One might otherwise seek tounderstand the role of time, or the temporal position of goods, onpreferences. To this end, outcomes are described in terms oftemporally-indexed bundles of goods, or consumption streams(for an early model of this kind see Ramsey 1928; a later influentialtreatment is Koopmans 1960). There may be systematic structure to anagent’s preferences over these consumption streams, over and above thestructure imposed by the EU axioms of preference.
Ifthe option set includes all kinds of states of affairs, thenCompleteness is not immediately compelling. For instance, it isquestionable whether an agent should be able to compare the optionwhereby two additional people in the world are made literate with theoption whereby two additional people reach the age of sixty. If, onthe other hand, all options in the set are quite similar to eachother, say, all options are investment portfolios, then Completenessis more compelling. By contrast, if preferences areunderstood rather as mental attitudes, typically considered judgmentsabout whether an option is better or more desirable than another, thenthe doubts about Completeness alluded to above are pertinent (forfurther discussion, see Mandler 2001). The work of Maurice Allais and Daniel Ellsberg showed that human behavior has systematic and sometimes important departures from expected-utility maximization (Allais paradox and Ellsberg paradox).11 The prospect theory of Daniel Kahneman and Amos Tversky renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. Let us conclude by summarising the main reasons why decision theory,as described above, is of philosophical interest.
What are preferences over prospects?
Therefore,in that case many people do think that the slight extra risk of $0 isworth the chance of a better prize. The roots of decision theory lie in probability theory, developed by Blaise Pascal and Pierre de Fermat in the 17th century, which was later refined by others like Christiaan Huygens. These developments provided a framework for understanding risk and uncertainty, which are central to decision-making. Formalizing and modeling different decision-making scenarios in AI involves a combination of decision theory, machine learning techniques, and AI-driven decision-making models. These models can be used to make more precise and valuable decisions, streamline processes, and enhance decision-making initiatives.
In particular, theirtheory can capture the intuition that the (un)desirability of winningnothing partly depends on whether or not one was guaranteed to winsomething had one chosen differently. Therefore, their theory canrepresent Allais’ preferences as maximising the value of anextended Jeffrey-desirability function. To the extent that decision theory can be reconciled with the fullrange of ethical theories, should we say that there decision theory is concerned with are no meaningfuldistinctions between these theories?
3 The von Neumann and Morgenstern (vNM) representation theorem
Notably, probabilistic decision theory can sometimes be sensitive to assumptions about the probabilities of various events, whereas non-probabilistic rules, such as minimax, are robust in that they do not make such assumptions. Expected utility theory has been criticised for not allowing for valueinteractions between outcomes in different, mutually incompatiblestates of the world. For instance, recall that when deciding betweentwo risky options you should, according to Savage’s version ofthe theory, ignore the states of the world where the two optionsresult in the same outcome. That seems very reasonable if we canassume separability between outcomes in different states ofthe world, i.e., if the contribution that an outcome in one state ofthe world makes towards the overall value of an option is independentof what other outcomes the option might result in. For then identicaloutcomes (with equal probabilities) should cancel each other out in acomparison of two options, which would entail that if two optionsshare an outcome in some state of the world, then when comparing theoptions, it does not matter what that shared outcome is.
Utility Theory and Decision Theory
- Bradley and Stefánsson (2017) also develop a new decisiontheory partly in response to the Allais paradox.
- The standard representation of a decision problem requires that we specify the alternatives available to the decision-maker, the possible outcomes of the decision, the values of these outcomes, and…
- In particular, economists Karni and Vierø (2013,2015) have recently extended standard Bayesian conditionalisation tosuch learning events.
- For instance, it isquestionable whether an agent should be able to compare the optionwhereby two additional people in the world are made literate with theoption whereby two additional people reach the age of sixty.
- In contrast, awareness of unawareness would seem to be of greatinterest from the perspective of decision-making.
- (Note that in this context,“desirability” and “value” should beunderstood as desirability/value according to the agent inquestion.) This simple maxim will be the focus of much of ourdiscussion.
As the reader will recall, Savage takes for granted a set of possibleoutcomes \(\bO\), and another set of possible states of the world\(\bS\), and defines the set of acts, \(\bF\), as the set of allfunctions from \(\bS\) to \(\bO\). Moreover, his representationtheorem has been interpreted as justifying the claim that a rationalperson always performs the act in \(\bF\) that maximises expectedutility, relative to a probability measure over \(\bS\) and a utilitymeasure over \(\bO\). Bradley and Stefánsson (2017) also develop a new decisiontheory partly in response to the Allais paradox. But unlike Buchak,they suggest that what explains Allais’ preferences is that thevalue of wining nothing from a chosen lottery partly depends on whatwould have happened had one chosen differently. To accommodate this,they extend the Boolean algebra in Jeffrey’s decision theory tocounterfactual propositions, and show that Jeffrey’sextended theory can represent the value-dependencies one often findsbetween counterfactual and actual outcomes.
- The only information contained in an ordinal utility representation ishow the agent whose preferences are being represented orders options,from least to most preferable.
- It can actually be seen as a weak version ofIndependence and the Sure Thing Principle, and it plays a similar rolein Jeffrey’s theory.
- In other words, thisindependence must be built into the decision model if it is tofacilitate appropriate measures of belief and desire.
- But this is toassume that we already have important information about the beliefs ofthe agent whose attitudes we are trying to represent; namely whatstate-partitions she considers probabilistically independent of heracts.
The static model has familiar tabular or normalform, with each row representing an available act/option, and columnsrepresenting the different possible states of the world that yield agiven outcome for each act. The sequential decision model, on theother hand, has tree or extensive form (such as in Figure 1). It depicts a series of anticipated choice points, where the branchesextending from a choice point represent the options at that choicepoint. Some of these branches lead to further choice points, oftenafter the resolution of some uncertainty due to new evidence.
Probability theory
For instance, theaforementioned authors considered and characterised preferences thatexhibit exponential time discounting. The above result may seem remarkable; in particular, the fact that aperson’s preferences can determine a unique probability functionthat represents her beliefs. On a closer look, however, it is evidentthat some of our beliefs can be determined by examining ourpreferences. Suppose you are offered a choice between two lotteries,one that results in you winning a nice prize if a coin comes up headsbut getting nothing if the coin comes up tails, another that resultsin you winning the same prize if the coin comes up tails but gettingnothing if the coin comes up heads.
Then assuming that thedesirability of the prize (and similarly the desirability of no prize)is independent of how the coin lands, your preference between the twolotteries should be entirely determined by your comparative beliefsfor the two ways in which the coin can land. For instance, if youstrictly prefer the first lottery to the second, then that suggestsyou consider heads more likely than tails. This chapter on normative decision theory is from the Stanford Encyclopedia of Philosophy, a dynamic reference work available online. Decision theory is concerned with the reasoning underlying an person’s choices, whether a mundane choice between taking the bus or getting a taxi, or a more far-reaching choice about whether to pursue a demanding political career.
Theseare intended as constraints on an agent’s preference relation,\(\preceq\), over a set of acts, \(\bF\), as described above. In our continuing investigation of rational preferences overprospects, the numerical representation (ormeasurement) of preference orderings will become important.The numerical measures in question are known as utilityfunctions. The two main types of utility function that will playa role are the ordinal utility function and the moreinformation-rich interval-valued (or cardinal)utility function.
Complex decisions
Brown (2011) and Dietrich andList (2017) demonstrate that in fact the choice-theoreticrepresentation of ethical theories better facilitates distinctionsbetween them; terms like “(non)consequentialism” can beprecisely defined, albeit in debatable ways. More generally, we cancatalogue theories in terms of the kinds of properties (whetherintrinsic or in some sense relational) that distinguish acts/outcomesand also in terms of the nature of the ranking of acts/outcomes thatthey yield (whether transitive, complete, continuous and so on). Forone thing, in many real-world decision circumstances, it is hard toframe the decision model in such a way that states are intuitivelyprobabilistically independent of acts. For instance, suppose an agentenjoys smoking, and is trying to decide whether to quit or not. But then it is obviousthat the options she is considering could, and arguably should, affecthow likely she finds each state of the world, since it is wellrecognised that life expectancy is reduced by smoking. Savage wouldthus require an alternative representation of the decisionproblem—the states do not reference life span directly, butrather the agent’s physiological propensity to react in acertain way to smoking.
For those who think that the only way to determine a person’scomparative beliefs is to look at her preferences, the lack ofuniqueness in Jeffrey’s theory is a big problem. Indeed, thismay be one of the main reasons why economists have largely ignoredJeffrey’s theory. Economists have traditionally been skepticalof any talk of a person’s desires and beliefs that goes beyondwhat can be established by examining the person’s preferences,which they take to be the only attitude that is directly revealed by aperson’s behaviour. For these economists, it is thereforeunwelcome news if we cannot even in principle determine thecomparative beliefs of a rational person by looking at herpreferences. So under what conditions can a preference relation \(\preceq\) on theset \(\Omega\) be represented as maximising desirability?
It then followsthat for any other proposition \(s\) that satisfies the aforementionedconditions that \(r\) satisfies, you should also be indifferentbetween \(p\cup s\) and \(q\cup s\), since, again, the two unions areequally likely to result in \(s\). As noted, a special case is when the content of\(p\) is such that it is recognisably something the agent can chooseto make true, i.e., an act. Another way to put this is that, when the above holds, the preferencerelation can be represented as maximising utility, since italways favours option with higher utility. There are several tools and platforms available that can assist in automating decision-making processes. For instance, GiniMachine is an AI-powered decision management platform that can process terabytes of historical data, building, validating, and deploying predictive models in minutes.