Equilibrium analysis underpins most of modern economic theory. Individuals are assumed to coordinate to equilibria where their actions and beliefs are mutually consistent. This assumption has profound consequences in terms of policy, from the idea of laissez-faire to the characterisation of the business cycle as an essentially exogenous phenomenon. But if major economic downturns such as the 2008 crisis are rather caused by endogenous forces, this opens up the possibility for early-warning signals and targeted interventions. So, when is the equilibrium assumption reasonable?

This project starts answering this question in the context of game theory, which is a transparent framework to study interdependent choices. For instance, the game Rock-Paper-Scissors can be represented as a 3x3 matrix, in which the cell where the first row and the second column meet (corresponding to the first player choosing Rock and the second player choosing Paper), contains the information that the second player wins.

The players are often assumed to coordinate on a Nash Equilibrium (NE), a state in which no player has an incentive to deviate from their current position. In Rock-Paper-Scissors, the only NE is the state where both players choose all three options with equal probability. If the first player would deviate and choose Rock with a higher probability, the second player would simply choose Paper with higher probability and win.

One seemingly convincing justification for convergence to an equilibrium is learning: the participants of the game learn the NE. We show that this is not necessarily the case: both in simple and complicated games the learning dynamics may be complex, even chaotic.

**Key findings: **

- We find that when the game is competitive, as the number of actions increases and the number of players gets larger, chaos tends to dominate. This is true in particular for ‘best-response dynamics’, where a player keeps choosing the action yielding the highest payoff.
- In multiplayer games where players use best-response dynamics, the order in which players update their action crucially determines whether the game converges to a Nash equilibrium.
- The dynamics of more complicated learning rules closely follow the dynamics of (clockwork) best-response dynamics.

**Key publications:**

- Heinrich, T., Jang, Y., Mungo, L., Pangallo, M., Scott, A., Tarbush, B., & Wiese, S. C. (2023).
**Best-response dynamics, playing sequences, and convergence to equilibrium in random games**.*International Journal of Game Theory.*

- Wiese, S. C. & Heinrich, T. (2021).
**The frequency of convergent games under best-response dynamics**.*Dynamic Games and Applications*.

- Pangallo, M., Heinrich, T. & Farmer, J. D. (2018).
**Best reply structure and equilibrium convergence in generic games**. INET Oxford Working Paper No. 2017-07. - Sanders, J.B.T., Farmer, J.D. & Galla, T. (2018)
**The prevalence of chaotic dynamics in games with many players**.*Scientific Reports***8,**4902.

- Pangallo, M., Sanders, J., Galla, T., & Farmer, J. D. (2017).
**A taxonomy of learning dynamics in 2 x 2 games**. arXiv:1701.09043

**Other outputs:**

**External collaborators:**

Tobias Galla, Bassel Tarbush, Alex Scott

**Funders:**

Baillie Gifford, James S McDonnell Foundation, Foundation of German Business

**Researchers involved:**

Doyne Farmer, Samuel Wiese, Marco Pangallo, Torsten Heinrich, Luca Mungo