Non-algebraic Convergence Proofs for Continuous-Time Fictitious Play

Berger, Ulrich

doi:10.1007/s13235-011-0033-4

Cited by 3 publications

(2 citation statements)

References 21 publications

(22 reference statements)

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“…Last but not least, there is some recent work that digs deeply into the di¤erential topology and projective geometry of …ctitious-play paths in two-person zero-sum games (van Strien, 2011;Berger, 2012). Exploring the potentially interesting implications of such approaches for zero-sum networks remains, however, beyond the scope of the present study.…”

Section: Discussionmentioning

confidence: 93%

Fictitious play in networks

Ewerhart

Valkanova

2020

Games and Economic Behavior

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This paper studies …ctitious play in networks of noncooperative twoperson games. We show that continuous-time …ctitious play converges to the set of Nash equilibria if the overall n-person game is zero-sum. Moreover, the rate of convergence is 1= , regardless of the size of the network. In contrast, arbitrary nperson zero-sum games with bilinear payo¤ functions do not possess the continuoustime …ctitious-play property. As extensions, we consider networks in which each bilateral game is either strategically zero-sum, a weighted potential game, or a twoby-two game. In those cases, convergence requires a condition on bilateral payo¤s or, alternatively, that the network is acyclic. Our results hold also for the discrete-time variant of …ctitious play, which implies, in particular, a generalization of Robinson's theorem to arbitrary zero-sum networks. Applications include security games, con ‡ict networks, and decentralized wireless channel selection.

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Section: Discussionmentioning

confidence: 93%

Fictitious play in networks

Ewerhart

Valkanova

2020

Games and Economic Behavior

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“…There are several results on the convergence of FP in games of strategic complements (see e.g. [67]), and it is believed that FP dynamics always converge in such games [68]. The game being a potential game is a sufficient condition for the empirical distribution of strategies to converge [69].…”

Section: Learning Dynamicsmentioning

confidence: 99%

Coordination problems on networks revisited: statics and dynamics

Dall’Asta¹

2021

J. Stat. Mech.

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Simple binary-state coordination models are widely used to study collective socio-economic phenomena such as the spread of innovations or the adoption of products on social networks. The common trait of these systems is the occurrence of large-scale coordination events taking place abruptly, in the form of a cascade process, as a consequence of small perturbations of an apparently stable state. The conditions for the occurrence of cascade instabilities have been largely analysed in the literature, however for the same coordination models no sufficient attention was given to the relation between structural properties of (Nash) equilibria and possible outcomes of dynamical equilibrium selection. Using methods from the statistical physics of disordered systems, the present work investigates both analytically and numerically, the statistical properties of such Nash equilibria on networks, focussing mostly on random graphs. We provide an accurate description of these properties, which is then exploited to shed light on the mechanisms behind the onset of coordination/miscoordination on large networks. This is done studying the most common processes of dynamical equilibrium selection, such as best response, bounded-rational dynamics and learning processes. In particular, we show that well beyond the instability region, full coordination is still globally stochastically stable, however equilibrium selection processes with low stochasticity (e.g. best response) or strong memory effects (e.g. reinforcement learning) can be prevented from achieving full coordination by being trapped into a large (exponentially in number of agents) set of locally stable Nash equilibria at low/medium coordination (inefficient equilibria). These results should be useful to allow a better understanding of general coordination problems on complex networks.

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