Steedman and Krause (1986) discussdifferent types of formation rules, which map a bundle (≽1, ≽2, …, ≽n)onto a single preference relation. The first formation rule describes avery cautious character, who considers an alternative at least as goodas another only if she considers it at least as good in everyaspect. In economics, and in other social sciences, preference refers to an order by which an agent, while in search of an “optimal choice”, ranks alternatives based on their respective utility. Preferences are evaluations that concern matters of value, in relation to practical reasoning.[1] Individual preferences are determined by taste, need, …, as opposed to price, availability or personal income.
3 Formal representation
- By this ismeant a function f that takes us from a pair⟨p,q⟩ of sentences to a setf(⟨p,q⟩) of pairs ofalternatives (perhaps possible worlds).
- TheBayesian decision maker is assumed to make her choices in accordancewith a complete preference ordering over the available options.However, in many everyday cases, we do not have, and do not need,complete preferences.
- In another famous example by Warren S. Quinn, a medical device hasbeen implanted into the body of a person (the self-torturer).
- Then \(\preceq\) satisfies axioms 1–4 ifand only if there exists a function \(u\), from\(\bO\) into the set of real numbers, that is unique up topositive linear transformation, and relative to which \(\preceq\) canbe represented as maximising expected utility.
- Preference at one point intime can refer to what happens or happened at other points in time.Furthermore, preferences can change over time, due to changes inbeliefs, values, tastes, or a combination of these.
- The most famous argument in favour of preference transitivity is themoney pump argument.
Jeffrey’s model can be generalised by introducing amore general probability updating rule (e.g., Jeffreyconditionalisation). It is based on relatively strong assumptions on the relationbetween prior and posterior unconditional preferences. Authors who recognize partial preferences usually give them priority,and consider total preferences to be completely determined by thepartial preferences. In other words, they assume that a totalpreference relation is uniquely determined by the partial preferencerelations through a process of aggregation.
2 On rational belief
Intuitively, Continuity guaranteesthat an agent’s evaluations of lotteries are appropriatelysensitive to the probabilities of the lotteries’ prizes. It alsoensures, as the name suggests, that a sufficiently rich preferenceordering over lotteries can be represented by a continuous cardinalfunction. Another way to put this is that, when the above holds, thepreference relation can be represented as maximising utility,since it always favours the option with highest utility. Let us nonetheless proceed by first introducing basic candidateproperties of (rational) preference over options and only afterwardsturning to questions of interpretation. As noted above, preferenceconcerns the comparison of options; it is a relation betweenoptions. For a domain of options we speak of anagent’s preference ordering, this being the ordering ofoptions that is generated by the agent’s preference between anytwo options in that domain.
Transitivity
Or, if one already has a dog, it may mean that one prefers just adog to both a cat and a dog. Combinative preferences are usually takento have states of affairs as their relata. It is usually assumed that logicallyequivalent expressions can be substituted for each other. Ramsey(1928a, 182), who pointed out that if a subject’s behaviourviolated certain axioms of probability and preference, then he would be willing to buy a bet that yields a gain to the seller, and a loss to the buyer, no matter what happens.
1 Savage’s theory
For a simple example,consider a person who prefers one apple today to two apples tomorrow,but yet (today) prefers two apples in 51 days to one apple in 50 days.Although this is a plausible preference pattern, it is incompatiblewith the discounted utility model. It can however be accounted for in abifactorial model with a declining discount rate. Pioneered financial anxiety following covid by Ainslie(1992), psychologists and behavioural economists have thereforeproposed to replace Samuelson’s exponential discounting model with amodel of hyperbolic discounting. The hyperbolic model discounts thefuture consumption with a parameter inversely proportional to the delayof the consumption, and hence covers examples like the above.
Most commonly used axioms
By means of social interactions, individual preferences can evolve without any necessary change to the utility.[23] This can be exemplified by taking the example of a group of friends having lunch together. Individuals in such a group may change their food preferences after being exposed to their friends’ preferences. Similarly, if an individual tends to be risk-averse but is exposed to a group of risk-seeking people, his preferences may change over time. First, it requires anunchanging evaluative function \(u\) defined over the atoms of thepropositional space, viz. Thus for all doxasticallychanged preference orderings, the preferences over worlds remainidentical. But it is plausible that one’s preference– say, for a vacation in Florida – changes just becauseone believes that it is more likely that there will be a hurricanenext week.
In this case it isgenuinely undetermined what will be the outcome of extending therelation to cover the previously uncovered cases. Wesay that alternative \(f\)“agrees with” \(g\) inevent \(E\) if, for any state inevent \(E\), \(f\) and \(g\) yieldthe same outcome. Based on this result, it would seem natural to search for anothervoting procedure that differs from the majority method in notgiving rise to this form of cycles. No such method was found, but itwas not until the 20th century that formal results thatrefer to all possible voting (or decision) procedures wereobtained. Preference aggregation and social decision processes were studiedalready in the 18th century.
OnBuchak’s interpretation, the explanation for Allais’preferences is not the different value that the outcome $0has depending on what lottery it is part of. To accommodate Allais’preferences (and other intuitively rational attitudes to risk thatviolate EU theory), Buchak introduces a risk function thatrepresents people’s willingness to trade chances of somethinggood for risks of something bad. Leonard Savage’s decision theory, as presented in his(1954) The Foundations of Statistics, is without a doubt thebest-known normative theory of choice under uncertainty, in particularwithin economics and the decision sciences.
Hansson (1995) suggest differentiating valuational change intopreference formation and preference change, and construct a formalisedmodel of preference change proper. If an agent forms a specificpreference as a result of some experience, further changes in heroverall preference state are often necessary to regainconsistency. The model of preference change proper shows which pathconsistency restoration will take, conditional on the previous stateand the available dynamic information, and it determines what theensuing state will look like. Two kinds of beliefs are especially important for doxastic models.The first is the belief that a state X is instrumental inbringing about a desired state Y. If she comes to disbelieve thisconnection, she may well abandon this preference. More generally, if X∧Y is preferred to X∧¬Y, then a rise of the probability that Y given X willresult in a rise in the desirability of X, and viceversa.
Otherwise, you would preferthe union that contains the one of \(p\) and \(q\) thatyou find less probable, since that gives you a higher chance of themore desirable proposition \(r\). Itthen follows that for any other proposition \(s\) that satisfies the aforementionedconditions \(r\) satisfies, youshould also be indifferent between \(p\cups\) and \(q\cup s\), since,again, the two unions are equally likely to resultin \(s\). These are intended as constraints on an agent’s preferencerelation, \(\preceq\), over a set ofacts, \(\bF\), as described above. In the second choicesituation, however, the minimum one stands to gain is $0 no matterwhich choice one makes. Therefore, in that case many people do thinkthat the slight extra risk of $0 is worth the chance of a betterprize.
