16.422
Human Supervisory Control
Classical Decision Theory
& Bayes’ Theorem
Decision Theory & Supervisory
Control
? Two broad areas of decision theory
– Normative
? Prescriptive
– Descriptive
? Why & how do we make decisions?
– How does technology support or detract from
“optimal” decision making
? Informing Design
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? Normative: How decisions should be made
? Prescriptive: How decisions can be made, given human limitations
? Descriptive: How decisions are made
Elements of Decision Making
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? The rational decision making process
– Define the problem
– Information gathering
– Identify alternatives
– Evaluate alternatives
– Select and implement decision
? Why decisions often go wrong:
– Certainty vs. uncertainty
– Bounded rationality
– Nonlinearity
– Habits & heuristics
– Path of least resistance
Classic Decision Theory
? Maximizing expected value of the outcome
– Primary assumption of rationality
? Mathematical models of human decision making
? Following assumptions are made about how people
make decisions:
– All alternatives are considered
– Information acquisition is perfect
– Probabilities are calculated correctly
? Problems:
– Humans are not rational decision makers
– No universal agreement on the worth associated with various
outcomes.
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Basic Concepts in Decision Analysis
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? State: A description of the world
? Events: A set of states {S
1
, S
2
,…, S
j
}
? Consequences: A set of states
? Acts: Results from decisions
S
1
S
2
Act 1 (A
1
)C
11
C
12
Act 2 (A
2
)C
21
C
22
Basic Terminology
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? Ordering of preferences
– Alternatives can be quantified and ordered
– A > B, A preferred to B
– A = B, A is equivalent to B
–A ≥ B, B is not preferred to A
? Transitivity of preference
–if A
1
≥ A
2
, & A
2
≥ A
3
, then A
1
≥ A
3
,
Decision Trees
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? Decisions over time and/or events
Decision
Node
S
1
S
2
C
11
C
12
C
21
C
22
A
2
A
1
A
1
A
2
Chance
Nodes
Decision Making Under Certainty
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? Each alternative leads to one and only one
consequence
– Consequences are known
? Lexicographic ordering
? Dominance
? Satisficing
? Maximin
? Minimax
Lexicographic Ordering
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? All options are first compared in terms of the
criterion deemed most important.
– If there is a unique best performing option, that option
is selected as the most preferred.
? In case of a tie, the selection process moves to the
second ranked criterion
– Seeks the remaining option which scores best on the
second criterion.
? In case of a tie on the second criterion, the process
is repeated for options tying in terms of both the
first and second ranked criteria,
– And so on until a unique option is identified or all
criteria have been considered.
Dominance
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? Effective for both quick qualitative and
quantitative comparisons
? In the real world, solutions rarely are outwardly
dominant
S
1
S
2
S
3
S
4
13
3
3
2
0
2
20
A
1
0
A
2
0
A
3
-1
Satisficing
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? Otherwise known as Minimum Aspiration
? Select any A
i
over all events j such that
C
ij
≥ aspiration level
? Cease to search for alternatives when you find an
alternative whose expected utility or level of
preference satisfaction exceeds some previously
determined threshold.
? Stopping rule
? Is this in keeping with the rational approach to
decision making?
Maximin
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? Select any A
i
over all events j such that you
minimize the maximum loss
– Maximum of the (row) minima.
– Conservative approach but overly pessimistic
S
1
S
2
S
3
15 -6
-2
1
9
3
A
1
3
A
2
4
A
3
2
Maximin
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? Select any A
i
over all events j such that you
minimize the maximum loss
– Maximum of the (row) minima.
– Conservative approach but overly pessimistic
S
1
S
2
S
3
Row
min
15 -6
-2
1
9
-6
-2
31
A
1
3
A
2
4
A
3
2
Minimax (Regret)
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? Avoid regrets that could result from making a non-
optimal decision.
? Regret: the opportunity loss if A
i
is chosen
– Opportunity loss is the payoff difference between the best
possible outcome under S
j
and the actual outcome from
choosing A
i
.
– Convert payoff matrix to opportunity loss (regret) table
S
1
S
2
S
3
15 -6
-2
A
3
321
Best 15 4 1
9
A
1
3
A
2
4
S
1
S
2
S
3
07
3
A
3
12 2 0
6
A
1
1
A
2
0
Maximin v. Minimax
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? Minimax contains more problem information
through opportunity losses (e.g. actual
monetary losses plus unrealized potential
profits)
– Still conservative but more middle-of-the-road
S
1
S
2
S
3
Maximin Minimax
-6
7
6
12
-2
1
-6
-2
1
15
9
3
A
1
3
A
2
4
A
3
2
The Uncertain State of the World
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Probability
Theory
Objective
Probability
Subjective
Probability
Statistical
Probability
Axiomatic
Probability
Decision Making Under Risk
? Each alternative has several possible
consequences
– The probability of occurrence for each consequence,
C, is known.
? Expected value: E (A
i
) = Σ
j
p
j
(C
ij
)
S
1
S
2
S
3
.3
-6
-2
1
P(S
j
).1 .6
A
1
15
9
3
3
A
2
4
A
3
2
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Decision Making Under Uncertainty
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? Alternatives are known
– Consequences are not known
– Probability of outcomes are not known
? Minimax/Maximin can still be used.
? Utility a key concept
Swerve Don’t
Swerve
Swerve 0,0
10, 0
0, 10
Don’t Swerve -1000, -1000
Other Driver
You
Utility Theory
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? Utility theory is an attempt to infer subjective
value, or utility, from choices.
– Applies to both decision making under risk & decision
making under uncertainty
? Two types
– Expected utility
? Same as EV but utility is the value associated with
some outcome, not necessarily monetary.
? Subjective Expected Utility (SEU)
– Subjective focuses on decision making behavior
? Risk neutral/adverse/seeking
– Multi-attribute utility
? Multiple objective extension of SEU
? Interval scale is critical
MAUT Example
? Want to buy a car based on price, fuel
economy, & reliability.
? Major drawback: Assumption of rationality
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Price Fuel Reliability
.7
156
Geo Metro 100 99 50 164.2
166.6
100
98
.5
20
60
Attribute
Weight
.8
BMW Z3 95
Subaru
Outback
85
Uncertainty, Conditional
Probability, & Bayes’ Theorem
? Bayes’ Theorem
– P(A) - prior probability of A
– P(A|B) - posterior probability
– P(B|A), for a specific value of B, is the likelihood
function for A
– P(B) - the prior probability of B
– P(A|B) ≠ P(B|A)
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)A)P(A|P(B A)P(A)|P(B
A)P(A)|P(B
B)|P(A
+
=
P(B)
A)P(A)|P(B
B)|P(A =
Absolute versus conditional
P(Tails) = .5 vs P(heart|red) = .5
Bayesian Networks
? Bayes nets are models which reflect the states of some
part of a world and describe how those states are related
by probabilities.
– Causal chains
? If it might rain today and might rain tomorrow, what is
the probability that it will rain on both days?
– Not independent events with isolated probabilities.
– Joint probabilities
? Useful for monitoring & alerting systems
? Problems
– Can’t represent recursive systems in a
straightforward manner
– Acquiring probabilities
– The number of paths to explore grows
exponentially with each node
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Good paper - Display of Information for Time-Critical (1995) Eric Horvitz, Matthew Barry
Human probability estimation is unreliable for infrequent events (tails)
References
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? Choices: An Introduction to Decision Theory,
Michael D. Resnick (1987)
? Making Hard Decisions: An Introduction to
Decision Analysis, 2nd ed., Robert T. Clemen,
(1996)
? Decision Analysis,
http://groups.msn.com/DecisionModeling/decision
analysis1.msnw
? Bayes Net Tutorial
http://www.norsys.com/tutorials/netica/nt_toc_A.htm