##### Marco Pangallo, Torsten Heinrich, J Doyne Farmer

The concept of equilibrium is central to economics. It is one of the core assumptions in the vast majority of economic models, including models used by policymakers on issues ranging from monetary policy to climate change, trade policy and the minimum wage. But is it a good assumption?

In a newly published Science Advances paper, we investigate this question in the simple framework of games, and show that when the game gets complicated this assumption is problematic. If these results carry over from games to economics, this raises deep questions about economic models and when they are useful to understand the real world.

Kids love to play noughts and crosses, but when they are about 8 years old they learn that there is a strategy for the second player that always results in a draw. This strategy is what is called an equilibrium in economics. If all the players in the game are rational they will play an equilibrium strategy.

In economics, the word rational means that the player can evaluate every possible move and explore its consequences to their endpoint and choose the best move. Once kids are old enough to discover the equilibrium of noughts and crosses they quit playing because the same thing always happens and the game is boring. One way to view this is that, for the purposes of understanding how children play noughts and crosses, rationality is a good behavioral model for eight year olds but not for six year olds.

In a more complicated game like chess, rationality is never a good behavioral model. The problem is that chess is a much harder game, hard enough that no one can analyze all the possibilities, and the usefulness of the concept of equilibrium breaks down. In chess no one is smart enough to discover the equilibrium, and so the game never gets boring. This illustrates that whether or not rationality is a sensible model of the behavior of real people depends on the problem they have to solve. If the problem is simple, it is a good behavioral model, but if the problem is hard, it may break down.

Theories in economics nearly universally assume equilibrium from the outset. But is this always a reasonable thing to do? To get insight into this question, we study when equilibrium is a good assumption in games. We don’t just study games like noughts and crosses or chess, but rather we study all possible gamesof a certain type (called normal form games).

We literally make up games at random and have two simulated players play them to see what happens. The simulated players use strategies that do a good job of describing what real people do in psychology experiments. These strategies are simple rules of thumb, like doing what has worked well in the past or picking the move that is most likely to beat the opponent’s recent moves.

We demonstrate that the intuition about noughts and crosses versus chess holds up in general, but with a new twist. When the game is simple enough, rationality is a good behavioral model: players easily find the equilibrium strategy and play it. When the game is more complicated, whether or not the strategies will converge to equilibrium depends on whether or not the game is competitive.

If the game is not competitive, or the incentives of the players are lined up, players are likely to find the equilibrium strategy, even if the game is complicated. But when the game is competitive and it gets complicated, they are unlikely to find the equilibrium. When this happens their strategies always keep changing in time, usually chaotically, and they never settle down to the equilibrium. In these cases equilibrium is a poor behavioral model.

A key insight from the paper is that cycles in the logical structure of the game influence the convergence to equilibrium. We analyze what happens when both players are myopic, and play their best response to the last move of the other player. In some cases this results in convergence to equilibrium, where the two players settle on their best move and play it again and again forever.

However, in other cases the sequence of moves never settles down and instead follows a best reply cycle, in which the players’ moves keep changing but periodically repeat – like the movie “ground hog day” – over and over again. When a game has best reply cycles, convergence to equilibrium becomes less likely. Using this result we are able to derive quantitative formulas for when the players of the game will converge to equilibrium and when they won’t, and show explicitly that in complicated and competitive games cycles are prevalent and convergence to equilibrium is unlikely.

When the strategies of the players do not converge to a Nash equilibrium, they perpetually change in time. In many cases the learning trajectories do not follow a periodic cycle, but rather fluctuate around chaotically. For the learning rules we study, the players never converge to any sort of “intertemporal equilibrium”, in the sense that their expectations do not match the outcomes of the game even in a statistical sense. For the cases in which learning dynamics are highly chaotic, no player can easily forecast the other player’s strategies, making it realistic that this mismatch between expectations and outcomes persists over time.

Are these results relevant for macroeconomics? Can we expect insights that hold at the small scale of strategic interactions between two players to also be valid at much larger scales?

While our theory does not directly map to more general settings, many economic scenarios – buying and selling in financial markets, innovation strategies in competing firms, supply chain management – are complicated and competitive. This raises the possibility that some important theories in economics may be inaccurate. Challenges to the behavioral assumption of equilibrium also challenge the predictions of the model. In this case, new approaches are required that explicitly simulate the behavior of economic agents and take into account the fact that real people are not good at solving complicated problems.