A subjective supply–demand model: the maximum Boltzmann/Shannon entropysolution

Abstract

The present authors have put forward a projective geometry model of rational trading. Theexpected (mean) value of the time that is necessary to strike a deal and the profit stronglydepend on the strategies adopted. A frequent trader often prefers maximal profit intensityto the maximization of profit resulting from a separate transaction because the grossprofit/income is the adopted/recommended benchmark. To investigate activities that havedifferent periods of duration we define, following the queuing theory, the profit intensity asa measure of this economic category. The profit intensity in repeated trading has a uniqueproperty of attaining its maximum at a fixed point regardless of the shape ofdemand curves for a wide class of probability distributions of random reversetransactions (i.e. closing of the position). These conclusions remain valid for ananalogous model based on supply analysis. This type of market game is oftenconsidered in research aiming at finding an algorithm that maximizes profit of atrader who negotiates prices with the Rest of the World (a collective opponent),possessing a definite and objective supply profile. Such idealization neglects thesometimes important influence of an individual trader on the demand/supplyprofile of the Rest of the World and in extreme cases questions the very idea ofdemand/supply profile. Therefore we put forward a trading model in which thedemand/supply profile of the Rest of the World induces the (rational) trader to(subjectively) presume that he/she lacks (almost) all knowledge concerning themarket but his/her average frequency of trade. This point of view introducesmaximum entropy principles into the model and broadens the range of economicphenomena that can be perceived as a sort of thermodynamical system. As aconsequence, the profit intensity has a fixed point with an astonishing connection withFibonacci classical works and looking for the quickest algorithm for obtainingthe extremum of a convex function: the profit intensity reaches its maximumwhen the probability of transaction is given by the golden ratio rule . This condition sets a sharp criterion of validity of the model and can be tested with realmarket data.