AGEC 352
Notes on Decision Analysis
April 2012
Decision analysis is an increasingly common tool used in strategic management. Decision
analysis is rooted in Game Theory and generally refers to cases where a decision maker is
optimizing in the presence of uncertainty (think of nature as the opponent) or against
another agent who is trying to optimize (competitive game). An example of decision making
under uncertainty might be the decision of whether or not to carry an umbrella given that
rain might occur.
In the table below, we have a payoff matrix. These can be thought of as the utility from
making either decision (carry an umbrella or not) when one of the two states of nature
occurs (it rains or it does not).
Umbrella
No Umbrella
Rain
5
0
No Rain
1
4
In decision analysis we want to come up with decision rules that allow us to assert that the
decision being made is optimal. In the above case first assume that the decision maker is told
with certainty that it either is raining or it is not raining. Considering either case we arrive at
the following decision rules given the payoffs.
Decision rules:
If I know it will rain I want to have my umbrella (5>0).
If I know it will not rain I want to not carry an umbrella (4>1).
This leads to a split-decision framework. If one event occurs (Rain) my choice is to make
one decision (carry an umbrella) if the other event occurs (No Rain) my choice is the
opposite. Split decisions are interesting because the decision rules associated with them
under uncertainty is complex.
Assume that I don’t know with certainty whether or not it will rain, I just know that it is
possible. Then we might want to adopt the Maxi-min (often called the safety-first rule) that
reflects which of the worst-case scenarios I prefer. We often speak of people using the Maximin rule as being risk averse, meaning they like to avoid risks.
The first step in Maxi-min decision making is to find the worst-case scenario for any
decision I could make (umbrella and no umbrella). The second step is to find the maximum
of the worst case scenarios. Finally, we make the decision that is associated with the best
worst case scenario.
Maxi-min decision making steps:
Step 1:
What are the worst outcomes for each decision I could make?
Umbrella  No rain = 1
Step 2:
No UmbrellaRain = 0
Find the maximum of the worst outcomes.
The maximum of the worst case payoffs is 1.
Step 3:
Which decision to make?
Carrying an umbrella means I will never get a payoff of less than 1.
Maxi-min is how a lot of decisions get made and it is common to hear a comparison of
worst-case scenarios used as justification for decision making. Still, it would be preferable to
have additional information on the worst outcomes and how likely they are to occur.
With some extra information such as the percentage chance that rain will occur as given by a
weather forecaster, we can adopt an alternative criterion for making decisions. This one is
called the Expected Value Criterion because it relies on making the decision that has the
best expected value or mean/average outcome given the probabilities of events.
E.g. Rain = 50% chance. (This also means there is a 50% chance it will not rain).
Assume the same payoffs as in the game matrix before and make the decision that offers the
highest expected value given the probabilities attached to different outcomes. The formula is
given below. Where EV(Decision) is the expected value of a given decision, p(i) is the
probability of the event i, and Y(i) is the payoff of a given decision and event i.
EV ( Decision )   piYi
i
Expected value of the umbrella/no umbrella decision = probability of rain*decision value
when it rains + probability of no rain*decision value when it does not rain.
EV (Umbrella) = .5*5 + .5*1 = 3
EV (No Umbrella) = .5*0 + .5*4 = 2
Comparing these two expected values and the probabilities as given by the rain forecast we
would make the decision to take our umbrella.
What if Rain forecast was 25% chance of rain (75% chance no rain).
EV(Umbrella) = .25*5 +.75*1 = 2
EV(No Umbrella) = .25*0 +.75*4 = 3
Comparing these two expected values and the probabilities as given by the rain forecast we
would make the decision to not take our umbrella.
What if we wanted a rule to always follow? This would entail solving for a breakeven rain
forecast percentage such that if the rain forecast were higher we know to take our umbrella
and if it is lower we do not.
Let x = probability of rain given by forecast
Then 1-x = probability of no rain
When would we be indifferent i.e. when would the expected value of taking our umbrella be
the same as the expected value of not taking our umbrella?
We just need to find out for what unknown rain probability (x) the expected values are
equal.
EV(Umbrella)=EV(No Umbrella)
x*5 + (1-x)*1 = x*0 + (1-x)*4
5x + 1 – x = 4 – 4x
4x + 1 = 4 – 4x
4x + 1 +(4x-1) = 4 – 4x +(4x-1)
8x = 3
X= .375
So if the chance of rain is less than 37.5% we never take our umbrella and if it is greater we
take our umbrella.
How could this be used in economics?
Forecasts on prices, profits, returns from investments, etc. are quite common. Consulting
and market information services exist to provide this type of information to aid in decision
making. So a probability for high versus low prices could be incorporated in a
straightforward manner into a decision where the events and payoffs depend on prices.
Competitive Decision Making
Think of the rain-umbrella decision as a game between two players, you and nature where
nature acts randomly. What if the following was your decision problem? What should you do
if it rains? If it does not rain?
Wear clothing
Wear no clothing
Rain
100
-100
No Rain
100
-50
Wearing clothing is always better. When we would always make the same decision regardless
of the move made by nature (or a competing player) we call that decision alternative a
Dominant Decision or Dominant Strategy. If it rains we prefer to wear clothing. If it
does not rain we prefer to wear clothing. Consider the previous case of the umbrella.
Umbrella
No Umbrella
Rain
5
0
No Rain
1
4
Here if it rains we prefer to have the umbrella. If it does not rain, we prefer not to have our
umbrella. The decisions depend on which move nature makes so no dominant decision
occurs, hence the definition above of a split-decision.
When working against another optimizing individual or company dominant decisions can
lead to sub-optimal outcomes for all involved, even though the decisions made by players
are consistent with individual optimization.
The famous prisoner’s dilemma is the example most often used to show that competition
and the use of dominant decisions or strategies being played by competitors can lead to
inferior outcomes.
Two people are arrested for a capital crime meaning they could face death sentences. Each is
questioned separately and probed for a confession that implicates both parties. Both face
misdemeanor charges as well for resisting arrest and public disorder with enough evidence to
convict and place them in jail for 1 year. The game setup is below:
Prisoner1
Confess
Don’t Confess
Prisoner 2
Confess
Don’t Confess
P1: Life sent.
P1: Free
P2: Life sent.
P2: Death
P1: Death
P1: 1 year
P2: Free
P2: 1 year
In each cell of the table there is a payoff for each prisoner (P1 and P2). The bold items
represent optimal choices for each player given a decision by another player. For example, if
prisoner 1 thinks that prisoner 2 will confess, then prisoner 1 is comparing the two
outcomes in the first column (life sentence and death). Assuming the life sentence is
preferable we can state that prisoner 1 will confess if he thinks that prisoner 2 is going to
confess. Moving to the second column, we can evaluate the decision of prisoner 1 if he
thinks that prisoner 2 will not confess. Here we are comparing P1’s second column
outcomes and see that if prisoner 1 confesses while prisoner 2 does not confess that prisoner
1 goes free. The alternative is serving 1 year which we assume to be worse than going free.
As a result, prisoner 1 should always confess since the payoffs are better (i.e. punishments
are lesser) when he does this. Confessing is a dominant decision for P1. It turns out since the
game is symmetric that P2 faces the same decisions and payoffs, and has the same dominant
decision to always confess.
Both prisoners play a dominant decision of confession and both receive a life sentence.
Clearly if the players could collude they could do better and each receive a 1 year sentence,
but in that situation each prisoner has clear incentive to confess and go free leaving the other
prisoner with a death sentence. In order to avoid the death sentence that each prisoner is
afraid will occur because of the opponent confessing the life sentence tends to be a
competitive equilibrium. A competitive equilibrium occurs when both agents behave
optimally and neither has any incentive to deviate because they fear something worse will
happen to them.
Company 1
Low Prices
High Prices
Company 2
Low Prices
High Prices
C1: 2000
C1: 13000
C2: 2000
C2: 0
C1: 0
C1: 10000
C2: 13000
C2: 10000
The prisoner dilemma can be applied to two companies selling the same product and
thinking of setting a high price. If one company sets a high price the other company can earn
more profits by setting a low price and cutting the other company out of the market. If they
both set high prices they share sales and earn large profits. If they both set low prices they
share sales and earn small profits. The fear of setting high prices and being undercut makes
the low price decisions and low profits for both firms outcome an equilibrium. Here again,
both companies are playing a dominant decision of low prices and reach an equilibrium. The
equilibrium is not the best outcome they could achieve but they would have to collude with
an enforceable agreement to reach any other outcome which is generally against the law.