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From simultaneous to leader-follower play in direct reciprocity.
Philip LaPorte1, Lenz Pracher2,3, Saptarshi Pal4
1Department of Mathematics, University of California, 970 Evans Hall, MC 3840, Berkeley, CA 94720, USA.
PNAS Nexus
|February 9, 2026
Summary
This study transforms repeated simultaneous-move games into leader-follower games, revealing new insights into cooperation and social behaviors. The leader-follower structure maintains stability for certain equilibria, offering a novel approach to evolutionary game theory.
Area of Science:
- Evolutionary Game Theory
- Behavioral Economics
- Game Theory
Background:
- Repeated interactions are fundamental to the evolution of social behaviors like cooperation and extortion.
- Traditional studies use simultaneous-move games, limiting analytical insights.
Purpose of the Study:
- To transform simultaneous-move games into leader-follower games for novel analytical insights.
- To investigate the stability of Nash equilibria under this new structure.
- To explore the role of prediction and anticipation in stable social behavior.
Main Methods:
- Adapted a standard two-action repeated game into a leader-follower format.
- Derived expressions for memory-1 Nash equilibria with discounted payoffs.
- Analyzed the stability of these equilibria under the leader-follower transformation.
Main Results:
- The leader-follower transformation yields useful analytical insights into repeated games.
- Simultaneous-move Nash equilibria of memory-1 remain stable in the leader-follower structure.
- Stability is not guaranteed for games with more than two actions or longer memory strategies.
Conclusions:
- A novel connection between repeated simultaneous-move and leader-follower games is established.
- The leader-follower approach provides a fruitful method for studying prediction and anticipation in social behavior.
- This framework enhances understanding of cooperation and other social dynamics in evolutionary game theory.
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