic for GG,c?
Then best-response dynamics are guaranteed to converge to a pure Nash equilibrium for every clustering game in GG,c,α if and only if α corresponds to a generalized weighted Shapley distribution rule.
Q3. What is the simplest way to determine if s is a coordination edge?
Using that the sum of the weights of all satisfied edges in s∗ is at most the sum of all edge weights, the authors obtainu(s∗) ≤ ∑i∈V qi(s∗i ) +∑e={i,j}∈E wij ≤∑ i∈V ui(s) + (1 + ᾱ) ∑ {i,j}∈E min{ui(s), uj(s)}.
Q4. What is the interesting thing about the study of the Braess paradox?
They study the Braess paradox on large Erdős-Rényi random graphs and show that for certain settings the Braess paradox occurs with high probability as the size of the network grows large.
Q5. What is the main goal of this paper?
Their main goal in this paper is to understand how structural properties of the network topology impact the Price of Anarchy in clustering games.
Q6. What is the price of anarchy for coordination games?
More specifically, the authors show that the Price of Anarchy is constant (with high probability) for coordination games on sparse randomgraphs (i.e., p = d/n for some constant d > 0) with equal-split distribution rule.
Q7. What is the price of anarchy for random clustering games?
The authors say that the Price of Anarchy for random clustering games is at most β with high probability (PoA(GGn) ≤ β,for short) if PGn∼G(n,p){PoA (GGn) ≤ β} ≥ 1−o(1).
Q8. Why do the authors turn to (, k)-equilibria?
The authors turn to ( , k)-equilibria because pure Nash equilibria are not guaranteed to exist in asymmetric coordination games (see, e.g., [4]).extends to the set of ( , k)-equilibria.
Q9. Why do the authors elaborate on their results for symmetric and asymmetric clustering games?
Due to space restrictions, the authors elaborate on their main findings for symmetric clustering games only below; their results for the asymmetric case are discussed in Sect.
Q10. What is the proof of the existence of pure Nash equilibria?
Prior to their work, the existence of pure Nash equilibria was known for certain special cases of coordination games only, namely for symmetric coordination games with individual preferences and c = 2 [3], and for symmetric coordination games without individual preferences [11].
Q11. What is the proof of the best-response convergence?
The authors provide a characterization of distribution rules that guarantee the convergence of best-response dynamics in symmetric clustering games.