Open Games Sheaves Prices
This meeting took place on the 23rd of November 2020. The slides are here. This meeting was a brief introduction to three interrelated topics: the theory of open games as developed by Jules Hedges et al., some ideas about sheaves of prices as developed in the text From Cognitive Mappings to Sheaves, and market equilibrium theory.
Open games are, briefly, a reformulation of game theory in category theoretical terms, specifically in the diagrammatic language of optics. Informally, open games differ from ordinary games in that they are composable with other games - we can wire games together to form a larger game. The idea is that we can subsume the notions of player, strategies, and payoffs in the wiring up of two or more open games, which implies that one game serves as the environment of the other. A key technical point is that each game comes equipped with a covariant pair of ports and a contravariant pair of ports. The covariant ports allow information from the game to flow to its environment (the player's actions) and the contravariant ports allow information from the environment (the payoffs of those actions) to flow back to the game. Unlike in ordinary game theory, an open game does not have a predefined payoff function, but depends on the other games it is composed with to determine its equilibria.
In the context of sheaves and prices, we have a similar phenomena seen from a different angle. We want to define exchanges as happening over a topological space, such that we can assign prices to commodities for any given open set of this space (and we can call these open sets "markets"). The question is whether there is a globally unique procedure of gluing open sets such that the assignment of prices will agree. To properly envisage this, we need to introduce some notion of temporality into the picture, and a process (like the dynamics of capitalism) which can make incompatible markets compatible. In this way, we come into contact with longstanding problems in microeconomics. However, from the standpoint of STP, microeconomics does not adequately treat these problems, given that their solutions are driven by mathematical simplicity rather than the actual forces at play in a real economy. However, it is still important to have an "open channel" with this tradition.
In ordinary market equilibrium theory, we start with traders which possess a basket of goods (usually represented as a vector) and a utility function (also represented as a vector). We then assign a price to every commodity (another vector), and each trader calculates their demand for each commodity at that price. When the total demand for each commodity is equal to the total supply, we reach "market clearing" prices. A Pareto optimal distribution of goods is one where the sum of utilities over all traders (the sum of the dot products of the commodity vector and the utility vector of every trader) is maximized. However, there may be non-optimal equilibria which are ones which are, in some sense, locally optimal for individual traders but not globally optimal. In the meeting, we discuss this in the context of an Edgeworth Box which is a model for a simple 2 player, 2 commodity economy. Any point in this box is an allocation of goods for both traders, and we draw curves to represent sets of allocations which are equal for a given trader (their indifference curve). We then show that optimal allocations are to be found when two indifference curves meet but do not cross.