# From simultaneous to leader–follower play in direct reciprocity

## TL;DR

This paper explores how changing the timing of moves in repeated games affects cooperation and social behaviors.

## Contribution

The paper introduces a novel transformation from simultaneous to leader–follower games to analyze stable social behaviors.

## Key findings

- Transforming games to leader–follower structures reveals stable Nash equilibria in discounted repeated games.
- Stability does not always hold in games with more actions or longer memory strategies.
- The approach connects simultaneous and leader–follower games in evolutionary game theory.

## Abstract

Repeated interaction is a key mechanism driving the evolution of cooperation, generosity, extortion, and other social behaviors. These behaviors are typically studied using two-player repeated games, where the two players act simultaneously in every round. In this article, we begin with a standard two-action repeated game, but transform it into a leader–follower game. We do this by forcing one player to indicate their impending move, or to openly commit first in each round. We show that this simple transformation provides new and useful analytical insights. In particular, we use this transformation to derive expressions for the main classes of simultaneous-move Nash equilibria of memory-1, when payoffs are discounted. We show that these equilibria also remain stable under the leader–follower structure. We prove that this reassuring property does not always extend to games with more than two actions or to equilibrium strategies with longer memory. Overall, we establish a novel connection between repeated simultaneous-move games and repeated leader–follower games in evolutionary game theory. This approach offers fruitful means of investigating how prediction and anticipation play a role in stable social behavior.