# Last-iterate Convergence for Symmetric, General-sum, 2 × 2 Games Under The Exponential Weights Dynamic

arXiv: https://arxiv.org/abs/2502.08063 · HTML: https://mlsanity.com/paper/2502.08063/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 12 Feb 2025.

## Builds on (most cited first, 20 of 20)

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- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
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- [Anticipating Performativity by Predicting from Predictions](https://mlsanity.com/paper/2208.07331/) (2022)
- [Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaos](https://mlsanity.com/paper/1703.01138/) (2017)
- [Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Games](https://mlsanity.com/paper/1905.08396/) (2019)
- [Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information](https://mlsanity.com/paper/2101.04667/) (2021)
- [Chaos, Extremism and Optimism: Volume Analysis of Learning in Games](https://mlsanity.com/paper/2005.13996/) (2020)
- [Efficient Stackelberg Strategies for Finitely Repeated Games](https://mlsanity.com/paper/2207.04192/) (2022)
- [Higher-Order Uncoupled Dynamics Do Not Lead to Nash Equilibrium -- Except When They Do](https://mlsanity.com/paper/2304.04282/) (2023)

## Abstract (arXiv)

We conduct a comprehensive analysis of the discrete-time exponential-weights dynamic with a constant step size on all general-sum and symmetric 2 × 2 normal-form games, i.e. games with 2 pure strategies per player, and where the ensuing payoff tuple is of the form (A,A^top) (where A is the 2 × 2 payoff matrix corresponding to the first player). Such symmetric games commonly arise in real-world interactions between 'symmetric" agents who have identically defined utility functions -- such as Bertrand competition and multi-agent performative prediction, and display a rich multiplicity of equilibria despite the seemingly simple setting. Somewhat surprisingly, we show through a first-principles analysis that the exponential weights dynamic, which is popular in online learning, converges in the last iterate for such games regardless of initialization with an appropriately chosen step size. For certain games and/or initializations, we further show that the convergence rate is in fact exponential and holds for any step size. We illustrate our theory with extensive simulations and applications to the aforementioned game-theoretic interactions. In the case of multi-agent performative prediction, we formulate a new "mortgage competition" game between lenders (i.e. banks) who interact with a population of customers, and show that it fits into our framework.