This paper combines two psychological posits, through Leibniz’s Principle of sufficient reason, to provide a probabilistic theory for intertemporal indifference. Leibniz’s principle, often associated with ex nihilo nihil fit¹, has been influential in the thinking of several philosophers and was recognised as one of the four laws of thought in the 18^th century. The principle states that the occurrence/existence/truthfulness of an event/entity/ statement implies the occurrence/existence/truthfulness of a sufficient explanation. In this paper, we will repeatedly make use of this principle, especially in condensing two psychological theories: the theory that we generally prefer to consume now than to consume in the future and the theory that our perception of time does not need to coincide with the “hourglass” passage of time. Through Leibniz’s principle, we aim to achieve face validation of a probabilistic theory for intertemporal indifference.

Patterns of intertemporal indifferences have captured the interest of psychologists and economists for several decades. Procrastination, addiction and willingness to save are a few observed behaviours involving time trade- offs [1] . It is generally observed that animals or human beings weigh the future with a discount weight, , compared to the present where the weight attached to a consumption now is exactly 1. A series of studies suggest that cognitive considerations might explain such empirical observations. For example, the primate, Saguinus oedipus, commonly referred to as the cotton-top tamarin, and common marmosets, Callithrix jacchus, are known to differ in their respective levels of temporal discounting [2] . Among the human species, age, health states, education level, to name a few, are endogeneous determinants of time preferences [3] .

A very general discount weight function is of the form, where is the time-perception function and is the time-preference rate function. Several authors have sought the proper functional form by eliciting either or or both. Albrecht and Weber [4] , for example, propose that individuals have a non-linear time preception function. The weight, , they proposed is given by and is the non-linear time perception function indicating how fast time is perceived to pass in an individual’s mind. Other authors assumed a linearly declining discount rate function, [5] or a discontinuous hyperbolic discount weight function [6] [7] or an implicit function for the weight such as Loewenstein and

Prelec’s [8] .

Empirical investigations on time preferences have, however, rarely converged [9] . Consequently, rather than claiming a certain functional form for time preference, in this paper, our main assumption shall be that

• An individual is expected to value a future quantity compared to a current quantity, q⁰, by placing a discount weight, , on the future quantity, i.e.

With an expectation in place, we, next, seek to identify the source of variability. Among the different senses of the human being, time perception is a sense that is increasingly gaining the interest of psychologists. A key issue in modern neuroscience is the association that an individual’s perception bears with her neural events [10] . Several studies have found that the speed of the internal clock is linked to the concentration of dopamine in the basal ganglia (see for example Meck [11] ). The sense of time is often argued to be a consequence of the dopaminergic state, influencing the sensory signal-to-noise ratio due to an effect of dopamine at the cellular level [12] [13] . As Einstein described relativity with the faster passage of time when one sits in the company of a beautiful woman, it would also not be implausible that, everything else being constant, cognitive processes allow the “present time” to be viewed as a random variable.

Strictly speaking, the present is an infinitely small-time interval separating the past and the future. However, it is argued that the perceived present, or the so-called specious present, is an interval which needs not be infinitely small—a concept first observed in 1882 [14] . Furthermore, since the passage of time allows any future time interval to be constructed by a sequence of the individual’s specious presents, then we have a time-inhomoge- neous stochastic process similar to Parouty [15] . So, similar to Samuelson [16] who condensed the disparate psychological motives towards time preference into the discount rate [9] , the source of variability that we identified effectively condenses the disparate psychological factors adding variability across an individual’s intertemporal indifference into a mathematically tractable parameter; i.e. into the variability of the specious present. Then, we can analyse the sentence: “In general, individuals prefer to consume now than to consume in the future” with focus on the word “in general” and reformulate the latter sentence into a more statistically interpretable form.

The assumption that an individual’s perception of a future interval is a sequence of random specious presents coupled with the assumption that the individual is expected to place a weight on future consumption are sufficient assumptions to formulate a probabilistic distribution for intertemporal indifference—a direct consequence of Leibniz’s principle. Given that perceived passage of time is not always equal to hour-glass passage of time and that individuals generally prefer to consume now than in the future, then from Leibniz’s Principle of sufficient reason, the existence of an intertemporal indifference probability distribution is implied.

The assignment of probability measures is done by deriving a maximum entropy distribution for the probability that an individual/animal is indifferent between some quantity at time and a given quantity now. First, we write our expectation in a more structured language, so that, given an initial amount, , now, the expected indifference amount at time t is

With Equation (1) being the unconditional expectation of, the Radon-Nykodym theorem asserts the existence of a conditional infinitesimal expectation [17] . Defining, the conditional infinitesimal expectation is given by

Next, it is important to note that an individual or animal will only express a time preference provided she perceives time to pass (or provided the interval of time that she perceives is bigger than her specious present). Since the specious presents are random time durations, we can assume that a given fixed indifference amount at a specific time point has a random duration being equal to the individual’s specious present at that time. Furthermore, at the instant that the next specious present begins, the indifference amount increases continuously with certainty. Thus, we ensure that an individual’s intertemporal indifference path is a continuous function of time. As a result the process adheres to the typical characterisations of diffusion processes; usually restricted by a jump constraint on their probability measure,. To be more specific, for a positive sufficiently small,

Equation (3) ensures that the indifference path is time-continuous. For example, an individual being given an amount q⁰ now will always be indifferent to the same amount, q⁰, in any future interval which lies in her specious present. It is at the instant the new specious present begins that the individual’s indifference path exits the state q⁰ moving, at all times, along a cádlág path².

An important assumption that we made is that we have assumed that the animal/individual lives forever implying that a choice can always be made. In case of investigations involving mortality, the latter can be rectified by taking into account mortality tables or parametric mortality laws and conditioning the probability of a choice on the probability that the choice exists. That specification is beyond this paper since it is more useful for us to assume that the individual lives forever as the latter assumption appears fairly well-founded for society. It is thus necessary that we discuss the assumptions under which the distribution is valid for society.

We will first assume that society lives forever⁸ and second, we will assume that society possesses a societal sense of time as with the “average” individuals comprising it. Individuals with Parkinson’s disease, attention deficit hyperactivity disorder (ADHD) and schizophrenia have a marked difference in time perception compared to an “average” individual [20] -[22] .

Consequently, even the “average” individual is bound to experience a marked alteration in time perception if drugs, such as psilocybin [23] , increasing the function of dopamine⁹, become common usage. Although the latter appears farfetched, an economic scenario entailing a general alteration in the “average” individual’s time perception appears reasonably realistic. Neurobiologists and psychologists often make the following associations: opioid = pleasure, dopamine = happiness, serotonin deficit = depression, oxytocin = love, nucleus accumbens = reward or amygdala = fear, etcetera [24] . Economists, on the other hand, would coin a bullish or a bearish sentiment, reflecting a positive or a negative feeling about market trends. A society with a bull market, a market with an upward trend, for example, might experience time to pass faster than one with a market with a downward trend—a bear market, say¹⁰.

As such, we assume some ethical¹¹ weights among the preferences of individuals comprising a temporal society so that we can assume that society, as a whole, also experiences the passage of time as a sequence of societal specious presents. Then a society will be indifferent to a single fixed quantity for the duration of its specious present, before feeling the need for some extra amounts, with a social rate of time preference, srtp, when it perceives the present to have become past. Moreover, we can also assume non-subjective rates.

Although, our considerations, thus far, have only concerned a subjective rate of time preference, in the case where rates are not subjective, such as the rate on government bonds, without loss of generality, the probability of intertemporal indifference for a society with an external social discount rate, SDR, can be equivalently seen to follow the same distributional laws as given in Equation (8) since the main source of variability is assumed to be in the perception of time. To be specific, the expectation is taken as given and it is only the occurrence times of events, identified by zeroes or ones, which are random and assigned probability measures through surprisal maximization. So, society’s probability of intertemporal indifference, can be represented by Equation (17).