Comparing Risks
by Acceptance and Rejection
Sergiu Hart
September 2009
This version: March 2010
c 2009 – p. 1
S ERGIU HART °
Comparing Risks
by Acceptance and Rejection
Sergiu Hart
Center for the Study of Rationality
Dept of Economics Dept of Mathematics
The Hebrew University of Jerusalem
hart@huji.ac.il
http://www.ma.huji.ac.il/hart
c 2009 – p. 2
S ERGIU HART °
Papers
Dean Foster and Sergiu Hart
"An Operational Measure of Riskiness"
Journal of Political Economy (2009)
www.ma.huji.ac.il/hart/abs/risk.html
c 2009 – p. 3
S ERGIU HART °
Papers
Dean Foster and Sergiu Hart
"An Operational Measure of Riskiness"
Journal of Political Economy (2009)
www.ma.huji.ac.il/hart/abs/risk.html
Dean Foster and Sergiu Hart
"A Reserve-Based Axiomatization of the
Measure of Riskiness" (2008)
www.ma.huji.ac.il/hart/abs/risk-ax.html
c 2009 – p. 3
S ERGIU HART °
Papers (continued)
Sergiu Hart
"A Simple Riskiness Order Leading to the
Aumann–Serrano Index of Riskiness" (2008)
www.ma.huji.ac.il/hart/abs/risk-as.html
c 2009 – p. 4
S ERGIU HART °
Papers (continued)
Sergiu Hart
"A Simple Riskiness Order Leading to the
Aumann–Serrano Index of Riskiness" (2008)
www.ma.huji.ac.il/hart/abs/risk-as.html
Sergiu Hart
"Comparing Risks by Acceptance and
Rejection" (2009)
www.ma.huji.ac.il/hart/abs/risk-u.html
c 2009 – p. 4
S ERGIU HART °
Gamble (“Risky Asset”)
1/2
+$120
g=
1/2
−$100
c 2009 – p. 5
S ERGIU HART °
Gamble (“Risky Asset”)
1/2
+$120
g=
1/2
−$100
Net gains and losses
c 2009 – p. 5
S ERGIU HART °
Gamble (“Risky Asset”)
1/2
+$120
g=
1/2
−$100
Net gains and losses
Positive expectation
c 2009 – p. 5
S ERGIU HART °
Gamble (“Risky Asset”)
1/2
+$120
g=
1/2
−$100
Net gains and losses
Positive expectation
Some losses
c 2009 – p. 5
S ERGIU HART °
Gamble (“Risky Asset”)
1/2
+$120
g=
1/2
−$100
Net gains and losses
Positive expectation
Some losses
Pure risk (known probabilities)
c 2009 – p. 5
S ERGIU HART °
Comparing Risks
c 2009 – p. 6
S ERGIU HART °
Comparing Risks
Let g and h be gambles
c 2009 – p. 6
S ERGIU HART °
Comparing Risks
Let g and h be gambles
Question:
c 2009 – p. 6
S ERGIU HART °
Comparing Risks
Let g and h be gambles
Question:
When is g
LESS RISKY THAN
h?
c 2009 – p. 6
S ERGIU HART °
Comparing Risks
Let g and h be gambles
Question:
When is g
LESS RISKY THAN
h?
Answer :
c 2009 – p. 6
S ERGIU HART °
Comparing Risks
Let g and h be gambles
Question:
When is g
LESS RISKY THAN
h?
Answer :
When RISK - AVERSE decision-makers
are LESS AVERSE to g than to h !
c 2009 – p. 6
S ERGIU HART °
Comparing Risks
“risk-averse decision-makers are
LESS AVERSE to g than to h”
=?
c 2009 – p. 7
S ERGIU HART °
Comparing Risks: Take 1
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
PREFERRED
=
to h
c 2009 – p. 7
S ERGIU HART °
Comparing Risks: Take 1
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
PREFERRED
=
to h
E [u(w + g)] ≥ E [u(w + h)]
for every (concave) utility u and wealth w
c 2009 – p. 7
S ERGIU HART °
Comparing Risks: Take 1
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
PREFERRED
=
to h
E [u(w + g)] ≥ E [u(w + h)]
for every (concave) utility u and wealth w
g
STOCHASTICALLY DOMINATES
h (2nd-degree)
c 2009 – p. 7
S ERGIU HART °
Comparing Risks: Take 1
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
PREFERRED
=
to h
E [u(w + g)] ≥ E [u(w + h)]
for every (concave) utility u and wealth w
g
STOCHASTICALLY DOMINATES
h (2nd-degree)
g >S h
c 2009 – p. 7
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (1st-degree)
g >S1 h
c 2009 – p. 8
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (1st-degree)
g >S1 h
1/2
1/2
+$200
+$150
>S1
1/2
−$100
1/2
−$100
c 2009 – p. 8
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (1st-degree)
g >S1 h
1/2
1/2
+$200
+$150
>S1
1/2
−$100
1/2
−$100
g ′ > h′
Distribution g = Distribution g ′
Distribution h = Distribution h′
c 2009 – p. 8
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (1st-degree)
g >S1 h
1/2
1/2
+$200
+$200
>S1
1/2
−$100
1/2
−$120
g ′ > h′
Distribution g = Distribution g ′
Distribution h = Distribution h′
c 2009 – p. 8
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (1st-degree)
g >S1 h
1/2
1/3
+$200
+$200
>S1
1/2
−$100
2/3
−$100
g ′ > h′
Distribution g = Distribution g ′
Distribution h = Distribution h′
c 2009 – p. 8
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (2nd-degree)
g >S2 h
c 2009 – p. 9
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (2nd-degree)
g >S2 h
1/2
1/4
+$200
1/4
>S2
1/2
−$100
1/2
+$250
+$150
−$100
c 2009 – p. 9
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (2nd-degree)
g >S2 h
1/2
1/4
+$200
1/4
>S2
1/2
−$100
from g to h: a
1/2
+$250
+$150
−$100
MEAN - PRESERVING SPREAD
c 2009 – p. 9
S ERGIU HART °
Stochastic Dominance
g
STOCHASTICALLY DOMINATES
h (2nd-degree)
g >S2 h
1/2
1/4
+$200
1/4
>S2
1/2
−$100
from g to h: a
1/2
+$250
+$150
−$100
MEAN - PRESERVING SPREAD
>S2 = >S1 + mean-preserving spreads
c 2009 – p. 9
S ERGIU HART °
Acceptance and Rejection
Let g be a gamble.
c 2009 – p. 10
S ERGIU HART °
Acceptance and Rejection
Let g be a gamble.
g is ACCEPTED by a decision-maker with
utility u at wealth w if
E [u(w + g)] > u(w)
c 2009 – p. 10
S ERGIU HART °
Acceptance and Rejection
Let g be a gamble.
g is ACCEPTED by a decision-maker with
utility u at wealth w if
E [u(w + g)] > u(w)
g is REJECTED by a decision-maker with
utility u at wealth w if
E [u(w + g)] ≤ u(w)
c 2009 – p. 10
S ERGIU HART °
Comparing Risks
“risk-averse decision-makers are
LESS AVERSE to g than to h”
=
c 2009 – p. 11
S ERGIU HART °
Comparing Risks: Take 2
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
REJECTED LESS
=
than h
c 2009 – p. 11
S ERGIU HART °
Comparing Risks: Take 2
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
IF
THEN
REJECTED LESS
=
than h
g is rejected by u at w
h is rejected by u at w
for every (concave) utility u and wealth w
c 2009 – p. 11
S ERGIU HART °
Comparing Risks: Take 2
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
IF
THEN
REJECTED LESS
=
than h
E [u(w + g)] ≤ u(w)
E [u(w + h)] ≤ u(w)
for every (concave) utility u and wealth w
c 2009 – p. 11
S ERGIU HART °
Comparing Risks: Take 2
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
IF
THEN
REJECTED LESS
=
than h
g is rejected by u at w
h is rejected by u at w
for every (concave) utility u and wealth w
c 2009 – p. 11
S ERGIU HART °
Comparing Risks: Take 2
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
IF
THEN
REJECTED LESS
=
than h
g is rejected by u at w
h is rejected by u at w
for every (concave) utility u and wealth w
g
ACCEPTANCE DOMINATES
h
c 2009 – p. 11
S ERGIU HART °
Comparing Risks: Take 2
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
IF
THEN
REJECTED LESS
=
than h
g is rejected by u at w
h is rejected by u at w
for every (concave) utility u and wealth w
g
ACCEPTANCE DOMINATES
h
g >A h
c 2009 – p. 11
S ERGIU HART °
Comparing Risks
“risk-averse decision-makers are
LESS AVERSE to g than to h”
=
c 2009 – p. 12
S ERGIU HART °
Comparing Risks: Take 3
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
WEALTH - UNIFORMLY REJECTED LESS
=
than h
c 2009 – p. 12
S ERGIU HART °
Comparing Risks: Take 3
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
WEALTH - UNIFORMLY REJECTED LESS
IF
THEN
=
than h
g is rejected by u at all w
h is rejected by u at all w
for every (concave) utility u
c 2009 – p. 12
S ERGIU HART °
Comparing Risks: Take 3
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
WEALTH - UNIFORMLY REJECTED LESS
IF
THEN
=
than h
E [u(w + g)] ≤ u(w) for all w
E [u(w + h)] ≤ u(w) for all w
for every (concave) utility u
c 2009 – p. 12
S ERGIU HART °
Comparing Risks: Take 3
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
WEALTH - UNIFORMLY REJECTED LESS
IF
THEN
=
than h
g is rejected by u at all w
h is rejected by u at all w
for every (concave) utility u
c 2009 – p. 12
S ERGIU HART °
Comparing Risks: Take 3
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
=
WEALTH - UNIFORMLY REJECTED LESS
IF
THEN
than h
g is rejected by u at all w
h is rejected by u at all w
for every (concave) utility u
g
WEALTH - UNIFORMLY DOMINATES
h
c 2009 – p. 12
S ERGIU HART °
Comparing Risks: Take 3
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
=
WEALTH - UNIFORMLY REJECTED LESS
IF
THEN
than h
g is rejected by u at all w
h is rejected by u at all w
for every (concave) utility u
g
WEALTH - UNIFORMLY DOMINATES
h
g >WU h
c 2009 – p. 12
S ERGIU HART °
Comparing Risks
“risk-averse decision-makers are
LESS AVERSE to g than to h”
=
c 2009 – p. 13
S ERGIU HART °
Comparing Risks: Take 4
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
UTILITY - UNIFORMLY REJECTED LESS
=
than h
c 2009 – p. 13
S ERGIU HART °
Comparing Risks: Take 4
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
UTILITY - UNIFORMLY REJECTED LESS
IF
THEN
=
than h
g is rejected by all u at w
h is rejected by all u at w
for every wealth w
c 2009 – p. 13
S ERGIU HART °
Comparing Risks: Take 4
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
UTILITY - UNIFORMLY REJECTED LESS
IF
THEN
=
than h
E [u(w + g)] ≤ u(w) for all u
E [u(w + h)] ≤ u(w) for all u
for every wealth w
c 2009 – p. 13
S ERGIU HART °
Comparing Risks: Take 4
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
UTILITY - UNIFORMLY REJECTED LESS
IF
THEN
=
than h
g is rejected by all u at w
h is rejected by all u at w
for every wealth w
c 2009 – p. 13
S ERGIU HART °
Comparing Risks: Take 4
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
=
UTILITY - UNIFORMLY REJECTED LESS
IF
THEN
than h
g is rejected by all u at w
h is rejected by all u at w
for every wealth w
g
UTILITY - UNIFORMLY DOMINATES
h
c 2009 – p. 13
S ERGIU HART °
Comparing Risks: Take 4
“risk-averse decision-makers are
LESS AVERSE to g than to h”
g is
=
UTILITY - UNIFORMLY REJECTED LESS
IF
THEN
than h
g is rejected by all u at w
h is rejected by all u at w
for every wealth w
g
UTILITY - UNIFORMLY DOMINATES
h
g >UU h
c 2009 – p. 13
S ERGIU HART °
Comparing Risks by Rejection
g is
LESS RISKY
than h
⇔ g is REJECTED LESS than h
c 2009 – p. 14
S ERGIU HART °
Comparing Risks by Rejection
g is
LESS RISKY
than h
⇔ g is REJECTED LESS than h
REJECTED
=
c 2009 – p. 14
S ERGIU HART °
Comparing Risks by Rejection
g is
LESS RISKY
than h
⇔ g is REJECTED LESS than h
g >A h
REJECTED
=
REJECTED
by u at w
c 2009 – p. 14
S ERGIU HART °
Comparing Risks by Rejection
g is
LESS RISKY
than h
⇔ g is REJECTED LESS than h
REJECTED
=
g >A h
REJECTED
by u at w
g >WU h
REJECTED
by u at
ALL
w
c 2009 – p. 14
S ERGIU HART °
Comparing Risks by Rejection
g is
LESS RISKY
than h
⇔ g is REJECTED LESS than h
REJECTED
=
g >A h
REJECTED
by u at w
g >WU h
REJECTED
by u at
ALL
g >UU h
REJECTED
by
u at w
ALL
w
c 2009 – p. 14
S ERGIU HART °
Comparing “Comparing Risks”
c 2009 – p. 15
S ERGIU HART °
Comparing “Comparing Risks”
g >S h
c 2009 – p. 15
S ERGIU HART °
Comparing “Comparing Risks”
g >S h
⇓
g >A h
c 2009 – p. 15
S ERGIU HART °
Comparing “Comparing Risks”
g >S h
⇓
⇓
g >WU h
g >A h
⇓
g >UU h
c 2009 – p. 15
S ERGIU HART °
Riskiness Orders: Results
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
UTILITY - UNIFORM DOMINANCE :
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
UTILITY - UNIFORM DOMINANCE :
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order :
for every g, h either g >WU h or h >WU g
UTILITY - UNIFORM DOMINANCE :
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
UTILITY - UNIFORM DOMINANCE :
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
UTILITY - UNIFORM DOMINANCE :
is a complete order
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
is equivalent to the order induced by the
Aumann–Serrano index of riskiness
UTILITY - UNIFORM DOMINANCE :
is a complete order
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
is equivalent to the order induced by the
Aumann–Serrano index of riskiness:
g >WU h ⇔ RAS (g) ≤ RAS (h)
UTILITY - UNIFORM DOMINANCE :
is a complete order
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
is equivalent to the order induced by the
Aumann–Serrano index of riskiness:
g >WU h ⇔ RAS (g) ≤ RAS (h)
UTILITY - UNIFORM DOMINANCE :
is a complete order
is equivalent to the order induced by the
Foster–Hart measure of riskiness
c 2009 – p. 16
S ERGIU HART °
Riskiness Orders: Results
WEALTH - UNIFORM DOMINANCE :
is a complete order
is equivalent to the order induced by the
Aumann–Serrano index of riskiness:
g >WU h ⇔ RAS (g) ≤ RAS (h)
UTILITY - UNIFORM DOMINANCE :
is a complete order
is equivalent to the order induced by the
Foster–Hart measure of riskiness:
g >UU h ⇔ RFH (g) ≤ RFH (h)
c 2009 – p. 16
S ERGIU HART °
AS
R
and
FH
R
c 2009 – p. 17
S ERGIU HART °
AS
R
and
FH
R
Aumann–Serrano index of riskiness RAS :
¶¸
·
µ
1
E 1 − exp − AS
g
=0
R (g)
c 2009 – p. 17
S ERGIU HART °
AS
R
and
FH
R
Aumann–Serrano index of riskiness RAS :
¶¸
·
µ
1
E 1 − exp − AS
g
=0
R (g)
Foster–Hart measure of riskiness RFH :
¶¸
·
µ
1
g
=0
E log 1 + FH
R (g)
c 2009 – p. 17
S ERGIU HART °
AS
R
and
FH
R
Aumann–Serrano index of riskiness RAS :
¶¸
·
µ
1
E 1 − exp − AS
g
=0
R (g)
( 1 / the
CRITICAL RISK - AVERSION
coefficient )
Foster–Hart measure of riskiness RFH :
¶¸
·
µ
1
g
=0
E log 1 + FH
R (g)
c 2009 – p. 17
S ERGIU HART °
AS
R
and
FH
R
Aumann–Serrano index of riskiness RAS :
¶¸
·
µ
1
E 1 − exp − AS
g
=0
R (g)
( 1 / the
CRITICAL RISK - AVERSION
coefficient )
Foster–Hart measure of riskiness RFH :
¶¸
·
µ
1
g
=0
E log 1 + FH
R (g)
( the
CRITICAL WEALTH LEVEL
)
c 2009 – p. 17
S ERGIU HART °
Riskiness Orders
g >S h
⇓
⇓
g >WU h
g >A h
⇓
g >UU h
c 2009 – p. 18
S ERGIU HART °
Riskiness Orders
g >S h
⇓
⇓
g >A h
g >WU h *
⇓
g >UU h *
* = complete order
c 2009 – p. 18
S ERGIU HART °
Riskiness Orders
g >S h
⇓
⇓
g >A h
g >WU h *
m
RAS (g) ≤ RAS (h)
⇓
g >UU h *
m
RFH (g) ≤ RFH (h)
* = complete order
c 2009 – p. 18
S ERGIU HART °
Technical Details
c 2009 – p. 19
S ERGIU HART °
Technical Details
GAMBLE
g:
c 2009 – p. 19
S ERGIU HART °
Technical Details
GAMBLE
g:
a real-valued random variable
E [g] > 0
P [g < 0] > 0
finitely many values
c 2009 – p. 19
S ERGIU HART °
Technical Details
GAMBLE
g:
a real-valued random variable
E [g] > 0
P [g < 0] > 0
finitely many values
UTILITY
u:
c 2009 – p. 19
S ERGIU HART °
Technical Details
GAMBLE
g:
a real-valued random variable
E [g] > 0
P [g < 0] > 0
finitely many values
u:
u : R+ → R (put u(x) = −∞ for x ≤ 0)
strictly increasing
concave
UTILITY
c 2009 – p. 19
S ERGIU HART °
Technical Details
UTILITY
u (continued):
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
rejection increases with scale:
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
rejection increases with scale:
g rejected at w ⇒
λg rejected at λw, for λ > 1
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
rejection increases with scale:
g rejected at w ⇒
λg rejected at λw, for λ > 1
or: IRRA (condition 1 of Arrow, 1965)
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
rejection increases with scale:
g rejected at w ⇒
λg rejected at λw, for λ > 1
or: IRRA (condition 1 of Arrow, 1965)
every gamble is sometimes rejected:
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
rejection increases with scale:
g rejected at w ⇒
λg rejected at λw, for λ > 1
or: IRRA (condition 1 of Arrow, 1965)
every gamble is sometimes rejected:
for every g there is w where g is rejected
UTILITY
c 2009 – p. 20
S ERGIU HART °
Technical Details
u (continued):
rejection decreases with wealth:
g rejected at w ⇒
g rejected at w′ , for w′ < w
or: DARA (condition 2 of Arrow, 1965)
rejection increases with scale:
g rejected at w ⇒
λg rejected at λw, for λ > 1
or: IRRA (condition 1 of Arrow, 1965)
every gamble is sometimes rejected:
for every g there is w where g is rejected
or: u(0+ ) = −∞
UTILITY
c 2009 – p. 20
S ERGIU HART °
Acceptance Dominance
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
g with probability p, and
0 with probability 1 − p
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
g with probability p, and
0 with probability 1 − p
g accepted ⇔ p ∗ g accepted
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
g with probability p, and
0 with probability 1 − p
g accepted ⇔ p ∗ g accepted
E[u(w+p∗g)] = pE[u(w+g)]+(1−p)u(w)
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
g with probability p, and
0 with probability 1 − p
g accepted ⇔ p ∗ g accepted
E[u(w+p∗g)] = pE[u(w+g)]+(1−p)u(w)
Theorem.
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
g with probability p, and
0 with probability 1 − p
g accepted ⇔ p ∗ g accepted
E[u(w+p∗g)] = pE[u(w+g)]+(1−p)u(w)
Theorem.
g
ACCEPTANCE DOMINATES
h
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
p ∗ g = the p-DILUTION of g =
g with probability p, and
0 with probability 1 − p
g accepted ⇔ p ∗ g accepted
E[u(w+p∗g)] = pE[u(w+g)]+(1−p)u(w)
Theorem.
g
ACCEPTANCE DOMINATES
⇔
h
there exist p, q ∈ (0, 1] such that
p ∗ g STOCHASTICALLY DOMINATES q ∗ h
c 2009 – p. 21
S ERGIU HART °
Acceptance Dominance
c 2009 – p. 22
S ERGIU HART °
Acceptance Dominance
For every g with E[g] > 0
g >A 2g
c 2009 – p. 22
S ERGIU HART °
Acceptance Dominance
For every g with E[g] > 0
g >A 2g
g ®S 2g
c 2009 – p. 22
S ERGIU HART °
Acceptance Dominance
For every g with E[g] > 0
g >A 2g
g ®S 2g :
E[g] < E[2g]
c 2009 – p. 22
S ERGIU HART °
Acceptance Dominance
For every g with E[g] > 0
g >A 2g :
2 u(w + x) ≥
u(w + 2x) + u(w)
g ®S 2g :
E[g] < E[2g]
c 2009 – p. 22
S ERGIU HART °
Acceptance Dominance
For every g with E[g] > 0
g >A 2g :
2 u(w + x) ≥ u(w + 2x) + u(w)
2 E[u(w + g)] ≥ E[u(w + 2g)] + u(w)
g ®S 2g :
E[g] < E[2g]
c 2009 – p. 22
S ERGIU HART °
Acceptance Dominance
For every g with E[g] > 0
g >A 2g :
2 u(w + x) ≥ u(w + 2x) + u(w)
2 E[u(w + g)] ≥ E[u(w + 2g)] + u(w)
IF
E[u(w + g)] ≤ u(w)
THEN E[u(w + 2g)] ≤ u(w)
g ®S 2g :
E[g] < E[2g]
c 2009 – p. 22
S ERGIU HART °
Summary
c 2009 – p. 23
S ERGIU HART °
Summary
g is LESS RISKY than h whenever risk-averse
agents are LESS AVERSE to g than to h
c 2009 – p. 23
S ERGIU HART °
Summary
g is LESS RISKY than h whenever risk-averse
agents are LESS AVERSE to g than to h
AVERSION
to a gamble:
REJECTION
c 2009 – p. 23
S ERGIU HART °
Summary
g is LESS RISKY than h whenever risk-averse
agents are LESS AVERSE to g than to h
AVERSION
to a gamble:
REJECTION
rejection of different gambles should be
compared whenever it is SUBSTANTIVE:
UNIFORM over a range of decisions
c 2009 – p. 23
S ERGIU HART °
Summary
g is LESS RISKY than h whenever risk-averse
agents are LESS AVERSE to g than to h
AVERSION
to a gamble:
REJECTION
rejection of different gambles should be
compared whenever it is SUBSTANTIVE:
UNIFORM over a range of decisions
g is less risky than h
whenever
g is uniformly rejected less than h
by risk-averse agents
c 2009 – p. 23
S ERGIU HART °
Summary
g >S h
⇓
⇓
g >A h
g >WU h *
m
RAS (g) ≤ RAS (h)
⇓
g >UU h *
m
RFH (g) ≤ RFH (h)
* = complete order
c 2009 – p. 24
S ERGIU HART °
Summary
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
OBJECTIVE: depends only on the gambles
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
OBJECTIVE: depends only on the gambles
(not on any specific decision-maker)
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
OBJECTIVE: depends only on the gambles
(not on any specific decision-maker)
STATUS QUO: current wealth w
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
OBJECTIVE: depends only on the gambles
(not on any specific decision-maker)
STATUS QUO: current wealth w
(in addition to the utility u)
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
OBJECTIVE: depends only on the gambles
(not on any specific decision-maker)
STATUS QUO: current wealth w
(in addition to the utility u)
g >A λg for every λ > 1
c 2009 – p. 25
S ERGIU HART °
Summary
ORDINAL approach to riskiness
(Aumann–Serrano and Foster–Hart: “cardinal")
OBJECTIVE: depends only on the gambles
(not on any specific decision-maker)
STATUS QUO: current wealth w
(in addition to the utility u)
g >A λg for every λ > 1
(ALL risk-averse agents reject λg more than g)
c 2009 – p. 25
S ERGIU HART °