Chapter 2: The Decision Matrix

Question 1

Before making a decision, you should decided what are the relevant ___________, __________, and _____________.

Answer 1

acts; states; outcomes

Question 2

Decision Matrix

Answer 2

Visualizes a formal representation of a decision problem, graphically.

Question 3

Decision Tree

Answer 3

• More convenient than Decision Matrix
• Visualizes a formal representation of a decision problem, graphically.

Question 4

Decision Tree Structure

Answer 4

1) Choice Node (square): You decide to go up or down.
2) Chance Node (circle): Acts
– intermittent: states–
3) Possible Outcomes (squares)

Question 5

Decision trees represent…

Answer 5

sequential decision problems.

Question 6

If you want to add more to the decision tree, you add to the rightmost __________ (leaves).

Answer 6

boxes

Question 7

3 Levels of Abstraction w/ Decision Matrices / Trees

Answer 7

1) Decision Problem
2) Formalization of Decision Problem
3) Visualization of the Formalization.

Question 8

Formalization of Decision Problem

Answer 8

• Made of information about decision being made
• Comprised of information on decision problem’s acts, states, & outcomes.

Question 9

How should we symbolize decision problems?

Answer 9

We should prefer to symbolize decision problems as Decisions Under Risk (we know the potential outcomes), NOT Decisions Under Ignorance.

Question 10

Vector

Answer 10

Ordered List of Mathematical Objects

1st Vector: Acts
2nd Vector: States
3rd Vector: Outcomes

Question 11

State

Answer 11

Part of the world that is not an act or an outcome.

Question 12

It is _________ a good idea to formalize your decision problem ___________ using ___________.

Answer 12

NOT; without; states

Question 13

**The __________ of an outcome should be ___________ ____________ of whether or not the __________ actually occurs.

Answer 13

value; casually independent; state

Question 14

Outcomes of decision problems HAVE to be ____________ from _________ to __________.

Answer 14

ranked; worst; best

Question 15

Utilities (Value)

Answer 15

(aka Values until Ch. 5)
Numbers that assign comparative evaluations of value.

Question 16

Ordinal Scale

Answer 16

• Measure objects without making comparisons to differences and ratios between measurement points across all transformations of scale.
• Numbers merely reflect preferfable qualitative ranking, not the degree of HOW MUCH you prefer one thing over another.
• Doesn’t specify the quantitative distance between numerical rankings.

Question 17

Ordinal Scales are ______________ up to ________________ ________________ _____________. This means ____________ of the _____________ in the original scale of the objects is ______________. This change is represented by “___(___) ≥ ____(___) if and only if ____ ≥ _____.”

Answer 17

invariant; positive monotone transformation; order; outcomes; preserved; f(x) ≥ f(y); x ≥ y.

Question 18

Cardinal Scales

Answer 18

• Embody more information than cardinal scales.
• Measure objects numerically, and all differences and ratios between measurement points are preserved across transformations of scale.

Question 19

Interval Scales

Answer 19

• Measure objects numerically, and preserves (& accurately reflects) all differences between measurement points across transformation of scale.

Question 20

Unlike ____________ scales, interval scales aren’t based on _____________ / _____________ ranking.

Answer 20

Ordinal; qualitative; preferential

Question 21

Interval Scales are ____________ up to _____________ ______________ ______________. This means the ____________ of the ___________ in the scale will be preserved across transformation if the scale values are multiplied by a _______________ number added by a ____________. This preserves the scale without ____________ or ______________ anything. This can be represented by “____=___ ____ + ____

Answer 21

invariant; positive linear transformation; ordering; outcomes; positive; constant; gaining; losing; y=ax+b

f’(x)= k * f(x) + m

Question 22

A function ____ that returns a real number _____ is an __________ ______________ iff. condition 1 [ __(__) ≥ ___(__) iff. ___ ≥ ___ ] is met AND for every function _____ there is a positive number ___ and a constant ____. This is represented by ___(__)= ___ * __(__) + __. This represents all transformations that are _________________ under positive linear transformation.

Answer 22

f; x; interval scale; f(x) ≥f (y); f’; k; m; f’(x)= k * f(x) + m; PERMISSABLE

Question 23

Interval Scales accurately reflect _____________, NOT _________.

Answer 23

differences; ratios

Question 24

For a positive linear transformation to be permissable for in interval scale, the ratio of ____________ between _____________ ____________ must be preserved.

Answer 24

differences; measurement points

Question 25

Examples of interval Scales:

Temperature is something that can be transformed on the interval scale–

New York– 32° F
Tokyo– 64° F

The claim _________ be made under positive linear transformation that “it’s _______ as warm in Tokyo as it is in New York.”

This is because though 32 x 2= 64, at Celcius their temperatures are different: 0°C and 17.8°C respectively. 0°C x 2 ≠ 17.8°C

Answer 25

cannot; twice

Question 26

Ratio Scale

Answer 26

• Measures objects numerically such that all ratios between measurement points are preserved across transformation process of scale.

Question 27

Ratio scales accurately reflect _____________, unlike interval scales.

Answer 27

ratios.

Question 28

Ratio Scales are __________________ up to ______________ by a ____________ ____________. The ______________ of _______________ will be preserved across transfromation of the scale when the scale values are multiplied by a __________ ______________. This can be represented by “___=____ ____.”

Answer 28

invariant; multiplication; positive constant; ordering of the outcome; positive constatn; y=mx

f’(x)= k * f(x)

Question 29

A function ____ that returns a real number _____ is an __________ ______________ iff. condition 1 [ __(__) ≥ ___(__) iff. ___ ≥ ___ ] is met AND for every function _____ there is a positive constant ____. This is represented by ___(__)= ___ * __(__). This represents all transformations that are _________________ under multiplication by a positive constant.

Answer 29

f; x; ratio scale; f(x) ≥f (y); f’; k; f’(x)= k * f(x); PERMISSABLE

Question 30

Why is it that Interval Scales uses the “+b” in their equation (y=mx+b) representing positive linear transformation, but Ratio Scales doesn’t use the “+b” in their equation representing multiplication by a positive constant?

Answer 30

In interval scales, the +b represents an arbitrary zero point on the graph in transformations. Meaning, across transformation from one scale to another, 0 could change into a completely different value (ARBITRARY).

+b for ratio scale transformations means the zero point is fixed. Meaning, across transformation to different scales, zero stays zero.

Question 31

Examples of Ratio Scales:

Mass and length can be transformed in ratio scales–

1 mile x 1.61 km/ 1mile = 1.61 km

10 mile x 1.61 km/ 1 mile = 16.1 km

20 miles x 1.61 km/ 1 mile = 32.2 km

Here, we take the scale values given to us, and multiply by a ____________ ______________.

Answer 31

positive constant

Question 32

Act

Answer 32

• Function from a set of states to a set of outcomes.
• Device that transforms the state of the world to another state.

Question 33

Generic Acts

Answer 33

Instantiated by different agents at different time intervals.

-anyone could do these actions at any given moment (i.e. walking, swimming, sailing etc)

• Columbus voyaging to the Americas and Cook to the Southern Hemisphere are both instantiations of the same generic act: sailing

Question 34

Particular Acts

Answer 34

Instantiated by specific agents at specific time intervals.

• Columbus voyaging specifically to the Americas at the time he did, whilst James Cook voyaged to the Southern Hemisphere at the time he did are separate instantiations of a particular act.

Question 35

Alternative Act:
The set A is an “alternative set” iff.

• every member of A is a particular act.
• A has at least ___ different members (has to be at least ___ available actions)
• the members of A should be…
• _________-identical (same __________should be doing the alternative)
• _________-identical (the acts should have taken place at the same ______)
• performable
• incompatible in _______ (can’t be done simoultaneously)
• jointly exhaustive (at least one of the events has to happen).

Answer 35

particular; 2; 2; agent; person; time; time; pairs;

Question 36

There are particular acts that are apart of a ______-____________ set.

Answer 36

non-identical

Question 37

Rival Formalizations

Answer 37

Arises when 2+ formalizations are equally reasonable & better than all other formalizations.