I have a few questions about utility(hopefully this will clear my confusion). Someone please answer.
Also, the following post contains math, viewer discretion is advised(the math is very simple however).
Suppose you have a choice between two games...
A: 1 game of 100% chance to win $1′000′000
B: 2 games of 50% chance to win $1′000′000 and 50% chance to win nothing
Which is better A, B or are they equivalent? Which game would you pick? Please answer before reading the rest of my rambling.
Lets try to calculate utility.
For A,
A: Utotal = 100%U[$1′000′000] + 0%U[$0]
For B, I see two possible ways to calculate it.
1)Calculate the utility for one game and multiply it by two
B-1: U1game = 50%U[$1′000′000] + 50%U[$0]
B-1: Utotal = U2games = 2U1game = 2{50%U[$1′000′000] + 50%U[$0]}
2)Calculate all possible outcomes of money possession after 2 games.
The possibilities are:
$0 , $0
$0 , $1′000′000
$1′000′000 , $0
$1′000′000 , $1′000′000
If we assume utility is linear:
U[$0] = 0
U[$1′000′000] = 1
U[$2′000′000] = 2
A: Utotal = 100%[$1′000′000] + 0%U[$0] = 100%1 + 0%0 = 1
B-1: Utotal = 2{50%U[$1′000′000] + 50%U[0]} = 2{50%1 + 50%0} = 1
B-2: Utotal = 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%U[$2′000′000] = 25%0 + 25%1 + 25%1 + 25%2 = 1
The math is so neat!
The weirdness begins when the utility of money is non linear. $2′000′000 isn’t twice as useful as $1′000′000 (unless we split that $2′000′000 between 2 people, but lets deal with one weirdness at a time). With the first million one can by a house, a car, quit their crappy job and pursue their own interests. The second million won’t change the persons’ life as much and the 3d even less.
Lets invent more realistic utilities(it has also been suggested that the utility of money is logarithmic but I’m having some trouble taking the log of 0):
U[$0] = 0
U[$1′000′000] = 1
U[$2′000′000] = 1.1 (reduced from 2 to 1.1)
Hmmmm… B-1 is not equal to B-2. Either I have to change around utility function values or discard one of them as the wrong calculation or some other mistake I didn’t think of. Maybe U[$0] != 0.
Starting with the assumption that B-1 = B-2 (U[$1′000′000] = 1
U[$2′000′000] = 1.1), then
2{50%U[$1′000′000] + 50%U[0]} = 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%*U[$2′000′000]
B-1 = B-2 = 0.7
Intuitively this kind of makes sense. Comparing:
A: 100%[$1′000′000] = 50%U[$1′000′000] + 50%U[$1′000′000]
to
B: 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%U[$2′000′000]
= 50%U[$1′000′000] + 25%U[$0] + 25%U[$2′000′000]
A (=/>/<)? B
50%U[$1′000′000] + 50%U[$1′000′000] (=/>/<)? 50%U[$1′000′000] + 25%U[$0] + 25%U[$2′000′000]
the first 50% is the same so it cancels out
50%U[$1′000′000] (=/>/<)? 25%U[$0] + 25%U[$2′000′000]
0.5 > 0.2
The chance to win 2 million doesn’t outweigh how much it would suck to win nothing so therefore the certainty of 1 million is preferable. The negative utility of U[$0] is absorbed by it’s 0 probability coefficient in A.
Or maybe calculation B-1 is just plain wrong, but that would mean we cannot calculate the utility of discrete events and add the utilities up.
Is any of this correct? What kind of calculations would you do?
I have a few questions about utility(hopefully this will clear my confusion). Someone please answer. Also, the following post contains math, viewer discretion is advised(the math is very simple however).
Suppose you have a choice between two games...
A: 1 game of 100% chance to win $1′000′000 B: 2 games of 50% chance to win $1′000′000 and 50% chance to win nothing
Which is better A, B or are they equivalent? Which game would you pick? Please answer before reading the rest of my rambling.
Lets try to calculate utility.
For A, A: Utotal = 100%U[$1′000′000] + 0%U[$0]
For B, I see two possible ways to calculate it.
1)Calculate the utility for one game and multiply it by two B-1: U1game = 50%U[$1′000′000] + 50%U[$0] B-1: Utotal = U2games = 2U1game = 2{50%U[$1′000′000] + 50%U[$0]}
2)Calculate all possible outcomes of money possession after 2 games. The possibilities are: $0 , $0 $0 , $1′000′000 $1′000′000 , $0 $1′000′000 , $1′000′000
B-2: Utotal = 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%U[$2′000′000]
If we assume utility is linear: U[$0] = 0 U[$1′000′000] = 1 U[$2′000′000] = 2 A: Utotal = 100%[$1′000′000] + 0%U[$0] = 100%1 + 0%0 = 1 B-1: Utotal = 2{50%U[$1′000′000] + 50%U[0]} = 2{50%1 + 50%0} = 1 B-2: Utotal = 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%U[$2′000′000] = 25%0 + 25%1 + 25%1 + 25%2 = 1 The math is so neat!
The weirdness begins when the utility of money is non linear. $2′000′000 isn’t twice as useful as $1′000′000 (unless we split that $2′000′000 between 2 people, but lets deal with one weirdness at a time). With the first million one can by a house, a car, quit their crappy job and pursue their own interests. The second million won’t change the persons’ life as much and the 3d even less.
Lets invent more realistic utilities(it has also been suggested that the utility of money is logarithmic but I’m having some trouble taking the log of 0): U[$0] = 0 U[$1′000′000] = 1 U[$2′000′000] = 1.1 (reduced from 2 to 1.1)
A: Utotal = 100%[$1′000′000] + 0%U[$0] = 100%1 + 0%0 = 1 B-1: Utotal = 2{50%U[$1′000′000] + 50%U[0]} = 2{50%1 + 50%0} = 1 B-2: Utotal = 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%U[$2′000′000] = 25%0 + 25%1 + 25%1 + 25%*1.1 = 0.775
Hmmmm… B-1 is not equal to B-2. Either I have to change around utility function values or discard one of them as the wrong calculation or some other mistake I didn’t think of. Maybe U[$0] != 0.
Starting with the assumption that B-1 = B-2 (U[$1′000′000] = 1 U[$2′000′000] = 1.1), then 2{50%U[$1′000′000] + 50%U[0]} = 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%*U[$2′000′000]
solving for U[$0]: 2{50%1 + 50%U[0]} = 25%U[$0] + 25%1 + 25%1 + 25%1.1 1 + U[$0] = 0.25U[$0] + 0.775 0.75*U[$0] = −0.225 U[$0] = −0.3
B-1 = B-2 = 0.7 Intuitively this kind of makes sense. Comparing: A: 100%[$1′000′000] = 50%U[$1′000′000] + 50%U[$1′000′000] to B: 25%U[$0] + 25%U[$1′000′000] + 25%U[$1′000′000] + 25%U[$2′000′000] = 50%U[$1′000′000] + 25%U[$0] + 25%U[$2′000′000]
A (=/>/<)? B 50%U[$1′000′000] + 50%U[$1′000′000] (=/>/<)? 50%U[$1′000′000] + 25%U[$0] + 25%U[$2′000′000] the first 50% is the same so it cancels out 50%U[$1′000′000] (=/>/<)? 25%U[$0] + 25%U[$2′000′000] 0.5 > 0.2 The chance to win 2 million doesn’t outweigh how much it would suck to win nothing so therefore the certainty of 1 million is preferable. The negative utility of U[$0] is absorbed by it’s 0 probability coefficient in A.
Or maybe calculation B-1 is just plain wrong, but that would mean we cannot calculate the utility of discrete events and add the utilities up.
Is any of this correct? What kind of calculations would you do?
A bird in the hand is indeed worth 2 in the bush.