Decision AnalysisApril 11, 2011
Game TheoryFrame WorkPlayers◦Decision maker: optimizing agent◦Opponent
Nature: offers uncertain outcome Competition: other optimizing agent
Strategies/actionsOutcomes
Payoff Matrix
• 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)
State 1 State 2
Act 1 Payoff 1,1 Payoff 2,1
Act 2 Payoff 1,2 Payoff 2,2
Decision TheoryNature is the opponentOne decision maker has to decide whether
or not to carry an umbrellaDecisions 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
Split Decision
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
Uncertainty
In know that rain is possible, but I have no idea how likely it is to occur.
How does the decision maker choose? Two Methods◦ Maximin: largest minimum payoff (caution)◦ Maximax: largest maximum payoff
(optimism)
Maximin (safety first rule)Maximize the minimums for each decision◦If I take my umbrella, what is the worst I can
do?◦If I don’t take my umbrella, what is the worst I
can 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
Framework implies that people are risk averse Focus on downside outcomes and try to
avoid the worst of these
Maximin (safety first rule)
MaximaxMaximize the maximums for each decision◦If I take my umbrella, what’s the best I can do?◦If I don’t take my umbrella, what’s the best I can
do?
Rain No Rain
Umbrella 5 1
No Umbrella 0 4
Maximax Comparing the two best case scenarios
Payoff of 5 for taking umbrella Payoff of 4 for not taking umbrella
An optimal choice under this framework is then to take the umbrella no matter what since 5 > 4
Both methods assume probabilistic knowledge of outcomes is not available or not able to be processed
Expected Value Criteria 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
Rain(p=0.5)
No Rain (p=0.5)
Umbrella 5 1No Umbrella 0 4
Fifty Percent Chance of Rain
Given the probability of rain, the EV for taking my umbrella is higher so that is the optimal decision
Rain(p=0.5)
No Rain(p=0.5)
EV(Sum over
row)Umbrella 5*0.5 1*0.5 3.0
No Umbrella
0*0.5 4*0.5 2.0
25 Percent Chance of Rain
Given the lower probability of rain, the EV for taking my umbrella is lower so no umbrella is my optimal decision
Rain(p=0.25)
No Rain(p=0.75)
EV(Sum over
row)Umbrella 5*0.25 1*0.75 2.0
No Umbrella
0*0.25 4*0.25 3.0
Common Rule for EV: a breakeven probability of rain
Probability (x) that event happened and probability (1-x) that something else happens
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.
Rain(p=x)
No Rain(p=1-x)
EV(Sum over
row)Umbrella 5*x 1*(1-x) 5x+(1-x)
No Umbrella
0*x 4*(1-x) 0x+4(1-x)
Common Rule for EV: a breakeven probability of rain
Umbrella: 4x + 1 No Umbrella: 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
Rain(p=x)
No Rain(p=1-x)
EV(Sum over row)
Umbrella 5*x 1*(1-x) 5x+(1-x)
No Umbrella
0*x 4*(1-x) 0x+4(1-x)
In PracticeThe 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.
Competitive Games: Bimatrix
Player 1Player 2
Action 1 Action 2
Action 1 P1, P2 P1, P2
Action 2 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.• Both players decide at once•Payoffs are listed in each intersecting cell for player 1 (P1) and player 2 (P2).
Prisoner’s DilemmaTwo criminals arrested for both murder
and illegal weapon possessionPolice have proof of weapon violation
(each get 1 year)Police need each prisoner to confess to
convict for murder (death penalty) If both keep quiet, each only get 1 yearIf either confesses, both could be
sentenced to death
Prisoner’s Dilemma
Prisoners are separated for questioningOutcomes range from going free to death
penalty
Prisoner 1Prisoner 2
Confess Don’t Confess
Confess P1 = Life jailP2 = Life jail
P1 = FreeP2 = Death
Don’t Confess P1 = DeathP2 = Free
P1 = 1 year jailP2 = 1 year jail
What will they do? Prisoner 1’s decision
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 1Prisoner 2
Confess Don’t Confess
Confess P1 = Life jail P1 = Free
Don’t Confess P1 = Death P1 = 1 year jail
What will they do? Prisoner 2’s decision
Prisoner 2 faces the same payoffs as prisoner 1Prisoner 2 has same dominant decision to
confess◦Optimally confesses no matter what prisoner 1 does
Prisoner 2Prisoner 1
Confess Don’t Confess
Confess P2 = Life jail P2 = Free
Don’t Confess P2 = Death P2 = 1 year jail
Both confess, Both get life sentences
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
Prisoner 1Prisoner 2
Confess Don’t Confess
Confess P1 = Life jailP2 = Life jail
P1 = FreeP2 = Death
Don’t Confess P1 = DeathP2 = Free
P1 = 1 year jailP2 = 1 year jail
Summary 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)
And for Agriculture…Objective: maximize gross product◦St.: resource availability and requirement
Decision variables:Cropping patterns
Size and equipment types Uncertainties:◦ Weather conditions◦ Market prices◦ Crop and animal disease