Net value of waiting for a larger-later outcome judged today and one period ahead

Signature

M_0 = b * w_1 - a * w_0; M_1 = b * w_2 - a * w_1

Function

Hyperbolic discount weight function with impatience that declines with delay

Computational function

• Computational function for a smaller-sooner against larger-later plan check with commitment thresholds under a chosen discount form

  Takes the two amounts, a choice of discount form and its parameters, builds the weights at delays 0, 1 and 2, and returns the two margins of HE-FM-HYPD-005, whether the plan reverses, the penalty on the sooner outcome felt now that would tip today's choice, the reward paid with the later outcome that would do the same, and the first whole delay at which waiting is preferred. The penalty is minus M_0 and the reward minus M_0 divided by w_1, so their ratio is the timing premium 1/w_1 in step 4 of the article's example. The inputs therefore differ from the formula's variables, which take the weights as given.

  Inputs and outputs: a and b are the smaller-sooner and larger-later amounts, one period apart, in units of value. form is exp, hyp, gen or qh for the constant-rate factor delta^t, the simple hyperbola, the generalised hyperbola or the quasi-hyperbolic form. alpha, beta and delta are that form's parameters. w_1 and w_2 are the weights one and two periods ahead. M_0 and M_1 are the margins. P is the smallest penalty felt now and R the smallest reward paid with the later outcome that would make waiting win today, both zero when it already wins. s_star is the first delay at which waiting is preferred, or none within the search range.

  Assumption: The amounts are fixed, the weights are seen from today and the later outcome arrives one period after the sooner one. The calculator line below follows the quasi-hyperbolic path, which gives constant-rate weights when beta = 1; the R and Python code cover all four forms.

  Worked example (Article's quasi-hyperbolic smoker): Smoking wins today by 0.991, quitting wins for next month, and the penalty and reward thresholds are 0.991 and about 1.430, as in the article. a = 10; b = 13; beta = 0.7; delta = 0.99; w_1 = 0.693; w_2 = 0.68607; M_0 = -0.991; M_1 = 1.98891; P = 0.991; R = 1.430014. The first delay at which waiting wins is 1.

  Worked example (Article's simple hyperbolic smoker): With alpha = 0.4 the plan also reverses, and the reward needed is exactly 1 unit, 10 divided by 0.714286 minus 13 (computed here for illustration). The values come from the R and Python code, since the calculator line follows the quasi-hyperbolic path. a = 10; b = 13; alpha = 0.4; w_1 = 0.714286; w_2 = 0.555556; M_0 = -0.714286; M_1 = 0.079365; P = 0.714286; R = 1. The first delay at which waiting wins is 1.

  Worked example (Impatient exponential smoker with delta of 0.75): Waiting never wins at any delay, so no plan to quit is formed and there is nothing to reverse. a = 10; b = 13; beta = 1; delta = 0.75; w_1 = 0.75; w_2 = 0.5625; M_0 = -0.25; M_1 = -0.1875; P = 0.25; R = 0.333333.

  Worked example (Patient exponential smoker with delta of 0.99): Waiting wins today, so both thresholds are zero, a limiting case. a = 10; b = 13; beta = 1; delta = 0.99; w_1 = 0.99; w_2 = 0.9801; M_0 = 2.87; M_1 = 2.8413; P = 0; R = 0.

  Excel: =AND(LaterAmount*QHBeta*QHDelta-SoonerAmount<0,LaterAmount*QHBeta*QHDelta^2-SoonerAmount*QHBeta*QHDelta>0) Returns TRUE when a quasi-hyperbolic plan reverses; =MAX(SoonerAmount/(QHBeta*QHDelta)-LaterAmount,0) returns the reward threshold.

  R: plan_check <- function(a, b, form, alpha = 0, beta = 1, delta = 1, s_max = 120) { w <- switch(form, exp = function(t) delta^t, hyp = function(t) 1/(1+alpha*t), gen = function(t) (1+alpha*t)^(-beta/alpha), qh = function(t) ifelse(t == 0, 1, beta*delta^t)); m0 <- b*w(1)-a*w(0); m1 <- b*w(2)-a*w(1); s <- 0:s_max; ok <- which(b*w(s+1) > a*w(s)); list(w_1 = w(1), w_2 = w(2), M_0 = m0, M_1 = m1, reversal = m0 < 0 & m1 > 0, P = max(-m0, 0), R = max(-m0/w(1), 0), s_star = if (length(ok)) s[ok[1]] else NA) } Returns a named list.

  Python: def plan_check(a, b, form, alpha=0.0, beta=1.0, delta=1.0, s_max=120): w = {'exp': lambda t: delta**t, 'hyp': lambda t: 1/(1+alpha*t), 'gen': lambda t: (1+alpha*t)**(-beta/alpha) if t else 1.0, 'qh': lambda t: beta*delta**t if t else 1.0}[form]; m0 = b*w(1)-a*w(0); m1 = b*w(2)-a*w(1); s_star = next((s for s in range(s_max+1) if b*w(s+1) > a*w(s)), None); return w(1), w(2), m0, m1, m0 < 0 < m1, max(-m0, 0), max(-m0/w(1), 0), s_star Plain Python with no imports.

  Test (Reward threshold equals the penalty divided by the one-period weight): The timing premium 1/w_1 links the two thresholds, 1.443 for the article's quasi-hyperbolic smoker. The expected result is TRUE. Excel check =ABS((10/(0.7*0.99)-13)-(10-13*0.7*0.99)/(0.7*0.99))<1E-9

  Test (Constant-rate weights never reverse): For delta = 0.75 and 0.99 the two margins share a sign. The expected result is TRUE. Excel check =AND((13*0.75-10)*(13*0.75^2-10*0.75)>0,(13*0.99-10)*(13*0.99^2-10*0.99)>0)

  Common error (Reward sized as if paid now): Offering the 0.991 penalty threshold as a reward paid with next month's health gain leaves today's choice unchanged, since 13.991 times 0.693 is about 9.70, still below 10; a reward paid then needs to exceed about 1.430 (computed here for illustration).

  Source: Frederick S, Loewenstein G, O'Donoghue T. Time discounting and time preference: a critical review. Journal of Economic Literature. 2002;40(2) 351-401. Section 5.1, pages 366 to 368, on time-inconsistent preferences under non-exponential discounting, precommitment, and commitment devices for sophisticated and naive people. Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7) 745-758. Section on discount models, on a preference between two programmes reversing within a year with nothing changed.

  w_1 = beta * delta; w_2 = beta * delta^2; M_0 = b * w_1 - a; M_1 = b * w_2 - a * w_1; P = max(a - b * w_1, 0); R = max(a / w_1 - b, 0)

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Implementations

• Excel

  Preference reversal check from named weights and amounts

  With the amounts in SoonerAmount and LaterAmount and the weights in Weight0, Weight1 and Weight2, the first two formulas return M_0 and M_1 and the third returns TRUE when the plan reverses, with the two margins in cells named WaitNow and WaitNext.

  =LaterAmount*Weight1-SoonerAmount*Weight0; =LaterAmount*Weight2-SoonerAmount*Weight1; =AND(WaitNow<0,WaitNext>0)

Assumptions

• All reversal-check weights from one function seen from today

  The three weights come from the same discount function evaluated at the present, so M_1 is today's view of next period's choice. Next period the person re-evaluates with w_0 and w_1 again, which is where a naive plan unravels.

• Reversal-check amounts in common units and fixed over time

  The amounts a and b are in the same units and do not change between the two judgements, so any change in the ranking comes from the weights alone. Attema and colleagues give the example of preferring one programme now and the other a year later with nothing changed.

Worked examples

• Article's quasi-hyperbolic smoker plans to quit next month

  With weights 1, 0.693 and 0.68607 from beta = 0.7 and delta = 0.99, quitting loses by 0.991 today but wins by about 1.989 when judged for next month, the article's preference reversal (illustrative values).

  a = 10; b = 13; w_0 = 1; w_1 = 0.693; w_2 = 0.68607; M_0 = -0.991; M_1 = 1.9889

• Article's simple hyperbolic smoker plans to quit next month

  With weights 1, 0.714286 and 0.555556 from alpha = 0.4, quitting loses by about 0.714 today and wins by about 0.079 for next month, as in the article's table (9.29 against 10, and 7.22 against 7.14).

  a = 10; b = 13; w_0 = 1; w_1 = 0.714286; w_2 = 0.555556; M_0 = -0.7143; M_1 = 0.0794

• Article's impatient exponential smoker never plans to quit

  With weights 1, 0.75 and 0.5625 from delta = 0.75, waiting loses both today and for next month, so the steep constant discounter is impatient but consistent.

  a = 10; b = 13; w_0 = 1; w_1 = 0.75; w_2 = 0.5625; M_0 = -0.25; M_1 = -0.1875

Common errors

• Steep constant discounting mistaken for present bias

  An impatient exponential discounter with delta = 0.75 has M_0 = minus 0.25 and M_1 = minus 0.1875, so the person never plans to wait. A reversal needs a rate that declines with delay, not a high rate, which is the article's second misreading.

• Reversal taken as proof of hyperbolic preferences

  A plan that unravels fits hyperbolic weights but does not establish them. Frederick and colleagues report Read's argument that rates falling with horizon could reflect subadditive discounting, and in their own review the link between horizon and discount factor disappears once studies with horizons of a year or less are excluded.

Sources

• Time-inconsistent preferences under any non-exponential discount function

  Frederick S, Loewenstein G, O'Donoghue T. Time discounting and time preference: a critical review. Journal of Economic Literature. 2002;40(2) 351-401. Section 5.1, page 366, which reports that Strotz recognised that any discount function other than exponential gives time-inconsistent preferences and describes the strategies of precommitment and consistent planning, and section 4.1, page 361, on subadditive discounting and the horizon correlation of almost exactly zero after excluding horizons of one year or less.

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• Preference between two programmes reversing over time under hyperbolic discounting

  Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7) 745-758. Section on discount models, which gives the example that under hyperbolic discounting one could prefer one programme over another now but the second over the first in one year, with the options unchanged and no new information.

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Canonical Identity