Decision Theory
Lecture 2
Decision Theory – the foundation of
modern economics
•
Individual decision making
– under Certainty
•
•
•
Choice functions
Revelead preference and ordinal utility theory
Operations Research, Management Science
– under Risk
•
•
•
•
Expected Utility Theory (objective probabilities)
Bayesian decision theory
Prospect Theory and other behavioral theories
Subjective Expected Utility (subjective probabilities)
– under Uncertainty
•
•
•
Interactive decision making
–
–
–
–
•
Decision rules
Uncertainty aversion models
Non-cooperative game theory
Cooperative game theory
Matching
Bargaining
Group decision making (Social choice theory)
– Group decisions (Arrow, Maskin, etc.)
– Voting theory
– Welfare functions
• Individual decision making
– under Certainty
• Revealed preference and utility theory
Choice
Choice function
Utility
U(The Truth) > U(The Matrix)
• Individual decision making
– under Certainty
• Choice functions
Weak axiom of revealed preference (WARP)
NOT ALLOWED
You go to a restaurant in while you are on vacation in Tuscany and
you are given the following menu:
• bistecca
• pollo
???
The cook anounces that he can also serve
• trippa alla fiorentina
• Individual decision making
– under Certainty
• Revelead preference and ordinal utility theory
Choice
Preference relation
≻
Utility function
U(The Truth) > U(The Matrix)
If u() is a utility
function, then any
strictly increasing
transformation g∘u() is
a utility function
representing the same
preferences
The doctrine of utilitarianism saw the maximization of
utility as a moral criterion for the organization of
society. According to utilitarians, such as Jeremy
Bentham (1748–1832) and John Stuart Mill (1806–
1873), society should aim to maximize the total utility
of individuals, aiming for "the greatest happiness for
the greatest number of people". Another theory
forwarded by John Rawls (1921–2002) would have
society maximize the utility of those with the lowest
utility, raising them up to create a more equitable
distribution across society
Choice function
Preference relation
≻
• Individual decision making
– under Certainty
• Operations Research, Management Science
Decision Theory – the foundation of
modern economics
•
Individual decision making
– under Certainty
•
•
•
Choice functions
Revelead preference and ordinal utility theory
Operations Research, Management Science
– under Risk
•
•
•
•
Expected Utility Theory (objective probabilities)
Bayesian decision theory
Prospect Theory and other behavioral theories
Subjective Expected Utility (subjective probabilities)
– under Uncertainty
•
•
•
Interactive decision making
–
–
–
–
•
Decision rules
Uncertainty aversion models
Non-cooperative game theory
Cooperative game theory
Matching
Bargaining
Group decision making (Social choice theory)
– Group decisions (Arrow, Maskin, etc.)
– Voting theory
– Welfare functions
• Individual decision making
– under risk
• Objective probabilities (Expected Utility)
• Subjective probabilities (Subjective Expected Utility)
• Expected utility Theory
– Cardinal utility function
If u(.) is a utility function, then any affine
transformation (au(.)+b, where a>0) is also
a utility function representing the same
preferences
– This is the foundation of game theory – mixed
strategies
– This is the foundation of decision theory under risk –
enables modeling risk attitudes
Normative vs positive decision theory
• Behavioral (positive) economics
– Experiments
– Psychology
– Empirical results
– Behavioral theories
• Traditional (normative) economics
– Mathematics
– Traditional Macro and Micro
Decision Theory – the foundation of
modern economics
•
Individual decision making
– under Certainty
•
•
•
Choice functions
Revelead preference and ordinal utility theory
Operations Research, Management Science
– under Risk
•
•
•
•
Expected Utility Theory (objective probabilities)
Bayesian decision theory
Prospect Theory and other behavioral theories
Subjective Expected Utility (subjective probabilities)
– under Uncertainty
•
•
•
Interactive decision making
–
–
–
–
•
Decision rules
Uncertainty aversion models
Non-cooperative game theory
Cooperative game theory
Matching
Bargaining
Group decision making (Social choice theory)
– Group decisions (Arrow, Maskin, etc.)
– Voting theory
– Welfare functions
• Individual decision making
– under uncertainty
• Decision Rules (in a while)
• Uncertainty/ambiguity aversion models, e.g. Multiple
prior/maximin model of Gilboa, Schmeidler
Subjective probability may
not exist
Decision Theory – the foundation of
modern economics
•
Individual decision making
– under Certainty
•
•
•
Choice functions
Revelead preference and ordinal utility theory
Operations Research, Management Science
– under Risk
•
•
•
•
Expected Utility Theory (objective probabilities)
Bayesian decision theory
Prospect Theory and other behavioral theories
Subjective Expected Utility (subjective probabilities)
– under Uncertainty
•
•
•
Interactive decision making
–
–
–
–
•
Decision rules
Uncertainty aversion models
Non-cooperative game theory
Cooperative game theory
Matching
Bargaining
Group decision making (Social choice theory)
– Group decisions (Arrow, Maskin, etc.)
– Voting theory
– Welfare functions
Individual decision theory vs game theory
Zero-sum games
• In zero-sum games, payoffs in each cell sum up to zero
• Movement diagram
Zero-sum games
• Minimax = maximin = value of the game
• The game may have multiple saddle points
Zero-sum games
• Or it may have no saddle points
• To find the value of such game, consider mixed
strategies
Zero-sum games
• If there is more strategies, you don’t know which one will be part of
optimal mixed strategy.
• Let Column mixed strategy be (x,1-x)
• Then Raw will try to maximize
Zero-sum games
• Column will try to choose x to minimize the upper envelope
Zero-sum games
• Tranform into Linear Programming
Fishing on Jamaica
• In the fifties,
Davenport studied
a village of 200
people on the
south shore of
Jamaica, whose
inhabitants made
their living by
fishing.
• Twenty-six fishing crews in sailing, dugout canoes fish this
area [fishing grounds extend outward from shore about 22
miles] by setting fish pots, which are drawn and reset,
weather and sea permitting, on three regular fishing days
each week … The fishing grounds are divided into inside and
outside banks. The inside banks lie from 5-15 miles offshore,
while the outside banks all lie beyond … Because of special
underwater contours and the location of one prominent
headland, very strong currents set across the outside banks
at frequent intervals … These currents are not related in any
apparent way to weather and sea conditions of the local
region. The inside banks are almost fully protected from the
currents. [Davenport 1960]
Jamaica on a map
Strategies
• There were 26 wooden canoes. The captains of
the canoes might adopt 3 fishing strategies:
– IN – put all pots on the inside banks
– OUT – put all pots on the outside banks
– IN-OUT) – put some pots on the inside banks, some
pots on the outside
Advantages and disadvantages of fishing
in the open sea
Disadvantages
• It takes more time to reach, so
fewers pots can be set
• When the current is running, it
is harmful to outside pots
– marks are dragged away
– pots may be smashed while
moving
– changes in temeperature may kill
fish inside the pots
Advanatages
• The outside banks produce
higher quality fish both in
variaties and in size.
– If many outside fish are available,
they may drive the inside fish off
the market.
• The OUT and IN-OUT
strategies require better
canoes.
– Their captains dominate the
sport of canoe racing, which is
prestigious and offers large
rewards.
Collecting data
• Davenport collected the data concerning the
fishermen average monthly profit depending on
the fishing strategies they used to adopt.
Fishermen\Current
FLOW
NO FLOW
IN
17,3
11,5
OUT
-4,4
20,6
IN-OUT
5,2
17,0
OUT Strategy
Zero-sum game? The current’s problem
• There is no saddle point
• Mixed strategy:
– Assume that the current is vicious and plays strategy FLOW with
probability p, and NO FLOW with probability 1-p
– Fishermen’s strategy: IN with prob. q1, OUT with prob. q2, IN-OUT
with prob. q3
– For every p, fishermen choose q1,q2 and q3 that maximizes:
– And the vicious current chooses p, so that the fishermen get min
Graphical solution of the current’s
problem
21
19
17
15
Solution: p=0.31
IN
13
11
OUT
IN-OUT
9
7
Mixed strategy of the
current
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
5
The fishermen’s problem
• Similarly:
– For every fishermen’s strategy q1,q2 and q3, the
vicious current chooses p so that the fishermen earn
the least:
– The fishermen will try to choose q1,q2 and q3 to
maximize their payoff:
Maximin and minimax
Maximize
13,31
Fishers' mixed strategy
q1
q2
q3
0,67
0,00
0,33
Expected payoff of the current when
FLOW
NO FLOW
probabilities
13,31
13,31
1,00
>=
>=
=
objective function
Value of the
game
minimize
Expected payoff from strategy:
IN
OUT
IN_OUT
probabilities
objective
function
13,31
13,31
12,79
13,31
1,00
Optimal
strategy for the
fishermen
13,31
13,31
1,00
Mixed strategy of the current
p
1-p
0,31
0,69
<=
<=
<=
=
13,31
13,31
13,31
1,00
Optimal
strategy for
the current
Forecast and observation
Game theory predicts
• No fishermen risks fishing
outside
• Strategy 67% IN, 33% IN-OUT
[Payoff: 13.31]
• Optimal current’s strategy 31%
FLOW, 69% NO FLOW
Observation shows
• No fishermen risks fishing
outside
• Strategy 69% IN, 31% IN-OUT
[Payoff: 13.38]
• Current’s „strategy”: 25%
FLOW, 75% NO FLOW
The similarity is striking
Davenport’s finding went unchallenged for several years
Until …
Current is not vicious
• Kozelka 1969 and Read, Read 1970 pointed out a
serious flaw:
– The current is not a reasoning entity and cannot adjust to
fishermen changing their strategies.
– Hence fishermen should use Expected Value principle:
• Expected payoff of the fishermen:
– IN: 0.25 x 17.3 + 0.75 x 11.5 = 12.95
– OUT: 0.25 x (-4.4) + 0.75 x 20.6 = 14.35
– IN-OUT: 0.25 x 5.2 + 0.75 x 17.0 = 14.05
• Hence, all of the fishermen should fish OUTside.
• Maybe, they are not well adapted after all
Current may be vicious after all
• The current does not reason, but it is very risky to fish outside.
• Even if the current runs 25% of the time ON AVERAGE, it might
run considerably more or less in the short run of a year.
• Suppose one year it ran 35% of the time. Expected payoffs:
– IN: 0.35 x 17.3 + 0.65 x 11.5 = 13.53
– OUT: 0.35 x (-4.4) + 0.65 x 11.5 = 11.85
– IN-OUT: 0.35 x 5.2 + 0.65 x 17.0 = 12.87.
• By treating the current as their opponent, fishermen
GUARANTEE themselves payoff of at least 13.31.
• Fishermen pay 1.05 pounds as insurance premium
Optimal
Actual
OUT
Actual (25%)
13.3125
13.291
14.35
Vicious (31%)
13.3125
13.31164
12.85
35%
13.3125
13.3254
11.85
Decision making under uncertainty
FLOW
NO FLOW
MAXIMIN
MAXIMAX
MINIMAX
REGRET
17,3
11,5
11,5
17,3
9,1
-4,4
5,2
13,3125
20,6
17
13,3125
-4,4
5,2
13,3125
20,6
17
13,3125
21,7
12,1
7,2875
Płynie
Nie płynie
0
9,1
OUT
21,7
0
IN-OUT
12,1
3,6
3,9875
7,2875
Fishermen\Current
IN
OUT
IN-OUT
0,67 IN+0,33 IN-OUT
Rybacy\Prąd
IN
0,67 IN+0,33 IN-OUT
Decision making under uncertainty
FLOW
NO FLOW
MAXIMIN
MAXIMAX
MINIMAX
REGRET
17,3
11,5
11,5
17,3
9,1
-4,4
5,2
13,3125
20,6
17
13,3125
-4,4
5,2
13,3125
20,6
17
13,3125
21,7
12,1
7,2875
Płynie
Nie płynie
0
9,1
OUT
21,7
0
IN-OUT
12,1
3,6
3,9875
7,2875
Fishermen\Current
IN
OUT
IN-OUT
0,67 IN+0,33 IN-OUT
Rybacy\Prąd
IN
0,67 IN+0,33 IN-OUT
Decision making under uncertainty
Fishermen\Current
FLOW
NO FLOW
MAXIMIN
MAXIMAX
MINIMAX
REGRET
17,3
11,5
11,5
17,3
9,1
-4,4
5,2
13,3125
20,6
17
13,3125
-4,4
5,2
13,3125
20,6
17
13,3125
21,7
12,1
7,2875
IN
OUT
IN-OUT
0,67 IN+0,33 IN-OUT
Regret matrix
Fishermen\Current
FLOW
NO FLOW
0
9,1
OUT
21,7
0
IN-OUT
12,1
3,6
3,9875
7,2875
IN
0,67 IN+0,33 IN-OUT
Decision making under uncertainty
Fishermen\Current
FLOW
NO FLOW
MAXIMIN
MAXIMAX
MINIMAX
REGRET
17,3
11,5
11,5
17,3
9,1
-4,4
5,2
13,3125
20,6
17
13,3125
-4,4
5,2
13,3125
20,6
17
13,3125
21,7
12,1
7,2875
IN
OUT
IN-OUT
0,67 IN+0,33 IN-OUT
Regret matrix
Fishermen\Current
FLOW
NO FLOW
0
9,1
OUT
21,7
0
IN-OUT
12,1
3,6
3,9875
7,2875
IN
0,67 IN+0,33 IN-OUT
Decision making under uncertainty
Fishermen\Current
IN
OUT
IN-OUT
0,67 IN+0,33 IN-OUT
FLOW
NO FLOW
MAXIMIN
MAXIMAX
Hurwicz
optimism/pessimism index
17,3
11,5
11,5
17,3
11,5α+17,3(1-α)
-4,4
5,2
13,3125
20,6
17
13,3125
-4,4
5,2
13,3125
20,6
17
13,3125
-4,4α+20,6(1-α)
5,2α+17(1-α)
13,3125
13
11
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
21
19
17
15
IN
OUT
IN-OUT
9
7
5
Father: “I want you to marry a girl of my choice”
Son: “I will choose my own bride!”
Father: “But the girl is Bill Gates’ daughter.”
Son: “Well, in that case…ok”
Next, the father approaches Bill Gates.
Father: “I have a husband for your daughter.”
Bill Gates: “But my daughter is too young to marry!”
Father: “But this young man is a vice‐president of the World
Bank.”
Bill Gates: “Ah, in that case…ok”
Finally the father goes to see the president of the World Bank.
Father: “I have a young man to be recommended as a
vicepresident.”
President: “But I already have more vice‐ presidents
than I need!”
Father: “But this young man is Bill Gates’s son‐in‐law.”
President: “Ah, in that case…”