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The intrinsic value of risky prospects

  • Decision-making and hypothetical reasoning
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Abstract

We study the representation of attitudes to risk in Jeffrey’s (The logic of decision, University of Chicago Press, Chicago, 1965) decision-theoretic framework suggested by Stefánsson and Bradley (Philos Sci 82(4):602–625, 2015; Br J Philos Sci 70(1):77–102, 2017) and Bradley (Econ Philos 32(2):231–248, 2016; Decisions theory with a human face, Cambridge University Press, Cambridge, 2017). We show that on this representation, the value of any prospect may be expressed as a sum of two components, the prospect’s instrumental value (the value the prospect has only in virtue of the outcomes it might lead to) and the prospect’s intrinsic value (the value the prospect has only in virtue of the way it assigns different probabilities to the different outcomes). Both components have an expectational form. We also make a distinction between a prospect’s overall intrinsic value and a prospect’s conditional intrinsic value given each one of its possible outcomes and argue that this distinction has great explanatory power. We explore the relation between these two types of intrinsic values and show that they are determined at the level of preferences. Finally, we explore the relation between the intrinsic values of different prospects and point to a strong restriction on this relation that is implicit in Jeffrey’s axioms. We suggest a natural interpretation to this restriction.

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Notes

  1. The expectational structure entails that the relationship between the value of a prospect and the chances it assigns to its possible outcomes is linear—the marginal value of chances of outcomes is constant.

  2. Prospect X is a mean-preserving contraction of prospect Y iff prospect Y is a mean-preserving spread of prospect X.

  3. As an anonymous referee commented, this is not a purely formal observation. Rather it has an interpretative element. For example, a utility function which is concave in money can be interpreted, within the orthodox framework, as representing the preferences of an agent who cares equally about gaining each extra dollar, but intrinsically disvalues risk. However, adopting such an interpretation makes it impossible to formally separate the agent’s intrinsic attitudes toward risk from her attitude toward gaining a marginal dollar. As is explained below, such a formal separation is exactly what Bradley’s and Stefánsson’s framework allows one to express. We thank an anonymous referee for pushing us on this point.

  4. The standard reply to this charge is that decision-theoretic representation theorems ensure that it is possible to represent a rational agent’s preferences as resulting from maximizing expected value with respect to a given credence function and a unique (up to affine transformation) value function, and so the linearity requirement should be understood as applying to this value function (i.e. the one that represents the agent’s preferences). However, Buchak (2013) convincingly argues that the axioms necessary and sufficient for expected value maximization, are no more intuitive than a weaker set of axioms that are necessary and sufficient for the maximization of some weighted average—not necessarily the expectation—of the values of the possible outcomes of prospects.

  5. According to a profile article in the Atlantic (Rich 2015) on rock-climber Alex Honnold, “When asked in public about the risk of falling to his death, he answers glibly: ‘It’ll be the worst four seconds of my life’”.

  6. For a recent discussion see Nissan-Rozen (2019).

  7. An alternative way of looking at it would be to argue that in Savage’s framework the independence conditions serve as constraints on re-individuation, but that these constraints are too strong: they do not allow for violations of the linearity requirement.

  8. The pair of value and credence functions \( \left\langle {V,C} \right\rangle \) is unique up to fractional liner transformations. See chapter 8 of Jefferey (1965). However, see our footnote 26 for a stronger result given the Principal Principle.

  9. Formally, and more generally, for any \( X \), if \( X,\neg X \ne \bot \) then \( V\left( X \right) > V\left( T \right) \leftrightarrow V\left( {\neg X} \right) < V\left( T \right) \).

  10. In the sense alluded to by the Representation condition above. Specifically, if \( { \succcurlyeq }_{A} \) denotes preferences conditional on \( A \) (in the sense articulated by Bradley (1999), see pp. 12–13), then for all \( X,Y \) consistent with \( A \), \( X{ \succcurlyeq }_{A} Y \) iff \( V\left( {X |A} \right) \ge V\left( {Y |A} \right) \).

  11. Formally, chance propositions are sets of probability functions over \( \Omega \) (more precisely, over the sub-algebra of propositions for which it is meaningful to assign chances. We will abstract this distinction away and assume that \( \Omega \) only includes such propositions). For example, \( A^{x} \) is the set of all possible probability functions that assign \( A \) a chance of \( x \). Stefánsson and Bradley (2015) apply the Jeffrey-Bolker axioms to preferences defined over the Cartesian product \( \Omega \times \Delta \) where \( \Omega \) is the original algebra of propositions and \( \Delta \) is the power set of the set of all possible probability functions on \( \Omega \). \( V \) and \( C \) are then defined on this product, that is, on ordered pairs \( \left\langle {X,Y} \right\rangle \) such that \( X \) is a factual proposition and \( Y \) is a chance proposition. Following Stefánsson and Bradley we simplify notation and write \( A \) instead of \( \left\langle {A,\Delta } \right\rangle \) and \( A^{x} \) instead of \( \left\langle {\Omega ,A^{x} } \right\rangle \). This is all fleshed out concisely in Stefánsson and Bradley (2017, pp. 18–19), and in Bradley (2017, chapter 7.3).

  12. The agent’s credence function satisfies the Principal Principle just in case her credence in any proposition conditional on the chance of that proposition being \( x \) equals \( x \). Formally: For any chance proposition \( A^{x} \), the agent’s credence function satisfies \( C(A |A^{x} ) = x \). This version of the principle lacks an admissibility clause and is therefore inconsistent with the probability axioms in some cases (i.e. after learning some propositions). Thus, the rest of the discussion should be understood as applying only to cases in which no inadmissible proposition (in Lewis’ sense) has been learnt.

  13. See Theorem 1 and its proof in Stefánsson and Bradley (2015, pp. 611–612).

  14. See “Appendix” for the proof.

  15. That is, the agent doesn’t care whether she wins or loses such a lottery. We thank an anonymous referee for pointing out this intuitive explication of the definition.

  16. See “Appendix” for the proof. Notice especially line 7 of the proof and footnote 26.

  17. Formally, \( x_{k} \ne y_{k} \) for some \( 1 \le k \le n \). Consequently, when there is such a \( k \), then there must be some \( j \ne k \) such that \( x_{j} \ne y_{j} \), because the \( x_{i} \)s and \( y_{i} \)s sum up to 1. That is, whenever two prospects disagree at all, they disagree regarding the chances of at least two elements of the partition.

  18. See “Appendix” for the proof.

  19. Notice that while Ruth values winning against the odds, she may also disvalue losing when chances of doing so are high. Therefore, the inequality in the text is quite consistent with \( V\left( {L_{2} |0\$ } \right) < 0 < V\left( {L_{1} |0\$ } \right) \).

  20. See “Appendix” for the proof.

  21. We leave the notion of “degrees of risk” imprecise here, as the chancy features of a prospect that render it intrinsically valuable may be more complex and nuanced than the standard mean-preserving spread notion used by economists. However, the example works also if this notion is adopted. That is, if \( R \) is a mean preserving spread of \( M \) and the latter is a mean preserving spread of \( S \).

  22. See “Appendix” for the proof.

  23. We thank an anonymous referee for emphasizing this consequence of Proposition 3.

  24. The proposition is derived by slightly altering the expressions in Stefánsson’s and Bradley’s (2015) proof of their Theorem 1 (pp. 611–612).

  25. See Jeffrey (1965, p. 132). For illuminating discussion see Bradley (2017, pp. 83–84).

  26. This is an interesting result: although the Jeffrey-Bolker representation theorem gives uniqueness only up to fractional linear transformation (except in the case of a desirability function which is unbounded above and below), here we see that in the case of credence functions that represent the same preferences and obey the Principal Principle we get uniqueness up to positive linear transformation of the desirability function and identity of the credence functions. That is, the beliefs of agents that satisfy the Principal Principle are revealed in their preferences.

  27. Notice that we assume here that \( V\left( T \right) = 0 \), not that \( V^{\prime}\left( T \right) = 0 \).

  28. All divisions are kosher since Lemmas 1 and 2 ensure that in the discussed case \( x_{i} ,y_{i} > 0 \) and \( C\left( {L_{1} } \right),C\left( {L_{2} } \right) > 0 \).

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Appendix

Appendix

1.1 Corollary 1

$$ V\left( L \right) = \mathop \sum \limits_{i = 1}^{n} x_{i} V\left( {A_{i} } \right) + \mathop \sum \limits_{i = 1}^{n} x_{i} V\left( {L|A_{i} } \right) - V\left( T \right) $$

Proof

  1. (1)

    \( V\left( L \right) = V\left( {A_{1} L} \right)C\left( {A_{1} |L} \right) + \cdots + V\left( {A_{n} L} \right)C(A_{n} |L) \), Desirability.

  2. (2)

    \( d\left( L \right) = x_{1} \left[ {V\left( {L|A_{1} } \right) + V\left( {A_{1} } \right) - V\left( T \right)} \right] + \cdots + x_{n} \left[ {V\left( {L|A_{n} } \right) + V\left( {A_{n} } \right) - V\left( T \right)} \right] \), PP, CD.

  3. (3)

    \( V\left( L \right) = \mathop \sum \limits_{i = 1}^{n} x_{i} V\left( {A_{i} } \right) + \mathop \sum \limits_{i = 1}^{n} x_{i} V\left( {L|A_{i} } \right) - V\left( T \right) \).

1.2 Theorem 1

Definition 1

A lottery \( L \) is trivial if for all \( i \) \( V\left( {A_{i} |L} \right) = V\left( T \right) \). This means that the news value of winning (or loosing) the lottery is the same as the news value of participating in it (\( V\left( {A_{i} L} \right) = V\left( L \right) \)).

Theorem 1

If \( \left\langle {V,C} \right\rangle \ne \left\langle {V^{\prime},C'} \right\rangle \) are any two pairs that represent the preference relation \( { \preccurlyeq } \), such that \( C \) and \( C' \) both satisfy the Principal Principle (PP), and \( L \) is a non-trivial lottery, then: \( I^{\prime}\left( L \right) < 0 \) iff \( I\left( L \right) < 0 \); \( I^{\prime}\left( L \right) > 0 \) iff I \( \left( L \right) > 0 \); \( I'\left( L \right) = 0 \) iff \( I\left( L \right) = 0 \).

Proof

Bolker’s Uniqueness TheoremFootnote 25 (UT) states that the representing pair \( \left\langle {V,C} \right\rangle \) is unique up to fractional linear transformation. That is, if \( \left\langle {V,C} \right\rangle \ne \left\langle {V^{\prime},C'} \right\rangle \) are two pairs of value and credence functions that represent the same preference relation, and if we adopt the convention that \( V\left( T \right) = 0 \), then there are real numbers \( \alpha ,\beta , \) and \( \gamma \) that satisfy the following five conditions:

  1. (a)

    \( \alpha - \beta \gamma > 0. \)

  2. (b)

    For all propositions \( A \ne \bot \), \( \gamma V\left( A \right) + 1 > 0. \)

  3. (c)

    For all propositions \( A \ne \bot \), \( V'\left( X \right) = \frac{\alpha V\left( X \right) + \beta }{\gamma V\left( X \right) + 1}. \)

  4. (d)

    For all propositions \( A \ne \bot \), \( C^{\prime}\left( X \right) = C\left( X \right)\left[ {\gamma V\left( X \right) + 1} \right]. \)

This theorem will be instrumental in the following proof of Theorem 1:

  1. (1)

    For all propositions \( X \ne \bot \), \( C^{\prime}\left( X \right) = C\left( X \right)\left[ {\gamma V\left( X \right) + 1} \right] \). UT condition (d).

  2. (2)

    For all outcomes \( A_{i} \) of \( L \), \( C\left( {A_{i} |L} \right) = C^{\prime}\left( {A_{i} |L} \right) = x_{i} \). By assumption both functions satisfy PP.

  3. (3)

    For all outcomes \( A_{i} \) of \( L \), \( C^{\prime}\left( {A_{i} |L} \right) = C\left( {A_{i} |L} \right)\left[ {\gamma V\left( {A_{i} |L} \right) + 1} \right] = x_{i} \). From (1) and (2).

  4. (4)

    For all outcomes \( A_{i} \) of \( L \), \( C^{\prime}\left( {A_{i} |L} \right) = x_{i} \left[ {\gamma V\left( {A_{i} |L} \right) + 1} \right] = x_{i} \). From applying PP to (3).

  5. (5)

    For all outcomes \( A_{i} \) of \( L \), \( \gamma V\left( {A_{i} |L} \right) = 0 \).

  6. (6)

    Since (by assumption) \( L \) is not trivial, there must be an outcome \( A_{j} \) such that \( V\left( {A_{j} |L} \right) \ne 0 \), and therefore (5) entails that \( \gamma = 0 \).

  7. (7)

    For all propositions \( X \), \( C^{\prime}\left( X \right) = C\left( X \right) \). From (1) and (6).Footnote 26

  8. (8)

    For all propositions \( X \), \( V'\left( X \right) = \alpha V\left( X \right) + \beta \). From UT condition (c) and (6).

  9. (9)

    \( \alpha > 0 \). From UT condition (a) and (6).

  10. (10)

    \( I^{\prime } \left( L \right) = \sum\nolimits_{i = 1}^{n} {x_{i} } V^{\prime}\left( {L|A_{i} } \right) - V^{\prime } \left( T \right) = \sum\nolimits_{i = 1}^{n} {x_{i} } \left[ {\alpha V\left( {L|A_{i} } \right) + \beta } \right] - \alpha V\left( T \right) - \beta \). From (8).Footnote 27

  11. (11)

    \( I^{\prime } \left( L \right) = \alpha \left[ {\sum\nolimits_{i = 1}^{n} {x_{i} } V\left( {L|A_{i} } \right)} \right] = \alpha I\left( L \right) \).

  12. (12)

    \( I^{\prime}\left( L \right) < 0 \) iff \( I\left( L \right) < 0 \); \( I^{\prime}\left( L \right) > 0 \) iff \( I\left( L \right) > 0 \); \( I^{\prime}\left( L \right) = 0 \) iff \( I\left( L \right) = 0 \). From (9) and (11).□

1.3 Proposition 1

Any two jointly exhaustive disagreeing prospects \( L_{1} = {\bigwedge }_{i = 1}^{n} A_{i}^{{x_{i} }} \) and \( L_{2} = {\bigwedge }_{i = 1}^{n} A_{i}^{{y_{i} }} \) such that \( C\left( {L_{1} } \right),C\left( {L_{2} } \right) > 0 \) satisfy the following condition:

For all \( A_{i} \in \left\{ {A_{1} , \ldots ,A_{n} } \right\} \), \( V(L_{1} |A_{i} ) \) and \( V(L_{2} |A_{i} ) \), when defined, satisfy the following conditions:

  1. (1)

    \( V\left( {L_{1} |A_{i} } \right) > 0 \) iff \( V\left( {L_{2} |A_{i} } \right) < 0. \)

  2. (2)

    \( V\left( {L_{1} |A_{i} } \right) < 0 \) iff \( V\left( {L_{2} |A_{i} } \right) > 0. \)

  3. (3)

    \( V\left( {L_{1} |A_{i} } \right) = 0 \) iff \( V\left( {L_{2} |A_{i} } \right) = 0. \)

Lemma 1

For all \( A_{i} \) , if \( V\left( {L_{1} |A_{i} } \right) \) is defined, then \( x_{i} > 0 \) (and similarly for \( L_{2} \) and \( y_{i} \) ).

Proof

Assume by way of negation that \( x_{i} = 0 \). Then it follows from the Principal Principle that \( C\left( {A_{i} |L_{1} } \right) = 0 \), and therefore, via Bayes theorem \( C\left( {A_{i} L_{1} } \right) = 0 \) and thus \( V\left( {A_{i} L_{1} } \right) \) is undefined and neither is \( V\left( {L_{1} |A_{i} } \right) \) (see Conditional Desirability). □

Lemma 2

For all \( A_{i} \), if \( x_{i} > 0 \) or \( y_{i} > 0 \) then \( C\left( {A_{i} } \right) > 0 \).

Proof

:it follows from the law of total probability that \( C\left( {A_{i} } \right) = C\left( {A_{i} |L_{1} } \right)C\left( {L_{1} } \right) + C\left( {A_{i} L_{2} } \right)C\left( {L_{2} } \right) \). Applying the Principal Principle yields: \( C\left( {A_{i} } \right) = x_{i} C\left( {L_{1} } \right) + y_{i} C\left( {L_{2} } \right) \). Since by stipulation \( C\left( {L_{1} } \right), C\left( {L_{2} } \right) > 0 \), if \( x_{i} > 0 \) or \( y_{i} > 0 \) then \( C\left( {A_{i} } \right) > 0 \).□

Proof

For all \( A_{i} \) such that \( V(L_{1} |A_{i} ) \) and \( V(L_{2} |A_{i} ) \) are defined:

  1. (1)

    \( V\left( {A_{i} } \right) = V\left( {A_{i} L_{1} } \right)C\left( {L_{1} |A_{i} } \right) + V\left( {A_{i} L_{2} } \right)C(L_{2} |A_{i} ) \). Desirability.

  2. (2)

    \( V\left( {A_{i} } \right) = \left[ {V\left( {L_{1} |A_{i} } \right) + V\left( {A_{i} } \right)} \right]C\left( {L_{1} |A_{i} } \right) + \left[ {V\left( {L_{1} |A_{i} } \right) + V\left( {A_{i} } \right)} \right]C(L_{2} |A_{i} ) \). Conditional Desirability.

  3. (3)

    \( 0 = x_{i} \frac{{C\left( {L_{1} } \right)}}{{C\left( {A_{i} } \right)}}V\left( {L_{1} |A_{i} } \right) + y_{i} \frac{{C\left( {L_{2} } \right)}}{{C\left( {A_{i} } \right)}}V\left( {L_{2} |A_{i} } \right) \).Footnote 28 PP and Bayes’ theorem.

  4. (4)

    \( V\left( {L_{1} |A_{i} } \right) = - V\left( {\left( {L_{2} |A_{i} } \right)} \right)\frac{{y_{i} }}{{x_{i} }}\frac{{C\left( {L_{2} } \right)}}{{C\left( {L_{1} } \right)}}. \)

  5. (5)

    Since by Lemma 1\( x_{i} ,y_{i} > 0 \), and therefore by Lemma 2\( C\left( {L_{1} } \right),C\left( {L_{2} } \right) \), Proposition 1 follows from (4). □

1.4 Proposition 2

Any two jointly exhaustive disagreeing prospects \( L_{1} = {\bigwedge }_{i = 1}^{n} A_{i}^{{x_{i} }} \) and \( L_{2} = {\bigwedge }_{i = 1}^{n} A_{i}^{{y_{i} }} \) such that \( C\left( {L_{1} } \right),C\left( {L_{2} } \right) > 0 \) satisfy the following condition:

  1. (1)

    \( I\left( {L_{1} } \right) > 0 \) iff \( I\left( {L_{2} } \right) < 0. \)

  2. (2)

    \( I\left( {L_{1} } \right) < 0 \) iff \( I\left( {L_{2} } \right) > 0. \)

  3. (3)

    \( I\left( {L_{1} } \right) = 0 \) iff \( I\left( {L_{2} } \right) = 0. \)

Proof

  1. (1)

    \( I\left( {L_{1} } \right) = \sum\nolimits_{i = 1}^{n} {x_{i} } V\left( {L_{1} |A_{i} } \right). \)

  2. (2)

    Recall that for all \( A_{i} \), \( V\left( {L_{1} |A_{i} } \right) = - V\left( {\left( {L_{2} |A_{i} } \right)} \right)\frac{{y_{i} }}{{x_{i} }}\frac{{C\left( {L_{2} } \right)}}{{C\left( {L_{1} } \right)}} \). Proposition 1.

  3. (3)

    \( I\left( {L_{1} } \right) = \sum\nolimits_{i = 1}^{n} - x_{i} V\left( {L_{2} |A_{i} } \right)\frac{{y_{i} }}{{x_{i} }}\frac{{C\left( {L_{2} } \right)}}{{C\left( {L_{1} } \right)}} = - \frac{{C\left( {L_{2} } \right)}}{{C\left( {L_{1} } \right)}}\sum\nolimits_{i = 1}^{n} {y_{i} } V\left( {L_{2} |A_{i} } \right) \). From (1) and (2).

  4. (4)

    \( I\left( {L_{1} } \right) = - \frac{{C\left( {L_{2} } \right)}}{{C\left( {L_{1} } \right)}}I\left( {L_{2} } \right) \)

  5. (5)

    Since by assumption \( c\left( {L_{1} } \right) \) and \( c\left( {L_{2} } \right) \) are positive, proposition 2 follows from (4).□

1.5 Proposition 3

Any \( m \) jointly exhaustive disagreeing prospects \( L_{1} = {\bigwedge }_{i = 1}^{n} A_{i}^{{x_{i}^{1} }} , \ldots ,L_{m} = {\bigwedge }_{i = 1}^{n} A_{i}^{{x_{i}^{m} }} \) such that \( C\left( {L_{j} } \right) > 0 \) for all \( 1 \le j \le m \) satisfy the following condition:

$$ \mathop \sum \limits_{j = 1}^{m} c\left( {L_{j} } \right)I\left( {L_{j} } \right) = 0 $$

Proof

For all \( A_{i} \):

  1. (1)

    For all \( A_{i} \): \( V\left( {A_{i} } \right) = \sum\nolimits_{j = 1}^{m} {V\left( {A_{i} L_{j} } \right)C\left( {L_{j} |A_{i} } \right)} \). Desirability.

  2. (2)

    For all \( A_{i} \): \( V\left( {A_{i} } \right) = \sum\nolimits_{j = 1}^{m} {\left[ {V\left( {L_{j} |A_{i} } \right) + V\left( {A_{i} } \right)} \right]C\left( {L_{j} |A_{i} } \right)} \). Conditional Desirability.

  3. (3)

    For all \( A_{i} \): \( \sum\nolimits_{j = 1}^{m} {V\left( {L_{j} |A_{i} } \right)\frac{{x_{i}^{j} C\left( {L_{j} } \right)}}{{C\left( {A_{i} } \right)}} = 0} \). Bayes’ theorem.

  4. (4)

    For all \( A_{i} \): \( \sum\nolimits_{j = 1}^{m} {x_{i}^{j} V\left( {L_{j} |A_{i} } \right)C\left( {L_{j} } \right) = 0} \). (We multiplied the equation in (3) by \( C\left( {A_{i} } \right) \)).

  5. (5)

    The equation in (4) is true for all \( A_{i} \), then we can sum over the \( A_{i} \) s and get the following expression that is still equal to zero:

    $$ \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{m} {x_{i}^{j} V\left( {L_{j} |A_{i} } \right)C\left( {L_{j} } \right) = 0} } $$
  6. (6)

    Some algebra (reversing the summations) yields:

    $$ \sum\limits_{j = 1}^{m} {C\left( {L_{j} } \right)\sum\limits_{i = 1}^{n} {x_{i}^{j} V\left( {L_{j} |A_{i} } \right) = 0} } $$
  7. (7)

    \( \sum\nolimits_{j = 1}^{m} {C\left( {L_{j} } \right)I\left( {L_{j} } \right) = 0} \). This follows from (6) and the definition of overall intrinsic value.

  8. (8)

    Since by assumption for all \( j \), \( C\left( {L_{j} } \right) > 0 \), proposition 4 follows from claim (7). □

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Goldschmidt, Z., Nissan-Rozen, I. The intrinsic value of risky prospects. Synthese 198, 7553–7575 (2021). https://doi.org/10.1007/s11229-020-02532-3

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