Do It Today Or Do It Tomorrow: Empirical Non-Exponential Discounting Explained by Symmetry and Fuzzy Ideas

At first glance, it seems to make sense to conclude that when a 1 dollar reward tomorrow is equivalent to a D \u3c 1 dollar reward today, the day-after-tomorrow\u27s 1 dollar reward would be equivalent to D * D = D2 dollars today, and, in general, a reward after time t is equivalent to D(t) = Dt dollars today. This exponential discounting function D(t) was indeed proposed by the economists, but it does not reflect the actual human behavior. Indeed, according to this formula, the effect of distant future events is negligible, and thus, it would be reasonable for a person to take on huge loans or get engaged in unhealthy behavior even when the long-term consequences will be disastrous. In real life, few people behave like that, since the actual empirical discounting function is different: it is hyperbolic D(t) = 1 / (1 + k * t). In this paper, we use symmetry and fuzzy ideas to explain this empirical phenomenon