Phil 343 Exam 1

What is descriptive decision theory?

Descriptive decision theory studies how people actually make decisions. It examines psychological processes, biases, heuristics, and inconsistencies that influence real-world choices, even when those choices violate principles of rationality.

What is rational choice theory?

Rational choice theory is a normative framework that examines how people ought to make decisions. It identifies principles such as transitivity, dominance, and consistency that choices should satisfy to be considered rational.

What is the difference between descriptive and rational choice theory?

Descriptive decision theory explains and predicts actual behavior, while rational choice theory evaluates whether decisions satisfy standards of rationality. A choice can be common in practice but still violate a normative principle.

What is the belief/desire model of practical reasons?

The belief/desire model explains practical reasoning through beliefs about how the world works and desires concerning which outcomes are valuable. Together, they provide reasons for taking actions that are expected to achieve desired outcomes.

What role do beliefs play in practical reasoning?

Beliefs represent what an individual thinks is true about the world, including the likely consequences of actions. They help determine which actions are expected to achieve desired outcomes.

What role do desires play in practical reasoning?

Desires represent the goals, outcomes, or states of affairs an individual values. They provide the motivation for action, while beliefs help determine how those goals can be achieved.

Give an example of the belief/desire model.

A student desires a high exam grade and believes studying increases the probability of earning one. Together, the desire and belief provide a practical reason to study.

Can an action be rational if it is based on a false belief?

An action may be rational relative to an agent's evidence and beliefs even if a belief turns out false. However, if the belief was unreasonable given the available evidence, the action may be criticized as epistemically irrational.

What is choiceworthiness?

Choiceworthiness describes how good or rationally justified an alternative is relative to other available alternatives. It concerns the reasons supporting a choice rather than simply what someone happens to select.

What is a choiceworthiness ranking?

A choiceworthiness ranking is a relation that compares alternatives according to how choiceworthy they are. It indicates whether one option is better, worse, or equally choiceworthy compared with another.

What does A ≽ B mean?

A ≽ B means A is at least as choiceworthy as B. This includes situations where A is strictly better than B and situations where they are equally choiceworthy.

What does A ≻ B mean?

A ≻ B means A is strictly more choiceworthy than B. In a standard weak-ranking framework, A ≽ B holds but B ≽ A does not.

What does A ~ B mean?

A ~ B means A and B are equally choiceworthy or indifferent. Both A ≽ B and B ≽ A hold, meaning neither is ranked strictly above the other.

What is a choice function?

A choice function assigns a selected alternative or set of alternatives to each possible menu of available options. It describes what is chosen from a particular set and allows researchers to evaluate consistency across menus.

What does C(S) represent?

C(S) represents the alternative or set of alternatives chosen from the available menu S. For example, if S={A,B,C} and C(S)={B}, the decision maker selects B when all three options are available.

What is a menu in choice theory?

A menu is the set of alternatives available to a decision maker. Rational choice principles examine whether choices remain consistent when alternatives are added or removed.

What is a set-valued choice function?

A set-valued choice function can select multiple alternatives from the same menu. This allows for ties when several options are equally best.

What does it mean for a choice function to be rationalizable?

A choice function is rationalizable when its selections can be explained by maximizing a stable preference relation. The chosen alternatives must be best according to that ranking within each available menu.

What is the difference between preference and choice?

Preference is a comparative evaluation of alternatives, while choice is the actual selection from a menu. Observed choices reveal stable preferences only under additional assumptions.

What is completeness?

Completeness requires that any two alternatives can be ranked. For any A and B, either A ≽ B or B ≽ A, or both, so the model does not permit pairs that cannot be weakly compared.

What is trichotomy?

Trichotomy states that for any two alternatives, exactly one relationship holds: A is strictly better than B, B is strictly better than A, or they are equally good. It assumes these three categories exhaust the possibilities.

What is transitivity?

Transitivity requires consistency across chains of comparisons. If A ≽ B and B ≽ C, then A must be at least as choiceworthy as C.

Why is transitivity considered a principle of rationality?

Transitivity prevents inconsistent preference cycles and allows alternatives to be arranged in a coherent ranking. Violations can expose decision makers to exploitation through sequences of costly trades.

Give an example of a transitivity violation.

Someone prefers A over B, B over C, and C over A. These preferences form a cycle because each option is ranked above another while eventually returning to the original option.

What is a preference cycle?

A preference cycle occurs when strict preferences form a loop, such as A ≻ B, B ≻ C, and C ≻ A. Such a pattern violates transitivity and prevents a consistent ordering of those alternatives.

What is a money-pump argument?

A money-pump argument shows how cyclic preferences can expose an agent to repeated financial losses. If someone pays small fees to exchange an option for one they prefer, they can eventually return to the original option with less money.

Give a step-by-step example of a money pump.

Suppose A ≻ B, B ≻ C, and C ≻ A. Starting with A, the agent pays to exchange A for C, then C for B, and finally B for A. They end with the original option but have lost the fees.

What does a money-pump argument attempt to prove?

It provides a practical reason to avoid certain intransitive preferences by showing how they can produce avoidable losses. The argument depends on assumptions about trading opportunities, stable preferences, and willingness to pay.

Does every transitivity violation necessarily produce a money pump?

No. A successful money pump requires opportunities to trade and a willingness to pay for each preferred exchange. A preference cycle alone does not guarantee exploitation.

What is dominance?

Dominance occurs when one option performs at least as well as another in every relevant state or dimension and better in at least one. A rational decision maker generally should not select an option dominated by an available alternative.

What is strict dominance?

Strict dominance occurs when one action produces a better outcome than another in every possible state of the world. Because the dominating action is always superior, the dominated alternative should be rejected.

What is weak dominance?

Weak dominance occurs when one action performs at least as well as another in every state and strictly better in at least one. It provides a reason to favor the dominating option, although the strength of that reason can depend on the model's assumptions.

Give an example of dominance.

Suppose two products are identical in quality and features, but Product A costs $50 while Product B costs $70. Product A dominates Product B because it provides identical benefits at a lower cost.

Why is dominance important in rational choice theory?

Dominance allows decision makers to eliminate inferior alternatives without calculating expected utility. If one option is never worse and sometimes better, the dominated alternative has no compensating advantage in the modeled comparison.

What is the sure-thing principle?

The sure-thing principle states that if an action is preferred to another whether an event occurs or does not occur, it should also be preferred before learning whether the event occurs. It requires consistency between conditional and unconditional decisions.

Give an example of the sure-thing principle.

Suppose a student prefers studying to watching television if an exam is difficult and also prefers studying if the exam is easy. Uncertainty about the exam's difficulty should not reverse that preference.

What is the common-consequence interpretation of the sure-thing principle?

If two lotteries share an identical consequence in the same circumstances, that common component should not determine which lottery is preferred. Replacing the shared consequence identically should preserve the ranking under the independence assumptions.

Why is the sure-thing principle important?

It prevents preference reversals caused solely by consequences that are identical across alternatives. It is closely connected to expected-utility independence and is challenged by Allais-type choices.

What is the independence axiom for lotteries?

If A is preferred to B, mixing both with the same lottery C in the same proportion should preserve the ranking. Formally, A ≽ B implies pA+(1-p)C ≽ pB+(1-p)C for 0<p<1.

Why does independence matter to expected utility?

Expected utility is linear in probabilities, so a shared lottery component contributes equally to both alternatives. Adding or removing that common component cannot reverse their ranking.

What is stake invariance?

Stake invariance is the idea that comparative probability judgments inferred from appropriately matched wagers should remain unchanged when the size of the prizes changes. The underlying likelihood of an event does not depend on the amount wagered.

Why can observed stake invariance fail?

People may respond differently to small and large wagers because of risk aversion, loss aversion, or changing marginal utility of money. Betting preferences therefore do not always reveal probability judgments independently of preferences over outcomes.

How does stake invariance relate to credences?

If betting preferences reliably reveal beliefs, changing the stakes of matched wagers should not change the inferred probability ordering. A change may indicate that utility considerations are influencing the observed betting preferences.

What is a representation theorem?

A representation theorem establishes that when preferences satisfy specified axioms, they can be represented mathematically by a numerical utility function. The function assigns values that reproduce the relevant preference ordering.

State the expected-utility representation theorem.

If preferences over lotteries satisfy completeness, transitivity, continuity, and independence in the standard von Neumann–Morgenstern framework, then there exists a utility function such that lotteries are ranked by their expected utilities.

What does a representation theorem tell us about rational choice?

It shows that satisfying certain consistency principles is sufficient for preferences to have a structured mathematical representation. It does not establish that real people satisfy those principles or that the principles are beyond criticism.

Does a representation theorem prove that its axioms are correct?

No. It is a conditional mathematical result: if the axioms hold, then a particular representation exists. Whether those axioms are normatively justified is a separate philosophical question.

What is knockout reasoning?

Knockout reasoning eliminates an alternative because it fails a criterion treated as decisive, without fully weighing its other advantages. This can simplify decisions but may overlook important tradeoffs if the criterion is not genuinely nonnegotiable.

Give an example of knockout reasoning.

A student rejects every apartment more than 15 minutes from campus without considering price, safety, or quality. Distance acts as a knockout criterion that prevents further evaluation.

How does knockout reasoning differ from dominance?

Dominance rejects an option because another performs at least as well in every relevant respect and better somewhere. Knockout reasoning may reject an option based on one feature even when it has significant advantages elsewhere.

What is indifference?

Indifference occurs when two alternatives are equally choiceworthy according to a preference ranking. Neither is considered better, although the alternatives may differ in their characteristics.

What does it mean for two options to be on par?

Two options are on par when neither is better, worse, nor equally good, yet they remain meaningfully comparable. Parity is often used to describe difficult choices involving qualitatively different values.

How is parity different from indifference?

Indifference means the alternatives are equally good, while parity means they are comparable without being equal in value. A small improvement can distinguish these relationships.

Give an example of options potentially on par.

A fulfilling artistic career and a fulfilling scientific career might be on par. They offer different valuable features, but neither must be better, worse, or exactly equally good.

What is the small improvements test?

The small improvements test distinguishes indifference from parity by slightly improving one alternative. If the alternatives were equally good, a genuine improvement should make the improved option better; if they were on par, the improvement might leave them on par.

How does the small improvements test identify parity?

If A and B initially seem neither better nor worse, slightly improve A. If improved A still is not clearly better than B, that supports the possibility that A and B were on par rather than indifferent.

How does parity challenge trichotomy?

Trichotomy assumes every pair must be better, worse, or equally good. Parity introduces a possible fourth relationship, challenging the claim that the traditional three categories are exhaustive.

What is Sen's Alpha?

Sen's Alpha is a contraction consistency condition requiring that an alternative chosen from a larger menu remain chosen when the menu shrinks, provided the alternative is still available. Removing other options should not disqualify a previously chosen alternative.

Give an example satisfying Sen's Alpha.

If A is chosen from {A,B,C}, then A must remain among the chosen alternatives when the menu shrinks to {A,B}. The removal of C should not cause A to lose its chosen status.

What is a violation of Sen's Alpha?

Suppose C({A,B,C})={A}, but C({A,B})={B}. This violates Alpha because A was previously chosen and remains available, yet removing C causes A to stop being chosen.

Why is Sen's Alpha important?

Alpha expresses the idea that removing alternatives should not undermine an option that was already selected. It helps test whether choices are consistent with a stable underlying ranking.

What is Sen's Beta?

Sen's Beta is a consistency condition concerning alternatives jointly chosen from a menu. If two alternatives are both chosen from a smaller menu, then in a larger menu containing both, selecting one requires selecting the other as well.

Give an example satisfying Sen's Beta.

Suppose A and B are both chosen from {A,B}. If a larger menu {A,B,C} selects A, it must also select B, preserving the tie between the two alternatives.

What is a violation of Sen's Beta?

Suppose C({A,B})={A,B}, but C({A,B,C})={A}. This violates Beta because the larger menu selects A while excluding B, even though the two were jointly chosen in the smaller menu.

How do Sen's Alpha and Beta differ?

Alpha focuses on whether a chosen alternative remains chosen when the menu contracts. Beta focuses on preserving the relationship between jointly chosen alternatives when the menu expands.

Why are Sen's Alpha and Beta important together?

Under standard assumptions about nonempty set-valued choice, satisfying Alpha and Beta is closely connected to rationalizability by a complete and transitive preference relation. Together, they impose consistency across menu changes.

What is a credence?

A credence is a numerical representation of an individual's subjective degree of belief in a proposition. Credences range from 0 to 1 and, in a coherent probability model, satisfy the laws of probability.

What does a credence of 0.7 mean?

It means the agent assigns a 70% subjective probability to the proposition. This expresses the strength of the person's belief, not necessarily the event's observed frequency.

What is comparative probability?

Comparative probability is a qualitative ranking of events according to which is judged more likely. It identifies relative likelihood without necessarily assigning exact numerical probabilities.

What does A >. B mean?

A >. B means A is judged strictly more probable than B. It is an ordinal comparison and does not specify how much more probable A is.

How does comparative probability differ from numerical probability?

Comparative probability ranks events by likelihood, while numerical probability assigns exact values between 0 and 1. Numerical probabilities determine comparative rankings, but comparative rankings alone need not uniquely determine numerical values.

What is the nonnegativity axiom?

Every event must have probability greater than or equal to zero. In ordinary probability theory, negative probabilities are not permitted.

What is the normalization axiom?

The probability of the entire sample space is 1. This means the exhaustive set of possible outcomes collectively receives total probability one.

What is the additivity axiom?

For mutually exclusive events A and B, P(A or B)=P(A)+P(B). The full probability axiom extends this principle to countably many pairwise disjoint events.

What is the complement rule?

P(not A)=1−P(A). An event and its complement are mutually exclusive and together cover every possible outcome.

What is the general addition rule?

P(A or B)=P(A)+P(B)−P(A and B). The intersection is subtracted because outcomes belonging to both events were counted twice.

What is monotonicity in probability?

If event A is contained within event B, then P(A) cannot exceed P(B). A more specific event cannot be more probable than an event that includes it.

What is the conjunction rule?

P(A and B) cannot exceed either P(A) or P(B). The conjunction is a subset of each individual event.

What is a partition?

A partition is a collection of mutually exclusive events that together cover the entire sample space. Their probabilities must add to 1.

What is conditional probability?

Conditional probability is the probability of event A given that event B has occurred. It is calculated as P(A|B)=P(A and B)/P(B), provided P(B)>0.

What does conditioning on B mean?

It restricts the relevant probability space to situations in which B occurs. Probabilities are then evaluated relative to that restricted set of possibilities.

Calculate P(A|B) if P(A and B)=0.18 and P(B)=0.30.

P(A|B)=0.18/0.30=0.60. This means A occurs in 60% of the probability mass associated with B.

What is the multiplication rule?

P(A and B)=P(A|B)P(B). It allows a joint probability to be calculated from a conditional probability and the probability of the conditioning event.

What is independence?

Two events are independent when learning that one occurred does not change the probability of the other. Mathematically, P(A and B)=P(A)P(B).

Are mutually exclusive events independent?

Generally no, if both events have positive probability. If one occurs, the other becomes impossible, so learning one occurred changes the probability of the other.

What is Bayes' theorem?

Bayes' theorem states P(A|B)=P(B|A)P(A)/P(B), provided P(B)>0. It allows probabilities to be updated using new evidence.

What is the law of total probability?

If B1 through Bn form a partition, P(A)=ΣP(A|Bi)P(Bi). It calculates the overall probability of A by combining its probability within each exhaustive case.

If P(A)=0.6 and P(B)=0.5, what are the bounds on P(A and B)?

The intersection must be at least max(0,0.6+0.5−1)=0.1 and at most min(0.6,0.5)=0.5. Therefore P(A and B) must lie between 0.1 and 0.5.

What is the conjunction fallacy?

The conjunction fallacy occurs when someone judges two events occurring together as more probable than one of those events alone. This violates the rule that a conjunction cannot exceed the probability of either component.

What is Ramsey's qualitative principle?

Ramsey's qualitative approach uses preferences between carefully matched wagers to infer which events an agent considers more probable. When the prizes and other consequences are held constant, preference for one wager can reveal a comparative probability judgment.

Why must wager prizes be matched in Ramsey's qualitative approach?

Matching prizes helps isolate beliefs about likelihood from preferences about the rewards. Otherwise, someone might choose a wager because its payoff is more attractive rather than because its event is considered more probable.

Give an example of qualitative Ramsey reasoning.

Suppose two wagers each pay $10 if their respective events occur and $0 otherwise. If an agent prefers the wager on A to the wager on B, that can indicate A is judged more probable under suitable assumptions.

What is Ramsey's quantitative principle?

Ramsey's quantitative principle states that credences can be inferred from betting preferences while accounting for utility. Under risk neutrality, credence = willingness to pay ÷ prize amount, assuming a fixed prize ($0 otherwise), consistent and truthful preferences, and no nonmonetary influences.

How can a fair wager reveal a numerical credence?

If a wager pays $1 when A occurs and $0 otherwise, and the agent has linear utility of money, a fair price of $0.40 implies P(A)=0.40. Without linear utility, the price need not equal the credence.

Why does risk aversion complicate Ramsey's quantitative approach?

Risk aversion makes willingness to pay an unreliable direct measure of credence because people may pay less for bets due to their dislike of risk, not because they believe the event is unlikely. Therefore, utility must be accounted for when inferring credences.

What is a fair price for a wager?

A fair price is the amount at which the agent is indifferent between purchasing a wager and declining it. It depends on beliefs, preferences over outcomes, and potentially the agent's initial wealth.

When does fair price equal expected monetary payoff?

When the agent has linear utility of money and there are no relevant fees or additional consequences. Under these assumptions, indifference occurs when the price equals the probability-weighted average payoff.

How do you normalize proportional probability weights?

Divide each nonnegative weight by the sum of all weights. For weights 2, 3, and 5, the total is 10, producing probabilities of 0.2, 0.3, and 0.5.

Can arbitrary willingness-to-pay amounts be normalized into credences?

Not necessarily. Prices can reflect risk attitudes, wealth, and the structure of wagers, so normalization is justified only when the values are genuinely proportional to probabilities.

What is utility?

Utility is a numerical representation of how an agent values or ranks outcomes. It does not necessarily measure happiness and may reflect preferences involving money, well-being, goals, or other values.