With the success of modern machine learning, it is becoming increasingly important to understand and control how learning algorithms interact. Unfortunately, negative results from game theory show there is little hope of understanding or controlling general n-player games. We therefore introduce smooth markets (SM-games), a class of n-player games with pairwise zero sum interactions. SM-games codify a common design pattern in machine learning that includes some GANs, adversarial training, and other recent algorithms. We show that SM-games are amenable to analysis and optimization using first-order methods. As artificial agents proliferate, it is increasingly important to analyze, predict and control their collective behavior (Parkes and Wellman, 2015; Rahwan et al., 2019) . Unfortunately, despite almost a century of intense research since von Neumann (1928) , game theory provides little guidance outside a few special cases such as two-player zero-sum, auctions, and potential games (Monderer and Shapley, 1996; Nisan et al., 2007; Vickrey, 1961; von Neumann and Morgenstern, 1944) . Nash equilibria provide a general solution concept, but are intractable in almost all cases for many different reasons (Babichenko, 2016; Daskalakis et al., 2009; Hart and Mas-Colell, 2003) . These and other negative results (Palaiopanos et al., 2017) suggest that understanding and controlling societies of artificial agents is near hopeless. Nevertheless, human societies -of billions of agents -manage to organize themselves reasonably well and mostly progress with time, suggesting game theory is missing some fundamental organizing principles. In this paper, we investigate how markets structure the behavior of agents. Market mechanisms have been studied extensively (Nisan et al., 2007) . However, prior work has restricted to concrete examples, such as auctions and prediction markets, and strong assumptions, such as convexity. Our approach is more abstract and more directly suited to modern machine learning where the building blocks are neural nets. Markets, for us, encompass discriminators and generators trading errors in GANs (Goodfellow et al., 2014) and agents trading wins and losses in StarCraft (Vinyals et al., 2019) . Machine learning has got a lot of mileage out of treating differentiable modules like plug-and-play lego blocks. This works when the modules optimize a single loss and the gradients chain together seamlessly. Unfortunately, agents with differing objectives are far from plug-and-play. Interacting agents form games, and games are intractable in general. Worse, positive feedback loops can cause individually well-behaved agents to collectively spiral out of control. It is therefore necessary to find organizing principles -constraints -on how agents interact that ensure their collective behavior is amenable to analysis and control. The pairwise zero-sum condition that underpins SM-games is one such organizing principle, which happens to admit an economic interpretation. Our main result is that SM-games are legible: changes in aggregate forecasts are the sum of how individual firms expect their forecasts to change. It follows that we can translate properties of the individual firms into guarantees on collective convergence, stability and boundedness in SM-games, see theorems 4-6. Legibility is a local-to-global principle, whereby we can draw qualitative conclusions about the behavior of collectives based on the nature of their individual members. Identifying and exploiting games that embed local-to-global principles will become increasingly important as artificial agents become more common.
We introduce a class of n-player games suited to gradient-based methods.
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