## Abstract

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.

### Similar content being viewed by others

## Notes

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).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.

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 \(\beta \).

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.

## References

Ackermann H, Berenbrink P, Fischer S, Hoefer M (2016) Concurrent imitation dynamics in congestion games. Distrib Comput 29(2):105–125. https://doi.org/10.1007/s00446-014-0223-6

Alós-Ferrer C, Netzer N (2010) The logit-response dynamics. Games Econ Behav 68(2):413–427

Alós-Ferrer C, Netzer N (2017) On the convergence of logit-response to (strict) Nash equilibria. Econ Theory Bull 5(1):1–8. https://doi.org/10.1007/s40505-016-0104-1 (ISSN 2196-1093)

Anshelevich E, Dasgupta A, Kleinberg J, Tardos É, Wexler T, Roughgarden T (2008) The price of stability for network design with fair cost allocation. SIAM J Comput 38(4):1602–1623

Asadpour A, Saberi A (2009) On the inefficiency ratio of stable equilibria in congestion games. In: Proceedings of the 5th international workshop on internet and network economics (WINE). LNCS, vol 5929, pp 545–552

Auletta V, Ferraioli D, Pasquale F, Persiano G (2013) Mixing time and stationary expected social welfare of logit dynamics. Theory Comput Syst 53(1):3–40

Auletta V, Ferraioli D, Pasquale F, Penna P, Persiano G (2015) Logit dynamics with concurrent updates for local interaction potential games. Algorithmica 73(3):511–546. https://doi.org/10.1007/s00453-014-9959-4

Blume LE (1993) The statistical mechanics of strategic interaction. Games Econ Behav 5(3):387–424

Blume LE (1998) Population games. Addison-Wesley, Reading

Christodoulou G, Koutsoupias E, Spirakis PG (2011) On the performance of approximate equilibria in congestion games. Algorithmica 61(1):116–140

Chung C, Pyrga E (2009) Stochastic stability in internet router congestion games. In: Proceedings of the 2nd international symposium on algorithmic game theory (SAGT). LNCS, vol 5814, pp 183–195

Chung C, Ligett K, Pruhs K, Roth A (2008) The price of stochastic anarchy. In: Proceedings of the 1st international symposium on algorithmic game theory (SAGT). LNCS, vol 4997, pp 303–314

Coucheney P, Durand S, Gaujal B, Touati C (2014) General revision protocols in best response algorithms for potential games. In: Network games, control and optimization (NetGCoop)

Fanelli A, Moscardelli L, Skopalik A (2012) On the impact of fair best response dynamics. In: Proceedings of the 37th international conference on mathematical foundations of computer science (MFCS). LNCS, vol 7464, pp 360–371

Finn G, Horowitz E (1979) A linear time approximation algorithm for multiprocessor scheduling. BIT Numer Math 19(3):312–320. https://doi.org/10.1007/BF01930985 (ISSN 0006-3835)

Fotakis D, Kaporis AC, Spirakis PG (2010) Atomic congestion games: fast, myopic and concurrent. Theory Comput Syst 47(1):38–59

Kawase Y, Makino K (2013) Nash equilibria with minimum potential in undirected broadcast games. Theor Comput Sci 482:33–47

Kleinberg R, Piliouras G, Tardos É (2009) Multiplicative updates outperform generic no-regret learning in congestion games. In: Proceedings of the 41st annual ACM symposium on theory of computing (STOC), pp 533–542

Koutsoupias E, Papadimitriou CH (2009) Worst-case equilibria. Comput Sci Rev 3(2):65–69

Mamageishvili A, Mihalák M (2015) Multicast network design game on a ring. In: Proceedings of the 9th international conference on combinatorial optimization and applications (COCOA). LNCS, vol 9486, pp 439–451

Mamageishvili A, Penna P (2016) Tighter bounds on the inefficiency ratio of stable equilibria in load balancing games. Oper. Res. Lett. 44(5):645–648. https://doi.org/10.1016/j.orl.2016.07.014

Marden JR, Shamma JS (2012) Revisiting log-linear learning: asynchrony, completeness and payoff-based implementation. Games Econ Behav 75(2):788–808

Pradelski BSR, Young HP (2012) Learning efficient Nash equilibria in distributed systems. Games Econ Behav 75(2):882–897

Roughgarden T (2009) Intrinsic robustness of the price of anarchy. In: Proceedings of the 41st ACM symposium on theory of computing (STOC), pp 513–522

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

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

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

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).

## Rights and permissions

## About this article

### Cite this article

Penna, P. The price of anarchy and stability in general noisy best-response dynamics.
*Int J Game Theory* **47**, 839–855 (2018). https://doi.org/10.1007/s00182-017-0601-y

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00182-017-0601-y