How would you like your gain in life expectancy to be provided? An experimental approach

Abstract

This paper reports the results of an empirical study investigating people’s preferences over three different types of perturbation to their survival function, each perturbation generating the same gain in life expectancy. Preferences over the three different perturbations were found to be distributed more or less evenly across the subject pool. Use of a novel experimental methodology generated economically consistent and intuitively plausible responses to (necessarily) hypothetical questions concerning improvements in life expectancy by first allowing respondents to gain experience while making similar choices in an incentivized setting involving financial risk. The results demonstrate the potential for economic experiments to contribute to the development of more robust methods for policy evaluation in domains where physical risk is an important factor.

Appendices

Appendix A

1.1 Assumption required for the calculations of life expectancy gains

Data stems from Government Actuary’s Department (mean 2003–2005) and reflects a simple average between female and male. The simulations are carried out from the age of 40 until the age of 90.

The hazard rates are calculated the following way: $\frac{{{l_{50}} - {l_{40}}}}{{{l_{40}}}} = {p_{40}}$

l_x is the number of survivors to exact age x of 100.000 live births who are assumed to be subject throughout their lives to the mortality rates experiences in the 3 year period to which the data relates.

Accordingly, in the money experiment, 1000p₄₀ is the number of green counters in the first bag and the number of white counters is 1000(1-p₄₀). In the money experiment this is the appropriate starting point for an estimation of E (tokens) since you either “survive” a whole bag or “die”. However, the life expectancy section should technically be continuous since there is both a probability of dying at the beginning of the decade, during the decade, and at the end of the decade. By assuming that the respondent needs to survive to the end of the decade, we have overestimated the probability of dying slightly and hence underestimated the current life expectancy.

Accordingly, it would have been more accurate to apply the following approximation instead of Eq. 6:

$$ {\hbox{L}}{{\hbox{E}}_{{4}0}} = 0.{5} * {1}0(({1} - {{\hbox{p}}_{{4}0}}) + ({1} - {{\hbox{p}}_{{4}0}})({1} - {{\hbox{p}}_{{5}0}}) + ({1} - {{\hbox{p}}_{{4}0}})({1} - {{\hbox{p}}_{{5}0}})({1} - {{\hbox{p}}_{{6}0}})) + { }0.{5} * {1}0({1} + ({1} - {{\hbox{p}}_{{4}0}}) + ({1} - {{\hbox{p}}_{{4}0}})({1} - {{\hbox{p}}_{{5}0}}) + ({1} - {{\hbox{p}}_{{4}0}})({1} - {{\hbox{p}}_{{5}0}})({1} - {{\hbox{p}}_{{6}0}})) $$

This would approximate the continuous curve more accurately. However, since the main purpose of this experiment is to show that a method works, it was decided to mirror the money game exactly in the life expectancy section. Also, since we wish to examine whether preferences shift across contexts, we must keep all other things constant.

Appendix B

2.1 Money experiment

The experiment applied was built around four different incentivised games—three option games and one indifference game—the purpose of which are described below. An additional baseline game was included which served as a practice game. Each game potentially contained five rounds and each round of a game consisted of a bag containing a total of 1,000 counters (a mix of green and white cards) where the specific mix reflected ‘game specific’ hazard rates for the next five decades. Each participant drew a counter from a bag. The counter was subsequently placed back into the bag before the next participant drew a counter.¹⁴

In the option games of the money experiment, participants were asked to make pair-wise comparisons of three games. All games (i.e. sets of bags) reflect a changed distribution of hazard rates according to the changes described in Table 1; however, in the money game they were labelled A, B and C instead of X, Y and Z. Following a choice of which ‘game’ (i.e. set of bags) to play, the participants either ‘survived’ a bag (drew a white counter), thereby collecting a token and able to continue in the game; arrived at the end of the game; or were eliminated prematurely (drew a green counter). Furthermore, it was explained to respondents that drawing a white counter has two beneficial effects: 1) by ‘surviving’ that bag they have obtained a token to enter the draw, and 2) they are in a position to progress to the next bag(s) and hence have the chance of winning additional tokens. Changing the distribution of the counters changes both of these effects and the two effects mirror the safety and survivor effects introduced to the respondents in the subsequent LE experiment.

The tokens collected in the games were entered into a draw for a prize¹⁵ which was adjusted on a per-session basis to make the expected pay-off across different group sizes equal. Each token was equivalent to one entry into the draw and the winner of the prize was determined by a random draw at the end of each session. A brief outline of the format for the money experiment is presented in Table 6. Notice though the expected number of tokens in the baseline game is 3.46, which is equivalent to 3.46 decades in the life expectancy survey, the current life expectancy for a 40-year-old British individual (truncated at the age of 90).

Table 6 Format for the money experiment

Appendix C

3.1 Extract from the experimental protocol

• Let us say that you were offered the opportunity to choose between three on-going programmes, X, Y and Z. All would cost more or less the same. All programmes would increase the average life expectancy by 6 months. Although in each case this would be achieved differently. By this I mean that the risks of dying in each decade would be different in each programme.

3.1.1 Handout 2

Handout 2 shows you your current risk. Programme X,Y and Z would change these in different ways.

• X VS Z

• i) We will now ask you to choose between programme X and programme Z. Let us now have a look at the two programs in details.

  • OVERHEAD

    • On the top of this overhead you see the risk of dying in each decade as they are before we make any changes. Underneath is programme X indicating the change in the risk of dying in each decade offered by programme X. In programme X we have reduced the risk of dying with 14/1,000 in the first decade. This gives us a new risk of dying in the first decade of 6/1,000. The risk of dying in the second decade is still 48/1,000, 121/1,000 in the third, 347/1,000 in the fourth and 652/1,000 in the fifth.

    • Underneath is programme Z indicating the change in risk of dying in each decade offered by programme Z. In programme Z we have changed the risk of dying differently in each decade. We have reduced the risk in each decade, starting with 1/1,000 in the first decade which gives a new risk of dying equal to 19/1,000. We reduce the risk of dying with 2/1,000 in the second period to 46/1,000. In the third decade we reduce the risk of dying with 5/1,000 to 116/1,000. In the fourth decade we reduce the risk by 15/1,000 to 332/1,000 and in the fifth decade with 28/1,000 to 624/1,000.

• ii) The change in life expectancy is 6 months in each programme. Note that the change in life expectancy is the same in both programmes even though we don’t have the same total risk reduction i.e. 14/1,000 in X vs 51/1,000 in Z.

  This is because you get a lot of survivor effect from reducing the risk in decade 1, since you have many periods ahead of you. On the other hand; reducing the risk in the fifth decade doesn’t give you any survivor effect. In order to get the same change in life expectancy from X and Z you need a bigger safety effect in later decades in Z. This mirrors the situation in the finance game where we changed different number of green cards in the different options but ended up with the same total effect as in the expected number of tokens.

• iii) Please take your time and think about what programme X and programme Z would mean to you. Please indicate whether you prefer X or Z in Q5 in the response sheet. If you would be equally happy with either i.e. you wouldn’t mind letting the policy makers choose which programme to implement; please tick the ‘equally happy’ box.

  After you have made your choice I will ask you to answer Q6 in your response sheet.

Appendix D

4.1 Sample demographics

Description	Variable name	N	Mean
Proportion of males	Male	130	0.43
Age (in years)	Age	130	41
Below 40	Below age 40 (=1 if below the age of 40, else 0)	130	0.55
Education (1=primary, 2= secondary, 3= higher)	Higher education (=1 if higher education, else 0)	130	2.5
Health (5 categories. The first question in the Short Form 36 (SF- 36) health survey 16 )	Good (=1 if health status is the respondent’s self-reported health is good or excellent)	130	3.8
Proportion with children	children	130	0.67
Proportion with an additional health insurance	insurance	130	0.28
Mean individual income per month (GBP) (8 response categories)	Individual income	129	1,800
Mean household income per month (GBP) (8 response categories)	Household income	124	2,850

Appendix E

5.1 Results of the three different comparison questions in the experiment

	LE	Money	Signed rank test ( p -value)
Comparison no 1 (X vs. Z)
X	43(13)	26	−2.6(0.01)**
Z	63(45)	78	2.1(0.04)**
Indifferent	12(2)	14	0.42(0.67)
Comparison no 2 (Y vs. Z)
Y	65(29)	47	−2.4(0.01)**
Z	43(27)	64	2.9(0.004)***
Indifferent	10(1)	7	−0.78(0.44)
Comparison no 3 (X vs. Y)
X	37(11)	27	−1.5(0.12)
Y	67(50)	82	2.1(0.03)**
Indifferent	14(2)	9	−1.1(0.25)

*,**,*** Significant at 0.1, 0.05 and 0.01 levels, respectively