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. This dynamic has been studied before in exchange markets by Wu and Zhang. We analyze the long term behavior of the dynamic and characterize which players grow in the long term as a function of the graph structure. At a high level, we find that a player grows in the long term if and only if it has a good self loop (i.e. is productive 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 finding a function that gives insight into the behavior of the dynamical system.