DECISION THEORY
We must often make decisions about what
to do even when we’re uncertain of what
will happen.
Suppose a friend offers you the following
choice:
1. Toss a coin: heads you win $2; tails you
lose $2.
OR:
2. Toss a coin twice in a row: two heads,
you win $10; otherwise you lose $1.
Which one should you choose?
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Game 1:
1. 50% chance of winning $2;
2. 50% chance of losing $2.
3. Each consequence “cancels the other
out”.
4. This game offers you no advantage.
Game 2:
1. 25% percent chance of winning:
Pr(H1 & H2) = Pr(H1) x Pr(H2) = ¼.
2. 75% chance of losing.
3. But, if you win, you get $10, while if
you lose you only lose $1.
4. This game seems more appealing.
Let’s get more precise than this.
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When considering an act, we must take into
account both the probability and the utility
(i.e. the benefit or value) of each possible
consequence.
We do this by multiplying the probability of
each consequence by its utility and then
adding them all up.
Game 1: Pr(Win) = 0.5
Pr(Lose) = 0.5
Utility(Win) = +2 Utility(Lose) = -2
Value = 0.5 x 2 + 0.5 x -2 = 0
Game 2: Pr(W) = 0.25
Pr(L) = 0.75
Utility(W) = +10
Utility(L) = -1
Value = 0.25 x 10 + 0.75 x -1 = 1.75
As we suspected, game 2 has a higher
“expected value”.
On average you will win $1.75 per game.
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Notation
Acts: bold capital letters: A, B, C …
Consequences: capital letters: A, B, C …
Utility (of a consequence): U(C)
Probability of C given A: Pr(C/A)
Expected value of an Act: Exp(A)
The Expected value of an act is the sum of
the products (utilities x probabilities)
In other words:
Exp(A) = [Pr(Ci)U(Ci)]
Where C1, C2, … Cn are the consequences
of A and i = 1, 2, … n.
Notice that we leave out the conditional
probability because it’s clear we’re talking
about act A. Technically, it should be there.
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Example: Suppose your friend charges you
$2 to enter game 2. What is the expected
value of the game now?
A: play game 2 for $2.
C1: win
U(C1): 8
C2: lose
U(C2): -3
Exp(A) = (0.25 x 8) + (0.75 x -3)
= -0.25
Now the game is advantageous to your
friend, i.e. in the long you will lose 25 cents
per game you play.
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Fair Price
A fair price for a risk is the price at which
neither buyer nor seller has an advantage,
i.e. the price at which:
Exp(A) = 0
What is the fair price for game 2?
B: play game 2 without entry fee
Exp(B) = 1.75.
To set Exp(B) = 0 we must set the price at
$1.75. In that case:
Exp(B) = (0.25 x 8.25)+(0.75 x –2.75)
=0
The fair price for game 2 is $1.75.
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Should you refuse to play for $2?
It depends on your values:
1. Perhaps you enjoy the game more than
the money you win.
2. Perhaps you enjoy spending time with
your friend.
3. Perhaps the thrill of gambling is
rewarding to you.
4. Etc…
If you value the time with your friend at, say,
$5, then even at a charge of $2, playing the
game once has an expected value of $4.75.
It is not necessarily irrational to act even
when the expected value of the act is
negative.
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St. Petersburg Paradox
“Paradox”: An argument that apparently
derives self-contradictory or absurd
conclusions by reasoning from apparently
acceptable premises.
The game: Toss a coin.
- If it falls heads, you win $2. Game stops.
- If not, you toss again.
- If it falls heads, win $4, game stops.
- If not, toss again.
- If heads, win $8, game stops.
- And so on (until heads is thrown).
(Notice you can’t lose money)
S: play the game
N: game stops on the nth toss.
U(N): 2n
Pr(N): (½)n
What is a fair price to enter this game?
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We need to calculate Exp(S):
Exp(S) = Pr(N1)U(N1) + Pr(N2)U(N2) + …
= ½ x 2 + ¼ x 4 + 1/8 x 8 + 1/16 x 16 + …
=1+1+1+1+1+…
= Infinite!
The fair price for this game is that for which
Exp(S) = 0, i.e. infinity.
You should be willing to pay all the money
you, your friends and your family have or
will have just to play this game.
Are you willing to pay this?
What’s gone wrong here?
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Possible answers:
1. The game is fair at any price. Pay
whatever is asked!
2. In reality the game would end in a finite
time so you don’t really need to pay an
infinite amount. Pay a small amount.
3. Exp(S) is not really defined because
there is no upper bound. No real problem
has been raised.
4. You shouldn’t be willing to pay an infinite
amount because after a finite number of
tosses, the value of further winning goes
down (diminished marginal utility). Pay a
lot, but not too much.
5. The utility of a large win is great, but the
chances are extremely small. Don’t pay
very much to play.
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Choosing among possible acts
Suppose I go to the store to buy some
chips. Why do I do this?
1. I believe that the store sells chips.
2. I want some chips.
Decisions depend on:
 What we believe.
 What we want.
So:
 We can represent beliefs by
probabilities.
 We can represent desires by utilities.
In this way we can calculate the expected
value of an act easily enough. How do we
use the result?
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Expected value rule:
Act so as to maximize expected value.
Lottery A:
Lottery B:
100 tickets sold
1,000,000 sold
Prize: $10,000
Prize: $100,000
Tickets: $150.00
Tickets: $1
A: buy a ticket for Lottery A
B: buy a ticket for Lottery B
Exp(A) = 0.01 x 9850 + 0.99 x -150
= -$50
Exp(B) = 0.000001 x 99,999 +
0.999999 x -1
= -0.9
The expected value rule says to buy a ticket
for Lottery B over one for Lottery A.
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Evaluating the rule
Consider a game where you toss a coin and
win $1000 if it comes up heads and lose
$1000 if it comes up tails.
The expected value of this game is 0.
The expected value of declining to play is 0.
The expected value rule says that each
choice is of equal value and you should not
prefer one to the other.
Is this right?
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What if you’re risk averse, i.e. more afraid
of risking money than gaining money?
Then you’d choose to decline the game.
If you aren’t very wealthy, then losing $1000
is far worse than gaining $1000. So, the
risk of losing might outweigh the benefit of
winning even though each is equally
probable.
Some say it is irrational to violate the
expected value rule. Is this just arrogance?
If not, what’s wrong with the rule?
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Utiles
There are other values than money, for
example, the pleasure of playing a game
and the desire to remain financially solvent.
We shouldn’t only measure utility in dollars.
We shall use the term utiles to indicate a
general measure of value.
For example, you might value $1000 at
+1000 utiles, but the risk of gambling might
be -100 utiles. In that case, the expected
value of the coin toss game is negative
(i.e. -200).
Perhaps, then, we should follow the
expected value rule but broaden our notion
of value to include risk aversion.
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Problem: Utility (valuing something) and
risk aversion are difficult if not impossible
to compare. It seems artificial to try to
combine them into one measure called
“utility”. How could you do it?
You could give numbers to each, but they
appear to be on different dimensions. Do
you think we can save the expected value
rule in this way?
Insurance companies assume that we
dislike risk enough to pay more money that
we will (on average) get back.
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We must be careful to consider all possible
consequences of an act.
If we don’t, and we make the wrong
decision, it does not follow that that
expected value rule is wrong.
We might also disagree about the
probability of an event or its utility.
Just because we do, it does not mean
that the expected value rule is wrong.
Accident
No Accident
Explorers
Probability
1/2500
2499/2500
Consequence
Not so bad
Tremendous
Accident
No Accident
Safety-first
Probability
1/430
429/430
Consequence
Many deaths
Not so great
Even a good rule can lead to disagreement.
What do you think of the expected value
rule?
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Pascal’s Wager
Q: Should you force yourself to believe in
God?
Pascal: Yes. Here’s why:
Either God exists or doesn’t exist (G v ~G)
1. If you believe and God exists, you are
rewarded with infinite bliss. 2. If you believe
and God doesn’t exist, you don’t lose very
much (a few Sunday mornings).
3. If you refuse to believe and God exists,
you are punished with infinite torture. 4. If
you refuse to believe and God doesn’t exist,
you don’t gain very much (a few Sunday
mornings).
Therefore, you gain more and lose less by
believing, whether or not God exists.
So, don’t try to convince yourself, just
believe!
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DECISION PROBLEMS:
 A partition of possible states of affairs.
 The possible acts one can undertake.
 The utilities of the consequences of
each act in each state of affair.
A partition is a set of exhaustive (one must
occur) and mutually exclusive (only one
may occur) possibilities.
Pascal’s Partition:
~B
B
G
-
+
~G
0
0
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If in at least one state of affairs, one act has
more utility than every other act, and if in no
state of affairs does it have less utility, then
this act dominates the others.
Clearly, B dominates ~B.
Dominance Rule:
If one act dominates the others, do it.
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Warning:
The Dominance Rule only works if your acts
don’t causally influence the states of affairs
on the partition. Why?
Well, imagine if refusing to believe in God
caused God to respect you and so reward
you a little bit. Assume also that believing in
God caused God to think you were
spineless and punish you a bit. Then:
~B
B
G
+
-
~G
0
0
Now ~B dominates. Clearly Pascal
assumes that your choice whether to
believe doesn’t causally influence the
possible states of affairs in the partition.
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Improved Dominance Rule:
If no act in the set of acts has any causal
influence on the states of affairs in the
partition, then if one act dominates, do it.
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But what if you aren’t indifferent about going
to church, praying, etc.?
That is, what if you value the fun you get
from free Sunday mornings and disvalue
the boredom of attending church? In that
case:
~B
B
G
-
+
~G
+
-
B doesn’t dominate: it has a better payoff in
one state of affairs but a lower payoff in the
other.
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Pascal’s response: use probability
assignments and the expected value rule:
No matter how small you think it is, there is
some finite probability that God exists:
Pr(G)>0
Exp(B) = Pr(G)U(B,G) + Pr(~G)U(B,~G)
But U(B,G) is infinite:
Exp(B) = +
Exp(~B) = Pr(G)U(~B,G) + Pr(~G)U(~B,~G)
No matter how high, U(~B,~G) is still
finite and U(~B,G) infinitely bad:
Exp(~B) = -
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So according to Pascal a decision problem
consists of:
 A partition of possible states of affairs.
 The possible acts one can undertake.
 The utilities of the consequences of
each act in each state of affair.
And:
 A class of admissible probability
assignments to the states of affairs in the
partition.
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Pascal claims:
If in every admissible probability distribution,
one act has greater expected value than
every other act, then this act dominates the
others in expected value:
Dominant Expected Value Rule
If one act dominates all the others in
expected value, DO IT!
What do you think?
 Is Pascal’s argument a good one?
 How would you criticize it?
 Is belief in God a “live possibility” for you?
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Homework
 Do the exercises at the end of chapters 8,
9 and 10.
 Go over the examples in those chapters to
get clear on the ideas from today’s
lecture.
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