Proof of Proposition 1: We have to show that for all lotteries = (1 1; ; ) obtaining the expected value of the lottery, (), for sure is weakly preferred to the lottery . That is,
(1 ())< for all lotteries .
Let be the smallest outcome of some arbitrary lottery. Substitution of CT into the previous preference gives
Observe that for any∈N the following identity holds
(())−() = we assumed that is continuously differentiable over R. Then, there exists 0 such that ∆+ ≥ [(+)−()] for all 0 . Note that the latter inequality is strict, unless is linear on [ ()]. It then follows that
(())−()≥ (+)−()
[()−] (10)
for all 0 with a strict inequality unless is linear on[ ()].
Let ∗ = min{ 0}. Then, the Inequalities (10) and (11) above hold for all 0 ∗. Recall that Inequality (3) in the main text also holds, i.e.,
(0+0)−(0)≥(0)−(0−0)
for all 0 ∈ R 0 0 0, and thus, it holds in particular for 0 = 0 = and 0 = for all 0 ∗. Therefore,
(+)−()
[()−]≥ ()−(−)
[()−]
holds for all 0 ∗. Invoking Inequality (10) for the term on the left and Inequality (11) for the term on the right implies Inequality (9) as desired. Further, Inequality (9) is strict unless is linear on [0max∈{1}{−}]or is linear on [ ()].
As was an arbitrary non-degenerate lottery, it follows that CT implies weak risk aversion for decision under risk. This completes the proof of Proposition 1. ¤
Proof of Proposition 2: We have to show that strong concavity of under CT is equivalent to = (1 1; ; ) Â 0 = (1 1; ; +; ;0 0 −0; ; ) whenever
0 and 0.
Step (i): Suppose that 0 −0 is not the minimal outcome of0, i.e. and 0 have the same minimal outcome. In this case CT reduces to an expected utility representation. As a result, it follows that is concave. Conversely, strict concavity of is sufficient for  0 if 0 −0 is not the minimal outcome of 0.
Step (ii): Suppose now that 0 is the minimal outcome of such that0 −0 is the minimal
outcome of0. We have to show that Â0 or, equivalently, after substitution of CT that
As is strictly concave it holds that
(+−0+0) +0(0) (−0 +) +0(0)
From condition (3) we know that the utility of wealth difference on the left side of the exceeds each chance utility difference on the right hand side of the inequality. Therefore, Â0 follows whenever
0 is the minimal outcome of.
Step (iii): It remains to consider the case that0 is not the minimal outcome of but0−0
is the minimal outcome of0. Suppose that is the minimal outcome of . We have 0 ≥
0 −0. If = 0 −0, then  0 follows from strict concavity of using arguments similar to those in Step (i). Alternatively, if0 0−0, then we can the mean-preserving
spread from to 0 can be obtained from two successive mean-preserving spreads follows. Define 0 such that =0 +00 and set00= (1 1; ; +; ;0 0−0; ; ). By construction00 is a mean-preserving spread of and both lotteries have the same minimal outcome.
Thus, Â00 follows as is concave. Further, 0 is a mean-preserving spread of 00 with minimal outcome 0 +00 and 0 +00 −0. By Step (ii) it then follows that 00 Â 0. Thus,
Â00,00 Â0 and transitivity yields implies Â0. Thus, Steps (i)—(iii), complete the proof of
Proposition 2. ¤
Proof of Proposition 3: The proof follows immediately from the arguments provided preceding
the proposition in the main text. ¤
follows (see above) that . Consider defined by
Suppose now that converges to zero. This implies that and converge to . This follows from the fact that CT is continuous in probabilities. Now can only hold if
0 ()() 0 ()(). As this argument is valid arbitrary 0 ∈R, it is also valid for all
0 ∈R.
This completes the proof of Proposition 4. ¤
Proof of Proposition 5: The proof follows immediately from the arguments provided preceding
the proposition in the main text. ¤
Proof of Lemma 6: First we note that by Debreu (1954) the preference conditions imply the existence of a strongly monotonic continuous function :R→Rthat represents the preference<
on F. Obviously, also represents <on each setF = 1 .
Let us fix an arbitrary state ∈ S. Without loss of generality let = 1. Continuity and monotonicity imply that for each act ∈ F1 there exists a unique outcome such that ∼ (1 ). Next we restrict the analysis to the set of quasi binary acts{ ∈F1|(1 )}. In our notation for acts (i.e., = (1 : 1; −1)) this set is isomorphic to the two dimensional set
1 := { ∈ F1|(1 : 1; −1)} (∼= R×R+). The restriction of the preference < to 1, which for simplicity we also denote <, inherits weak order, continuity and monotonicity from < on F1. Ad-ditionally, it satisfies the triple cancellation condition formulated with indifferences on 1. In the presence of weak order, monotonicity and continuity, and the structural richness that we have in 1, the indifference version of triple cancellation is equivalent to the preference version of the triple can-cellation (see Köbberling and Wakker, 2003, for a similar argument showing that their indifference version of tradeoffconsistency is equivalent to the preference version of tradeoffconsistency). Hence, by Corollary 3.6 and Remark 3.7 of Wakker (1993a) it follows that there exists (jointly cardinal) continuous and strictly increasing functions 1 : R → R and −1 : R+ → R such that < on 1 is represented by
1() =1(1) +−1( −1)
Note that continuity and monotonicity imply that −1(0) = 0. This means that, for 0 and
real, 1 can be replaced by 1 + whenever −1 is replaced by −1. Next we extend −1
from R+ to {0} ×R+−1. Using the indifference ∼ (1 : 1; 0 −1 −1) we can define
−1 :{0} ×R+−1→R through
−1(0 −1) :=−1(0 −1 −1)≡−1(0 −1).
This way continuity and strong monotonicity of −1 is inherited through . Then, for ∈F1, we have
< ⇔ (1 −1)<(1 −1)
⇔ 1(1) +−1(−1)≥1(1) +−1(−1)
⇔ 1(1) +−1( −1)≥1(1) +−1(−1)
demonstrating that 1() = 1(1) +−1( −1) represents < on F1. Obviously, the uniqueness results for1 and −1 are maintained through this extension of the representation.
In the preceding analysis we have fixed state = 1. The proof for any arbitrary state ∈ S is completely analogous. Hence, we can conclude that for each state ∈ S there exists strongly monotonic and continuous functions :R→ R − :{0} ×R+−1 → R such that on each set F the preference<is represented by
() =() +−(−)
The functions − satisfy the corresponding uniqueness results for the representation onF for all ∈S.
Next we apply once more WOS. We take two arbitrary but distinct states 0 ∈S. Locally, in a small neighborhood, we canfind for all , and , such that the following three indifferences hold: (:;)∼(:;),(:;)∼(:;) and(0 :;0)∼(0 :;0)and all acts are from
F∩F0. By WOS it follows that(0 :;0)∼(0 :;0)with these acts being from F∩F0. As both and0 represent preferences onF∩F0 these functions must be ordinal transformations of each other. Further, substitution of in the former two indifferences and taking differences of the resulting equations and cancelling common terms, implies()−() =()−(). Similarly, substitution of 0 in the latter two indifferences, implies 0()−0() =0()−0(). As,
and were arbitrary, it follows that and 0 are (first locally and by continuity also globally) proportional. As= and 0 =0 for all constant acts, and the latter are included in F ∩F0, it follows that and 0 are, actually, cardinally related. Hence, we can choose them identical on the set of common acts F∩F0. In particular this means that − =−0 on (R+−× {0}×→R+−1)∩(R+−0× {0}×→R+0−1). As and0 were arbitrary chosen, we can set
:= for all∈S. That is
() =() +−( −)
holds on each setF. Further, uniqueness results are maintained for and− ∈S. This completes
the proof of Lemma 6. ¤
Proof of Theorem 7: First we prove that statement (ii) implies statement (i). Monotonicity of
<follows from the strong monotonicity of. Continuity of<follows from the continuity of. The fact that is representing < on F implies that < is a weak order. Notice that on each set F the act is evaluated by () =() +(−) where( −) =P
∈S\{}(−), which is an additively separable representation of <on F and has independent of. Hence, substitution of
for the indifferences in the definition of WOS immediately shows that <satisfies WOS. To derive tradeoff consistency for chance assume that the acts in the following indifferences are all from the
same set F for some state∈S, and that
(+) ∼ (+)
(+) ∼ (+) and (0 +)00 ∼ (0 +)00 hold
but(0 +)00 Â (0 +)00
for some 0 6=. Substitution of into thefirst two indifferences implies that
() + X
∈S\{}
(−) +() = () + X
∈S\{}
(−) +() and
() + X
∈S\{}
(−) +() = () + X
∈S\{}
(−) +()
hold. Subtracting the second equation from thefirst and cancelling common terms gives:
[()−()] =[()−()]
or
()−() =()−()
as the probability is positive.
Substitution of into the third indifference and the latter preference gives, by using similar calculus,
()−() ()−()
a contradiction.
If in the previous analysis, instead of (0 +)00 Â (0 +)00, we assume (0 +)00 ≺
(0 +)00, a similar contradiction (i.e., ()−() =()−() and ()−() ()−()) is obtained. Hence, (0 +)00 ∼ (0 +)00 must hold. As ∈S was arbitrary, it follows that tradeoffconsistency for chance holds on each set F. Hence, it holds onF.
Finally, the property that for all ∈R 0
(+)−() ()−(−)
holds follows from the strong monotonicity property of and Lemma 6.
Next we assume statement (ii) and derive statement (i). The assumptions of Lemma 6 are satisfied, hence, there exists continuous strongly monotonic functions :R → R − : R+−× {0}× → R+−1
with−(0 0) = 0 ∈S, such that the preference <is represented by
() =() +−(−)
for ∈ F. Further, for 0 and real-valued the function can be replaced by +whenever
− is replaced by −,∈{1 }.
Take any arbitrary state ∈S. Tradeoff consistency for chance implies that if
(+) ∼ (+)
(+) ∼ (+)
and(0 +)00 ∼ (0 +)00 hold,
then(0 +)00 ∼(0 +)00 follows, provided that 0 6= and all acts involved are from the set
F. Substituting =+− we obtain that onR+−× {0}×→R+−1 the equalities
−((+) −) = −((+)−)
−((+) −) = −((+)−) and −((0 +)00−0 ) = −((0 +)00−0 ) imply−((0 +)00−0 ) = −((0 +)00−0 )
This condition is analogous to tradeoff consistency (see Köbberling and Wakker 2003) for the func-tion − (representing a continuous monotonic preference) on R+−× {0}× → R+−1. Following Köbberling and Wakker (2003) this implies that there exist positive numbers− ∈S\{} and a continuous strictly increasing utility function −:R+→R such that
−(−) = X
∈S\{}
−−(−).
From−(0) = 0it follows that −(0) = 0. This means that <on F is represented by
() =() + X
∈S\{}
−−(−)
We now set (0) = 0 tofix the location of.
Recall that ∈S was arbitrary chosen. Hence, we conclude that for each state∈S there exist strictly increasing and continuous function − : R+ → R with −(0) = 0, and positive numbers
− ∈S\{}such that() :=() +P
∈S\{}−−(−)represents the preference on the set of actsF. Further the functionsand − are unique up to multiplication by a positive constant (i.e., they are ratio scales).
Take any two distinct states 0 ∈S and consider the restriction of<on the set of actsF{0} :=
F∩F0. On this set both and 0 represent <. Uniqueness results imply that −0 = − (as
both have in common) and that − = −0 =: for all ∈ S\{ 0}. As 0 ∈ S was chosen arbitrary it follows that the positive numbers − are independent of ∈ S, such that we have positive numbers ∈S.
We define ˜:=−. Further, we set =P
∈S and define
:=
for each ∈S
and obtain a subjective probability distribution over the states in S. Finally, we define :=˜.
Hence, we have shown that the representations of <on F is of the form
() =() + X
∈S\{}
(−)for ∈F ∈S
Hence, we have derived statement (i) of the theorem.
By construction the probabilities ∈S are uniquely determined. By construction, for positive
and some constant, we can replace andby+and, respectively. That there is no further flexibility in the choice of these functions follows from Lemma 6. This concludes the proof of Theorem
7. ¤
References
Allais, Maurice (1953), “Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Américaine,” Econometrica 21, 503—546.
Alon, Shiri (2013), “Derivation of a Cardinal Utility Through a Weak Trade-offConsistency Require-ment,”Mathematics of Operations Research, forthcoming (moor.2013.0615).
Andreoni, J, and C. Sprenger (2010), “Certain and Uncertain Utility: The Allais Paradox and Five Decision Theory Phenomena,” Working Paper.
Andreoni, J, and C. Sprenger (2012), “Risk Preferences Are Not Time Preferences,”American Eco-nomic Review 102, 3357-3376.
Andreoni, J. and W. Harbaugh (2010), “Unexpected Utility: Five Experimental Tests of Preferences For Risk,” Working Paper.
Arrow, K.J. (1965), “The theory of risk aversion”. In: Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin Saatio, Helsinki. Reprinted in: Essays in the Theory of Risk Bearing (1971, pp.
90—109). Markham Publ. Co., Chicago.
Arrow, Kenneth J., and Hurwicz, Leonid (1972) “Decision Making Under Ignorance,” in C. F. Carter and J.L. Ford (eds.), Uncertainty and Expectations in Economics. Essays in Honour of G.L.S.
Shackle. Oxford: Basil Blackwell.
Bernoulli, Daniel (1954), “Exposition of a New Theory on the Measurement of Risk,” Econometrica 22, 23—36. Translated version of Bernoulli, Daniel (1738), “Specimen Theoriae Novae de Mensura Sortis,” Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V [Papers of the Imperial Academy of Sciences in Petersburg, Vol. V], 175—192.
Bleichrodt, Han and Ulrich Schmidt (2002), “A Context-Dependent Model of the Gambling Effect,”
Management Science 48, 802—812.
Camerer, Colin F. (1992), “Recent Tests of Generalizations of Expected Utility Theory.” In Ward Edwards (ed.) Utility Theories: Measurement and Applications, 207—251, Kluwer Academic Publishers, Dordrecht.
Cerreia-Vioglio, Simone, David Dillenberger, and Pietro Ortoleva (2013), “Cautious Expected Utility and the Certainty Effect,”Working Paper.
Chateauneuf, Alain, Jürgen Eichberger and Simon Grant (2007), “Choice under Uncertainty with the Best and Worst in Mind: NEO-Additive Capacities,”Journal of Economic Theory137, 538—567.
Cohen, Michèle (1992), “Security Level, Potential Level, Expected Utility: A Three-Criteria Decision Model under Risk,” Theory and Decision 33, 101—134.
Cohen, M. and J.Y. Jaffray (1988) “Preponderance of the Certainty Effect over Probabillity Distortion in Decision Making under Risk,” in B.R. Munier (ed),Risk, Decision, and Rationality. D. Reidel:
Dordrecht.
Conlisk, John (1989), “Three Variants on the Allais Example,” American Economic Review 79, 392—407.
Conlisk, John (1993), “The Utility of Gambling,” Journal of Risk and Uncertainty 6, 255—275.
Debreu, Gerard (1954), “Representation of a Preference Ordering by a Numerical Function,” in R.M.
Thrall, C.H. Coombs, R.L. Davis (Eds.),Decision Processes 159—165, Wiley, New York.
Debreu, Gerard (1960), “Topological Methods in Cardinal Utility Theory,” in K.J. Arrow, S. Karlin, P. Suppes (Eds.),Mathematical Methods in the Social Sciences, 16—26, Stanford University Press, Stanford, CA.
Diecidue, Enrico, Ulrich Schmidt, and Peter P. Wakker (2004),“The Utility of Gambling Reconsid-ered,”Journal of Risk and Uncertainty 29, 241—259.
Dillenberger, David (2010), “Preferences for One-Shot Resolution of Uncertainty and Allais-Type Behavior,”Econometrica 78, 1973—2004.
Dyer, James S., and Rakesh K. Sarin (1982), “Relative Risk Aversion,” Management Science 28, 875—886.
Ellsberg, Daniel (1961), “Risk, Ambiguity and the Savage Axioms,”Quarterly Journal of Economics 75, 643—669.
Fishburn, Peter C. (1977), “Mean-Risk Analysis with Risk Associated with Below-Target Returns,”
American Economic Review 67, 116—126.
Fishburn, Peter C. (1980), “A Simple Model for the Utility of Gambling,”Psychometrika45, 435—448.
Friedman, Milton, and Leonard J. Savage (1948), “The Utility Analysis of Choices Involving Risk,”
Journal of Political Economy 56, 279—304.
Gilboa, Itzhak (1988), “A Combination of Expected Utility and Maxmin Decision Criteria,”Journal of Mathematical Psychology 32, 405—420.
Gilboa, Itzhak and David Schmeidler (1989), “Maxmin Expected Utility with a Non-Unique Prior,”
Journal of Mathematical Economics 18, 141—153.
Gneezy, Uri, John List, and George Wu (2006), “The Uncertainty Effect: When a Risky Prospect Is Valued Less than Its Worst Possible Outcome,”Quarterly Journal of Economics121, 1283—1309.
Gul, Faruk (1991), “A Theory of Disappointment Aversion,”Econometrica 59, 667—686.
Harless, and Camerer (1994), “The Predictive Utility of Generalized Expected Utility Theories,”
Econometrica 62, 1251—1289.
Hey, John and Chris Orme (1994), “Investigating Generalizations of Expected Utility Theory Using Experimental Data,”Econometrica 62, 1291—1326.
Jaffray, Jean Yves (1988), “Choice Under Risk and the Security Factor: An Axiomatic Model,”
Theory and Decision, 24, 169—200.
Kahneman, Daniel and Amos Tversky (1979), “Prospect Theory: An Analysis of Decision under Risk,” Econometrica 47, 263—291.
Kimball, Miles S. (1990): “Precautionary Saving in the Small and in the Large”, Econometrica 58, 53—73.
Köbberling, Veronika and Peter P. Wakker, (2003), “Preference Foundations for Nonexpected Utility:
A Generalized and Simplified Technique.”Mathematics of Operations Research 28, 395—423.
Krantz, David H., R. Duncan Luce, Patrick Suppes, and Amos Tversky, (1971), Foundations of Measurement, Vol. I (Additive and Polynomial Representations). Academic Press, New York.
2nd edn. 2007, Dover Publications, New York.
Lopes, Lola L. (1987), “Between Hope and Fear: The Psychology of Risk. In Leonard Berkowitz (Ed.), Advances in Experimental and Social Psychology (pp. 255—295), San Diego, Academic Press.
Luce, R. Duncan, and Anthony A.J. Marley (2000) “On Elements of Chance,” Theory and Decision 49, 97—126.
Luce, R. Duncan and Howard Raiffa (1957),Games and Decisions, Wiley, New York.
Luce, R. Duncan and John W. Tukey (1964), “Simultaneous Conjoint Meassurement: A New Type of Fundamental Measurement,”Journal of Mathematical Psychology 1, 1—27.
Payne, John W., Dan J. Laughhunn, and Roy L. Crum (1980), “Translation of Gambles and Aspi-ration Level Effects in Risky Choice Behavior,” Management Science 26, 1039—1060.
Payne, John W., Dan J. Laughhunn, and Roy L. Crum (1981), “Further Tests of Aspiration Level Effects in Risky Behavior,”Management Science 27, 953—958.
Pratt, J.W. (1964). “Risk Aversion in the Small and in the Large,” Econometrica 32, 122—136.
Quiggin, John (1981), “Risk Perception and Risk Aversion among Australian Farmers,” Australian Journal of Agricultural Economics 25, 160—169.
Quiggin, John (1982), “A Theory of Anticipated Utility,” Journal of Economic Behaviour and Or-ganization 3, 323—343.
Rabin, Matthew (2000), “Risk Aversion and Expected-utility Theory: A Calibration Theorem,”
Econometrica 68, 1281—1292.
Rawls, J (1971),A Theory of Justice. Cambridge, Mass.
Rothschild, Michael and Joseph E. Stiglitz (1970), “Increasing Risk: I. A Definition,” Journal of Economic Theory 2, 225—243.
Roy, Andrew D. (1952) “Safety First and the Holding of Assets,” Econometrica 20, 431—449.
Savage, Leonard J., (1954), The Foundations of Statistics. New York: Wiley.
Schmeidler, David (1989), “Subjective Probability and Expected Utility without Additivity,” Econo-metrica 57, 571—587.
Schmidt, Ulrich (1998), “A Measurement of the Certainty Effect,”Journal of Mathematical Psychol-ogy 42, 32—47.
Segal, Uzi and Avia Spivak (1990), “First Order Versus Second Order of Risk Aversion,”Journal of Economic Theory 51, 111—125.
Simonsohn, Uri (2009), “Direct-Risk-Aversion: Evidence from Risky Prospects Valued Below Their Worst Outcome,”Psychological Science 20, 686—692.
Spinu, Vitalie (2012), “From Simple to Complex: A General Method for Extending Behavioral Foundations to Infinitely Many Outcomes,” Working Paper, Erasmus University Rotterdam, the Netherlands.
Starmer, Chris (2000), “Developments in Non- Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk,”Journal of Economic Literature 38, 332—382.
Starmer, Chris and Robert Sugden (1989), “Violations of the Independence Axiom in Common Ratio Problems: An Experimental Test of some Competing Hypotheses,” Annals of Operations Research 19, 79—102.
Tversky, Amos and Daniel Kahneman (1992), “Advances in Prospect Theory: Cumulative Repre-sentation of Uncertainty,”Journal of Risk and Uncertainty 5, 297—323.
Wakker, Peter P., (1989), Additive Representations of Preferences, A New Foundation of Decision Analysis, Kluwer Academic Publishers, Dordrecht.
Wakker, Peter P., (1993a), “Additive Representations on Rank-ordered Sets II. The Topological Approach,”Journal of Mathematical Economics 22, 1—26.
Wakker, Peter P., (1993b), “Unbounded Utility for Savage’s “Foundation of Statistics,” and Other Models.”Mathematics of Operations Research 18, 446—485.
Wakker, Peter P. (2010), Prospect Theory for Risk and Ambiguity, Cambridge University Press, Cambridge, UK.
Webb, Craig S., and Horst Zank (2011). “Accounting for Optimism and Pessimism in Expected Utility,”Journal of Mathematical Economics 47, 706—717.
Yaari, Menahem E. (1987), “The Dual Theory of Choice under Risk,” Econometrica 55, 95—115.