On the discrete-time origins of the replicator dynamics: from convergence to instability and chaos

Fryderyk Falniowski () and Panayotis Mertikopoulos ()
Additional contact information
Fryderyk Falniowski: Krakow University of Economics
Panayotis Mertikopoulos: Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG

International Journal of Game Theory, 2025, vol. 54, issue 1, No 7, 29 pages

Abstract: Abstract We consider three distinct discrete-time models of learning and evolution in games: a biological model based on intra-species selective pressure, the dynamics induced by pairwise proportional imitation, and the exponential/multiplicative weights algorithm for online learning. Even though these models share the same continuous-time limit—the replicator dynamics—we show that second-order effects play a crucial role and may lead to drastically different behaviors in each model, even in very simple, symmetric $$2\times 2$$ 2 × 2 games. Specifically, we study the resulting discrete-time dynamics in a class of parametrized congestion games, and we show that (i) in the biological model of intra-species competition, the dynamics remain convergent for any parameter value; (ii) the dynamics of pairwise proportional imitation exhibit an entire range of behaviors for larger time steps and different equilibrium configurations (stability, instability, and even Li–Yorke chaos); while (iii) in the exponential/multiplicative weights algorithm, increasing the time step (almost) inevitably leads to chaos (again, in the formal, Li–Yorke sense). This divergence of behaviors comes in stark contrast to the globally convergent behavior of the replicator dynamics, and serves to delineate the extent to which the replicator dynamics provide a useful predictor for the long-run behavior of their discrete-time origins.

Keywords: Replicator dynamics; Discrete-time dynamics; Revision protocols; Exponential weights; Li-Yorke chaos (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s00182-025-00929-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jogath:v:54:y:2025:i:1:d:10.1007_s00182-025-00929-3

Ordering information: This journal article can be ordered from
http://www.springer. ... eory/journal/182/PS2

DOI: 10.1007/s00182-025-00929-3

Access Statistics for this article

International Journal of Game Theory is currently edited by Shmuel Zamir, Vijay Krishna and Bernhard von Stengel

More articles in International Journal of Game Theory from Springer, Game Theory Society
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().