Online Appendix

to

HETEROGENEITY IN RISK PREFERENCES:EVIDENCE FROM A REAL-WORLD BETTING MARKET

Angie Andrikogiannopoulou Filippos PapakonstantinouLondon School of Economics Imperial College London

August 2016

C.1 Betting slip and lottery calculation

All of the bets available in the sportsbook are combinations of various types of accumulators, where an

accumulator of type k ≥ 1 is a bet that involves k individual wagers and pays money only if all k wagers

win.1 An accumulator of type k can be represented as a lottery with two possible prizes: a positive prize

that equals the stake times the product of the gross returns associated with each of the k selections minus

one, and a negative prize with absolute value equal to the stake.2 Since only unrelated selections can be

combined into accumulators of type k > 1, the probability of the positive prize in such a bet is simply the

product of the probabilities associated with each of the k selections involved.

A bet involving N ≥ 1 individual wagers may belong to one of two broad categories of bet types: it could

be a) a permutation bet, which includes all accumulators of type k that could be constructed from the N

wagers, for some k ≤ N , i.e., it includes(N

k

)accumulators of type k, or b) a full-cover bet, which includes all

accumulators of type k that could be constructed from these N wagers, for all 1 ≤ k ≤ N or 2 ≤ k ≤ N , i.e.,

it includes∑

1≤k≤N

(Nk

)or∑

2≤k≤N

(Nk

)accumulators of type 1 or 2 through N . The amount staked on permu-

tation and full-cover bets is divided evenly among the number of accumulator bets they involve. For example,

a bet that stakes amount S and involves N = 3 wagers could belong to one of the following five bet types:

• a singles bet that stakes amount S3 on each of the

(31

)= 3 accumulators of type 1, each of which

corresponds to one of the wagers and pays money if that wager wins,

• a doubles bet that stakes amount S3 on each of the

(32

)= 3 accumulators of type 2, each of which corre-

sponds to one of the pairwise combinations of wagers and pays money if both wagers in the combination

win,

• a treble bet that stakes amount S on the(3

3

)= 1 accumulator of type 3, which pays money if all three

wagers win,

• a trixie bet which stakes amount S4 on each of the aforementioned double bets (3 accumulators of type

2) and the aforementioned treble bet (1 accumulator of type 3), or

• a patent bet that stakes amount S7 on each of the aforementioned single bets (3 accumulators of type 1) and

each of the bets in the aforementioned trixie bet (3 accumulators of type 2 and 1 accumulator of type 3).

Note that the singles, doubles, and treble bets belong to the category of permutation bet types, while the

1The simplest bet type is an accumulator of type 1. Accumulators of type 1, 2, and 3 are usually called single, double, andtreble bets, respectively.

2For example, a bet that combines a wager to back Barcelona to draw Real Madrid at odds 4.00 with a wager to back Federer tobeat Hartfield in the first set at odds 1.60 is an accumulator bet of type 2. This accumulator would have a gross (net) payoff equalto the amount of money staked times 4.00× 1.60 = 6.40 (6.40− 1 = 5.40) if both wagers won, and a gross (net) payoff equal to0 (minus the stake) otherwise.

C-1

trixie and patent bets belong to the category of full-cover bet types.

Lottery representations of permutation and full-cover bets can be constructed by combining the lottery

representations of the elemental accumulator bets. Several bet calculators are available on the web that help

bettors compute the prizes and probabilities of their chosen bets following the rules described above.

In Figure C.1, we present a sample betting slip that individuals submit to an online sportsbook. In

Figure C.2, we compute all possible prizes and corresponding probabilities of the lotteries associated with

each of the three bets contained in the betting slip in Figure C.1. While not shown on the betting slip, the

individual placing these bets observes the odds for all possible outcomes of each event, which he can use

to calculate the implied win probability of each outcome. For example, in the event “USA vs. Germany

(FIBA Worlds)”, the quoted odds for Germany to win are 1.33 and for USA to win are 2.80, so the implied

win probability for the wager backing Germany is 1/1.331/1.33+1/2.80

≈ 0.68. The probabilities written in red color

beside the branches of the lottery representations in Figure C.2 have been calculated in this manner.

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Event Sport Selection Bet Type Odds Total StakeUSA vs Germany (FIBA Worlds)Match Winner Basketball Germany  Single 1.33 50Arsenal vs Chelsea (EPL)                          Total Goals Over/Under 2.5 Soccer Over 2.5  Doubles 2.25NY Yankees vs Toronto Blue Jays (MLB) Total Match Runs Odd/Even Baseball Odd  Doubles 1.67Taylor vs Baxter (World GP, 1st round) Match Winner Darts Baxter  Doubles 1.10 30Barcelona vs Real Madrid (LFP)Match Winner Soccer Draw  Double 4.00Federer vs Hartfield (Australian Open)1st set winner Tennis Federer  Double 1.60 2

Figure C.1: Sample betting slip.

Single:

50 · (1.33− 1) = 16.5

-50

0.68

0.32

Double:

2 · (4.00 · 1.60− 1) = 10.8

-2

0.13

0.87

Doubles:

303 · (2.25 · 1.67− 1) = 27.58

303 · (2.25 · 1.10− 1) = 14.75

303 · (1.67 · 1.10− 1) = 8.37

27.58+ 14.75+ 8.37 = 50.7

-30

0.04

0.15

0.18

0.26

0.37

Figure C.2: Lottery representations of the bets contained in the betting slip in Figure C.1.

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C.2 Probability weighting function

Here, we discuss the effects of the curvature parameter γ and the elevation parameter κ on the probability

weighting function suggested by Lattimore, Baker and Witte (1992) and on its derivative. For this purpose,

in Figure C.3 we plot the probability weighting function (in Panels a and c) and its derivative (in Panels

b and d) for several values of γ and κ . The probability weighting function transforms the cumulative (over

losses) or the decumulative (over gains) distribution function. Its derivative is the weighting factor applied

to the density — for continuous distributions — or, roughly, the probability per unit interval — for discrete

distributions; the plots of the derivative in Panels (b) and (d) show, for gains, how this weighting factor

depends on the location of the outcome in the distribution.

Letting κ = 1, for γ < 1 the probability weighting function has an inverse-S shape (see Panel a), which

corresponds to an overweighting of outcomes in the tails of the distribution (see Panel b). The lower the

value of γ , the more pronounced is this shape. For γ > 1 (not shown), the function becomes S-shaped,

which corresponds to an underweighting of outcomes in the tails. Letting γ = 1, for κ > 1 the probability

weighting function is concave (see Panel c), which corresponds to an overweighting of the right (left) tail of

the distribution of positive (negative) outcomes (see Panel d). As κ gets higher, the concavity of the function

increases. For κ < 1 (not shown), the function becomes convex, which corresponds to an underweighting

of the right (left) tail of the distribution of positive (negative) outcomes.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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1

p

w(p)

γ = 1.0, κ = 1.0γ = 0.8, κ = 1.0γ = 0.5, κ = 1.0

(a) Probability weighting for different values of γ .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

F

w′(F

)

γ = 1.0, κ = 1.0γ = 0.8, κ = 1.0γ = 0.5, κ = 1.0

(b) Density weighting for different values of γ .

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p

w(p)

γ = 1.0, κ = 1.0γ = 1.0, κ = 1.5γ = 1.0, κ = 2.0

(c) Probability weighting for different values of κ .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

F

w′(F

)

γ = 1.0, κ = 1.0γ = 1.0, κ = 1.5γ = 1.0, κ = 2.0

(d) Density weighting for different values of κ .

Figure C.3: The Lattimore, Baker and Witte (1992) probability weighting function (in Panels a and c)and its derivative (in Panels b and d) for several values of the curvature parameter γ and the elevationparameter κ . In Panels (a) and (b), we present plots for κ = 1 and several values of γ ≤ 1, and in Panels(c) and (d) we present plots for γ = 1 and several values of κ ≥ 1.

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C.3 Directed acyclic graphs of the model

In this section, we present Directed Acyclic Graph (DAG) representations of our model. In these graphs,

squares represent quantities that are fixed or observed, e.g., prior parameters and data, and circles represent

unknown model parameters that need to be estimated.

In Figure C.4, we present the DAG representation of the model in its basic form (Panel a) and in its

hierarchical form (Panel b).

hq

en

μ V

ρn

π V

π n

x ,ntynt

ρ~~

τn

μ τ~

,

π~ τ~ ρ~q q

q

(a) Graph representation of the model.

h

hq

en

μ V

ρn

π V

π n

x ,ntynt

ρ~~

τn

μ τ~

,

π~ τ~ ρ~

κ Κ λ Λ κ Κ λ Λ

q

qq

(b) Graph representation of the hierarchical model.

Figure C.4: Directed acyclic graph representation of the model. Squares represent quantities that are fixedor observed, e.g., prior parameters and data, and circles represent unknown model parameters that needto be estimated.

The only difference between the two figures is that the latter takes as given the prior distributions for the

population parameters. This is necessary because classical estimation would be intractable, hence we use

Bayesian estimation which requires the specification of priors.

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C.4 Additional results on utility types

This section presents additional results from the estimation of the utility types, which supplement those

presented in Section 5.1 of the paper. First, in Figure C.5 we plot histograms of the individual-level posterior

mean of the probability of belonging to each utility type: the type that exhibits neither probability weighting

nor loss aversion, the type that exhibits probability weighting but no loss aversion, the type that exhibits

loss aversion but no probability weighting, and the unrestricted type that exhibits both loss aversion and

probability weighting. Second, in Table C.1, we present results on the posterior means and variances of

the model parameters, separately for each utility type.

0 0.5 10

100

200

300

400λ = γ = κ = 1 Type

0 0.5 10

50

100

150λ = 1 Type

0 0.5 10

100

200

300

400γ = κ = 1 Type

0 0.5 10

50

100

150Unrestricted Type

Figure C.5: Histograms of individuals’ posterior probability of belonging to the 4 types.

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Table C.1: Posterior Estimates of Population Mean and Variance — 4 Types

Summary statistics of the posterior estimates for the population mean and variance of the elements of βn for the four utility types estimatedusing our mixture model. Note that π and τ are common parameters across all types, hence their means and variances take the same valuesfor all types.

Panel A : λ = γ = κ = 1 Type Panel B : λ = 1 Type Panel C : γ = κ = 1 Type Panel D : Unrestricted Type

Mean Median Std.Dev. Mean Median Std.Dev. Mean Median Std.Dev. Mean Median Std.Dev.

Means

π 0.25 0.25 0.01 0.25 0.25 0.01 0.25 0.25 0.01 0.25 0.25 0.01τ 1.93 1.93 0.10 1.93 1.93 0.10 1.93 1.93 0.10 1.93 1.93 0.10α 0.88 0.69 0.88 0.64 0.64 0.02 0.29 0.29 0.06 0.67 0.67 0.01λ 1 1 0 1 1 0 3.73 3.44 1.32 1.68 1.67 0.08γ 1 1 0 0.88 0.88 0.03 1 1 0 0.81 0.81 0.01κ 1 1 0 1.91 1.91 0.10 1 1 0 3.19 3.18 0.21

Variances

π 0.05 0.05 0.00 0.05 0.05 0.00 0.05 0.05 0.00 0.05 0.05 0.00τ 2.12 2.11 0.26 2.12 2.11 0.26 2.12 2.11 0.26 2.12 2.11 0.26α 0.01 0.00 0.13 0.01 0.01 0.00 0.00 0.00 0.00 0.02 0.02 0.00λ 0 0 0 0 0 0 0.35 0.32 0.17 0.46 0.46 0.08γ 0 0 0 0.01 0.01 0.00 0 0 0 0.02 0.02 0.00κ 0 0 0 0.17 0.16 0.06 0 0 0 3.58 3.55 0.54

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C.5 Trace plots of MCMC draws

As explained in Section 4 of the paper, to estimate our model we derive the joint posterior distribution of

the model parameters conditional on the data. Since this joint posterior cannot be calculated analytically,

we obtain information about it by drawing from it using a Markov chain Monte Carlo (MCMC) algorithm.

Section 4 of the paper and Section B of the paper’s appendix provide details about our MCMC.

Using this algorithm, we make 5 million draws from which we discard the first 10% as burn-in and

retain every 50th after that to mitigate serial correlation. These draws form a Markov chain with stationary

distribution equal to the joint posterior. In Figure C.6, we present trace plots — i.e., plots of the retained draws

against the iteration number — of the proportions of individuals that belong to each behavioral type. In Figure

C.7, we present trace plots of the population means and variances of the model parameters; for convenience of

presentation, we present these trace plots for the case in which heterogeneity in the (transformed) parameters

is simply modeled through a single normal distribution, rather than for the case in which individuals are

allocated to four different utility types. Both these sets of plots indicate no problems with convergence.

0

0.2

0.4

0.6

0.8

1

Figure C.6: Trace plots of the MCMC draws for the population proportions of individuals belonging toi) the type that exhibits no loss aversion and no probability weighting (at the bottom, using black dots),ii) the type that exhibits no loss aversion (second from the top, using red dots), iii) the type that exhibitsno probability weighting (second from the bottom, using green dots), and iv) the type that exhibits bothloss aversion and probability weighting (at the top, using blue dots).

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−1.8

−1.2

µπ̃

−2

−1.4

µτ̃

−0.75

−0.6

µα̃

−2

−1.8

µλ̃

−0.4

−0.25

µγ̃−3.6

−3.2

µκ̃

2

3

Vπ̃

0.8

1.6

Vτ̃

0.06

0.18

Vα̃

0.15

0.25

Vλ̃

0.04

0.1

Vγ̃

0.6

1.2

Vκ̃

Figure C.7: Trace plots of the MCMC draws for the population mean and variance of the elements of β̃n .

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C.6 Preferences and demographics

In this section, we examine the relationship between individuals’ observed demographic characteristics and

their estimated CPT parameters. In particular, in Table C.2 we present the results of Ordinary Least Squares

(OLS) regressions of the posterior mean estimates for the individual-level CPT preference parameters (α,

λ, γ , and κ) on individuals’ demographic characteristics (gender, age, level of education, income, and area

of residence).

First, we note that very little of the heterogeneity in preferences is explained by socio-demographic

characteristics, as the R2s of our regressions are quite small, ranging from 1% to 5%. Furthermore, we find

few statistically significant relations: i) women exhibit lower diminishing sensitivity to gains/losses (i.e.,

they have higher α), ii) older people exhibit a slightly higher degree of loss aversion λ and slightly stronger

diminishing sensitivity to deviations from certainty and impossibility (they have lower γ and higher κ),

and iii) individuals with higher income are less loss averse. To be specific, i) α is 0.05 higher for a woman

than for a man; ii) for every 10 additional years of age, λ increases by 0.10, γ decreases by 0.02, and κ

increases by 0.14; and iii) for every AC10,000 extra in annual income, λ decreases by 0.15.

While there is not much comparable evidence from the field, several experimental studies have estimated

the relation between demographics and CPT parameters. The results from these studies are quite mixed,

with some being consistent with ours and some not. For example, unlike us, Donkers, Melenberg and van

Soest (2001) find a smaller α and von Gaudecker, van Soest and Wengstrom (2011) find a higher λ for

women. On the other hand, consistent with what we find, Gachter, Johnson and Herrmann (2007) find that λ

increases with age, and Booij, van Praag and van de Kuilen (2010) find that λ decreases with higher income

and furthermore that γ decreases with age. For a more detailed review of the relevant evidence, see Booij,

van Praag and van de Kuilen (2010).

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Table C.2: Demographics and Estimated Preference Parameters

Ordinary Least Squares (OLS) estimates from regressing the posterior means of the individual-level CPTpreference parameters α (Panel A), λ (Panel B), γ (Panel C), and κ (Panel D) on individuals’ demographiccharacteristics. The explanatory variables include gender and rural/urban dummies, age, income, education,and a constant. Age is in years. Income and Education are proxies based on census data. Income is inthousands of euros. Education is between 1 and 3; values 1, 2, and 3 correspond to middle-, high-, andpost-high-school education. Specifications (1) through (5) differ only in the included explanatory variables.Robust t-statistics are reported in parentheses. ∗ /∗∗ /∗∗∗ indicate significance at the 10% /5% /1% levels.

Panel A : α Panel B : λ

(1) (2) (3) (4) (5) (1) (2) (3) (4) (5)

Age -0.000 -0.000 -0.000 -0.000 -0.001 0.009∗∗ 0.009∗∗ 0.009∗∗ 0.009∗∗ 0.010∗∗∗(-0.709) (-0.739) (-0.706) (-0.703) (-0.836) (2.257) (2.251) (2.422) (2.260) (2.619)

Female 0.048∗∗ 0.047∗∗ 0.048∗∗ 0.048∗∗ 0.047∗∗ 0.137 0.136 0.136 0.146 0.133(2.239) (2.189) (2.244) (2.235) (2.220) (0.929) (0.922) (0.914) (1.006) (0.901)

Rural -0.019 -0.016 -0.026 -0.090(-0.940) (-0.684) (-0.197) (-0.631)

Education 0.004 -0.005 0.018 0.145(0.195) (-0.214) (0.136) (0.840)

Income 0.001 0.001 -0.010 -0.015∗(0.719) (0.521) (-1.622) (-1.844)

R2 0.015 0.016 0.015 0.016 0.017 0.022 0.022 0.022 0.029 0.033

Panel C : γ Panel D : κ

(1) (2) (3) (4) (5) (1) (2) (3) (4) (5)

Age -0.002∗∗∗ -0.002∗∗∗ -0.002∗∗∗ -0.002∗∗∗ -0.002∗∗∗ 0.013 0.013 0.014∗ 0.013 0.014∗(-4.037) (-4.036) (-3.839) (-4.006) (-3.870) (1.634) (1.625) (1.670) (1.630) (1.727)

Female 0.001 0.000 -0.001 -0.001 -0.001 0.009 0.008 0.006 0.014 0.004(0.041) (0.023) (-0.090) (-0.034) (-0.074) (0.034) (0.032) (0.022) (0.054) (0.015)

Rural -0.007 0.008 -0.017 -0.050(-0.384) (0.426) (-0.070) (-0.172)

Education 0.025 0.014 0.036 0.122(1.515) (0.739) (0.108) (0.324)

Income 0.002 0.001 -0.006 -0.010(1.536) (1.071) (-0.405) (-0.560)

R2 0.045 0.045 0.050 0.052 0.054 0.009 0.009 0.009 0.009 0.010

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C.7 Application to portfolio choice

In this section, we provide more details on how we solve the portfolio choice problem described in Section

6 of the paper and we present some additional results.

In the asset-allocation problem we consider, an individual solves

maxwS ,wF

U(rp;α, λ, γ, κ

)(C.1)

s.t. rp = r f + wSrS + wFrF ,

where all quantities are defined in Section 6 of the paper. We solve this problem for each individual in our

sample, using nominal annual returns for r f , rS , and rF , and the posterior mean estimates of his preference

parameters (αn , λn , γn , and κn). To solve this problem, we construct a two-dimensional grid spaced at 1%

increments over the range of values for the weights wS and wF , and we pick the weights that maximize the

objective U for each individual. We approximate the objective by generating draws for the rates of return

r f , rS , and rF , and numerically calculating the integrals.

We generate 1 million draws for r f , rS , and rF as follows. We obtain, for the period January 1975 to

December 2011, data on the risk-free (the one-month Treasury Bill) rate and on stock returns from the Center

for Research in Securities Prices (CRSP), and on U.S. equity mutual fund returns (net of fees, expenses, and

transaction costs) from the CRSP Survivorship-Bias-Free U.S. Mutual Fund Database.3 To generate each

draw, we randomly pick a month t in the period January 1975 to January 2011, and then we randomly pick

a stock among all the stocks and a fund among all the funds in our data in month t .4 For the year starting

in t , we calculate the rate of return for the risk-free asset and the excess rate of return for the chosen stock

and fund. We reduce survivorship bias as follows: If a stock’s/fund’s return is missing from the data starting

at some point during the year, we assume that with probability 0.5 it lost all its value hence its return for

the year is −1, and with probability 0.5 its return information was removed from the data for a different

reason hence we replace it with another stock/fund drawn randomly. The mean/median/standard deviation

of our draws for r f , rS , rF are 5.51%/5.23%/3.26%, 12.07%/4.69%/79.60%, and 11.49%/12.54%/24.52%,

3Since CRSP treats different share classes of the same fund as separate funds, we identify funds’ share classes (using ourown fund-name matching algorithm as well as the MF Links database from Wharton Research Data Services) and we computeeach fund’s monthly return as the weighted average of the returns of its component share classes, with weights equal to thebeginning-of-month total net asset value of each class.

4The random draws of month t , the stock, and the fund are done with replacement. The probability of drawing each month isthe same for all months, and the probability of drawing each stock (fund) equals the market capitalization (net asset value) of eachstock (fund) divided by the total for all stocks (funds) active in month t . Alternatively, we could draw among all active stocks andfunds with equal probability; our results using this alternative generating process are very similar to the ones we present here.

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respectively. Using these draws, we can calculate the corresponding draws of the portfolio return rp for

each combination of weights wS and wF . Then, we can approximate the objective U(rp; ·

)by calculating

the weighted distribution function Fp on a linear grid of 10,000 values from slightly below the minimum

to slightly above the maximum draw of rp, calculating the corresponding numerical derivatives at these

values, and finally calculating the numerical integral over the whole range.

To understand how our estimated distribution of preference parameters translates to the distribution of

optimal portfolios we present in Figure 8, we analyze how each preference parameter affects the optimal

direct and indirect investment in equity. In Figures C.8 and C.9, we present the optimal weights on the

stock and fund as a function of the measure α of value-function curvature and of loss aversion λ, for various

combinations of values for the measures of the curvature γ and elevation κ of the probability weighting

function. We see that a higher α and a lower λ increase the total investment in equity and tilt the equity

investment toward an undiversified portfolio and away from a well-diversified one. Furthermore, while the

effect of γ and κ on total investment in equity is not monotonic, comparing Figure C.9a with C.9b and

Figure C.9c with C.9d, we see that both a lower γ and a higher κ unambiguously tilt the equity investment

toward an undiversified portfolio and away from a well-diversified one. Thus, the considerable heterogeneity

we estimate in α and λ implies that half of the individuals have a combination of high enough α and/or low

enough λ such that they optimally invest a significant fraction of their financial wealth in equity. Furthermore,

the heterogeneity we estimate in all preference parameters accounts for the wide heterogeneity in the optimal

split of individuals’ equity investment between the stock and the fund.

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0.6

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1

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1

1.5

2

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3

0

0.5

1

α

λ

Figure C.8: Optimal weights on the stock (in the left graph) and fund (in the right graph) as a functionof α and λ, for the median values of γ (i.e., γ = 0.84) and κ (i.e., κ = 2.14).

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λ

(d) Median γ , High κ

Figure C.9: Optimal weights on the stock (in the left graph of each panel) and fund (in the right graph of each panel)as a function of α and λ, for various combinations of values of γ and κ across Panels (a) through (d). Panel (a) plotsthe optimal weights for γ = 0.76 (the 25th percentile of the estimated γ distribution) and κ = 2.14 (the median ofthe estimated κ distribution); Panel (b) for γ = 0.92 (the 75th percentile of the estimated γ distribution) and κ = 2.14;Panel (c) for γ = 0.84 (the median of the estimated γ distribution) and κ = 1.61 (the 25th percentile of the estimatedκ distribution); and Panel (d) for γ = 0.84 and κ = 3.38 (the 75th percentile of the estimated κ distribution).

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C.8 Prior sensitivity analysis

In this section we present some results that are complementary to the prior sensitivity analysis presented

in Section 7.1 of the paper.

First, we present results from the sensitivity analysis in which we vary the scale3 of the prior distribution

for Vβ̃ , from our baseline value (I ), to a low value (0.1I ), to a high value (10I ). In Figure C.10, we plot the

estimated population densities for βn , for each of these values for 3. We see that the prior has some effect

on the estimated densities for the CPT parameters. However, the variances of all parameters are bounded far

away from 0 under all prior specifications, suggesting that the heterogeneity we estimate is not driven by the

choice of our priors. Moreover, while we vary the prior variances by an order of magnitude, the posterior

variances vary by much less than that, indicating that the posteriors should be close to our baseline.

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Figure C.10: Prior sensitivity analysis, for different values of the scale 3 of the the prior distribution for Vβ̃ .We plot the estimated population densities from our baseline model, which has 3 = I (in blue solid lines),from a model with small-variance priors, which has 3 = 0.1I (in black dash-dotted lines), and from a modelwith large-variance priors, which has 3 = 10I (in red dotted lines). The proportion with no loss aversion is36% for the baseline priors, 27% for the small-variance priors, and 34% for the large-variance priors. Thecircles at λ = 1 and at γ = 1, κ = 1 represent point masses that correspond to the proportions of individualswith no loss aversion and with no probability weighting, respectively. The proportion with no probabilityweighting is 1% for the baseline priors, 7% for the small-variance priors, and 1% for the large-variance priors.

In Table C.3, we also present summary statistics of the posterior estimates of the population means and

variances of the parameters βn , for the two alternative values of 3: 3 = 0.1I and 3 = 10I ; the results

that correspond to the baseline prior, 3 = I , are presented in Table 4 in the paper.

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Table C.3: Posterior Estimates of Population Mean and Variance — Prior Sensitivity for 3

Summary statistics of the posterior estimates for the population mean and variance of the elements of βn

obtained from the low-variance prior (Panel A) and the high-variance prior (Panel B).

Panel A : 3 = 0.1I Panel B : 3 = 10I

Mean Median Std.Dev. 95% HPDI Mean Median Std.Dev. 95% HPDI

Means

π 0.25 0.25 0.01 [0.23 , 0.28] 0.25 0.25 0.01 [0.23 , 0.28]

τ 1.97 1.97 0.10 [1.79 , 2.17] 1.81 1.81 0.10 [1.63 , 2.00]

α 0.67 0.67 0.01 [0.66 , 0.69] 0.66 0.66 0.01 [0.64 , 0.69]

λ 1.37 1.37 0.04 [1.30 , 1.45] 1.54 1.54 0.05 [1.44 , 1.66]

γ 0.83 0.83 0.01 [0.82 , 0.84] 0.88 0.88 0.02 [0.85 , 0.91]

κ 2.38 2.38 0.10 [2.20 , 2.58] 2.99 2.98 0.14 [2.72 , 3.26]

Variances

π 0.05 0.05 0.00 [0.04 , 0.06] 0.05 0.05 0.00 [0.04 , 0.06]

τ 2.20 2.18 0.26 [1.72 , 2.73] 2.06 2.05 0.26 [1.62 , 2.63]

α 0.02 0.01 0.00 [0.01 , 0.02] 0.03 0.03 0.00 [0.03 , 0.04]

λ 0.30 0.29 0.05 [0.22 , 0.40] 0.71 0.68 0.17 [0.49 , 1.12]

γ 0.01 0.01 0.00 [0.00 , 0.01] 0.05 0.05 0.01 [0.04 , 0.06]

κ 1.57 1.55 0.25 [1.14 , 2.12] 3.67 3.65 0.44 [2.87 , 4.56]

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Next, we present results from the sensitivity analysis that replaces the symmetric D (1) prior with the

asymmetric D (1, 1, 1, 13) prior, which underweights component probability vectors with higher values for

the proportion of individuals with no loss aversion and overweights component probability vectors with

higher values for the proportion of individuals who exhibit all features of prospect theory. In Figure C.11,

we present the estimated population densities of the model parameters that correspond to each of these two

prior specifications.

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Figure C.11: Estimated population densities of the model parameters, for the baseline prior specificationthat assumes a symmetric Dirichlet prior D (1) for the population proportions h, and for an alternative priorspecification that assumes an asymmetric Dirichlet prior D (1, 1, 1, 13). The estimated densities are plottedin solid blue for the baseline prior, and in dash-dotted red for the alternative prior. The circles at λ = 1 and atγ = 1, κ = 1 represent point masses that correspond to the proportions of individuals with no loss aversionand with no probability weighting, respectively. The proportion with no loss aversion is 36% for the baselineprior and 33% for the alternative prior, and the proportion with no probability weighting is 1% for both priors.

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Next, we discuss in more detail the results for the analysis that checks the sensitivity of the posterior

predictive distributions of the preference parameters to their prior predictive distributions. To achieve the

desired change in the prior predictive distribution, we vary some of the baseline hyperprior parameters for

the population mean and variance of the preference parameters. In particular, in the baseline priors we use

parameters κ = 0, K = 100, λ = 6, 3 = I , and in the alternative specification for which we present results

here we use K = 0.5 (and κ = 0, λ = 6, 3 = I , as in the baseline). The former parameter values (combined

with our other prior parameters) imply that the prior predictive density is almost flat everywhere, except for

values close to the endpoints for bounded distributions, while the latter imply that the prior predictive density

is more concentrated over a small range of values away from the boundaries. As we show in Figure C.12, the

change in the prior predictive density is drastic but the effect on the posterior predictive density is very small.

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(a) Prior predictive density, baseline vs. alternatives.

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(b) Posterior predictive density, baseline vs. alternatives.

Figure C.12: Plots of the prior and posterior predictive densities of β = (π, τ, α, λ, γ, κ) for the baseline priorspecification (κ = 0, K = 100, λ = 6, 3 = I ) and for an alternative (κ = 0, K = 0.5, λ = 6, 3 = I ). In Panel (a),we plot the prior predictive densities for the baseline prior specification (in solid blue) and for the alternative priorspecification (in dash-dotted red). The proportion with no loss aversion is 50% for both priors, and the proportionwith no probability weighting is also 50% for both priors. In Panel (b), we plot the posterior predictive densities for thebaseline prior specification (in solid blue) and for the alternative prior specification (in dash-dotted red). In both panels,the circles at λ = 1 and at γ = 1, κ = 1 represent point masses that correspond to the proportions of individuals withno loss aversion and with no probability weighting, respectively. The proportion with no loss aversion is 36% forboth posteriors, and the proportion with no probability weighting is 1% for both posteriors.

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C.9 Betting activity on local teams

In this section, we present some histograms relating to our discussion in Section 7.4 of the paper with respect

to whether the betting choices we observe are driven by attitudes toward local teams. The histograms in Panels

(a) and (b) of Figure C.13 present, respectively, the percentage of total wagers placed by each individual in

favor of his local teams and on matches in which his local teams are participating. It is clear from these graphs

that betting activity in relation to local teams is low. Panels (c) and (d) of Figure C.13 show, respectively, that

the majority of our individuals almost never select day lotteries that involve solely bets that back their local

teams to win, neither do they select lotteries that involve only bets in which their local teams are participating.

(a) %age of wagers that back the local teams to win. (b) %age of wagers that involve the local teams.

0 10 20 30 40 50 60 70 80 90 100

(c) %age of lottery selections that include onlywagers that back the local teams to win.

0 10 20 30 40 50 60 70 80 90 100

(d) %age of lottery selections that include onlywagers that involve the local teams.

Figure C.13: Histograms, across individuals, of the percentage of betting activity relating to local teams.

We note that we get similar results when the analysis is performed in terms of money wagered rather

than number of wagers.

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References

Booij, A, B van Praag, and G van de Kuilen. 2010. “A parametric analysis of prospect theory’s functionals for the

general population.” Theory and Decision, 68(1): 115–148.

Donkers, B, B Melenberg, and A van Soest. 2001. “Estimating risk attitudes using lotteries: A large sample approach.”

Journal of Risk and Uncertainty, 22(2): 165–95.

Gachter, S, E Johnson, and A Herrmann. 2007. “Individual-level loss aversion in riskless and risky choices.” Unpub-

lished Paper.

von Gaudecker, H, A van Soest, and E Wengstrom. 2011. “Heterogeneity in risky choice behavior in a broad popula-

tion.” American Economic Review, 101(2): 664–694.

Lattimore, P, J Baker, and A Witte. 1992. “The influence of probability on risky choice: A parametric examination.”

Journal of Economic Behavior & Organization, 17(3): 377–400.

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