Risk and Tradeoffs

The orthodox theory of instrumental rationality, expected utility (EU) theory, severely restricts the way in which risk-considerations can figure into a rational individual's preferences. It is argued here that this is because EU theory neglects an important component of instrumental rationality. This paper presents a more general theory of decision‐making, risk-weighted expected utility (REU) theory, of which expected utility maximization is a special case. According to REU theory, the weight that each outcome gets in decision‐making is not the subjective probability of that outcome; rather, the weight each outcome gets depends on both its subjective probability and its position in the gamble. Furthermore, the individual's utility function, her subjective probability function, and a function that measures her attitude towards risk can be separately derived from her preferences via a Representation Theorem. This theorem illuminates the role that each of these entities plays in preferences, and shows how REU theory explicates the components of instrumental rationality.

1. To say that two utility functions u(x) and u′(x) are equivalent up to positive affine transformation means that there are some constants a and b where a is positive and au(x) + b = u′(x).

2. In all these examples, I will assume that probabilities are given, to simplify the discussion. But the probabilities involved should be assumed to be the agent’s subjective probabilities. Moreover, I will use the term “risk-averse” neutrally: an agent is risk-averse with respect to some good (say, money) iff she prefers a sure-thing amount of that good to a gamble with an equivalent mathematical expectation of that good. For a more general definition of risk aversion that is compatible with what I say here and that captures the idea that a risk-averse person prefers a gamble that is less spread out, see Rothschild and Stiglitz (1970).

3. A real-valued function f is concave in some interval C iff \( \forall x,y \in C \) and all \( \alpha \in [\begin{array}{*{20}c} {0,} & 1 \\ \end{array} ],\;{\text{f}}(\alpha {\text{x}} + (1 - \alpha ){\text{y}}) \ge \alpha {\text{f}}({\text{x}}) + (1 - \alpha ){\text{f}}({\text{y}}) \). A continuous function is concave in some interval C iff \( \forall x,y \in C \), f((x + y)/2) ≥ (f(x) + f(y))/2. Concavity is strict if the inequality is strict for x ≠ y and \( \alpha \in (0, \; 1) \). Definitions of convexity and strict convexity are given by reversing the inequalities. Throughout this paper, I will continue to use “concave” to refer to the definition that uses weak inequalities, so that linearity is a degenerate case of concavity. Jensen’s Inequality (Jenson 1906) tells us that if u is concave, then the expected utility of a random variable (in this case, amounts of money) is no greater than the utility of its expected monetary value: if f is a gamble, then EU(f) ≤ u(E(f)), where E(f) is the expected monetary value of f. And if u is convex, EU(f) ≥ u(E(f)). Chateauneuf and Cohen (1994: 82) note that a preference for a guaranteed E(f) rather than f itself implies a concave utility function. But I do not know where this was originally shown.

4. Particularly clear expositions of this view appear in Maher (1993), Broome (1999), and Dreier (2004).

5. Rabin (2000).

6. Rabin states his results in terms of changes from initial wealth levels because he hypothesizes that part of the correct explanation for people’s risk aversion in modest stakes is loss aversion of the kind discussed in Kahneman and Tversky (1979).

7. Example due to Allais (1953). Amounts of money used in the presentation of this paradox vary.

8. For if L₁ is preferred to L₂, then we have 0.1(u($5 m)) + 0.9(u($0)) > 0.11(u($1 m)) + 0.89(u($0)). Equivalently, 0.1(u($5 m)) + 0.01(u($0)) > 0.11(u($1 m)). And if L₄ is preferred to L₃, then we have u($1 m) > 0.89(u($1 m)) + 0.1(u($5 m)) + 0.01(u($0)). Equivalently, 0.11(u($1 m)) > 0.1(u($5 m)) + 0.01(u($0)). These two contradict; so there is no utility assignment that allows for the common Allais preferences.

9. This talk may faze a certain kind of constructivist. We could recast it in terms that are acceptable to the constructivist as follows. If risk-preferences are based only on local considerations so that the agent obeys the axioms of EU theory, then the utility function as determined by EU theory will reflect these even if it doesn’t correspond to anything ‘real.’ If risk-preferences are based on both kinds of considerations so that the agent doesn’t obey the axioms of EU theory, then constructivist EU theory will read the agent as not having a utility function. However, if we can define the utility function from suitable preference axioms that these preferences do obey, then the utility function will again reflect the local considerations, as we will see in Sect. 5.

10. For the skeptical constructivist of the previous footnote: local considerations might point in the direction of one act, and considerations about the likelihood of realizing various ends might point in the direction of another.

11. For the constructivist: if two decision-makers share the same local considerations and agree on the probabilities involved, they must have identical preference orderings.

12. If contra our supposition, u(x₂) ≤ u(x₁), then the value of the gamble would be r(p(\( {\bar{\text{E}}} \)))u(x₁) + (1 − r(p(\( {\bar{\text{E}}} \)))u(x₂), i.e., r(1 − p(E))u(x₁) + (1 − r(1 − p(E))u(x₂), which need not be equivalent to r(p(E))u(x₂) + (1 − r(p(E))u(x₁).

13. For further discussion of this point, see Buchak (2013).

14. L₁ > L₂ ⇔ r(.1)[u($5 m) − u($0)] > r(.11)[u($1 m) − u($0)].

    L₄ > L₃ ⇔ (1 − r(.99))[u($1 m) − u($0)] > r(.1)[u($5 m) − u($1 m)].

    These inequalities hold jointly only if r(0.11) − r(0.1) < 1 − r(0.99).

15. Schmeidler (1989), Gilboa (1987).

16. Quiggin (1982).

17. Schmeidler’s (1989) version includes some objective probabilities to derive the decision weights.

18. For further discussion of the relationship of REU theory to these theories and other non-expected utility theories present in the economics and psychology literature, see Buchak (2013). For other surveys of non-expected utility theories, see Sugden (2004) and Starmer (2000).

19. See Savage (1954). See also Ramsey (1926). For a survey of representation theorems for EU theory, see Fishburn (1981). A different sort of representation theorem is due to Jeffrey (1965). For Jeffrey, the state space and the outcome space are the same, and each outcome is a gamble over other outcomes, i.e., there are no “final outcomes.” As a result, his uniqueness result for the utility function is weaker.

20. A third use is the descriptive use, which I won’t discuss here.

21. Recall that the utility function is only unique up to positive affine transformation. Therefore, only the facts that are common to all of these utility functions, e.g., the relative size of the utility intervals between outcomes, are rightly called facts about the agent’s utilities.

22. The proof of the theorem is found in Buchak (2013).

23. Köbberling and Wakker (2003), Machina and Schmeidler (1992).

24. Comonotonic Tradeoff Consistency follows directly from Unrestricted Tradeoff Consistency. In the presence of the other axioms of EU or REU theory, UTC is strictly stronger than the combination of CTC and Strong Comparative Probability.

25. In this discussion, x and y stand for outcomes and f and g for gambles (including the degenerate gamble which yields the same outcome in every state), but the difference won’t matter for understanding the discussion here.

26. The axiom is actually framed as: if $110 rather than $100 is tradeoff equal to $10 rather than $0 (there is some event and some gamble pair in which they are both appropriate tradeoffs), and we have some outcome y′ that is preferred to $110, then $y′ rather than $100 is not tradeoff equal to $10 rather than $0 (there is no event and no gamble pair in which they are both appropriate tradeoffs). But the difference is not important for our informal discussion.

27. It rules them out, in the presence of the Ordering Axiom, because if the agent is indifferent in the first three pairs of gambles, UTC implies that she is indifferent in the fourth pair, but she in fact has a strict preference.

28. As above, the axiom is actually framed as: if $110 rather than $100 is comonotonic tradeoff equal to $10 rather than $0 (there is some event and some gamble pair in which they are both appropriate tradeoffs, and all four gambles are comonotonic), and we have some outcome y′ that is preferred to $110, then y′ rather than $100 is not comonotonic tradeoff equal to $10 rather than $0 (there is no event and no gamble pair in which they are both appropriate tradeoffs and all four gambles are comonotonic). But, again, the difference is not important for our informal discussion.

29. To clarify: substituting y rather than x into a gamble will make a different value difference depending on the event the substitution occurs in, merely because the more probable the event, the bigger value difference it will make; however, if substituting y rather than x for some event in some gamble makes the same difference as substituting w rather than z for that same event in that same gamble, then for any event and any gamble, substituting y rather than x in that event in that gamble makes the same value differences as substituting w rather than z in that event and that gamble.

30. Köbberling and Wakker (2003). Machina and Schmeidler (1992). See Buchak (2013) for a discussion of related results. Particularly noteworthy is a similar theorem due to Nakamura (1995).

31. In the denotation of the spaces, I follow Machina and Schmeidler (1992).

32. Machina and Schmeidler (1992: 749).

33. Köbberling and Wakker (2003: 400). On p. 403, Köbberling and Wakker (2003) point out that we can also define a comoncone on an infinite state space, although this is not necessary for our purposes.

34. Köbberling and Wakker (2003: 398). The concept of a standard sequence, however, does not originate with them.

35. Köbberling and Wakker (2003: 396–7).

36. Köbberling and Wakker (2003: 397). Note the similarity to the four-place relation “=” in Ramsey’s (1926) axiomatization of EU theory. In Ramsey’s axiomatization, “xy = zw” holds when the agent is indifferent between {E, x; \( {\bar{\text{E}}} \), w} and {E, y; \( {\bar{\text{E}}} \), z} for any “ethically neutral” proposition E believed to degree 0.5.

37. Mostly following Köbberling and Wakker (2003). I alter their axioms slightly, to make the comparison clear. See Appendix B of Buchak (2013).

Authors and Affiliations

1. University of California, Berkeley, CA, USA

   Lara Buchak

Corresponding author

Correspondence to Lara Buchak.

This article is excerpted in large part from Lara Buchak (2013), Risk and Rationality, Oxford University Press.

Appendix: Representation Theorem and Technical Discussion

The theorem here draws on two other results, one by Veronika Köbberling and Peter Wakker, and the other by Mark Machina and David Schmeidler.^{Footnote 30} The set of axioms I use in the REU representation theorem are a combination of Köbberling and Wakker’s and Machina and Schmeidler’s axioms, strictly stronger than either set of axioms.

I start by explaining the spaces and relations we are dealing with.^{Footnote 31} The state space is a set of states \( \fancyscript{S} \) = {…, s, …}, whose subsets are called events. The event space, \( \fancyscript{E} \), is the set of all subsets of \( \fancyscript{S} \). Since we want to represent agents who have preferences over not just monetary outcomes but discrete goods and, indeed, over outcomes described to include every feature of the world that the agent cares about, it is important that the outcome space be general. Thus, the outcome space is a set of outcomes \( \fancyscript{X} \) = {…, x, …}. I follow Savage (1954/1972) in defining the entities an agent has preferences over as “acts” that yield a known outcome in each state. The act space \( \fancyscript{A} \) = {…, f(.), g(.), …} is thus the set of all finite-valued functions from \( \fancyscript{S} \) to \( \fancyscript{X} \), where the inverse of each outcome f⁻¹(x) is the set of states that yields that outcome, so f⁻¹(x) ∈ \( \fancyscript{E} \). So for any act f ∈ \( \fancyscript{A} \), there is some partition of the state space \( \fancyscript{S} \) into {E₁, … E_n} and some finite set of outcomes Y ⊆ \( \fancyscript{X} \) such that f can be thought of as a member of Yⁿ. And as long as f(s) is the same for all s ∈ E_i, we can write f(E_i) as shorthand for “f(s) such that s ∈ E_i.”

For any fixed finite partition of events M = {E₁, …, E_n}, all the acts on those events will form a subset A_M ⊆ \( \fancyscript{A} \). Thus, A_M is defined to contain all the acts that each yield for each event in the partition, the same outcome for all states in that event: A_M = {f ∈ \( \fancyscript{A} \) | (∀E_i ∈ M)(∃x ∈ \( \fancyscript{X} \))(∀s ∈ E_i)(f(s) = x)}. An upshot is that for all acts in A_M, we can determine the outcome of the act by knowing which event in M obtains: we needn’t know the state of the world in a more fine-grained way.

The preference relation ≥ is a two-place relation over the act space. This gives rise to the indifference relation and the strict preference relation: f ~ g iff f ≥ g and g ≥ f; and f > g if f ≥ g and ¬(g ≥ f).

For all x ∈ \( \fancyscript{X} \), x denotes the constant act {f(s) = x for all s ∈ \( \fancyscript{S} \)}. The relata of the preference relation must be acts, but it will be useful to talk about preferences between outcomes. Thus, we will define an auxiliary preference relation over outcomes:

where indifference and strict preferences are defined as above. It will be useful to talk about preferences between outcomes of particular acts, so, following the above definition, f(s) ≥ g(s) holds iff f(s) ≥ g(s), the constant act that yields f(s) in every state is weakly preferred to the constant act that yields g(s) in every state. Furthermore, x_Ef denotes the act that agrees with f on all states not contained in E, and yields x on any state contained in E: x_Ef(s) = {x if s ∈ E; f(s) if s ∉ E}. Likewise, for disjoint E ₁ and E ₂ in \( \fancyscript{E} \), x_E1y_E2f is the act that agrees with f on all states not contained in E ₁ and E ₂, and yields x on E ₁ and y on E ₂: x_E1y_E2f(s) = {x if s ∈ E₁, y if s ∈ E₂, f(s) if s ∉ E₁∪ E₂}. Similarly, g_Ef is the act that agrees with g on all states contained in E and agrees with f on all states not contained in E: g_Ef(s) = {g(s) if s ∈ E; f(s) if s ∉ E}. We say that an event E is null on F ⊆ \( \fancyscript{A} \) just in case the agent is indifferent between any pair of acts which differ only on E: g_Ef ~ f for all f, g_Ef ∈ F.^{Footnote 32}

Several concepts are important in Köbberling and Wakker’s result. The first, comonotonicity, was introduced by Schmeidler (1989). Two acts f and g are comonotonic if there are no states s₁, s₂ ∈ \( \fancyscript{S} \) such that f(s₁) > f(s₂) and g(s₁) < g(s₂). This is equivalent to the claim that for any partition A_M of acts such that f, g ∈ A_M, there are no events E₁, E₂ ∈ M such that f(E₁) > f(E₂) and g(E₁) < g(E₂). The acts f and g order the states (and, consequently, the events) in the same way: if s₁ leads to a strictly preferred outcome to that of s₂ for act f, then s₁ leads to a weakly preferred outcome to that of s₂ for act g. We say that a subset C of some A_M is a comoncone if all the acts in C order the events in the same way: for example, the set of all acts on coin-flips in which the heads outcome is as good as or better than the tails outcome forms a comoncone. Formally, as Köbberling and Wakker define it, take any fixed partition of events M = {E₁, …, E_n}. A permutation ρ from {1, …, n} to {1, …, n} is a rank-ordering permutation of f if f(E_ρ(1)) ≥ ··· ≥ f(E_ρ(n)). So a comoncone is a subset C_ρ of A_M that is rank-ordered by a given permutation: C_ρ = {f  ∈  A_M | f(E_ρ(1)) ≥ ··· ≥ f(E_ρ(n))} for some ρ. For each fixed partition of events of size n, there are n! comoncones.^{Footnote 33}

Here is an example to illustrate the idea of a comoncone. Consider the following gambles:

The set [f, g, j] forms a comoncone, because for each gamble in the set, the heads outcome is weakly preferred to the tails outcome. The set [h, j] forms a comoncone, because for each gamble in the set, the tails outcome is weakly preferred to the heads outcome.

We say that outcomes x¹, x², … form a standard sequence on F ⊆ \( \fancyscript{A} \) if there exist an act f ∈ F, events E_i ≠ E_j that are non-null on F, and outcomes y, z with ¬(y ~ z) such that for all k, (x^k+1)_Ei(y)_Ejf ~ (x^k)_Ei(z)_Ejf, with all acts (x^k)_Ei(y)_Ejf, (x^k)_Ei(z)_Ejf ∈ F.^{Footnote 34} The intended interpretation is that the set of outcomes x¹, x², x³, …, will be “equally spaced.” Since the agent is indifferent for each pair of gambles, and since each pair of gambles differs only in that the “left-hand” gamble offers y rather than z if E_j obtains, and offers x^k+1 rather than x^k if E_i obtains, the latter tradeoff must exactly make up for the former. And since the possibility of x^k+1 rather than x^k (if E_i) makes up for y rather than z (if E_j) for each k, the difference between each x^k+1 and x^k must be constant. Note that a standard sequence can be increasing or decreasing, and will be increasing if z > y and decreasing if y > z. A standard sequence is bounded if there exist outcomes v and w such that ∀i(v ≥ xⁱ ≥ w).

We are now in a position to define a relation that is important for Köbberling and Wakker’s result and that also makes use of the idea that one tradeoff exactly makes up for another. For each partition M, we define the relation ~*(F) for F ⊆ A_M and outcomes x, y, z, w ∈ \( \fancyscript{X} \) as follows:

where all four acts are contained in F.^{Footnote 35}

Köbberling and Wakker explain the relation ~*(F) as follows: “The interpretation is that receiving x instead of y apparently does the same as receiving z instead of w; i.e., it exactly offsets the receipt of the [f’s] instead of the [g’s] contingent on [\( {\bar{\text{E}}} \)].”^{Footnote 36} The idea here is that if one gamble offers f if \( {\bar{\text{E}}} \) obtains, whereas another gamble offers g if \( {\bar{\text{E}}} \) obtains, then this is a point in favor of (let’s say) the first gamble. So in order for an agent to be indifferent between the two gambles, there has to be some compensating point in favor of the second gamble: it has to offer a better outcome if E obtains. And it has to offer an outcome that is better by the right amount to exactly offset this point. Now let’s assume that offering y rather than x (on E), and offering w rather than z (on E) both have this feature: they both exactly offset the fact that a gamble offers f rather than g (on \( {\bar{\text{E}}} \)). That is, if one gamble offers f if \( {\bar{\text{E}}} \), and a second gamble offers g if \( {\bar{\text{E}}} \), then this positive feature of the first gamble would be exactly offset if the first offered x if E and the second offered y if E—and it would be exactly offset if instead the first offered z if E and the second offered w if E. If this is the case, then there is some important relationship between x and y on the one hand and z and w on the other: there is a situation in which having the first member of each pair rather than the second both play the same compensatory role. This relationship ~* is called tradeoff equality. We write xy ~*(C) zw if there exists a comoncone F ⊆ A_M such that xy ~*(F) zw: that is, if x and y play the same compensatory role as z and w in some gambles f and g where all of the modified gambles after x, y, z, and w have been substituted in are in the same comoncone.

The relation ~*(F), and particularly ~*(C), feature centrally in the representation theorem, because one important axiom places restrictions on when it can hold: on when pairs of outcomes can play the same compensatory role. This relation also plays a crucial role in determining the cardinal utility difference between outcomes using ordinal preferences. What we are interested in is the utility contribution each outcome makes to each gamble it is part of: this will help us determine the utility values of outcomes. More precisely, since utility differences are what matter, we are interested in the utility contribution that x rather than y makes to each gamble. And tradeoff equality gives us a way to begin to determine this: if getting y rather than x in event E and getting z rather than w in event E both exactly compensate for getting f rather than g in event \( {\bar{\text{E}}} \), then y rather than x and z rather than w make the same difference in utility contribution in event E in those gamble pairs. In order to get from these differences in utility contributions to utility full stop, we need to fix when it is that two pairs making the same difference in utility contribution means that they have the same difference in utility. And to do this, we identify the conditions under which if two pairs have the same difference in utility (full stop), they must make the same difference in utility contribution; and we constrain the rational agent to treat a pair consistently in these situations—to consistently make tradeoffs. Tradeoff consistency axioms provide such a constraint.

With the preliminaries out of the way, I can now present the axioms of REU theory, side-by-side with those of the analogous representation theorem for EU theory that Köbberling and Wakker spell out.^{Footnote 37}

Any agent whose preferences obey the axioms in the left-hand column maximizes expected utility relative to a unique probability function and a utility function unique up to positive affine transformation. Furthermore, in the presence of (A3), any agent who maximizes expected utility will satisfy the remaining axioms.

Analogously, if a preference relation ≥ on \( \fancyscript{A} \) satisfies (A1) through (A7), then there exist

1. (i)

   a unique finitely additive, non-atomic probability function p: \( \fancyscript{E} \) → [0, 1];

2. (ii)

   a unique risk function r: [0, 1] → [0, 1]; and

3. (iii)

   a utility function unique up to positive affine transformation

such that REU represents the preference relation ≥. If there are three such functions so that REU(f) represents the preference relation, we say that REU holds. Thus, if ≥ satisfies (A1) through (A7), then REU holds.

Furthermore, in the presence of (A3), if REU holds with a continuous r-function, then the remaining axioms are satisfied.

Therefore, if we assume preference richness (A3), we have:

• (A1), (A2), (A4), (A5), (A6), (A7) are sufficient conditions for REU.

• (A1), (A2), (A4), (A5), (A6), (A7) are necessary conditions for REU with continuous r-function.

The proof of this theorem, with references to details found in Köbberling and Wakker (2003) and in Machina and Schmeidler (1992), can be found in Buchak (2013).

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