PHIL 321 - Midterm
Bad-outcome-implies-bad-decision-fallacy
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Causation vs. correlation
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Clues to behavioral dispositions
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Coherence
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Commitment devices & how emotions act as commitment devices
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Conditionalization
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Emotions as self-control devices
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Frank: emotions and decisions
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How do principle of maximizing expected utility and principle of dominance appear to conflict?
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Incomparable preferences
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Indifference classes
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Interpretations of probabilities: subjective vs. objective interpretations
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Matching law
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Maximizing expected value
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Ordering condition
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Ordinal transformation
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Preference orderings
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Probabilistic independence
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Problem of mimicry
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Problem of priors
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Problems with interpretations of probability
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Rationally required vs. rationally permitted vs. irrational
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State-act dependance
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The cheating problem
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Dutch book theorem
A Dutch book is a set of odds and bets which guarantees the bookie a profit, regardless of the outcome of the gamble. It is associated with probabilities implied by the odds not being coherent. A Dutch Book cannot be made against you, however, if all your bets adhere to the probability axioms.
Positive linear transformation
A function that changes the values of a scale, but keeps their relative magnitudes the same. Two linear scales count as equivalent if and only if they can be obtained from each other by means of positive linear transformation.
Newcomb's paradox
ALSO: The Predictor Paradox One red box, one blue. The blue has nothing in. The red has 1,000. You leave the room and the predictor fills the blue box with 1M, or he doesn't. Which do you choose? Many would simply take the blue box as there's a greater chance that there will be a million in there. But is there? POINT: If you had a friend who could see what was in the boxes, he'd always tell you to take both.
Hatfield and McCoy example
An example of the effect of emotion on decision making. The Hatfield's and McCoy's kept getting vengeance on each other, but had they resolved the issue after the first death they would have saved themselves much more grief.
How ignorance can help in decision making (Gigerenzer)
Approach with a clean slate. No previous biases. No particular mindset or preferences, thus more objective decision making.
Axioms of probability theory
Axiom 1: (a) P(p) is greater than or equal to 0 and less than or equal to 1 (b) P(p/q) is greater than or equal to 0 and less than or equal to 1 Axiom 2: If p is certain, then P(p) = 1 Axiom 3: If p and q are mutually exclusive, then P(p or q) = P(p) + P(q)
Decisions under risk
Decisions in which the probabilities of the given outcomes are known, however these probabilities are not certain
Decisions under ignorance
Decisions in which there nothing, or too little to be useful, known about the potential outcomes except the outcomes themselves. There are 4 methods to make these decisions Practice: What are the 4?
Problems with hypothetical relative frequency
Defines probability in terms of itself. Not a strong argument - subject to the "circularity objection"
Indifferent preferences
Denoted pIq. An agent is indifferent to p or q if he has no preference for one over the other
St. Petersburg paradox
Depends on the assumption that there is no upper bound to the utility scale. You get paid $2 for every flip of the coin, until it lands on heads. Calculating utility, you get: (1/2)2 + (1/4)4 +...+ (1/2n)n = infinity It appears as though the expected utility (or EMV if you like) is infinity. The St. Petersburg game is an infinite lottery.
Diminishing marginal utility of money
EMV does not always suffice for making decisions under risk as the value of money often diminishes given external factors. i.e. paying more than its worth for a house in the mountains b/c you like the solitude. e.x. The basketball ticket example
Ordinal utility functions
Equations for calculating the utility of ordered utilities. While the utilities have to remain in the same order, their 'degree' of difference is allowed to change. (i.e. 6,5,2,1 is equivalent to 20,5,2,1) THESE DO NOT SUFFICE FOR DECISIONS UNDER RISK - magnitude is important in these utilities
How frequency formats make a difference
Example - Roulette results board. i.e. since it's been black for the last 10, it's due to be red.
Problems with maximin
Falls apart with very large values. (ex. In Act 1 you can win either $1 or $1M, and in Act 2 you can win either $3 or $5. Maximin would say pick Act 2)
Satisficing
Finding the first adequate solution, and using that rather than searching for better alternatives.
Reduction-of-compound-lotteries condition
For any lotteries x and y and any numbers a, b, c, d (again btwn 0 and 1 inclusively), if d = ab + (1 -a)c, then L(a, L(b, x, y), L(c,x,y)) I L(d,x,y) In lotteries with more than two stages, there might be several ways to reach a prize. This condition says that an agent must be indifferent btwn the compound lottery and simple one so long as the probabilities of getting prizes are the same.
Continuity condition
For any lotteries x, y and z, if xPy and yPz, then there is some real number a such that 0<a<1(equal too) and y I L(a,x,z). Less formally, this states that if the agent ranks y btwn x and z, there is some lottery with x and z as prizes that the agent ranks along with y.
Mixture condition
If a rational agent is indifferent btwn two acts, the agent will be indifferent between them and the third act of flipping a fair coin and doing the first on heads and the second on tails.
Better-prizes condition
If all other things are equal, the agent prefers one lottery to another just in case the former involves better prizes.
Maximin
In a decision under ignorance, choose the Act that has the highest minimum reward of all the acts
Minimax regret
In a decision under ignorance, choose the Act that has the least highest regret of all the acts. Regrets are found by comparing all rewards to the highest reward for that given act. Displayed in a regret table
Rawls' defense of minimax
In a society, minimizing the maximum creates equality over all the states, kinda close to communism. Rawls says, that behind a veil of ignorance, one should want to live in the society where the worst off people are better off than the worsts of other society.
Optimism-pessimism rule
In decisions under ignorance, the optimism-pessimism rule tells us to calculate the number aMAX + (1-a)min for each act (this is the a-index) and then pick the act who's a-index is maximal. "a" is the "optimism index"
Principle of insufficient reason
In decisions under ignorance, treating all states as equally probable and independent of our choices. Therefore, simply pick the one that has the greatest expected utility.
Rawls' veil of ignorance
In the Rawls v. Harsanyi debate, the "veil of ignorance" is a hypothetical situation in which, when choosing a society one wants to live in, the subject does not know who they are, their social status, their ancestry or their talents.
Calvinism example
In the terms of Newcomb's paradox, God has already given you a place in heaven, or not. You can choose to sin, or not; a life of sin, in this case, is more fun.
Problems with relative frequency view
It obviously has problems with indefinite and infinite totalities, since no exact proportion can every be determined in such cases.
Problems with EMV
Leaves out preferences, possibility of practical problems, as well as no considerations of "on average" or previous cases of the same situation. i.e. in the problem where Emily is buying a car. New or used? Used has best EMV, but what about her preference to drive a new car? What if the old car breaks down etc.
Causal decision theory
M3aintains that an account of rational choice must use causality to identify the considerations that make a choice rational. The probabilities depend on the option. Causal decision theory takes the dependence to be causal rather than merely evidential. Causal decision theory can solve the predictor theory by applying unconditional probabilities of the states.
Interval utility scales
Necessary for decisions under risk as, in addition to knowing whether you prefer one outcome to another, you'll want to know if an outcome is ENOUGH to take the risks involved in obtaining it.
Problems with optimism-pessimism rule
No consistency. Since it's up to the agent to decide the optimism index, it is to be expected that different agents will have different indexes. Furthermore, it isn't clear that the OA would be the same in groups, or even the same for one agent over time.
Cost and time of decision making issues
Often, the utility of a given outcome is greater than the cost and time of making the decision, and therefore any Act is of greater utility then sitting around and drawing up decision tables.
Interval scales
Ordered scales that convey the degree to which one outcome is desired over another. i.e. 4 is only slightly better than 2, but 100 is waaay better than 2. In ordinal scales, it would suffice to say "100 is better than 4 which is better than 2"
Problems with principle of insufficient reason
Ordinal scales do not suffice. They have to be interval. There is also a significant philosophical objection: If there is no reason for assigning one set of probabilities rather than another, there is no justification for assuming that the states are equiprobable either.
Better-chances condition
Other things being equal, the agent prefers one lottery to another just in case the former gives a better chance at the better prize.
Mutually exclusive
P and Q are mutually exclusive if it is impossible for both to be true
Absolute vs. conditional probability
P(p) = 0.9 vs P(p|q) = 0.5
Inverse Probability Law
P(p|q) = [P(p) x P(q|p)] / P(q)
Bayes Theorum
P(p|q) = P(p) x P(q|p) / [P(p) x P(q|p)] + [P(not p) x P(q|not p)]
Maximizing expected utility
Principle that finds that the best action is one that yields the greatest utility.
Relative frequency
Probabilities are proportions (or relative frequencies) of events of one kind to those of others. Views all probabilities as implicitly conditional.
Hypothetical relative frequency
Probabilities that would, theoretically, be arrived at if relative frequency
Rawls vs. Harsnayi (which principles each favors and why)
Rawls favors the DIFFERENCE PRINCIPLE (choose the society in which the worst-off people do better than the worst-off people of other societies - sort of like Maximin) Harsanyi says pick the one with the greatest overall utility (utilitarianism). Decisions must be made behind the veil of ignorance
Tversky and Kahneman's studies of how we reason about probabilities
Reliance on heuristics - Representativeness Availability, Adjustment and Anchoring especially. Gambler's fallacy's in there,
Irrelevant expansion condition
Rule that states "The addition of a new act, which is not regarded as better than the original ones, will not change a rational agent's ranking of the original acts." Though, by the other rules, the order may change.
Ratio scales
Scales used to represent the ratio of a thing measured to some standard unit of measurement. They all have natural zero points. Two ratio scales are equivalent to each other if and only if they may be obtained from each other by multiplying by positive constants. Special case of positive linear transformation -> tighter kind of interval scale
Ellsberg's paradox
See text for clear explanation. This version of the paradox. 30 yellow balls. 60 balls that are either Red or Blue. In the first decision table, you want to pick A. In the second you want to C. But if you calculate it, the expected utilities suggest that you are contradicting yourself if you pick A and then also pick C. Practice: draw out Ellsberg's paradox
Smoking/heart disease example
Smoking is highly correlated with heart disease, but heart disease is also hereditary. The Type A person, where genetic predisposition is highest, are also more likely to smoke. So, essentially, smoking is genetic too. When deciding to smoke, you would find a dilemma similar to the Predictor paradox. Practice: set up the decision table for the smoking/heart disease paradox
Conditional causal probability
The PROPENSITY for the act to produce or prevent the state. NOTE: if an act has a propensity to bring about a state, the conditional causal probability of that act will be greater than the unconditional probability of the state, and vice versa. If an act has no propensity to affect the state, the probabilities will be the same.
Expected monetary value
The amount of money you can expect, on average, by choosing an act. You can calculate EMV for an act from a decision table by: multiplying the monetary value in that square by the probability in that same square, and then summing across the row.
Voting paradox
The generation of recursive cycles by majority rule. Occurs when Maximin, Minimax regret, and Optimism-pessimism all recommend different acts.
Probability calculus
The mathematical theory that enables us to calculate additional probabilities from those we already know. This can take several forms. (i.e. inverse probability law, Bayes theorem, etc.)
Allais' paradox
The paradox is as such: u(a) = u(1M) u(b) = 0.1u(5M) + 0.89u(1M) + 0.01u(0) u(c) = 0.1u(5M) + 0.9(0) u(d) = 0.11(1M) + 0.89u(0) And it follows that u(a) - u(b) = 0.11u(1M) - [0.1u(5M) + 0.1u(0)] u(d) - u(c) = 0.11u(1M) - [0.1u(5M) + 0.1u(0)] Therefore if you prefer a to b, you HAVE to prefer d to c to conform to utility. Practice: Set up Allais' paradox in a decision tree
Bayesianism
The position that all decisions can be made by Bayesian methods, using your best hunch as a prior probability.
State-act independence
The probability of a state is unchanged regardless of whether the act which the state is in is picked.
Unconditional probability
The probability that an event will occur, not contingent on any prior or related results. An unconditional probability is the independent chance that a single outcome results from a sample of possible outcomes. To find the unconditional probability of an event, sum the outcomes of the event and divide by the total number of possible outcomes.
Utility
The reward of a given state, multiplied by the probability of that state occurring if the act is picked.
Problems with minimax regret
To use minimax regret, agents must have more refined preference structures because of the linear scales (maximin does not need this). Also, it is susceptible to irrelevant expansion (addition of another act, that is not better than any of the others, might change the act you should pick). Also, it's not always appropriate: Practice: Draw out the table where it is not - (99 cases of 99.9 vs. 1 of 100)
Degree of belief interpretation
View that probabilities can be assigned given how the agent 'feels' about the chances, or how he believes they 'ought to be'.
Dominance
We say the Act A dominates Act B if, in a state-by-state comparison, Act A yields results at least as good as B, and in some cases better than B. The DOMINANCE PRINCIPLE tells us to rule out dominated acts
Decisions Under Certainty
a decision in which you are sure your action will produce a certain outcome. Here, you simply need to decide which outcome you like best, since you know what act (or acts) are certain to produce it
Conditional probability
the probability of an event, assuming a particular set of circumstances. Read as, the probability of A, given B - P(A | B)
Decision Theory
trying to make sense of how individuals and groups make or should make decisions
