R. Keeney November 28, 2012. A decision maker wants to behave optimally but is faced with an...
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Transcript of R. Keeney November 28, 2012. A decision maker wants to behave optimally but is faced with an...
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)