Tit-for-tat dynamics and market volatility

Abstract

We consider tit-for-tat dynamics in production markets, where there is a set of n players connected via a weighted graph. Each player i can produce an eponymous good using its linear production function, given as input various amounts of goods in the system. In the tit-for-tat dynamic, each player i shares its good with its neighbours in fractions proportional to how much they helped player i’s production in the last round. Our contribution is to characterize the asymptotic behaviour of the dynamic as a function of the graph structure, finding that the fortune of a player grows in the long term if and only if the player has a good self loop (i.e. the player works well alone) or works well with at least one other player. We also consider a generalized damped update, where the players may update their strategies with different speeds, and obtain a lower bound on their rate of growth by identifying a function that gives insight into the behaviour of the dynamical system. The model can capture circular economies, where players use each other’s products, and organizational partnerships, where fostering long-term growth of an organization hinges on creating relationships in which reciprocal exchanges between the agents in the organization are paramount.