Tit-for-Tat Dynamics and Market Volatility

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 neighbors in fractions proportional to how much they helped player $i$'s production in the last round. Our contribution is to characterize the asymptotic behavior 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 behavior 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.