Modelling intertemporal preferences through soliton dynamics: A new application of an integrable system to distinguish stationarity and dynamic consistency

The present manuscript intends to demonstrate how an intertemporal preference defined ‘today’ develops over time. The dynamics defining the development of intertemporal preference over time can be seen as a combination of dispersion and nonlinearity: dispersion represents a conservative redistribution of preference intensity across the contextual dimension, while nonlinearity captures all the cognitive, emotional and contextual influences that shape the decision-making process in a nonlinear way. For the purpose of describing these two characteristics of intertemporal preference trends, for the first time in the literature, the KdV equation is applied to examine the evolution of intertemporal preferences. The Painlevé approach is used to discuss the integrability of the considered equation. The soliton solution is derived using an Ansatz method. The process of making decisions at different points in time, or intertemporal decision-making, is shaped by a few factors, including uncertainty and behavioural dynamics. It is challenging for traditional models to fully capture observed behaviours, particularly in relation to concepts such as stationarity and dynamic coherence. The paper draws an original analogy between preferences and solitons. In particular, the difference between intertemporal stationarity and dynamic consistency is investigated using the solitons’ properties of nonlinearity and dispersion. After obtaining a soliton solution to represent the evolution of the intertemporal preference over time, a comparison of the same preference at different decision points is investigated, illustrating the distinction between profile shape preservation and dynamic coherence. The model provides a formal representation of how intertemporal preferences may evolve while preserving their shape and may generate dynamic inconsistency when the introduction of a new decision moment leads to a reversal of the induced ranking of feasible alternatives. We introduce logarithmic sensitivities of the soliton preference-intensity profile. The introduction of two logarithmic sensitivities – one with respect to evaluation time 𝑡 and the other with respect to the contextual coordinate 𝑥 – describes local changes in the preference-intensity profile.