R. Keeney
April 4, 2012

A decision maker wants to behave optimally
but is faced with an opponent
 Nature – offers uncertain outcomes
 Competition – another optimizing decision maker

We focus on simple examples using payoff
matrix
 Decisions for one actor are the rows and for the
other are the columns
 Intersecting cells are the payoffs
▪ Bimatrix (two payoffs in the cells)


One decision maker has to decide whether or
not to carry an umbrella
Decisions are compared for each column
 If it rains, Umbrella is best (5>0)
 If no rain, No Umbrella is best (4>1)
Rain No Rain
Umbrella
5
1
No Umbrella
0
4

The play made by nature (rain, no rain)
determines the decision maker’s optimal
strategy
 Assume I have to make the decision in advance of
knowing whether or not it will rain
Rain No Rain
Umbrella
5
1
No Umbrella
0
4


I know that rain is possible, but no idea how
likely it is to occur
Maxi-min decision making helps us formulate
a plan in an optimal fashion
 Maximize the minimums for each decision
▪ If I take my umbrella, what is the worst I could do?
▪ If I don’t take my umbrella, what is the worst I could do?
Rain No Rain
Umbrella
5
1
No Umbrella
0
4

Comparing the two worst case scenarios
 Payoff of 1 for taking umbrella
 Payoff of 0 for not taking umbrella

An optimal choice under this framework is
then to take the umbrella no matter what
since 1 > 0
Rain No Rain
Umbrella
5
1
No Umbrella
0
4

A lot of decisions are made this way
 Identify the worst that could happen, choose a
course that has a “worst case scenario” that is
least detrimental

Framework implies that people are risk
averse
 Focus on downside outcomes and try to avoid the
worst of these

Assumes probabilistic knowledge of
outcomes is not available or not able to be
processed

What if I know probabilities of events?
 Wake up and check the weather forecast, tells me
50% chance of rain

Take a weighted average (i.e. the expected
value) of outcomes for each decision and
compare them
Umbrella
No Umbrella
Rain
(p=0.5)
5
0
No Rain
(p=0.5)
1
4
Rain No Rain
EV
(p=0.5) (p=0.5) (Sum over row)
Umbrella 5*0.5
1*0.5
3.0
No
0*0.5
4*0.5
2.0
Umbrella

Given the probability of rain, the EV for
taking my umbrella is higher so that is the
optimal decision
Rain
No Rain
EV
(p=0.25) (p=0.75) (Sum over row)
Umbrella 5*0.25
1*0.75
2.0
No
0*0.25
4*0.25
3.0
Umbrella

Given the lower probability of rain, the EV for
taking my umbrella is lower so no umbrella is
my optimal decision
Umbrella
No
Umbrella

Rain
(p=x)
5*x
0*x
No Rain
EV
(p=1-x) (Sum over row)
1*(1-x)
5x+(1-x)
4*(1-x)
0x+4(1-x)
Setting the two values in the last column
equal gives me their EV’s in terms of x.
Solving for x gives me a breakeven
probability.
Umbrella
No
Umbrella


Rain
(p=x)
5*x
0*x
Umbrella:
No Umbrella:




No Rain
EV
(p=1-x) (Sum over row)
1*(1-x)
5x+(1-x)
4*(1-x)
0x+4(1-x)
4x + 1
4 – 4x
Setting equal: 4x + 1 = 4 – 4x -> 8x – 3 =0
X = 0.375
If rain forecast is > 37.5%, take umbrella
If rain forecast is < 37.5%, do not take umbrella

The tough work is not the decision analysis it
is in determining the appropriate
probabilities and payoffs
 Probabilities
▪ Consulting and market information firms specialize in
forecasting earnings, prices, returns on investments etc.
 Payoffs
▪ Economics and accounting provide the framework here
▪ Profits, revenue, gross margins, costs, etc.
Rain
No Rain
Wear clothing
100
100
Wear no clothing
-100
-50
 Decision is whether or not to wear clothing
 If it rains prefer to wear clothing
▪ Get sick from rain and get arrested
 If it doesn’t rain prefer to wear clothing
▪ Don’t get sick but still arrested

Wearing clothing is a Dominant Decision
 Nature’s play has no influence on the decision
 Weather effects how much and what type of clothing just
as it effects our decision on umbrella (where we saw a split
decision)
Player 1
Action 1
Action 2
Player 2
Action 1
Action 2
P1, P2
P1, P2
P1, P2
P1, P2
Each player has two actions and each player’s action has an
impact on their own and the opponent’s payoff.
Payoffs are listed in each intersecting cell for player 1 (P1)
and player 2 (P2).
Prisoner 2
Prisoner 1
Confess
Don’t Confess



Confess
P1 = Life jail
P2 = Life jail
P1 = Death
P2 = Free
Don’t Confess
P1 = Free
P2 = Death
P1 = 1 year jail
P2 = 1 year jail
Two criminals apprehended with enough evidence
to prosecute for 1 year sentences
Suspected of also committing a murder
Outcomes range from going free to death penalty
Prisoner 2
Prisoner 1
Confess
Don’t Confess



Confess
P1 = Life jail
P1 = Death
Don’t Confess
P1 = Free
P1 = 1 year jail
If Prisoner 2 confesses then prisoner 1 optimally
confesses since: Life jail > Death
If Prisoner 2 does not confess then prisoner 1
optimally confesses since: Free > 1 year in jail
Confession is a dominant decision for prisoner 1
 Optimally confesses no matter what prisoner 2 does
Prisoner 1
Prisoner 2
Confess
Don’t Confess


Confess
P2 = Life jail
P2 = Death
Don’t Confess
P2 = Free
P2 = 1 year jail
Prisoner 2 faces the same payoffs as prisoner 1
Prisoner 2 has same dominant decision to
confess
 Optimally confesses no matter what prisoner 1 does
Prisoner 2
Prisoner 1
Confess
Don’t Confess

Confess
P1 = Life jail
P2 = Life jail
P1 = Death
P2 = Free
Don’t Confess
P1 = Free
P2 = Death
P1 = 1 year jail
P2 = 1 year jail
This is far from the best outcome overall for the prisoners
 If neither confesses, they get only one year in jail
 But, if either does not confess, the other can go free just by
confessing while the other gets the death penalty

Incentive is to agree to not confess, then confess to go free
Company 2
Company 1
Low Prices
High Prices

Low Prices
C1 = 2000
C2 = 2000
C1 = 0
C2 = 13000
High Prices
C1 = 13000
C2 = 0
C1 = 10000
C2 = 10000
Two companies set prices and earn profits
 If C2 sets low price, C1 sets low price 2000>0
 If C2 sets high price, C1 sets low price 13000>10000
▪ Low prices are a dominant decision for C1
Company 2
Company 1
Low Prices
High Prices

Low Prices
C1 = 2000
C2 = 2000
C1 = 0
C2 = 13000
High Prices
C1 = 13000
C2 = 0
C1 = 10000
C2 = 10000
C2 faces the same payoffs
 Also has low prices as a dominant decision

Both earn 2000
 If they collude (with a contract) they could both earn 10000
▪ Illegal contract in most cases

Decision analysis is a more complex world for
looking at optimal plans for decision makers
 Uncertain events and optimal decisions by competitors
limit outcomes in interesting ways
 In particular, the best outcome for both decision makers
may be unreachable because of your opponent’s
decision and the incentive to deviate from a jointly
optimal plan when individual incentives dominate
 Broad application: Companies spend a lot of time
analyzing competition
▪ Implicit collusion: Take turns running sales (Coke and Pepsi)