Decision Theory Stanford Encyclopedia of Philosophy
22 de setembro de 2022
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 revival of subjective probability theory, from the work of Frank Ramsey, Bruno de Finetti, Leonard Savage and others, extended the scope of expected utility theory to situations where subjective probabilities can be used. At the time, von Neumann and Morgenstern’s theory of expected utility10 proved that expected utility maximization followed from basic postulates about rational behavior.
The decision theories of Savage and Jeffrey, as well as those of theircritics, apparently concern a single or “one shot only”decision; at issue is an agent’s preference ordering, andultimately her choice of act, at a particular point in time.
And suppose now we find a proposition \(r\), thatis pairwise incompatible with both \(p\) and \(q\), and which you findmore desirable than both \(p\) and \(q\).
Is there anyprobability \(p\) such that you would be willing to accept a gamblethat has that probability of you losing your life and probability\((1-p)\) of you gaining $10?
Decision theory can accommodate determinism by incorporating counterfactuals and hypothetical scenarios, allowing individuals to evaluate alternative outcomes and their implications.
The revival of subjective probability theory, from the work of Frank Ramsey, Bruno de Finetti, Leonard Savage and others, extended the scope of expected utility theory to situations where subjective probabilities can be used.
Decision theory provides a number of suggestions for how to estimate complex probabilities under uncertainty, most of which are derived from Bayesian inference. AI technologies such as machine learning, natural language processing, and computer vision are trusted aspects of business today, used to increase profits and reach set goals. Decision automation relies on prescriptive or predictive analytics, benefiting from its scalability, speed, and consistency in decision-making.
AI-Driven Decision-Making Models
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. 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.
Future Directions in Decision Theory
Companies use the decision theory in operation research because it considers several outcomes, rational reasoning, and influencing factors to understand how a person thinks logically rather than idealistically. Decision theory refers to a range of econometric and statistical tools for analyzing an individual’s choices. In other words, it lets the entity make the best logical decision possible when dealing with uncertain and unknown circumstances. It can actually be seen as a weak version ofIndependence and the Sure Thing Principle, and it plays a similar rolein Jeffrey’s theory.
Decision-Making Models
Is there anyprobability \(p\) such that you would be willing to accept a gamblethat has that probability of you losing your life and probability\((1-p)\) of you gaining $10? However,the very same people would presumably cross the street to pick up a$10 bill they had dropped. But that is just taking a gamble that has avery small probability of being killed by a car but a much higherprobability of gaining $10! More generally, although people rarelythink of it this way, they constantly take gambles that have minusculechances of leading to imminent death, and correspondingly very highchances of some modest reward. Game theory occupies about a sixth of the book, with the principal topics being zero-sum games, the prisoner’s dilemma, Nash equilibrium strategy sets, and the Nash solution to bargaining problems. Peterson also provides some helpful sections on the influence of game theory on evolutionary theory and ethical theory.
Start with the Completeness axiom, which says that an agent cancompare, in terms of the weak preference relation, all pairs ofoptions in \(S\). Whether or not Completeness is a plausiblerationality constraint depends both on what sort of options are underconsideration, and how we interpret preferences over these options. 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.
First, the book presupposes a high level of technical sophistication on the part of the reader. For example, when deriving one mathematical expression from another, Peterson frequently skips important steps, and it is likely that some students will get stuck. Similarly, in proving important propositions such as theorems, Peterson frequently leaves important claims unstated. In many cases, filling in the gaps will require more insight and persistence than many students have; in some cases, they may require more than many instructors have.
Alternatives
The orthodox normative decision theory, expectedutility (EU) theory, essentially says that, in situations ofuncertainty, one should prefer the option with greatestexpected desirability or value. (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. Perhaps there is always a way to contrive decision models such thatacts are intuitively probabilistically independent of states. Recall that Savage was tryingto formulate a way of determining a rational agent’s beliefsfrom her preferences over acts, such that the beliefs can ultimatelybe represented by a probability function. If we are interested inreal-world decisions, then the acts in question ought to berecognisable options for the agent (which we have seen isquestionable). 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.
While Ulysses isrational at the first choice node by static decisionstandards, we might regard him irrational overall bysequential decision standards, understood in terms of the relativevalue of sequences of choices. It would have beenbetter were he able to sail unconstrained and continue on home toIthaca. This sequence could have been achieved if Ulysses werecontinuously rational over the extended time period; say, ifat all times he were to act as an EU maximiser, and change his beliefsand desires only in accordance with Bayesian norms (variants ofstandard conditionalisation). On this reading, sequentialdecision models introduce considerations of rationality-over-time. One may well wonder whether EU theory, indeed decision theory moregenerally, is neutral with respect to normative ethics, or whether itis compatible only with ethical consequentialism, given thatthe ranking of an act is fully determined by the utility of itspossible outcomes. Such a model seems at odds withnonconsequentialist ethical theories for which thechoice-worthiness of acts purportedly depends on more than the moralvalue of their consequences.
Decision theory is a multidisciplinary field that combines insights from economics, philosophy, psychology, and statistics to understand how individuals make decisions.
Furthermore, assigning values or utilities to outcomes can be subjective and vary significantly between individuals or organizations, affecting the decision-making process.
On a closer look, however, it is evidentthat some of our beliefs can be determined by examining ourpreferences.
For instance, if youstrictly prefer the first lottery to the second, then that suggestsyou consider heads more likely than tails.
In other words, thisindependence must be built into the decision model if it is tofacilitate appropriate measures of belief and desire.
Types of Decision-Making: Under Certainty, Risk, and Uncertainty
On Buchak’s interpretation, the explanation forAllais’ preferences is not the different value that theoutcome $0 has depending on what lottery it is part of. However, the contribution that $0makes towards the overall value of an option partly depends on whatother outcomes are possible, she suggests, which reflects the factthat the option-risk that the possibility of $0 generates depends onwhat other outcomes the option might result in. To accommodateAllais’ preferences (and other intuitively rational attitudes torisk that violate EU theory), Buchak introduces a riskfunction that represents people’s willingness to tradechances of something good for risks of something bad.
After all, an apt model of preference issupposedly one that captures, in the description of final outcomes andoptions, everything that matters to an agent. In that case, however,EU theory is effectively vacuous or impotent as a standard ofrationality to which agents can aspire. Moreover, it stretches thenotion of what are genuine properties of outcomes that can reasonablyconfer value or be desirable for an agent. The above result may seem remarkable; in particular, the fact that aperson’s preferences can determine decision theory is concerned with 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.
2 On rational desire
But it is not directly inconsistent withAllais’ preferences, and its plausibility does not depend on thetype of probabilistic independence that the STP implies. The postulaterequires that no proposition be strictly better or worse than all ofits possible realisations, which seems to be a reasonable requirement.When \(p\) and \(q\) are mutually incompatible, \(p\cup q\) impliesthat either \(p\) or \(q\) is true, but not both. Hence, it seemsreasonable that \(p\cup q\) should be neither strictly more nor lessdesirable than both \(p\) and \(q\). Then since \(p\cup q\) is compatiblewith the truth of either the more or the less desirable of the two,\(p\cup q\)’s desirability should fall strictly between that of\(p\) and that of \(q\). However, if \(p\) and \(q\) are equallydesirable, then \(p\cup q\) should be as desirable as each of thetwo. In the second choice situation, however, the minimumone stands to gain is $0 no matter which choice one makes.
Decision Theory Stanford Encyclopedia of Philosophy
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 revival of subjective probability theory, from the work of Frank Ramsey, Bruno de Finetti, Leonard Savage and others, extended the scope of expected utility theory to situations where subjective probabilities can be used. At the time, von Neumann and Morgenstern’s theory of expected utility10 proved that expected utility maximization followed from basic postulates about rational behavior.
Decision theory provides a number of suggestions for how to estimate complex probabilities under uncertainty, most of which are derived from Bayesian inference. AI technologies such as machine learning, natural language processing, and computer vision are trusted aspects of business today, used to increase profits and reach set goals. Decision automation relies on prescriptive or predictive analytics, benefiting from its scalability, speed, and consistency in decision-making.
AI-Driven Decision-Making Models
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. 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.
Future Directions in Decision Theory
Companies use the decision theory in operation research because it considers several outcomes, rational reasoning, and influencing factors to understand how a person thinks logically rather than idealistically. Decision theory refers to a range of econometric and statistical tools for analyzing an individual’s choices. In other words, it lets the entity make the best logical decision possible when dealing with uncertain and unknown circumstances. It can actually be seen as a weak version ofIndependence and the Sure Thing Principle, and it plays a similar rolein Jeffrey’s theory.
Decision-Making Models
Is there anyprobability \(p\) such that you would be willing to accept a gamblethat has that probability of you losing your life and probability\((1-p)\) of you gaining $10? However,the very same people would presumably cross the street to pick up a$10 bill they had dropped. But that is just taking a gamble that has avery small probability of being killed by a car but a much higherprobability of gaining $10! More generally, although people rarelythink of it this way, they constantly take gambles that have minusculechances of leading to imminent death, and correspondingly very highchances of some modest reward. Game theory occupies about a sixth of the book, with the principal topics being zero-sum games, the prisoner’s dilemma, Nash equilibrium strategy sets, and the Nash solution to bargaining problems. Peterson also provides some helpful sections on the influence of game theory on evolutionary theory and ethical theory.
Start with the Completeness axiom, which says that an agent cancompare, in terms of the weak preference relation, all pairs ofoptions in \(S\). Whether or not Completeness is a plausiblerationality constraint depends both on what sort of options are underconsideration, and how we interpret preferences over these options. 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.
First, the book presupposes a high level of technical sophistication on the part of the reader. For example, when deriving one mathematical expression from another, Peterson frequently skips important steps, and it is likely that some students will get stuck. Similarly, in proving important propositions such as theorems, Peterson frequently leaves important claims unstated. In many cases, filling in the gaps will require more insight and persistence than many students have; in some cases, they may require more than many instructors have.
Alternatives
The orthodox normative decision theory, expectedutility (EU) theory, essentially says that, in situations ofuncertainty, one should prefer the option with greatestexpected desirability or value. (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. Perhaps there is always a way to contrive decision models such thatacts are intuitively probabilistically independent of states. Recall that Savage was tryingto formulate a way of determining a rational agent’s beliefsfrom her preferences over acts, such that the beliefs can ultimatelybe represented by a probability function. If we are interested inreal-world decisions, then the acts in question ought to berecognisable options for the agent (which we have seen isquestionable). 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.
While Ulysses isrational at the first choice node by static decisionstandards, we might regard him irrational overall bysequential decision standards, understood in terms of the relativevalue of sequences of choices. It would have beenbetter were he able to sail unconstrained and continue on home toIthaca. This sequence could have been achieved if Ulysses werecontinuously rational over the extended time period; say, ifat all times he were to act as an EU maximiser, and change his beliefsand desires only in accordance with Bayesian norms (variants ofstandard conditionalisation). On this reading, sequentialdecision models introduce considerations of rationality-over-time. One may well wonder whether EU theory, indeed decision theory moregenerally, is neutral with respect to normative ethics, or whether itis compatible only with ethical consequentialism, given thatthe ranking of an act is fully determined by the utility of itspossible outcomes. Such a model seems at odds withnonconsequentialist ethical theories for which thechoice-worthiness of acts purportedly depends on more than the moralvalue of their consequences.
Types of Decision-Making: Under Certainty, Risk, and Uncertainty
On Buchak’s interpretation, the explanation forAllais’ preferences is not the different value that theoutcome $0 has depending on what lottery it is part of. However, the contribution that $0makes towards the overall value of an option partly depends on whatother outcomes are possible, she suggests, which reflects the factthat the option-risk that the possibility of $0 generates depends onwhat other outcomes the option might result in. To accommodateAllais’ preferences (and other intuitively rational attitudes torisk that violate EU theory), Buchak introduces a riskfunction that represents people’s willingness to tradechances of something good for risks of something bad.
After all, an apt model of preference issupposedly one that captures, in the description of final outcomes andoptions, everything that matters to an agent. In that case, however,EU theory is effectively vacuous or impotent as a standard ofrationality to which agents can aspire. Moreover, it stretches thenotion of what are genuine properties of outcomes that can reasonablyconfer value or be desirable for an agent. The above result may seem remarkable; in particular, the fact that aperson’s preferences can determine decision theory is concerned with 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.
2 On rational desire
But it is not directly inconsistent withAllais’ preferences, and its plausibility does not depend on thetype of probabilistic independence that the STP implies. The postulaterequires that no proposition be strictly better or worse than all ofits possible realisations, which seems to be a reasonable requirement.When \(p\) and \(q\) are mutually incompatible, \(p\cup q\) impliesthat either \(p\) or \(q\) is true, but not both. Hence, it seemsreasonable that \(p\cup q\) should be neither strictly more nor lessdesirable than both \(p\) and \(q\). Then since \(p\cup q\) is compatiblewith the truth of either the more or the less desirable of the two,\(p\cup q\)’s desirability should fall strictly between that of\(p\) and that of \(q\). However, if \(p\) and \(q\) are equallydesirable, then \(p\cup q\) should be as desirable as each of thetwo. In the second choice situation, however, the minimumone stands to gain is $0 no matter which choice one makes.