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