# Last-Iterate Convergence of No-Regret Learning for Equilibria in Bargaining Games

arXiv: https://arxiv.org/abs/2507.03150 · HTML: https://mlsanity.com/paper/2507.03150/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 27 references that are in the map.
- First posted on arXiv 3 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Adaptive Learning in Continuous Games: Optimal Regret Bounds and Convergence to Nash Equilibrium](https://mlsanity.com/paper/2104.12761/) (2021)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information](https://mlsanity.com/paper/2101.04667/) (2021)
- [The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities](https://mlsanity.com/paper/2107.01906/) (2021)
- [It's Not Fairness, and It's Not Fair: The Failure of Distributional Equality and the Promise of Relational Equality in Complete-Information Hiring Games](https://mlsanity.com/paper/2209.05602/) (2022)
- [Equal Merit Does Not Imply Equality: Discrimination at Equilibrium in a Hiring Market with Symmetric Agents](https://mlsanity.com/paper/2412.15162/) (2024)

## Abstract (arXiv)

Bargaining games, where agents attempt to agree on how to split utility, are an important class of games used to study economic behavior, which motivates a study of online learning algorithms in these games. In this work, we tackle when no-regret learning algorithms converge to Nash equilibria in bargaining games. Recent results have shown that online algorithms related to Follow the Regularized Leader (FTRL) converge to Nash equilibria (NE) in the last iterate in a wide variety of games, including zero-sum games. However, bargaining games do not have the properties used previously to established convergence guarantees, even in the simplest case of the ultimatum game, which features a single take-it-or-leave-it offer. Nonetheless, we establish that FTRL (without the modifications necessary for zero-sum games) achieves last-iterate convergence to an approximate NE in the ultimatum game along with a bound on convergence time under mild assumptions. Further, we provide experimental results to demonstrate that convergence to NE, including NE with asymmetric payoffs, occurs under a broad range of initial conditions, both in the ultimatum game and in bargaining games with multiple rounds. This work demonstrates how complex economic behavior (e.g. learning to use threats and the existence of many possible equilibrium outcomes) can result from using a simple learning algorithm, and that FTRL can converge to equilibria in a more diverse set of games than previously known.