  We introduce a decentralized mechanism for pricing and exchanging
alternatives constrained by transaction costs. We characterize the
time-invariant solutions of a heat equation involving a (weighted) Tarski
Laplacian operator, defined for max-plus matrix-weighted graphs, as approximate
equilibria of the trading system. We study algebraic properties of the solution
sets as well as convergence behavior of the dynamical system. We apply these
tools to the "economic problem" of allocating scarce resources among competing
uses. Our theory suggests differences in competitive equilibrium, bargaining,
or cost-benefit analysis, depending on the context, are largely due to
differences in the way that transaction costs are incorporated into the
decision-making process. We present numerical simulations of the
synchronization algorithm (RRAggU), demonstrating our theoretical findings.
