Decision Theory: A Formal Philosophical Introduction SpringerLink

Under certain assumptions, the overallor aggregate preference ordering is compatible with EUtheory. Dietrich and List (2013 & 2015) have proposed aneven more general framework for representing the reasons underlyingpreferences. In their framework, preferences satisfying some minimalconstraints are representable as dependent on the bundle of propertiesin terms of which each option is perceived by the agent in a givencontext. Properties can, in turn, be categorised as either optionproperties (which are intrinsic to the outcome), relationalproperties (which concern the outcome in a particular context),or context properties (which concern the context of choiceitself). Such a representation permits more detailed analysis of thereasons for an agent’s preferences and captures different kindsof context-dependence in an agent’s choices.

For instance, it may be that Bangkok isconsidered almost as desirable as Cardiff, but Amsterdam is a long waybehind Bangkok, relatively speaking. Or else perhaps Bangkok is onlymarginally better than Amsterdam, compared to the extent to whichCardiff is better than Bangkok. This kind of information about therelative distance between options, in terms of strength of preferenceor desirability, is precisely what is given by an interval-valuedutility function. 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 $\bO$. Consider first an ordering over three regular options, e.g., the threeholiday destinations Amsterdam, Bangkok and Cardiff, denoted $A$,$B$ and $C$ respectively.

This information suffices to ordinally representyour judgement; recall that any assignment of utilities is thenacceptable as long as $C$ gets a higher value than $B$ which getsa higher value than $A$. But perhaps we want to know more than canbe inferred from such a utility function—we want to know howmuch $C$ is preferred over $B$, compared to how much $B$ ispreferred over $A$. Decision theory is the study of how choices are and should be made.in a variety of different contexts. Here we look at the topic from a formal-philosophical point of view with a focus on normative and conceptual issues. After considering the question of how decision problems should be framed, we look at the both the standard theories of chance under conditions of certainty, risk and uncertainty and some of the current debates about how uncertainty should be measured and how agents should respond to it.

Then there is a desirability measure on $\Omega\setminus \bot $ decision theory is concerned with and a probability measure on $\Omega$ relative towhich $\preceq$ can be represented as maximisingdesirability. 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.

This brings us to the Transitivity axiom, which says that if an option $B$ is weakly preferred to $A$, and$C$ weakly preferred to $B$, then $C$ is weakly preferred to$A$. A recent challenge to Transitivity turns on heterogeneous setsof options, as per the discussion of Completeness above. But here adifferent interpretation of preference is brought to bear on thecomparison of options.

Broader significance of Expected Utility (EU) theory

In their framework, preferencessatisfying some minimal constraints are representable as dependent onthe bundle of properties in terms of which each option is perceived bythe agent in a given context.
The agent will use her beliefs about states to select an act that provides the best means for securing a desirable consequence.
Instead of adding specific belief-postulates to Jeffrey’stheory, as Joyce suggests, one can get the same uniqueness result byenriching the set of prospects.
In what follows, thes standard interpretation of sequential decision models will be assumed, and moreover, it will be assumed that rational agents reason aboutsuch decisions in a sophisticated manner (as per Levi 1991, Maher1992, Seidenfeld 1994, amongst others).

Moreover, now we see that one of Savage’srationality constraints on preference—the Sure ThingPrinciple—is plausible only if the modelled acts areprobabilistically independent of the states. 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. If we are interested inreal-world decisions, then the acts in question ought to berecognisable options for the agent (which we have seen isquestionable). The central goal of rational choice theory is to identify the conditions under which a decision maker’s beliefs and desires rationalize the choice of an action.

Even this limited characterisation iscontentious, however, and points to divergent interpretations of“preferences over prospects/options”.
Therefore,you should prefer to stake the prize $Y$ on $g$ rather than $g’$since the prize itself does not affect the probability of theevents.
Theorem 2 (von Neumann-Morgenstern)Let $\bO$ be a finite set of outcomes, $\bL$ a set ofcorresponding lotteries that is closed under probability mixture and$\preceq$ a weak preference relation on $\bL$.

Ethical and Social Issues

As noted above, preferenceconcerns the comparison of options; it is a relation between options.For a domain of options we speak of an agent’s preferenceordering, this being the ordering of options that is generated bythe agent’s preference between any two options in thatdomain. In the 18th century, Daniel Bernoulli introduced the concept of “expected utility” in the context of gambling, which was later formalized by John von Neumann and Oskar Morgenstern in the 1940s. Their work on Game Theory and Expected Utility Theory helped establish a rational basis for decision-making under uncertainty. In decision theory, a decision problem is situation in which a decision maker, (a person, a company, or a society) chooses what to do from a set of alternative acts, where the outcomeof the… 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.

1 On risk and regret attitudes

For instance, theaforementioned authors considered and characterised preferences thatexhibit exponential time discounting. When the above holds, we say that there is an expected utilityfunction that represents the agent’s preferences; in otherwords, the agent can be represented as maximising expectedutility. Thenthere is an ordinal utility function that represents $\preceq$ justin case $\preceq$ is complete and transitive.

The Representation of Decisions

Bradley and Stefánsson (2016) also develop a new decision theorypartly in response to the Allais paradox. But unlike Buchak, theysuggest that what explains Allais’ preferences is that the valueof wining nothing from a chosen lottery partly depends on what wouldhave happened had one chosen differently. To accommodate this, theyextend the Boolean algebra in Jeffrey’s decision theoryto counterfactual propositions, and show that Jeffrey’sextended theory can represent the value-dependencies one often findsbetween counterfactual and actual outcomes.

It should moreover be evident, given the discussion of the Sure ThingPrinciple (STP) in Section 3.1, that Jeffrey’s theory does not have this axiom. Since statesmay be probabilistically dependent on acts, an agent can berepresented as maximising the value of Jeffrey’s desirabilityfunction while violating the STP. Moreover, unlike Savage’s,Jeffrey’s representation theorem does not depend on anythinglike the Rectangular Field Assumption.

Compare the extra chance of outcome $0that $L_1$ hasover $L_2$ with the same extrachance of $0 that $L_3$ hasover $L_4$. Many people think thatthis extra risk counts more heavily in the first comparison than thelatter; i.e., that an extra 0.01 chance of $0 contributes a greaternegative value to $L_1$ thanto $L_3$. In eithercase, the value that $0 contributes towards the overall value of anoption clearly depends on what other outcomes the option might resultin.

3 The von Neumann and Morgenstern (vNM) representation theorem

Furthermore, itpermits explicit restrictions on what counts as a legitimate reasonfor preference, or in other words, what properties legitimatelyfeature in an outcome description; such restrictions may help toclarify the normative commitments of EU theory. The two central concepts in decision theoryare preferences and prospects (orequivalently, options). Roughly speaking, we say that anagent “prefers” the “option” $A$ over $B$ justin case, for the agent in question, the former is more desirable orchoice-worthy than the latter. This rough definition makes clear thatpreference is a comparative attitude; it is one of comparing optionsin terms of how desirable/choice-worthy they are.

Probability theory

Additionally, artificial intelligence systems leverage these principles to enhance machine learning algorithms and improve predictive accuracy. Suppose, however, that there is probabilisticdependency between the states of the world and the alternatives we areconsidering, and that we find $Z$ to be better than both $X$ and$Y$, and we also find $W$ to be better than both $X$ and$Y$. Moreover, suppose that $g$ makes $\neg E$ more likely than$f$ does, and $f’$ makes $\neg E$ more likely than $g’$does. The intuition behind the STP is that if $g$ is weakly preferredto $f$, then that must be because the consequence $Y$ isconsidered at least as desirable as $X$, which by the same reasoningimplies that $g’$ is weakly preferred to $f’$. Wesay that alternative $f$“agrees with” $g$ inevent $E$ if, for any state inevent $E$, $f$ and $g$ yieldthe same outcome. AI decision-making strategies involve balancing the benefits of improved accuracy, efficiency, and bias reduction against the challenges of explainability, privacy, complexity, and ethical considerations.