Decision Theory at a Glance

Note: The following glossary has been excerpted from Peterson, Martin. An Introduction to Decision Theory (Cambridge Introductions to Philosophy). Cambridge University Press, 2017. All copy/paste errors are mine.

Decision theory: The theory of rational decision making. Decision theory is an interdisciplinary project to which philosophers, economists, psychologists, computer scientists and statisticians contribute their expertise. However, decision theorists from all disciplines share a number of basic concepts and distinctions. To start with, everyone agrees that it makes sense to distinguish between descriptive and normative decision theory. Descriptive decision theories seek to explain and predict how people actually make decisions. This is an empirical discipline, stemming from experimental psychology.

Normative theories seek to yield prescriptions about what decision makers are rationally required – or ought – to do. Descriptive and normative decision theory are, thus, two separate fields of inquiry, which may be studied independently of each other. For example, from a normative point of view it seems interesting to question whether people visiting casinos in Las Vegas ought to gamble as much as they do. In addition, no matter whether this behavior is rational or not, it seems worthwhile to explain why people gamble (even though they know they will almost certainly lose money in the long run). The focus of this book is normative decision theory.

Axiom: An axiom is a fundamental premise of an argument for which no further justification is given. Example: According to the asymmetry axiom, no rational agent strictly prefers x to y and y to x.

Bargaining problem: The bargaining problem is a cooperative game with infinitely many Nash equilibria, which serves as a model for a type of situation that arises in many areas of society: A pair of players are offered to split some amount of money between the two of them. Each player has to write down his or her demand and place it in a sealed envelope. If the amounts they demand sum to more than the total amount available the players will get nothing; otherwise each player will get the amount he or she demanded. The players are allowed to communicate and form whatever binding agreements they wish. A general solution to this problem was offered by Nash, who based his proposal on a small set of intuitively plausible axioms.

Bayes’ theorem: Bayes’ theorem is an undisputed mathematical result about the correct way to calculate conditional probabilities. It holds that the probability of B given A equals the probability of B times the probability of A given B, divided by the following two terms: the probability of B times the probability of A given B and the probability of not-B times the probability of A given not-B.

Bayesianism: The term ‘Bayesianism’ has many different meanings in decision theory, statistics and the philosophy of those disciplines. Most Bayesian accounts of decision theory and statistics can be conceived of as claims about the correct way to apply Bayes’ theorem in various real-life contexts. In decision theory, Bayesianism is particularly closely associated with the view that probabilities are subjective and that rational decision makers seek to maximize subjective expected utility.

Cardinal scale: Cardinal scales are used when we measure objects numerically and differences or ratios between measurement points are preserved across all possible transformations of the scale. Example: Time, mass, money and temperature can be measured on cardinal scales. Cardinal scales can be divided into two categories, namely interval scales and ratio scales.

Decision matrix: A decision matrix is used for visualizing a formal representation of a decision problem graphically. All decision matrices can be converted into decision trees (but the converse is not true).

Decision tree: A decision tree is used for visualizing a formal representation of a decision problem graphically. All decision matrices can be converted into decision trees, but some decision trees (e.g. trees with more than one choice node) cannot be converted into decision matrices.

Dominance: Act A strictly dominates act B if and only if the outcome of A will be strictly better than B no matter which state of the world happens to be the true one. Act A weakly dominates act B if and only if the outcome of A will be as good as that of B no matter which state of the world happens to be the true one, and strictly better under at least one state.

Equilibrium: In game theory, a set of strategies is in equilibrium if and only if it holds that once these strategies are chosen, none of the players could reach a better outcome by unilaterally switching to another strategy. This means that each player’s strategy is optimal given that the opponents stick to their chosen strategies. (See also ‘Nash equilibrium’.)

Expected utility: The expected utility of an act can be calculated once we know the probabilities and utilities of its possible outcomes, by multiplying the probability and utility of each outcome and then summing all terms into a single number representing the average utility of the act.

Function (mathematical): A function is a device that takes something (such as a number) as its input and for each input returns exactly one output (such as another number). f(x) = 3x + 7 is a function, which returns 7 if x = 0 and 10 if x = 1, etc.

Impossibility theorem: An impossibility theorem is a formal result showing that a set of seemingly plausible premises (desiderata, axioms, etc.) imply a contradiction, and hence that no theory or principle can satisfy all premises (desiderata, axioms, etc.).

Interval scale: Interval scales measure objects numerically such that differences between measurement points are preserved across all possible transformations of the scale. Example: The Fahrenheit and Centigrade scales for measuring temperature are interval scales (but the Kelvin scale is a ratio scale). In decision theory, the most frequently used interval scale is the von Neumann–Morgenstern utility scale.

Law of large numbers: The law of large numbers is a mathematical theorem showing that if a random experiment (such as rolling a die or tossing a coin) is repeated n times and each experiment has a probability p of leading to a predetermined outcome, then the probability that the percentage of such outcomes differs from p by more than a very small amount ε converges to 0 as the number of trials n approaches infinity. This holds true for every ε > 0, no matter how small.

Lemma: A lemma is an intermediate result in a proof of some more complex theorem. Example: Instead of directly proving Arrow’s impossibility theorem it can be broken down into several lemmas, which together entail the theorem.

Logical consequence: A conclusion B is a logical consequence of a set of premises A if and only if it can never be the case that A is true while B is false. Example: B is a logical consequence of P-and-not-P, because it can never be the case that P-and-not-P is true (so in this case the truth value of B is irrelevant).

Maximin: Maximin is a decision rule sometimes used in decisions under ignorance, which holds that one should maximize the minimal value obtainable in each decision. Hence, if the worst possible outcome of one alternative is better than that of another, then the former should be chosen.

Mixed strategy: In game theory, rational players sometimes decide what to do by tossing a coin (or by letting some other random mechanism make the decision for them). By using mixed strategies players sometimes reach equilibria that cannot be reached by playing pure strategies, i.e. nonprobabilistic strategies.

Nash equilibrium: The Nash equilibrium is a central concept in game theory. A set of strategies played by a group of players constitutes a Nash equilibrium if and only if “each player’s … strategy maximizes his pay-off if the strategies of the others are held fixed. Thus each player’s strategy is optimal against those of the others” (Nash 1950). The definition of the term equilibrium given above is equivalent, although other definitions are also discussed in the literature.

Ordinal scale: Ordinal scales measure objects without making any comparisons of differences or ratios between measurement points across different transformations of the scale. Ordinal scales may very well be represented by numbers, but the numbers merely carry information about the relative ordering between each pair of objects. Example: I like Carmen more than The Magic Flute, and The Magic Flute more than Figaro. This ordering can be represented numerically, say by the numbers 1, 2 and 3, but the numbers do not reveal any information about how much more I like one of them over the others.

Paradox: A paradox is a false conclusion that follows logically from a set of seemingly true premises. Examples: The St. Petersburg paradox, the Allais paradox, the Ellsberg paradox, and the two-envelope paradox are all well-known examples of paradoxes in decision theory.

Posterior probability: The posterior probability is the probability assigned to an event after new evidence or information about the event has been received. Example: You receive further evidence that supports your favorite scientific hypothesis. The posterior probability is the new, updated probability assigned to the hypothesis. (See also ‘Bayes’ theorem’ and ‘prior probability’.)

Preference: If you prefer A to B, i.e. if A ≻ B, then you would choose A rather than B if offered a choice between the two. Some philosophers explain this choice-disposition by claiming that a preference is a mental disposition to choose in a certain way.

Prior probability: The prior probability is the (unconditional) probability assigned to an event before new evidence or information about the event is taken into account. Example: You receive new evidence that supports your favorite scientific hypothesis. You therefore update the probability that the hypothesis is true by using Bayes’ theorem, but to do this you need to know the prior (unconditional) probability that the theory was true before the new evidence had been taken into account. (See also ‘Bayes’ theorem’ and ‘posterior probability’.)

Prisoner’s dilemma: The prisoner’s dilemma is a noncooperative game in which rational players must choose strategies that they know will be suboptimal for all players. The prisoner’s dilemma illustrates a fundamental clash between individual rationality and group rationality.

Probability: The probability calculus measures how likely an event (proposition) is to occur (is to be true). Philosophers disagree about the interpretation of the probability calculus. Objectivists maintain that probabilities are objective features of the world that exist independently of us, whereas subjectivists maintain that probabilities express statements about the speaker’s degree of belief in a proposition or event.

Randomized act: A randomized act is a probabilistic mixture of two or more (randomized or nonrandomized) acts. Example: You have decided to do A or not-A, but instead of performing one of these acts for sure you toss a coin and perform A if and only if it lands heads up, otherwise you perform not-A.

Ratio scale: Ratio scales measure objects numerically such that ratios between measurement points are preserved across all possible transformations of the scale. Example: The widely used scales for measuring mass, length and time; in decision theory, the most frequently used ratio scale is the probability calculus, which measures how likely an event (proposition) is to occur (is to be true).

Real number: A real number is any number that can be characterized by a finite or infinite decimal representation. All rational and irrational numbers are real numbers, e.g. 2 and 4.656576786, and π. However, imaginary numbers are not real numbers.

Representation theorem: A representation theorem is a mathematical result showing that some nonnumerical structure, such as preferences over a set of objects, can be represented by some mathematical structure. Example: According to the ordinal utility theorem preferences over a set of objects, {x, y, …} can be represented by a real-valued function u such that x is preferred to y if and only if u( x) > u( y).

Risk: The term “risk” has several meanings. In decision theory, a decision under risk is taken if and only if the decision maker knows the probability and utility of all possible outcomes. In other contexts, the term “risk” sometimes refers to the probability of an event or the expected (dis)utility of an act.

Theorem: A theorem summarizes a conclusion derived from a specified set of premises. Example: The von Neumann–Morgenstern theorem holds that if a set of preferences over lotteries satisfies certain structural conditions (axioms), then these preferences can be represented by a certain mathematical structure.

Uncertainty: The term “uncertainty” has several meanings. In decision theory, a decision under uncertainty is taken if and only if the decision maker knows the utility of all possible outcomes, but not their probabilities. In other contexts, the term “uncertainty” is sometimes used in a wider sense to refer to any type of situation in which there is some lack of relevant information.

Utilitarianism: Utilitarianism is the ethical theory prescribing that an act is morally right if and only if it maximizes overall well-being. This theory was originally developed by Bentham and Mill in the nineteenth century, and is currently one of the most influential ethical theories among professional philosophers, economists and others interested in what individuals or groups of individuals ought to do and not to do.

Utility: The more you desire an object, the higher is its utility. Utility is measured on some utility scale, which is either ordinal or cardinal, and if it is cardinal it is either an interval scale or a ratio scale. (See “cardinal scale”, “interval scale”, “ordinal scale” and “ratio scale”.)

Zero-sum game: In a zero-sum game each player wins exactly as much as the opponent( s) lose. Most casino games and parlor games such as chess are zero-sum games, because the total amount of money or points is fixed. Games that do not fulfill this condition, such as the prisoner’s dilemma, are called nonzero-sum games.

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