Decision Theory

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Decision theory provides a general, mathematically rigorous account of decision making under uncertainty. The subject includes rational choice theory, which seeks to formulate and justify the normative principles that govern optimal decision making, and descriptive choice theory, which aims to explain how human beings actually make decisions. Within both these areas one may distinguish individual decision theory, which concerns the choices of a single agent with specific goals and knowledge, and game theory, which deals with interactions among individuals. This entry will focus on rational choice theory for the single agent, but some descriptive results will be mentioned in passing.

Decision Problems

It is standard to portray decision makers as facing choices among acts that cause desirable or undesirable consequences when performed in various states of the world. Acts characterize those aspects of the world that an agent can directly control. States specify contingencies beyond her control that might influence the consequences of acts. Each combination of an act A and state S fixes a unique consequence A (S ) that describes the result of doing A in S. When there are only finitely many acts and states the decision situation can be represented as a matrix:


The agent decides the row, the world decides the column, and these together determine the consequence.

In any well-formed decision problem (1) the value of each consequence is independent of the act and state that bring it about, (2) each consequence is sufficiently detailed to settle every matter about which the agent intrinsically cares, (3) neither acts nor states have any value except as a means for producing consequences, and (4) the agent will not believe that she has the ability to causally influence which state obtains. When these conditions are met, the agent's goals and values affect her decision only via her desires for consequences, and her beliefs influence her choice via her uncertainty about which state obtains. The agent will use her beliefs about states to select an act that provides the best means for securing a desirable consequence.

For theoretical purposes, it is useful to idealize the decision setting by assuming that the repertoire of actions is rich. Specifically, for each consequence c there is a constant act [c ] that produces c in every state of the world, and, for any acts A and B, and any disjunction of states E, there is a mixed act AE B~E that produces A 's consequence when E holds and B 's consequence when ~E holds. While real agents will typically be unable to realize such recherché prospects as these, imagining that decision makers have attitudes toward them often helps one determine which realistic acts should be performed.

This model applies to one-choice decisions made at a specific time. Early decision theorists believed that sequences of decisions made over time could be reduced to one-shot decisions among contingency plans, or strategies, but this view now has few adherents. The topic of dynamic decision making lies beyond the scope of this entry. For relevant discussions, see Peter Hammond (1988), Edward McClennen (1990), and James M. Joyce (1995).

Subjective Expected Utility

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. According to the standard model of decision-theoretic rationality, an action is rational just in case, relative to the agent's beliefs and desires, it has the highest subjective expected utility of any available option. This subjective expected utility (SEU) theory has its roots in the work of Blaise Pascal, Daniel Bernoulli, Vilfredo Pareto, and Frank P. Ramsey, and finds its fullest expression in Leonard J. Savage's Foundations of Statistics (1972). According to SEU a rational agent's basic desires can be represented by a utility function u that assigns a real number u (c ) to each consequence c. The value of u (c ) measures the degree to which c would satisfy the agent's desires and promote his or her aims.

Likewise, the agent's beliefs can be characterized by a subjective probability function P whose values express the agent's subjective degrees of confidence, or credences, in the states of the world. P is assumed to be unique, and u is unique once the choice of a unit and a zero for measuring utilities are fixed. Given P and u , the expected utility of each act A is a weighted average of the utilities of its consequences, so that ExpP,u (A ) = i = 1 P (Si )u (A (Si )). According to the core doctrine of SEU, the choice of an act is rational only if it maximizes the chooser's subjective expected utility, so that ExpP,u (A ) ExpP,u (B ) for all acts B. This should not be taken to suggest that the agent sees herself as maximizing expected utility, or even that she has the concept of expected utility. SEU does not propose expected utility maximization as a decision procedure, but as a way of assessing the results of such procedures. Rational decision makers merely act as if they maximize subjective expected utility; they need not explicitly do so.

Representation of Rational Preference

A central challenge for SEU is to find a principled way of characterizing credences and utilities. Following the lead of Ramsey (1931), the standard solution involves proving a representation theorem that shows how an agent's beliefs about states and desires for outcomes are related to her all-things-considered preferences for acts. The agent is assumed to make three sorts of comparative evaluations between acts: She might strictly prefer A to B, written A > B, weakly prefer A to B, A > B, or be indifferent between them, A B. These relations hold, respectively, just in case the agent judges that, on balance, A will do more than, at least as much as, or exactly as much as, B will to satisfy her desires and promote her aims. The totality of such evaluations is the agent's preference ranking.

Early decision theorists, motivated by a misguided scientific methodology, thought of preferences as operationally defined in terms of overt choices, so that, by definition, an agent prefers A to B if and only if (iff) she will incur a cost to choose A over B. Even though this sort of behaviorism remains firmly ensconced in some areas of economics, it has been widely and effectively criticized (Sen 1977, Joyce 1999). In the end, preferences are best thought of as subjective judgments of the comparative merits of actions as promoters of desirable outcomes. While such judgments are closely tied to overt choice behavior, the relationship between the two is nowhere near as direct and unsophisticated as behaviorism suggests.

The representation theorem approach seeks to justify SEU by (1) imposing a system of axiomatic constraints on preference rankings, (2) arguing that these express requirements of rationality, and then (3) proving that any preference ranking that satisfies the axioms can be associated with a probability P and a utility u such that each of A >, > , B hold iff, respectively, ExpP,u (A ) >, , = ExpP,u (B ). An agent whose preferences can be represented in this way evaluates acts as if she were aiming to maximize expected utility relative to P and u .

Frame Invariance

All versions of SEU share a common set of core principles. The first says that logically equivalent redescriptions of prospects should not alter preferences.

SEU 1 Frame Invariance. The evaluation of an act should not depend on how its consequences happen to be described.

People often violate this constraint. Consider the following two decision framings due to E. Shafir and A. Tversky (1995):

  • You receive $300, and are then given a choice between getting another $100 for sure or getting $200 or $0 depending on the toss of a fair coin.
  • You receive $500, but are then forced to choose between returning $100 for sure or returning $200 or $0 depending on the toss of a fair coin.

Since both decisions offer a sure $400 or a fifty-fifty chance of $300 or $500, SEU1 requires agents to make the same choice in each case (though it does not tell them which choice to make). As it turns out, most people make the safe choice in the first case and take the sure $400, but they make risky choice in the second case by taking the fifty-fifty gamble. Cognitive psychologists attribute this violation of SEU1 to the following two irrational tendencies of human decision makers:

Divergence from Status Quo. People are more concerned with incremental gains and losses, seen as changes in the status quo, than with total well-being or overall happiness.

Asymmetrical Risk Aversion. People eschew risk when pursuing gains, but to seek risk when avoiding losses.

Under the first description, where the status quo is $300, people see themselves as trying to secure an additional gain, and so opt for the safe alternative. Under the second description, where the status quo is $500, people see themselves avoiding losses, and so incline toward the risky choice. These divergent attitudes are irrational given that the options are effectively identical.

Value Independence

The second principle requires each act to have a value that depends only on the values and probabilities of the outcomes it might cause.

SEU2 Value Independence. If the agent prefers A to B in a decision where C is not an option, then she should still prefer A to B even if C is an option, provided that C 's inclusion does not provide any information about state probabilities.

Apparent counterexamples to SEU2 as a requirement of rationality always involve violations of the proviso. For example, R. Duncan Luce and Howard Raiffa (1957) discuss a diner who, thinking he is in a greasy spoon, prefers salmon to steak, but then orders steak when told that snails are on the menu. SEU2 is vindicated by the observation that the availability of snails provides the diner with evidence for thinking that he is in fine restaurant, and this alters his views about the comparative merits of the salmon and steak. Other common violations of SEU2 are clearly irrational. For example, D. Redelmeier and E. Shafir (1995) show that physicians are less likely to prescribe ibuprofen to patients in pain when they have the option of prescribing the inferior drug piroxicam than when piroxicam is unavailable. While this sort of behavior does not discredit SEU2 as a normative principle, it does show that it is inaccurate as a description of human behavior.


The third principle rules out preference cycles in which A > B, B > C, but A > C, and it requires that the preference ranking be complete in the sense that exactly one of A > B, A B or B > A always hold.

SEU3 Ordering. Preference rankings completely order the set of acts.

Though some dispute anticyclicality, and Peter C. Fishburn (1991) even develops an acyclic decision theory, the prohibition against cycles remains among the most widely accepted principles of rational preference. On views that equate preferences and choices, preference cycles are irrational because they leave the agent open to exploitation as a "money pump": she will freely trade C for B and B for A, and then pay a fee to exchange C for A, thereby getting nothing for something. Even if choice is not equated with preference, cycles are still problematic. Many seemingly rational cycles treat preferences as partial, rather than all-things-considered evaluations. For instance, one might prefer an expensive shirt to a moderately priced one on the basis of style, and prefer the moderately priced shirt to a cheap shirt on the basis of durability, but prefer the cheap shirt to the expensive one on the basis of price. Here what seems to be a rational preference cycle is really a failure to integrate considerations of style, durability, and price into an all-things-considered value judgment.

Failures of evaluative discrimination can also seem to generate rational preference cycles. Suppose a vinophile, who cares only about how his wine tastes, cannot taste any difference between wine A and wine B, or between wine B and wine C, but can taste that C is better than A. It is tempting to think that the vinophile should be indifferent between A and B and between B and C, but should prefer C to A. A clearer understanding of the situation shows that this is incorrect. A person should only be indifferent between prospects when he lacks any reason, on balance, for preferring one to the other. The vinophile, however, has reason to favor B over A since B is indistinguishable in taste from a wine superior to A. He also has reason to favor C over B since B is indistinguishable from a wine inferior to C. Properly speaking, then, the vinophile is not indifferent between A and B or between B and C : his preferences run B > A, C > B, and C > A, but neither A B nor B C is true.

One might worry that the vinophile's reasons seem insufficient to justify strict preferences. It would, for example, be silly for him to pay anything to trade a bottle of A for a bottle of B (unless he could convert the latter into a bottle of C for a small enough fee). While this is a legitimate concern, it tells against completeness rather than anticyclicality. When an agent cannot precisely discriminate the qualities of prospects on which his evaluations depend, or when these qualities are themselves vague or indeterminate, his preference ranking will be incomplete: for certain options, all three of A > B, A B, and B > A will fail. Sometimes both A > B and B > A will fail as well, in which case the agent has no views about the comparative merits of A and B. Alternatively, as in the vinophile example, the agent might determinately weakly prefer B to A even though he neither strictly prefers B to A nor is indifferent between them. So, while A B and B > A each entail B > A, the latter is consistent with the falsity of both A B and B > A. Besides indeterminacy or vagueness in values, incompleteness in preferences can arise via an imprecision in credences. In both sorts of cases it can be perfectly rational to have an incomplete preference ranking.

One response to these considerations, which is advocated in Isaac Levi (1980), Richard Jeffrey (1983), and Mark Kaplan (1983), is to construe SEU3's completeness clause as a requirement of coherent extendibility. Instead of asking an agent to completely order acts, one demands merely that there be at least one complete preference ranking (usually there will be many) that satisfies all other requirements of rationality, and that agrees with the agent's preferences whenever she has definite preferences. One then represents vague or indeterminate preferences by giving up the idea that the agent's attitudes can be modeled by a single probability/utility pair (given a unit and zero for utility). Rather, there will be a representing set R of (P, u ) pairs that agree with the agent's preferences in the sense that, for any options A and B, each of A >, > , B hold iff, respectively, ExpP,u (A ) >, , = ExpP,u (B ) holds for every (P, u ) pair in R . Act A is unambiguously choice worthy only if maximizes expected utility relative to every (P, u ) pair in R . It is admissible when it maximizes expected utility relative to some such pair. There is no generally accepted procedure for handling situations where no admissible act is unambiguously choice worthy. Some theorists would say that the agent's beliefs and desires are too indefinite to justify any choice as rational. Others, most notably Levi (1980), maintain that principles of decision making that outrun expected utility maximization come into play in this situation. For example, Levi allows agents to decide among admissible options using maximin, that is, by selecting the act whose worst consequence is at least as good as the worst consequence of any alternative.

Comparative Probability

The next principle of SEU forges a link between rational preference and rational belief. A wager on event E is an act of the form [c ]E [d ]~E where [c ] > [d ]. Such a wager produces the desirable consequence c in every state consistent with E and the undesirable consequence d in every state consistent with ~E. Intuitively, a person should prefer such a wager more strongly the more likely she takes E to be. More precisely, given any events E and F, [c ]E [d ]~E should be preferred to [c ]F . [d ]~F exactly if E as more probable than F. The following axiom is meant to ensure that this is so.

SEU4 Comparative Probability. Assuming [c ] > [d ], if the agent prefers [c ]E [d ]~E to [c ]F [d ]~F , she must also prefer [c* ]E [d* ]~E to [c* ]F [d* ]~F for any consequences such that [c* ] > [d* ].

SEU4 can seem implausible when the values of consequences vary with the world's state. Suppose, for example, that c and d are monetary fortunes that one might have in ten years, say c = $500,000 and d = $400,000. Let E and F be hypotheses about the cumulative rate of inflation over the decade: E puts the figure at 60 percent, while F puts it at 10 percent. Even if one regards E as the more probable hypothesis, one might still prefer to wager on F since one's fortune will be worth more if F is true.

There are two standard responses to this problem. Savage (1972) maintains that decision problems of this sort, in which the values of consequences depend on states, are ill formed. He argues any such problem could be transformed into a well-formed decision by a suitable subdivision of consequences. In the previous example, c would be split into c 1 = "$500,000 after cumulative inflation of 60 percent," and c 2 = "$500,000 after cumulative inflation of 10 percent." Alternatively, one might opt for a state-dependent utility theory, which replaces SEU4 by a weaker condition and allows the values of consequences of vary with states (for details, see Karni 1993; Schervish, Seidenfeld, and Kadane 1990).

Independence and the Sure-Thing Principle

The most controversial tenet of SEU is the independence axiom:

SEU5 Independence. Preference among acts that have exactly the same consequences when E is false should depend exclusively on whathappens when E is true. If AE C~E is preferred to BE C~E forsome act C, then AE D~E is preferred to BE D~E for all acts D.

To illustrate, consider the following act types, where c, d, c* and d* are known consequences, and x ranges over possible consequences.


SEU5 says that an agent's preference between Ax and Bx should not depend on x 's value. More generally, it requires agents to have well-defined conditional preferences: A is preferred to B in the event of E just in case AE C~E > BE C~E for some (hence any) C.

SEU5 has the following intuitive consequence:

Sure-Thing Principle : Let E 1, E 2, , En be mutually exclusive, collectively exhaustive events. If A is weakly preferred to B conditional on each Ei, then A is weakly preferred to B simpliciter. Moreover, if A is strictly preferred to B conditional on some event that is not judged certainly false, then A is strictly preferred to B.

Independence and the sure-thing principle have been quite controversial. Some apparent failures of SEU5 arise in ill-formed decision problems whose states are not independent of acts. For example, imagine a man who has to drive home from a party where alcohol is being served. He likes to drink, but worries about getting home safely. Suppose he frames his decision like this:

Car accidentNo accident

Since the consequences of drinking are better that those of refraining both in the event of an accident and otherwise, it looks as if the sure-thing principle advocates drinking, which is clearly bad advice given that drinking increases the probability of an accident. Problems of this sort led Jeffrey (1983) to develop an evidential version of decision theory in which independence is only valid for decisions in which acts provide no evidence about the occurrence of any state. Reflections on Newcomb problems, in which acts and states are causally independent but evidentially correlated, led causal decision theorists like Robert Stalnaker (1981), Allan Gibbard and William Harper (1978), and Brian Skyrms (1980) to insist that the two principles be restricted to decisions in which the choice of an act has no causal influence over states.

The most famous objections to SEU5 are the paradoxes of Maurice Allais (1953) and Daniel Ellsberg (1961), which seem to show that SEU rules out certain rational attitudes toward risk and uncertainty. An act involves risk when the agent knows the objective probabilities with which its consequences will obtain. It involves uncertainty when the agent's information allows a range of possible risk profiles for consequences. SEU5 entails that, insofar as decision making is concerned, all legitimate considerations of risk and uncertainty are fully captured in expected utilities. The Allais and Ellsberg paradoxes suggest, to the contrary, that risk and uncertainty are nonseparable quantities: one cannot express them as weighted averages of their values conditional on disjoint events. If this is correct, then an agent need not have any fixed preference between the act types Ax and Bx because x 's value might provide information about the relative risk or uncertainty of the two options, and this information might justifiably influence the agent's preferences.

The Allais paradox envisions an agent who chooses between A and A* and then between B and B* (with the know probabilities listed).

Empirical studies show that people systematically violate independence when presented with such choices. They "play it safe" and select A over A* in the first choice, but favor the riskier option B* over B in the second. The standard rationale for these choices assumes (1) that there is more risk involved in choosing A* over A than there is in choosing B* over B, and (2) that it is rational minimize this risk even when doing so violates independence.

Ellsberg's paradox shows something similar with respect to judgments of uncertainty. Suppose a ball will be drawn at random from an urn that holds thirty red balls and sixty white or blue balls in an unknown proportion. One chooses between C and C* and then between D and D*.


Here most people prefer C to C* and D* to D. Interestingly, when gains are replaced by losses, people still violate independence, but both choices are reversed. People thus seem to prefer risk to uncertainty when they have something to gain, but prefer uncertainty to risk when they have something to lose. Those who regard Ellsberg's paradox as a counterexample to SEU maintain that such nonseparable preferences for risk over uncertainty or uncertainty over risk are entirely rational.

Some proponents of SEU (see Broome 1991) respond by arguing that the consequences in the Allais and Ellsberg paradoxes are underdescribed. For example, the standard pattern of preferences in Allais can be rationalized by noting that, when the 0.01 event occurs, agents who choose A* over A may feel regret (because they passed up a sure thing), while those who choose B* over B will feel no regret (because they probably would have ended up with nothing anyhow). For such agents, the decision matrix really looks like this:
A*$5,000,000$0 with regret$1,000,000
B*$5,000,000$0 with regret$0

Likewise, if an agent feels uneasy when gains ride on uncertain prospects (or losses ride on risky prospects), then the correct description of the Ellsberg problem is this:

C*$0 with uneasiness$100 with uneasiness$0 with uneasiness
D$100 with uneasiness$0 with uneasiness$100 with uneasiness

If these matrices accurately describe the decisions, then neither the Allais or Ellsberg paradoxes provide a genuine counterexample to SEU5.

These sorts of rationalizing responses are weakened by their dependence on substantive assumptions about the psychology of risk, uncertainty, and regret that are not universally accepted (see Loomes and Sudgen 1982, Weber 1998). An alternative is to argue that the usual preferences in the Allais and Ellsberg paradoxes are simply irrational. In Allais, for example, agents assume that the disparity in risk between A and A* exceeds the disparity in risk between B and B*. This may be a mistake. One way to determine differences in risk is to consider the costs of insuring against the incremental risk one incurs by trading one option for another. Someone who switches from A* to A in Allais can offset this risk by purchasing an insurance policy that pays out $1,000,000 contingent on the 0.01 event. Notice, however, that the risk incurred by switching from B* to B can be offset by the same policy. Since a single policy eliminates both risks there is reason to think that the actual change in risk is the same in each case. Similar things can be said about the Ellsberg choosers, who implicitly assume that they decrease their uncertainty more by switching from C* to C than they do by switching from D* to D. So, if one measures disparities in risk or uncertainty by the costs of insuring against it, then SEU is safe from the Allais and Ellsberg examples.

Opponents of SEU will, of course, deny that risks should be measured by the costs of insuring against them. Ultimately, the issue will be resolved by the development of a convincing measure of risk. While there is a well-known theory of risk aversion within SEU, there is no universally accepted method for quantifying risk itself. The best work in this area, which builds on M. Rothschild and J. E. Stiglitz (1970), suggests that risk is indeed separable.

Alternatives to SEU

While subjective expected utility theory remains firmly ensconced as the standard model of rational decision making for individuals, a number of alternatives have been developed. One kind of approach seeks to relax independence while preserving most other aspects of SEU. Especially noteworthy here is the "generalized expected utility analysis" of Mark Machina (1982), and the "weighted utility model" of Soo-Hong Chew and Kenneth R. MacCrimmon (1979). Alternatively, one can reject maximizing conceptions of rationality altogether and see decision making as matter of satisficing relative to fixed constraints. For example, G. Gigerenzer et al. (1999) seek to replace the single all-purpose prescription to maximize expected utility by an ecological model of rationality in which decision makers employ a set of simple, highly localized decision heuristics. These heuristics efficiently generate choices that produce desirable consequences in the contexts where they tend to be employed, but they can go badly awry when used in out of context. For discussion of further nonstandard decision theories, see Robert Sugden (2004).

Interesting though these alternatives are, none has seriously challenged the normative status of SEU. Though highly idealized, and far from adequate as a description of human behavior, SEU remains the best overall account of rational decision making.

See also Bayes, Bayes' Theorem, Bayesian Approach to Philosophy of Science; Game Theory; Pareto, Vilfredo; Pascal, Blaise; Probability and Chance; Ramsey, Frank Plumpton; Savage, Leonard; Sen, Amartya; Statistics, Foundations of.


Allais, Maurice. "Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'École Americaine." Econometrica (1953).

Broome, John. Weighing Goods: Equality, Uncertainty, and Time. Cambridge, MA: Basil Blackwell, 1991.

Chew, Soo-Hong, and Kenneth R. MacCrimmon. "Alpha-nu Theory: A Generalization of Expected Utility Thoery." Working Paper 669 (1979). University of British Columbia, Vancouver.

Ellsberg, Daniel. "Risk, Ambiguity, and the Savage Axioms." Quarterly Journal of Economics 75 (1961): 643669.

Fishburn, Peter C. 1991. "Nontransitive Preferences in Decision Theory." Journal of Risk and Uncertainty 4 (1991): 113134.

Gibbard, Allan, and William Harper. "Counterfactuals and Two Kinds of Expected Utility." In Foundations and Applications of Decision Theory, edited by C. Hooker, J. Leach, and E. McClennen. Dordrecht, Netherlands: D. Reidel, 1978.

Gigerenzer, G, et al. Simple Heuristics that Make Us Smart. New York: Oxford University Press, 1999.

Hammond, Peter. "Consequentialist Foundations for Expected Utility." Theory and Decision 25 (1988): 2578.

Jeffrey, Richard. The Logic of Decision. 2nd ed. Chicago: University of Chicago Press, 1983.

Joyce, James M. The Foundations of Causal Decision Theory. New York: Cambridge University Press, 1999.

Kaplan, Mark. "Decision Theory as Philosophy." Philosophy of Science 50 (1983): 549577.

Karni, E. "Subjective Expected Utility Theory with State-Dependent Preferences." Journal of Economic Theory 60 (1993): 428438.

Levi, Isaac. The Enterprise of Knowledge. Cambridge, MA: MIT Press, 1980.

Loomes, Graham, and Robert Sugden. "Regret Theory: An Alternative Theory of Rational Choice under Uncertainty." Economic Journal 92 (1982): 805824.

Luce, R. Duncan, and Howard Raiffa. Games and Decisions. New York: Wiley, 1957.

Machina, Mark. "'Expected Utility' Analysis without the Independence Axiom." Econometrica 50 (1982): 277323.

McClennen, Edward. Rationality and Dynamic Choice: Foundational Explorations. New York: Cambridge University Press, 1990.

Ramsey, Frank P. "Truth and Probability." In The Foundations of Mathematics and Other Logical Essays, edited by Richard Braithwaite. London: Kegan Paul, 1931.

Redelmeier, D., and E. Shafir. "Medical Decision Making in Situations that Offer Multiple Alternatives." Journal of the American Medical Association 273 (4) (1995): 302305.

Rothschild, M., and J. E. Stiglitz. "Increasing Risk: I. A Definition." Journal of Economic Theory 2 (1970): 225243.

Savage, Leonard J. The Foundations of Statistics. 2nd ed. New York: Dover, 1972.

Schervish, M., T. Seidenfeld, and J. Kadane. "State-Dependent Utilities." Journal of the American Statistical Association 85 (1990): 840847.

Sen, Amartya. "Rational Fools: A Critique of the Behavioral Foundations of Economic Theory." Philosophy and Public Affairs 6 (1977): 317344.

Shafir, E., and A. Tversky. "Decision Making." In An Invitation to Cognitive Science. Vol. 3, Thinking. 2nd ed., edited by Edward Smith and Daniel Osherson, 77100. Cambridge, MA: MIT Press, 1995.

Skyrms, Brian. Causal Necessity. New Haven, CT: Yale University Press, 1980.

Stalnaker, Robert. "Letter to David Lewis, May 21, 1972." In IFs: Conditionals, Belief, Decision, Chance, and Time, edited by Robert Stalnaker, William Harper, and Glen Pearce. Dordrecht, Netherlands: D. Reidel, 1981.

Sugden, Robert. "Alternatives to Expected Utility: Formal Theories." In Handbook of Utility Theory. Vol. 2, edited by Peter Hammond, Salvador Barberà, and Christian Seidl. Dordrecht, Netherlands: Kluwer Academic, 2004.

Weber, Michael. "The Resilience of the Allais Paradox." Ethics 109 (1998): 94118.

James M. Joyce (2005)