Decision AnalysisR. Keeney November 28, 2012

Game Theory

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)

Nature is the opponent

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

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 and Maxi-min / Safety First Rule

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

What’s the best worst case scenario?

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

Maxi-min (Safety First)

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

Expected Value Criteria (Mixed strategy)

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

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 practice

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.

Dominant Decisions

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)

Rain No Rain

Wear clothing 100 100

Wear no clothing -100 -50

Competitive Games: Bimatrix

Player 1Player 2

Action 1 Action 2

Action 1 P1, P2 P1, P2

Action 2 P1, P2 P1, P2Each 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’s Dilemma

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 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 1

Prisoner 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

They 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

Price Setting Competitors

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 1Company 2

Low Prices High Prices

Low Prices C1 = 2000C2 = 2000

C1 = 13000C2 = 0

High Prices C1 = 0C2 = 13000

C1 = 10000C2 = 10000

Price Setting Competitors

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

Company 1Company 2

Low Prices High Prices

Low Prices C1 = 2000C2 = 2000

C1 = 13000C2 = 0

High Prices C1 = 0C2 = 13000

C1 = 10000C2 = 10000

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)