Cultural Neuroeconomics of Intertemporal Choice

C Appendix

Researchers with a professional background that is different from (statistical)
physics cannot easily spot the correspondence between the two basic equa-
tions that were crucial in the formulation of Tsallis’ entropy: The standard
Boltzmann-Gibbs-Shannon (BGS) entropy formula and the generalized nonex-
tensive entropy formula. It is therefore our intention here to mathematically
clarify this relationship. More specifically, it will be shown that the Tsallis’

1    ¹~V Pi

entropy S_q = —q—i ⁱ (q ∈ R) reduces to BGS entropy SBGS = — ɪɪiPi lnPi
as the nonextensive entropic index q approaches unity. The proof is based on
(Bernoulli-)L’HOpital’s rule.

The rule named after the French mathematician Guillaume de l’Hopital em-
ploys derivatives to calculate limits with indeterminate forms. In this sense,
using this rule, one can convert an indeterminate form (e.g. 0 or ∞) into a
determinate form with an easy computation of the limit.

When q → 1, the numerator of Sq tends to 1 — _i Pi = 0 ( _i Pi = 1). Since
the denominator also tends to 0, S_q has the indeterminate behavior 0 as q → 1.
Therefore, L'H^pital's Rule can be applied to the limit lim_q→₁ S_q:

where ' indicates the derivative with respect to q, i.e. ' = dqq. Since we can
differentiate term by term, we obtain

б—∑ ι>q∖ '=(i)'—∑ (pq )'=

and

(q — 1)' =(q)' — (1)' = 1.

We use above the following differentiation rules: (1)' = 0, (q)' = 1, and the rule
for differentiating the exponential function a^q (the base a is a constant and the
variable q is in the exponent), which reads (a^q)' = a^qlna (in our case a = pi).
Therefore, from (1) we get

lim Sq = lim  ∑^ipi lnpi = — ∑pi lnpi.

q→1      q→1      1

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