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.TITLE "Calibrating Methods for Decision Making Under Uncertainty"
.ce 6
Robert Marks
Economics, UNSW Australia,
.sp .5
The International Conference on Decision Economics (DECON 2019),
\('Avila, Spain,
June 26\-28, 2019
.sp
robert.marks@gmail.com
An engineering approach: normative (should), not positive (how actually).
.sp .5
An ongoing research program.
.SK
.TITLE "Calibrating Methods for Decision Making Under Uncertainty"
.AL
.LI
Introduction
.LI
Example: Four Lotteries, Five Methods
.LI
Simulating Choice under Uncertainty
.LI
Choosing the Best Lottery \(em Three Utility Functions
.LI
Discussion
.LE
.SK
.SUBTITLE "1. Introduction"
What is the best choice among lotteries (when the prizes and their
probabilities are known) in a risky world?
.\".PAUSE
"Best" means that the agent's average net winnings from choosing successive
lotteries is highest: this is risky decision making.
By "risky" is meant that both the possible outcomes and probabilities
are known.
.SK
.SUBTITLE "2. Example: Four Lotteries"
Figure 1:
.DS CB
.S 14
.PS 7 3
linethick = 2
circlerad = 0.25i
A: circle at (-4,0) "A"
B: circle invis at (-4.5, -2) "$25"
C: circle invis at (-4, -2) "$150"
D: circle invis at (-3.5, -2) "$600"
line from A.c to B.c chop ".7" rjust
line from A.c to C.c chop ".2" ljust
line from A.c to D.c chop ".1" ljust
E: circle at (-2.5,0) "B"
F: circle invis at (-3, -2) "$80"
G: circle invis at (-2.5, -2) "$90"
H: circle invis at (-2, -2) "$98"
line from E.c to F.c chop ".2" rjust
line from E.c to G.c chop ".58" ljust
line from E.c to H.c chop ".22" ljust
I: circle at (-.75,0) "C"
J: circle invis at (-1.5, -2) "\-$20"
K: circle invis at (-1, -2) "$0"
L: circle invis at (-.5, -2) "$100"
M: circle invis at (0, -2) " $1000 "
line from I.c to J.c chop ".6" rjust
line from I.c to K.c chop ".1" rjust
line from I.c to L.c chop ".2" ljust
line from I.c to M.c chop ".1" ljust
N: circle at (.75,0) "D"
O: circle invis at (.5, -2) " $105"
P: circle invis at (1, -2) " \-$100"
line from N.c to O.c chop ".95" rjust
line from N.c to P.c chop ".05" ljust
.PE
.S 18
.DE
.sp .5
Which is the best lottery?
.sp .5
The agent's choice depends on her method of choice.
.sp .5
But after the lottery is \f4realised\fP, which method is best?
.SK
\m[blue6]The agent's choice varies with method:\m[]
.\".PAUSE
1. \f4Expected Value\fP \(-> C
.br
A: 107.5 B: 89.76 C: \m[green]\f4108.0\fP\m[] D: 94.75
.sp .5
.\".PAUSE
2. \f4Laplace\fP (equal likelihoods) \(-> C
.br
A: 258.33 B: 89.33 C: \m[green]\f4270\fP\m[] D: 2.5
.sp .5
.\".PAUSE
3. \f4Max-Max\fP (ignore probabilities) \(-> C
.br
A: 600 B: 98 C: \m[green]\f41000\fP\m[] D: 105
.sp .5
.\".PAUSE
4. \f4Max-Min\fP (ignore probabilities) \(-> B
.br
A: 25 B: \m[green]\f480\fP\m[] C: \-20 D: \-100
.sp .5
.\".PAUSE
5. Risk-averse CARA (`gamma` = 0.00111) \(-> A
.br
A: \m[green]\f40.098\fP\m[] B: 0.095 C: 0.086 D: 0.074
.br
(see equation (1) below)
.SK
.SUBTITLE "2. Simulating the Choice"
Instead of four lotteries (as above), we generate eight lotteries, each with
six prizes, chosen uniformly between $10 and \-$10, with their probabilities
chosen at random.
.sp .5
.\".PAUSE
Then run 10,000 experiments where each time, for each of the 5 methods, a
lottery is chosen, and then, using the known probabilities of that lottery's
prizes, the lottery is \f4realised\fP, and the method's score is added (or
subtracted, if a negative realisation) to its previous score.
.sp .5
.\".PAUSE
The \m[blue6]Clairvoyant\m[] is the benchmark: if the agent knew the outcome for each of
the lotteries, she would choose the lottery with the highest outcome.
With simulation, we can determine each lottery's realised outcome
\f4before\fP the Clairvoyant chooses.
.br
.\".PAUSE
See R code at
\s-3\f[CR]http://www.agsm.edu.au/bobm/papers/riskmethods.r\fP\s0
.SK
\m[blue6]Table 1: Simulations, the mean scores by method:\m[]
.TS
center;
llll
lnnn.
Method Payoff ($) % Clairvoyant % EV
_
Clairvoyant 7.7880 100
Expected Value 3.8718 49.7143 100
Laplace 3.3599 43.1425 86.7800
Max-Max 1.3917 17.8702 35.9500
Max-Min 2.4279 31.1752 62.7100
Random 0.02 0 0
.TE
.sp .5
.\".PAUSE
The Random is zero, as it should be, given the prizes are chosen randomly.
.sp .5
.\".PAUSE
The Expected Value dominates the 4 methods,
although the Laplace method is not too bad (almost 87% of EV).
But, surprisingly, the Max-Min (choosing the lottery with the highest worst
possible prize) is almost twice as good (63%) as the Max-Max method (36%):
.br
Does pessimism dominate optimism?
.SK
.SUBTITLE "3.1 CARA Utility Functions"
The exponential CARA utility function is
.sp .5
.DS CB
.EQ (1)
U(x)~=~ 1~-~e sup {- gamma x},
.EN
.DE
.sp .5
where `U(0)` = 0 and `U( inf )` = 1, and
.br
.\".PAUSE
where `gamma` is the \f4risk aversion coefficient\fP:
.sp .5
.DS CB
.EQ (2)
gamma == - {U^ prime opprime (x) } over { U^ prime (x) }.
.EN
.DE
.TS
center;
ccc.
Sign of `gamma` Risk profile Curvature
.sp .2
_
`gamma` = 0 risk neutral `U^ prime opprime (x)~=~0`
`gamma` > 0 risk averse `U^ prime opprime (x)~<~0`
`gamma` < 0 risk preferring `U^ prime opprime (x)~>~0`
_
.TE
.SK
\m[blue6]Table 2: Simulations of CARA, mean payoffs, varying `gamma`\m[]
.TS
center;
llll
nnnn.
gamma `gamma` Payoff ($) % Clairvoyant % EV
_
\-0.2000 3.4714 44.5739 89.6600
\-0.1600 3.6111 46.3670 93.2669
\-0.1200 3.7005 47.5160 95.5782
\-0.0800 3.8196 49.0448 98.6532
\-0.0400 3.8582 49.5405 99.6503
`approx 0` 3.8718 49.7151 100
0.0400 3.8330 49.2163 98.9982
0.0800 3.7840 48.5873 97.7330
0.1200 3.7290 47.8818 96.3138
0.1600 3.6534 46.9111 94.3613
0.2000 3.5615 45.7301 91.9858
.TE
.sp .5
It is clear that the best (mean payoffs) occur with `gamma approx` 0: risk
neutral.
.SK
.SUBTITLE "3.2 CRRA Utility Functions"
The Constant Elasticity of Substitution (CES) CRRA utility function:
.sp -.5
.DS CB
.EQ (3)
U ( w) = {w sup {1 - rho}} over { 1 - rho } , ~~~ w > 0,
.EN
.DE
.sp .25
where `w` is agent's wealth, and `rho` is the
Arrow-Pratt measure of relative risk aversion (RRA):
.DS CB
.EQ (4)
rho (w) = - w {U prime prime (w)} over {U prime (w)} = w gamma
.EN
.DE
.sp .25
.\".PAUSE
This introduces wealth `w` into the agent's risk preferences, so that lower
wealth can be associated with higher risk aversion.
The coefficient `gamma` is as in (2).
.sp .5
As `rho` \(-> 1, (3) becomes logarithmic:
`u(w) = ln (w)`, risk averse.
.\".PAUSE
With `w` > 0, `rho` > 0 is equivalent to \f4risk averse\fP, while
`rho` < 0 is equivalent to \f4risk preferring\fP;
`rho` = 0: \f4risk neutral\fP.
.SK
\m[blue6]Table 3: Simulations of CRRA, mean payoffs, varying `rho`\m[]
.TS
center;
llll
nnnn.
\f3rho `rho` Payoff ($) % Clairvoyant % EV
_
\-2.5000 3.7570 48.2408 97.0360
\-2.0000 3.8114 48.9391 98.4407
\-1.5000 3.8350 49.2426 99.0512
\-1.0000 3.8490 49.4222 99.4124
\-0.5000 3.8665 49.6475 99.8656
`approx 0` 3.8718 49.7143 100
0.5000 3.8577 49.5343 99.6379
1.0000 3.8284 49.1581 98.8812
1.5000 3.8056 48.8655 98.2926
2.0000 3.7773 48.5012 97.5598
2.5000 3.7521 48.1780 96.9098
.TE
.sp .5
As with CARA, the simulations show that CRRA performs best when `rho approx`
0, the risk-neutral, Expected Value method.
.SK
.SUBTITLE "3.3 The DRP function."
This function is inspired by Prospect Theory (Kahnemann & Tversky 1979):
.DS CB
.EQ (5)
V = {1 - e sup {- beta X}} over {1 - e sup {-100 beta}} , ~~~~~0 <= X <= 100
.EN
.DE
.DS C
.EQ (6)
V = - delta {1 - e sup { beta X}} over {1 - e sup {-100 beta}} , ~~~~~-100 <= X <= 0
.
.EN
.DE
.sp .5
`beta > 0` models the curvature of the function, and
`delta >= 1`, the asymmetry associated with losses.
The DRP function is not wealth-independent.
.sp .5
.\".PAUSE
DRP exhibits the S-shaped asymmetric function of Prospect Theory.
It exhibits risk seeking (loss aversion) when `X` is negative with respect to
the reference point, `X` = 0, and risk aversion when `X` is positive.
.SK
.DS CB
.SUBTITLE "Figure 2: Dual-Risk-Profile DRP Functions"
.G1
frame invis ht 3 wid 4.5 left solid bot solid
label left "Value"
label bot """`X`"
coord x -100, 100 y -2, 1
a=1
b=.019
c=100
d=1.75
adj=.985
for i = 1 to c by .1 do X
"\s-8.\s0" at (i,((a-exp(-b*i))/(a-(exp(-b*c)))))
X
for i = -c to -1 by .1 do Y
"\s-8.\s0" at (i,(-(a-exp(b*i))/(a-(exp(-b*c))))*d*adj)
Y
"`beta = 0.019`" rjust at -21,-1
b=.05
for i = 1 to c by .1 do Z
"\s-8.\s0" at (i,((a-exp(-b*i))/(a-(exp(-b*c)))))
Z
for i = -c to -1 by .1 do W
"\s-8.\s0" at (i,(-(a-exp(b*i))/(a-(exp(-b*c))))*d*adj)
W
"`beta = 0.05`" rjust at 0,.5
b=.00005
for i = 1 to c by .5 do Z
"\s-8.\s0" at (i,((a-exp(-b*i))/(a-(exp(-b*c)))))
Z
for i = -c to -1 by .5 do W
"\s-8.\s0" at (i,(-(a-exp(b*i))/(a-(exp(-b*c))))*d*adj)
W
"`beta approx 0`" rjust at -48,-.7
.G2
\f3A DRP Function (`delta` = 1.75).\fP
.DE
.sp .5
.\".PAUSE
As `delta \(-> 1` and `beta \(-> 0`, the value function
asymptotes to a linear, risk-neutral function.
.SK
\m[blue6]Table 4: Simulations of DRP, % of EV, varying `delta` and `beta`\m[]
.TS
center;
l|lll
n|nnn.
beta `beta` `delta` = 1.001 `delta` = 1.2 `delta` = 1.4
_
0.0010 100 99.8069 99.4421
0.1000 99.5989 98.5836 98.6017
0.2000 98.2308 97.9890 97.2238
0.4000 96.9848 95.9122 95.2202
.TE
Again, the best results from the DRP functions occur when `beta approx 0` and
`delta approx 1`: this is risk-neutral, Expected Value decision making.
.SK
.SUBTITLE "Discussion"
Previous views:
"Risk aversion is one of the most basic assumptions underlying
economic behavior" (Szpiro 1997), perhaps because "a dollar that helps us
avoid poverty is more valuable than a dollar that helps us become
very rich" (Rabin 2000).
.sp .5
But is risk aversion the best risk profile?
Even with bankruptcy as a possibility?
.sp .5
.\".PAUSE
Previous researchers' answers:
.BL
.LI
Szpiro (1997): risk averse,
.LI
Chen et al. (2008): risk averse (log utility), and
.LI
DellaVigna & LiCalzi (2001) model Kahneman-Tversky agents which learn to
make risk-neutral choices.
.LE
Our answer: NO. RISK NEUTRAL IS BEST.