Chapter 2: The Decision Matrix Flashcards
Before making a decision, you should decided what are the relevant ___________, __________, and _____________.
acts; states; outcomes
Decision Matrix
Visualizes a formal representation of a decision problem, graphically.
Decision Tree
- More convenient than Decision Matrix
- Visualizes a formal representation of a decision problem, graphically.
Decision Tree Structure
1) Choice Node (square): You decide to go up or down.
2) Chance Node (circle): Acts
– intermittent: states–
3) Possible Outcomes (squares)
Decision trees represent…
sequential decision problems.
If you want to add more to the decision tree, you add to the rightmost __________ (leaves).
boxes
3 Levels of Abstraction w/ Decision Matrices / Trees
1) Decision Problem
2) Formalization of Decision Problem
3) Visualization of the Formalization.
Formalization of Decision Problem
- Made of information about decision being made
- Comprised of information on decision problem’s acts, states, & outcomes.
How should we symbolize decision problems?
We should prefer to symbolize decision problems as Decisions Under Risk (we know the potential outcomes), NOT Decisions Under Ignorance.
Vector
Ordered List of Mathematical Objects
1st Vector: Acts
2nd Vector: States
3rd Vector: Outcomes
State
Part of the world that is not an act or an outcome.
It is _________ a good idea to formalize your decision problem ___________ using ___________.
NOT; without; states
**The __________ of an outcome should be ___________ ____________ of whether or not the __________ actually occurs.
value; casually independent; state
Outcomes of decision problems HAVE to be ____________ from _________ to __________.
ranked; worst; best
Utilities (Value)
(aka Values until Ch. 5)
Numbers that assign comparative evaluations of value.
Ordinal Scale
- 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.
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 ____ ≥ _____.”
invariant; positive monotone transformation; order; outcomes; preserved; f(x) ≥ f(y); x ≥ y.
Cardinal Scales
- Embody more information than cardinal scales.
- Measure objects numerically, and all differences and ratios between measurement points are preserved across transformations of scale.
Interval Scales
- Measure objects numerically, and preserves (& accurately reflects) all differences between measurement points across transformation of scale.
Unlike ____________ scales, interval scales aren’t based on _____________ / _____________ ranking.
Ordinal; qualitative; preferential
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 “____=___ ____ + ____
invariant; positive linear transformation; ordering; outcomes; positive; constant; gaining; losing; y=ax+b
f’(x)= k * f(x) + m
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.
f; x; interval scale; f(x) ≥f (y); f’; k; m; f’(x)= k * f(x) + m; PERMISSABLE
Interval Scales accurately reflect _____________, NOT _________.
differences; ratios
For a positive linear transformation to be permissable for in interval scale, the ratio of ____________ between _____________ ____________ must be preserved.
differences; measurement points
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
cannot; twice
Ratio Scale
- Measures objects numerically such that all ratios between measurement points are preserved across transformation process of scale.
Ratio scales accurately reflect _____________, unlike interval scales.
ratios.
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 “___=____ ____.”
invariant; multiplication; positive constant; ordering of the outcome; positive constatn; y=mx
f’(x)= k * f(x)
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.
f; x; ratio scale; f(x) ≥f (y); f’; k; f’(x)= k * f(x); PERMISSABLE
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?
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.
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 ____________ ______________.
positive constant
Act
- Function from a set of states to a set of outcomes.
- Device that transforms the state of the world to another state.
Generic Acts
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
Particular Acts
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.
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).
particular; 2; 2; agent; person; time; time; pairs;
There are particular acts that are apart of a ______-____________ set.
non-identical
Rival Formalizations
Arises when 2+ formalizations are equally reasonable & better than all other formalizations.
