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We discuss a possibility of deriving an H-theorem for nonlinear discrete timeevolution equation that describes random wealth exchanges. In such kineticmodels economical agents exchange wealth in pairwise collisions just asparticles in a gas exchange their energy. It appears useful to reformulate theproblem and represent the dynamics as a combination of two processes. The firstis a linear transformation of a two-particle distribution function during theact of exchange while the second one corresponds to new random pairing ofagents and plays a role of some kind of feedback control. This representationleads to a Clausius-type inequality which suggests a new interpretation of theexchange process as an irreversible relaxation due to a contact with areservoir of a special type. Only in some special cases when equilibriumdistribution is exactly a gamma distribution, this inequality results in theH-theorem with monotonically growing `entropy' functional which differs fromthe Boltzmann entropy by an additional term. But for arbitrary exchange rulethe evolution has some features of relaxation to a non-equilibrium steady stateand it is still unclear if any general H-theorem could exist.

Clausius inequality and H-theorems for some models of random wealth exchange