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The price of anarchy and stability in general noisy best-response dynamics

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Logit-response dynamics (Alós-Ferrer and Netzer in Games Econ Behav 68(2):413–427, 2010) are a rich and natural class of noisy best-response dynamics. In this work we revise the price of anarchy and the price of stability by considering the quality of long-run equilibria in these dynamics. Our results show that prior studies on simpler dynamics of this type can strongly depend on a sequential schedule of the players’ moves. In particular, a small noise by itself is not enough to improve the quality of equilibria as soon as other very natural schedules are used.

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  1. In this work we consider pure Nash equilibria, that is, states in which each player chooses one strategy and unilateral deviations are not beneficial. In general games, stochastically stable states are not necessarily Nash equilibria, even for the class of potential games (which is the main focus of this work).

  2. Asadpour and Saberi (2009) call the PoA restricted to potential minimizers Inefficiency Ratio of Stable Equilibria (IRSE), while Kawase and Makino (2013) use the terms Potential Optimal Price of Anarchy (POPoA) and of Stability (POPoS). In this work we prefer to use the terms logit PoA and logit PoS to emphasize the comparison between logit and independent-learning logit-response dynamics.

  3. It is well known that the dynamics described above define an ergodic Markov chain over the possible strategy profiles for every \(0< \beta < \infty \). That is, the stationary distribution \(\mu ^{\beta }\) exists for all \(\beta \).

  4. In this work we deal only with weighted potential games, that is, games that admit a vector w and a potential function \(\phi \) such that \(u_i(s)-u_i(s')=(\phi (s')-\phi (s))w_i\) for all i and for all s and \(s'\) that differ in exactly player i’s strategy.


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I am grateful to Francesco Pasquale for comments on an earlier version of this work, and to Carlos Alós-Ferrer for bringing Coucheney et al. (2014) and Alós-Ferrer and Netzer (2017) to my attention.

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Correspondence to Paolo Penna.

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Part of this work has been done while at LIAFA, Université Paris Diderot, supported by the French ANR Project DISPLEXITY.

Appendix A: Postponed proofs and additional details

Appendix A: Postponed proofs and additional details

1.1 A.1 Proof of Theorem 1.3

Observe that for any state \({{\bar{s}}}\) which is not a Nash equilibrium there exists a sequence of best-response moves that reaches some Nash equilibrium s. That is, there exists a zero-waste path from any non-Nash \({{\bar{s}}}\) to some Nash s.

This implies that \(W({{\bar{s}}}) \ge W(s)\) because any waste tree directed to \({{\bar{s}}}\) can be transformed into a tree of no larger waste directed into s (by adding the path mentioned above and removing other transitions). This shows that the waste \(W({{\bar{s}}})\) cannot be smaller than the waste of all Nash equilibria.

1.2 A.2 Postponed details for the proof of Theorem 2.3

We need to prove that the set of all Nash equilibria is just the union of \({\textit{OPT}}\) and \({\textit{APX}}\), that is, the allocations of the form (3) and (4), respectively.

We first show that any allocation with a machine containing a \(\Delta \)-job and another large job (size \(\Delta \) or \(\Delta - \delta \)) cannot be a Nash equilibrium. To be a Nash equilibrium, the load of every other machine should be at least \(\Delta \). This will not be possible given the remaining jobs: If we have a machine with a \(\Delta \)-job and \(\Delta -\delta \)-job, then the jobs that are left for the other \(m-1\) machines are

$$\begin{aligned} \left\{ \Delta - \delta , \underbrace{\Delta ,\ldots ,\Delta }_{m-3}, \underbrace{\delta ,\ldots ,\delta }_{lm}\right\} \end{aligned}$$

where the \(\delta \)-jobs sum up to \(\Delta \). Thus the total load we can distribute is \((m-1)\Delta - \delta \), and some machine will have load less than \(\Delta \). The same holds if we start with a machine having two \(\Delta \)-jobs.

Finally, we show that every allocation with one large job per machine has a zero-waste path to some allocation in \({\textit{OPT}}\). Indeed, allocations with one large job per machine are of the form

$$\begin{aligned} \left[ \Delta - \delta , \underbrace{\delta ,\ldots ,\delta }_{k_1} \right] ,\left[ \Delta - \delta , \underbrace{\delta ,\ldots ,\delta }_{k_2}\right] ,\left[ \Delta ,\underbrace{\delta ,\ldots ,\delta }_{k_3}\right] ,\ldots ,\left[ \Delta ,\underbrace{\delta ,\ldots ,\delta }_{k_m}\right] .\nonumber \\ \end{aligned}$$

To obtain an allocation in \({\textit{OPT}}\) it is enough to move the \(\delta \)-jobs from higher load machines to lower load machines (or to machines with the same load), until we obtain \(k_1=\cdots =k_m=l\), that is (3).

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Penna, P. The price of anarchy and stability in general noisy best-response dynamics. Int J Game Theory 47, 839–855 (2018).

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