A simple framework for the axiomatization of exponential and quasi-hyperbolic discounting
aa r X i v : . [ q -f i n . E C ] N ov A simple framework for the axiomatization ofexponential and quasi-hyperbolic discounting
Nina Anchugina ∗ Abstract
The main goal of this paper is to investigate which normative re-quirements, or axioms, lead to exponential and quasi-hyperbolic formsof discounting. Exponential discounting has a well-established ax-iomatic foundation originally developed by Koopmans (1960, 1972)and Koopmans et al. (1964) with subsequent contributions by sev-eral other authors, including Bleichrodt et al. (2008). The papersby Hayashi (2003) and Olea and Strzalecki (2014) axiomatize quasi-hyperbolic discounting. The main contribution of this paper is to pro-vide an alternative foundation for exponential and quasi-hyperbolicdiscounting, with simple, transparent axioms and relatively straight-forward proofs. Using techniques by Fishburn (1982) and Harvey(1986), we show that Anscombe and Aumann’s (1963) version of Sub-jective Expected Utility theory can be readily adapted to axiomatizethe aforementioned types of discounting, in both finite and infinitehorizon settings.
Keywords:
Axiomatization; Exponential discounting; Quasi-hyperbolic discounting; Anscombe-Aumann model.
JEL Classification:
D90.
The axiomatic foundation of intertemporal decisions is a fundamental ques-tion in economics and generates considerable research interest. Despite the ∗ Department of Mathematics, The University of Auckland, Auckland, New Zealand;e-mail: [email protected]
Assume that the objectives of a decision-maker can be expressed by a pref-erence order < on the set of alternatives X n , where n may be ∞ . Think ofthese alternatives as dated streams, for time periods t ∈ { , , . . . , n } . Wesay that a utility function U : X n → R represents this preference order, iffor all x , y ∈ X n , x < y if and only if U ( x ) ≥ U ( y ).We assume that X is a mixture set . That is, for every x, y ∈ X andevery λ ∈ [0 , xλy ∈ X satisfying: • x y = x , • xλy = y (1 − λ ) x , • ( xµy ) λy = x ( λµ ) y .Since X is a mixture set, the set X n is easily seen to be a mixture set underthe following mixture operation: x λ y = ( x λy , . . . , x n λy n ), where x , y ∈ X n and λ ∈ [0 , u : X → R is called mixture linear (or just linear,where no confusion is likely to arise) if for every x, y ∈ X we have u ( xλy ) = λu ( x ) + (1 − λ ) u ( y ) for every λ ∈ [0 , It should also be mentioned that our setting considers a discrete time space. Acontinuous-time framework can be found, for example, in the above-mentioned paperby Fishburn and Rubinstein (1982) and in a generalized model of hyperbolic discountingintroduced by Loewenstein and Prelec (1992). Harvey (1986) analyzes discrete sequencesof timed outcomes with a continuous time space. < on X n induces a binary relation (also denoted < )on X in the usual way: for any x, y ∈ X the preference x < y holds if andonly if ( x, x, . . . , x ) < ( y, y, . . . , y ).The function U is called a discounted utility function if U ( x ) = n X t =1 D ( t ) u ( x t ) , for some non-constant u : X → R and some D : N → R with D (1) = 1.The function D is called the discount function . If u is linear (and non-constant), then the function U is called a discounted expected utilityfunction .There are two types of discount functions which are commonly used inmodeling of time preferences: • Exponential discounting : D ( t ) = δ t − , where δ ∈ (0 , • Quasi-hyperbolic discounting : D ( t ) = (cid:26) t = 1 ,βδ t − if t ≥ . for some δ ∈ (0 ,
1) and β ∈ (0 , t = 1) to the immediate future ( t = 2) can change thepreferences of a decision-maker between these consumption streams.The results of the recent experiments by Chark et al. (2015) show thatdecision-makers are decreasingly impatient within the near future, howeverthey discount the remote future at a constant rate. In other words, presentbias may extend over the present moment ( t = 1) to the near future t > T . This gives a further gen-eralization of quasi-hyperbolic discounting, which we call semi-hyperbolicdiscounting : D ( t ) = t = 1 , t − Y i =1 β i δ if 1 < t ≤ T,δ t − T T − Y i =1 β i δ if t > T.
4e use SH ( T ) to denote this discount function (for given δ, β , . . . , β T − ).This form of discounting was previously applied to model the time preferencesof a decision-maker in a consumption-savings problem (Young, 2007). Our SH ( T ) specification is not quite the same as the notion of semi-hyperbolicdiscounting used in Olea and Strzalecki (2014). They apply the term to anydiscount function which satisfies D ( t ) = δ t − T D ( T ) for all t > T (for some T ). This class includes SH ( T ), but is wider. The possibility of generalizingquasi-hyperbolic discounting was earlier suggested by Hayashi (2003). Theform of the discount function he proposed is: D ( t ) = t = 1 , t − Y i =1 β ′ i if 1 < t ≤ T,δ t − T T − Y i =1 β ′ i if t > T. By substituting δβ t = β ′ t for all t ≤ T − SH ( T ) coincides with the form suggested by Hayashi(2003). It is worth mentioning that he does not provide an axiomatizationof this form of discounting, pointing out that this case is somewhat com-plicated. In our framework, however, the axiomatization of semi-hyperbolicdiscounting can be obtained as a relatively straightforward extension of theaxiomatization of quasi-hyperbolic discounting.The evidence of Chark et al. (2015) on extended present bias suggests thefollowing restrictions on the coefficients in SH ( T ): β < β < . . . < β T − .In our version of SH ( T ) we will impose the weaker requirements β ≤ β ≤ . . . ≤ β T − , and β t ∈ (0 ,
1] for all t ≤ T − δ ∈ (0 , SH (1) is the exponential discount function, whereas SH (2) is the quasi-hyperbolic discount function.5 AA representations
We say that the preference order < on X n has an Anscombe and Aumann(AA) representation , if for every x , y ∈ X n : x < y if and only if n X t =1 w t u ( x t ) ≥ n X t =1 w t u ( y t ) , where u : X → R is non-constant and linear and w t ≥ t with at leastone w t >
0. We also say that the pair ( u, w ) provides an AA representationfor < .A pre-condition for obtaining discounting in an exponential or quasi-hyperbolic form is additive separability. In the framework of preferencesover streams of lotteries, Anscombe and Aumann’s (1963) theorem providesaxioms which give an additively separable representation when n < ∞ .Anscombe and Aumann formulated their result for acts rather than tem-poral streams. Here, states of the world are replaced by time periods. ( n < ∞ ) For n < ∞ the following axioms are necessary and sufficient for an AArepresentation: Axiom F1. (Weak order). < is a weak order on X n . Axiom F2. (Non-triviality). There exist some a, b ∈ X such that( a, a, . . . , a ) ≻ ( b, b, . . . , b ) . Axiom F3. (Mixture independence). x < y if and only if x λ z < y λ z forevery λ ∈ (0 ,
1) and every x , y , z ∈ X n . Axiom F4. (Mixture continuity). For every x , y , z ∈ X n the sets { α : x α z < y } and { β : y < x β z } are closed subsets of the unit in-terval. Axiom F5. (Monotonicity). For every x , y ∈ X n if x t < y t for every t then x < y . 6 heorem 1 (AA) . The preferences < on X n satisfy axioms F1-F5 if andonly if there exists an AA representation for < on X n . If ( u, w ) and ( u ′ , w ′ ) both provide AA representations for < on X n , then u = Au ′ + B for some A > and some B , and w = C w ′ for some C > . The proof of the theorem for the general mixture set environment caneasily be constructed by combining the arguments in Fishburn (1982) andRyan (2009). Evidently, the key axiom here is the condition of mixtureindependence. It is a strong axiom which imposes an additive structure. n = ∞ ) Anscombe and Aumann’s result may be extended to the infinite horizoncase. One possible extension is given by Fishburn (1982). However, we givea slightly modified version which incorporates ideas from Harvey (1986).Fix some x ∈ X . We refer to the same x throughout the rest of thepaper. A consumption stream x is called ultimately constant if there exists T such that x = ( x , . . . , x T , x , x , . . . ). Note that there is a difference fromthe usage of this definition in Bleichrodt et al. (2008) and Olea and Strzalecki(2014), where x can be arbitrary. Let X T be the set of ultimately constantconsumption streams of length T . Denote the union of the sets X T over all T as X ∗ . Let X ∗∗ be the union of X ∗ and all constant streams. It is nothard to see that both X ∗ , X ∗∗ ⊂ X ∞ are mixture sets.We must mention that the fixed x serves two purposes: firstly, it will beneeded to state the convergence axiom; and secondly, it allows us to definethe class X ∗ of ultimately constant streams in a way that makes them a strictsubset of the usually defined class. Since some of the axioms only restrictpreferences over X ∗∗ this second aspect confers some advantages. Axiom I1. (Weak order). < is a weak order on X ∞ . Axiom I2. (Non-triviality). There exist some a, b ∈ X such that a ≻ x ≻ b .Axiom I2 implies that x is an interior point with respect to preference. Itrestricts both < and the choice of the fixed element x . Axiom I3. (Mixture independence). x < y if and only if x λ z < y λ z forevery λ ∈ (0 ,
1) and every x , y , z ∈ X ∗∗ .7 xiom I4. (Mixture continuity). For every x , z ∈ X ∗∗ and every y ∈ X ∞ the sets { α : x α z < y } and { β : y < x β z } are closed subsets of the unitinterval. Axiom I5. (Monotonicity). For every x , y ∈ X ∞ : if x t < y t for every t then x < y .We have applied a weaker version of the monotonicity axiom in comparisonwith the interperiod monotonicity used by Fishburn. However, Axiom I5 issufficient to obtain an AA representation.For the statement of the next axiom we need to introduce some notation.Let [ a ] k = ( x , . . . , x , a, x , . . . ) where a ∈ X is in the k th position. Usingthis notation, we state the following axiom: Axiom I6. (Convergence). For every x = ( x , x , . . . ) ∈ X ∞ , every x + , x − ∈ X and every k : • if [ x + ] k ≻ [ x k ] k there exists T + ≥ k such that x x + k,T for all T ≥ T + , where x + k,T = ( x , x , . . . , x k − , x + , x k +1 , . . . , x T , x , x , . . . ); • if [ x − ] k ≺ [ x k ] k there exists T − ≥ k such that x < x − k,T for all T ≥ T − , where x − k,T = ( x , x , . . . , x k − , x − , x k +1 , . . . , x T , x , x , . . . ).Our convergence axiom differs from Axiom B6, that was used by Fishburn: Axiom B6.
For some ˆ x ∈ X , every x , y ∈ X ∞ and every λ ∈ (0 , • if x ≻ y , then there exists T such that ( x , . . . , x n , ˆ x, ˆ x, . . . ) < x λ y for all n ≥ T ; • if x ≺ y , then there exists T such that ( x , . . . , x n , ˆ x, ˆ x, . . . ) x λ y for all n ≥ T .Instead, Axiom I6 adapts ideas from Harvey (1986) . Axiom I6 is more ap-pealing for our purposes as it not only guarantees the convergence of the AA It is worth mentioning that Fishburn’s motivation for the convergence axiom B6 lookssomewhat contrived in the context of acts (Fishburn, 1982, p. 113). However, it becomesvery natural in the context where states of the world are re-interpreted as periods of time. X ∞ .We thus obtain the following representation: Theorem 2 (Infinite AA) . The preferences < on X ∞ satisfy axioms I1-I6if and only if there exists an AA representation for < on X ∞ . If ( u, w ) and ( u ′ , w ′ ) both provide AA representations for < on X ∞ , then u = Au ′ + B for some A > and some B , and w = Cw ′ for some C > . The proof of Theorem 2 is given in the Appendix. It combines elementsof the arguments in Fishburn (1982), Harvey (1986) and Ryan (2009). n < ∞ ) Recall that a preference < on X n is represented by an exponentially dis-counted utility function if there exists a non-constant function u : X → R and a parameter δ ∈ (0 ,
1) such that U ( x ) = n X t =1 δ t − u ( x t ) . If u is linear (and non-constant), then we say that the pair ( u, δ ) provides anexponentially discounted expected utility representation.Based on Theorem 1 it is easy to obtain such a representation. In order todo so an adjustment of non-triviality and two additional axioms - impatienceand stationarity - are required. Axiom F2 ′ . (Essentiality of period 1). There exist some a, b ∈ X and some x ∈ X n such that ( a, x , . . . , x n ) ≻ ( b, x , . . . , x n ). Axiom F6. (Impatience). For all a, b ∈ X if a ≻ b , then for all x ∈ X n ( a, b, x , . . . , x n ) ≻ ( b, a, x , . . . , x n ) . Axiom F7. (Stationarity). The preference ( a, x , . . . , x n ) < ( a, y , . . . , y n )holds if and only if ( x , . . . , x n , a ) < ( y , . . . , y n , a ) for every a ∈ X andevery x , y ∈ X n . 9t is not hard to see that essentiality of each period t follows from theessentiality of period 1 and the stationarity axiom.Now the following result can be stated: Theorem 3 (Exponential discounting) . The preferences < on X n satisfyaxioms F1, F2 ′ , F3-F7 if and only if there exists an exponentially discountedexpected utility representation for < on X n . If ( u, δ ) and ( u ′ , δ ′ ) both provideexponentially discounted expected utility representations for < on X n , then u = Au ′ + B for some A > and some B , and δ = δ ′ .Proof. It is straightforward to show that the axioms are implied by the rep-resentation. Conversely, suppose the axioms hold. Note that non-trivialityfollows from essentiality of period 1 and monotonicity.By Theorem 1 we therefore know that < has an AA representation ( u, w ).Define < ′ on X n − as follows:( x , . . . , x n − ) < ′ ( y , . . . , y n − ) ⇔ ( x , x , . . . , x n − ) < ( x , y , . . . , y n − ) . Then < ′ is represented by: U ′ ( x ) = w u ( x ) + . . . + w n u ( x n − ) . Next, define < ′′ on X n − as follows:( x , . . . , x n − ) < ′′ ( y , . . . , y n − ) ⇔ ( x , . . . , x n − , x ) < ( y , . . . , y n − , x ) . Then < ′′ is represented by: U ′′ ( x ) = w u ( x ) + . . . + w n − u ( x n − ) . According to stationarity, these preferences are equivalent ( < ′ ≡ < ′′ ) with twodifferent AA representations ( U ′ and U ′′ ). Preference orders < ′ ≡ < ′′ satisfythe AA axioms on X n − . Recall that w t are unique up to a scale. Hence, w t +1 = δw t for some δ > w n = δw n − = δ w n − = . . . = δ n − t w t = . . . = δ n − w . Since all periods are essential it is without loss of generality to set w = 1.Then we obtain the following representation for < on X n : U ( x ) = n X t =1 δ t − u ( x t ) , where δ > . a ≻ b , then( a, b, x , . . . , x n ) ≻ ( b, a, x , . . . , x n ) . From the representation it follows that: u ( a ) + δu ( b ) > u ( b ) + δu ( a ) , or, equivalently, (1 − δ )( u ( a ) − u ( b )) > . As u ( a ) > u ( b ) , it is possible to conclude that δ ∈ (0 , u, δ ) and ( u ′ , δ ′ ) both provide exponentially discountedexpected utility representations for < on X n . Since ( u, δ ) and ( u ′ , δ ′ ) bothprovide AA representations for < it follows that u = Au ′ + B for some A > B , and there is some C > δ t − = C ( δ ′ ) t − for all t .Taking t = 1 we obtain C = 1, and hence δ = δ ′ . A preference < on X n has a SH ( T ) discounted utility representation if thereexists a non-constant function u : X → R and parameters β ≤ β ≤ . . . ≤ β T − , and β t ∈ (0 ,
1] for all t ≤ T − δ ∈ (0 ,
1) such that the followingfunction represents < : U ( x ) = u ( x ) + β δu ( x ) + β β δ u ( x ) + . . . + β β · · · β T − δ T − u ( x T − )+ β β · · · β T − n X t = T δ t − u ( x t ) . If u is linear (and non-constant), then the function U is called a SH ( T ) discounted expected utility representation . In this case, we say that( u, β , δ ) provides a SH ( T ) discounted expected utility representation, where β = ( β , β , . . . , β T − ).To obtain this form of discounting a number of modifications to the setof axioms is required. A stronger essentiality condition should be used: Axiom F2 ′′ . (Essentiality of periods 1 , . . . , T ). There exist some a, b ∈ X and some x ∈ X n such that for every t = 1 , . . . , T :( x , x , . . . , x t − , a, x t +1 , . . . , x n ) ≻ ( x , x , . . . , x t − , b, x t +1 , . . . , x n ) . δ ∈ (0 , T and T + 1: Axiom F6 ′ . (Impatience). For every a, b ∈ X if a ≻ b , then for every x ∈ X n :( x , . . . , x T − , a, b, x T +2 , . . . , x n ) ≻ ( x , . . . , x T − , b, a, x T +2 , . . . , x n ) . The generalization requires relaxing the axiom of stationarity to stationarityfrom period T . Axiom F7 ′ . (Stationarity from period T). The preference( x , . . . , x T − , a, x T +1 , . . . , x n ) < ( x , . . . , x T − , a, y T +1 , . . . , y n )holds if and only if( x , . . . , x T − , x T +1 , . . . , x n , a ) < ( x , . . . , x T − , y T +1 , . . . , y n , a )for every a ∈ X and every x ∈ X n .The addition of the early bias axiom is needed, so that present bias may arisebetween any periods { t, t + 1 } , where t ≤ T . Axiom F8. (Early bias) For every a, b, c, d ∈ X such that a ≻ c, b ≺ d , forall x ∈ X n and every t ≤ T if( x , . . . , x t − , a, b, x t +2 , . . . , x n ) ∼ ( x , . . . , x t − , c, d, x t +2 , . . . , x n ) , then( x , . . . , x t − , a, b, x t +2 , . . . , x n , x t − ) < ( x , . . . , x t − , c, d, x t +2 , . . . , x n , x t − ) . The early bias axiom is also referred to as the extended present biasaxiom.
Theorem 4 (Semi-hyperbolic discounting) . The preferences < on X n sat-isfy axioms F1, F2 ′′ , F3, F4, F5, F6 ′ , F7 ′ , F8 if and only if there exists a SH ( T ) discounted expected utility representation for < on X n . If ( u, β , δ ) and ( u ′ , β ′ , δ ′ ) both provide SH ( T ) discounted expected utility representationsfor < on X n , then u = Au ′ + B for some A > and some B , and δ = δ ′ , β = β ′ . roof. It can be easily seen that the axioms are implied by the representation.Suppose that the axioms hold. As for Theorem 3, the conditions of AArepresentation are satisfied, so it follows that < has an AA representation( w , u ). Define < ′ on X n − T as follows:( x , . . . , x n − T ) < ′ ( y , . . . , y n − T ) ⇔ ( x , . . . , x , x , . . . , x n − T ) < ( x , . . . , x , y , . . . , y n − T ) . Then < ′ is represented by: U ′ ( x ) = w T +1 u ( x ) + . . . + w n u ( x n − T ) . Next, define < ′′ on X n − T as follows:( x , . . . , x n − T ) < ′′ ( y , . . . , y n − T ) ⇔ ( x , . . . , x , x , . . . , x n − T , x ) < ( x , . . . , x , y , . . . , y n − T , x ) . Then < ′′ is represented by: U ′′ ( x ) = w T u ( x ) + . . . + w n − u ( x n − T ) . According to stationarity from period T , the preferences are equivalent( < ′ ≡ < ′′ ) with two different AA representations ( U ′ and U ′′ ).Preference orders < ′ ≡ < ′′ satisfy the AA axioms on X n − T . Recall that w t are unique up to a scale. Hence, as essentiality holds for all t (which followsfrom Axiom F2 ′ and Axiom F7 ′ ), we have w t +1 = δw t for some δ > w n = δw n − = δ w n − = . . . = δ n − t w t = . . . = δ n − T w T . Therefore, w t = δ t − T w T for all t ≥ T + 1. We therefore obtain the followingrepresentation for < : U ( x ) = w u ( x ) + . . . + w T − u ( x T − ) + w T n X t = T δ t − T u ( x t ) . Because of the essentiality of the first period and uniqueness of u up to affinetransformations:ˆ U ( x ) = u ( x ) + w w u ( x ) + . . . + w T − w u ( x T − ) + w T w n X t = T δ t − T u ( x t ) . w w = w w · w w , · · · ,w T w = w T w T − · w T − w T − · . . . · w w . Let γ t − = w t w t − for all t ≤ T . Therefore, w w = γ ,w w = γ γ , · · · ,w T w = γ γ . . . γ T − . With this notation:ˆ U ( x ) = u ( x ) + γ u ( x ) + . . . + γ · · · γ T − u ( x T − ) + γ · · · γ T − n X t = T δ t − T u ( x t ) . It is necessary to show that γ t − = β t − δ with β t − ∈ (0 ,
1] for all t ≤ T .Suppose that t = T . Choose a, b, c, d ∈ X such that u ( b ) < u ( d ), u ( a ) > u ( c ) and γ · · · γ T − u ( a ) + γ · · · γ T − δu ( b ) = γ · · · γ T − u ( c ) + γ · · · γ T − δu ( d ) . (1)Since essentiality is satisfied for each period we can rearrange the equation(1): δ = u ( a ) − u ( c ) u ( d ) − u ( b ) . (2)From (1) it also follows that( x , . . . , x T − , a, b, x T +2 , . . . , x n ) ∼ ( x , . . . , x T − , c, d, x T +2 , . . . , x n ) , Therefore, by the early bias axiom:( x , . . . , x T − , a, b, x T +2 , . . . , x n , x T − ) < ( x , . . . , x T − , c, d, x T +2 , . . . , x n , x T − ) . γ · · · γ T − u ( a ) + γ · · · γ T − u ( b ) ≥ γ · · · γ T − u ( c ) + γ · · · γ T − u ( d ) . Since the essentiality condition is satisfied for each period we can rearrangethis inequality: γ T − ≤ u ( a ) − u ( c ) u ( d ) − u ( b ) . (3)Comparing (2) to (3) we conclude that δ ≥ γ T − , therefore, γ T − = β T − δ ,where β T − ∈ (0 , t = T −
1. Choose a ′ , b ′ , c ′ , d ′ ∈ X such that u ( b ′ ) < u ( d ′ ) and u ( a ′ ) > u ( c ′ ). Using present bias axiom and essentiality ofeach period we obtain γ T − = u ( a ′ ) − u ( c ′ ) u ( d ′ ) − u ( b ′ ) , (4)and γ T − ≤ u ( a ′ ) − u ( c ′ ) u ( d ′ ) − u ( b ′ ) . (5)It follows from (4) and (5) that γ T − ≤ γ T − . Therefore, γ T − = β ′ T − γ T − ,where β ′ T − ∈ (0 , γ T − = β T − δ . Hence, γ T − = β ′ T − β T − δ = β T − δ, where β T − = β ′ T − β T − and β T − ∈ (0 ,
1] as both β ′ T − ∈ (0 ,
1] and β T − ∈ (0 , β T − ≤ β T − .Using the early bias axiom repeatedly for t < T − γ t − = β t − δ with β t − ∈ (0 ,
1] for all t ≤ T and β ≤ β ≤ . . . ≤ β T − . Hence,ˆ U ( x ) = u ( x ) + β δu ( x ) + β β δ u ( x ) + . . . + β β · · · β T − δ T − u ( x T − )+ β β · · · β T − n X t = T δ t − u ( x t ) . To show that δ ∈ (0 ,
1) the impatience axiom should be applied. Forevery a, b ∈ X if a ≻ b , then for every x ∈ X n ( x , . . . , x T − , a, b, x T +2 , . . . , x n ) ≻ ( x , . . . , x T − , b, a, x T +2 , . . . , x n ) . Then β · · · β T − δ T − u ( a )+ β · · · β T − δ T u ( b ) > β · · · β T − δ T − u ( b )+ β · · · β T − δ T u ( a ) . − δ )( u ( a ) − u ( b )) > . Hence, δ ∈ (0 , u, β , δ ) and ( u ′ , β ′ , δ ′ ) both provide SH ( T ) discounted ex-pected utility representations for < on X n . Let D ( t ) and D ′ ( t ) be semi-hyperbolic discount functions for given β , δ and β ′ , δ ′ , respectively. Since( u, β , δ ) and ( u ′ , β ′ , δ ′ ) both provide AA representations for < it follows that u = Au ′ + B for some A > B , and there is some C > D ( t ) = C · D ′ ( t ) for all t . Taking t = 1 we obtain C = 1, and hence, letting t = 2 , , . . . , T we get β t δ = β ′ t δ ′ for all t ≤ T . Finally, letting t = T + 1 weconclude that δ = δ ′ . Therefore, β = β ′ . n = ∞ ) Based on the AA representation for the preferences over infinite consumptionstreams (Theorem 2), with some strengthening of non-triviality (Axiom I2)and the addition of a suitable stationarity axiom, discounting functions inan exponential form can be obtained. The impatience axiom is not neededsince convergence (Axiom I6) plays its role.
Axiom I2 ′ . (Essentiality of period 1). There exist some a, b ∈ X such that[ a ] ≻ x ≻ [ b ] . Axiom I7. (Stationarity). The preference ( a, x , x , . . . ) < ( a, y , y , . . . )holds if and only if ( x , x , . . . ) < ( y , y , . . . ) for every a ∈ X andevery x , y ∈ X ∞ . Theorem 5 (Exponential discounting) . The preferences < on X ∞ satisfyaxioms I1, I2 ′ , I3-I7 if and only if there exists an exponentially discountedexpected utility representation for < on X ∞ . If ( u, δ ) and ( u ′ , δ ′ ) both provideexponentially discounted expected utility representations for < on X ∞ , then u = Au ′ + B for some A > and some B , and δ = δ ′ .Proof. The necessity of the axioms is straightforward. The proof of suffi-ciency follows the steps of the proof of Theorem 3 with n = ∞ . Applying16heorem 2 to the preferences satisfying the stationarity axiom we obtain therepresentation: U ( x ) = ∞ X t =1 δ t − u ( x t ) , where δ > x ∈ X ∞ .Next, instead of using the impatience axiom as it is done in the finite case,the convergence axiom is applied. Take a constant stream a = ( a, a, . . . ), suchthat u ( a ) = 0. Then, U ( a ) = ∞ X t =1 δ t − u ( a ) = u ( a ) ∞ X t =1 δ t − , We know that U ( a ) should converge by Theorem 2. It follows that δ < The extension of semi-hyperbolic discounting to the case where n = ∞ iseasily obtained. Axiom I2 ′′ . (Essentiality of periods 1 , . . . , T ). For some a, b ∈ X we have[ a ] t ≻ x ≻ [ b ] t for every t = 1 , . . . , T .The generalization requires relaxing the axiom of stationarity to stationarityfrom period T . Axiom I7 ′ . (Stationarity from period T). The preference( x , . . . , x T − , a, x T +1 , . . . ) < ( x , . . . , x T − , a, y T +1 , . . . )holds if and only if( x , . . . , x T − , x T +1 , . . . ) < ( x , . . . , x T − , y T +1 , . . . )for every a ∈ X , and every x ∈ X ∞ .As in the finite case the addition of the early bias axiom allows present biasbetween { t, t + 1 } , where t ≤ T . 17 xiom I8. (Early bias) For every a, b, c, d ∈ X such that a ≻ c, b ≺ d , andfor all x ∈ X ∞ and every t ≤ T if ( x , . . . , x t − , a, b, x t +2 , . . . ) ∼ ( x , . . . , x t − , c, d, x t +2 , . . . ) , then( x , . . . , x t − , a, b, x t +2 , . . . ) < ( x , . . . , x t − , c, d, x t +2 , . . . ) . Theorem 6 (Semi-hyperbolic discounting) . The preferences < on X ∞ sat-isfy axioms I1, I2 ′′ , I3-I6, I7 ′ , I8 if and only if there exists a SH ( T ) dis-counted expected utility representation for < on X n . If ( u, β , δ ) and ( u ′ , β ′ , δ ′ ) both provide SH ( T ) discounted expected utility representations for < on X n ,then u = Au ′ + B for some A > and some B , and δ = δ ′ , β = β ′ .Proof. The necessity of the axioms is obviously implied by the representation.The proof of sufficiency is analogous to the finite case. Applying Theorem 2and stationarity from period T we get the representation: U ( x ) = w u ( x ) + . . . + w T − u ( x T − ) + w T ∞ X t = T δ t − T u ( x t ) . Next, dividing by w > w t w t − = γ t − > t ≤ T , the representation becomesˆ U ( x ) = u ( x ) + γ u ( x ) + . . . + γ · · · γ T − u ( x T − ) + γ · · · γ T − ∞ X t = T δ t − T u ( x t ) . Using essentilaity of each period and the early bias axiom repeatedly,we demonstrate that γ t − = β t − δ with β t − ∈ (0 ,
1] for all t ≤ T and β ≤ β ≤ . . . ≤ β T − . Therefore,ˆ U ( x ) = u ( x ) + β δu ( x ) + β β δ u ( x ) + . . . + β β · · · β T − δ T − u ( x T − )+ β β · · · β T − ∞ X t = T δ t − u ( x t ) . Finally, to show that δ ∈ (0 , a = ( a, a, . . . ),such that u ( a ) = 0. Then,ˆ U ( a ) = u ( a ) + β δu ( a ) + . . . + β · · · β T − δ T − u ( a ) + β · · · β T − ∞ X t = T δ t − u ( a )= u ( a ) β δ + . . . + β · · · β T − δ T − + β · · · β T − ∞ X t = T δ t − ! . U ( a ) converges, therefore, δ < A number of axiomatizations of exponential and quasi-hyperbolic discountinghave been suggested by different authors. In fact, all the axiomatizations usedifferent assumptions and there is no straightforward transformation fromone type of discounting to another. In this paper we provided an alter-native approach to get a time separable discounted utility representation,showing that Anscombe and Aumann’s result can be exploited as a commonbackground for axiomatizing exponential and quasi-hyperbolic discountingin both finite and infinite time horizons. In addition, we demonstrated thatthe axiomatization of quasi-hyperbolic discounting can be easily extended to SH ( T ).A key distinguishing feature of our set-up is the mixture set structure for X and the use of the mixture independence condition. An essential question,however, is whether mixture independence is normatively compelling in atime preference context, because states are mutually exclusive whereas timeperiods are not. It is worth mentioning that the temporal interpretation ofthe AA framework was also used by Wakai (2008) to axiomatize an entirelydifferent class of preferences, which exhibit a desire to spread bad and goodoutcomes evenly over time.Commonly, the condition of joint independence is used to establish ad-ditive separability in time-preference models. Given A ⊆ T , where T = { , . . . , n } , and x , y ∈ X n , define x A y as follows: ( x A y ) t is x t if t ∈ A and y t otherwise. The preference order < satisfies joint independence if for every A ⊆ T and for every x , x ′ , y , y ′ ∈ X n : x A y < x ′ A y if and only if x A y ′ < x ′ A y ′ . Joint independence is used to obtain an additively separable representa-tion by Debreu (1960), so we will sometimes refer it as a Debreu-type in-dependence condition. It is known that mixture independence implies jointindependence (Grant and Van Zandt, 2009), but whether joint independence(with some other plausible conditions) implies mixture independence is yetto be determined. 19n fact, we are not the first to use a mixture-type independence conditionin the context of time preferences. Wakai (2008) also does so, though he usesthe weaker form of constant independence introduced by Gilboa and Schmeidler(1989).A version of the mixture independence condition can also be formulatedin a Savage environment (Savage, 1954) without objective probabilities, asdiscussed in Gul (1992). Olea and Strzalecki (2014) use precisely this versionof mixture independence in one of their axiomatizations of quasi-hyperbolicdiscounting. For every x, y ∈ X let us write ( x, y ) for ( x, y, y, . . . ) ∈ X ∞ .Let m ( x , y ) denote some c ∈ X satisfying ( x , y ) ∼ ( c, c ). For any streams( x , x ) and ( z , z ) the consumption stream ( m ( x , z ) , m ( x , z )) is calleda subjective mixture of ( x , x ) and ( z , z ). Olea and Strzalecki’s versionof the mixture independence axiom (their Axiom I2) is as follows: for every x , x , y , y , z , z ∈ X if ( x , x ) < ( y , y ), then( m ( x , z ) , m ( x , z )) < ( m ( y , z ) , m ( y , z ))and ( m ( z , x ) , m ( z , x )) < ( m ( z , y ) , m ( z , y )) . In other words, if a consumption stream ( x , x ) is preferred to a stream( y , y ), then subjectively mixing each stream with ( z , z ) does not affectthe preference.In their axiomatization of quasi-hyperbolic discounting Olea and Strza-lecki invoke their mixture independence condition (Axiom 12) as well asDebreu-type independence conditions. The latter are used to obtain a rep-resentation in the form x < y if and only if u ( x ) + ∞ X t =2 δ t − v ( x t ) ≥ u ( y ) + ∞ X t =2 δ t − v ( y t ) , then their Axiom 12 is used to ensure v = βu . Hayashi (2003) and Epstein (1983) considered preferences over lotteriesover consumption streams. In their framework X ∞ is the set of non-stochastic As pointed out above, mixture independence stated for n periods implies joint inde-pendence for n periods. Hence, this raises the obvious question of whether it is possibleto use an n-period version of the subjective mixture independence axiom to obtain a timeseparable discounted utility representation without the need for the Debreu-type indepen-dence conditions. X is required to be a compact connected separa-ble metric space. Denote the set of probability measures on Borel σ -algebradefined on X ∞ as ∆( X ∞ ). It is useful to note that our setting is the re-striction of the Hayashi and Epstein set-up to product measures, i.e., to∆( X ) ∞ ⊂ ∆( X ∞ ). The axiomatization systems by Hayashi and Epstein arebased on the assumptions of expected utility theory. The existence of a con-tinuous and bounded vNM utility index U : ∆( X ∞ ) → R is stated as one ofthe axioms. A set of necessary and sufficient conditions for this is providedby Grandmont (1972), and includes the usual vNM independence conditionon ∆( X ∞ ): for every x , y , z ∈ ∆( X ∞ ) and any α ∈ [0 , x ∼ y implies α x + (1 − α ) z ∼ α y + (1 − α ) z .Obviously, this independence condition is not strong enough to deliverjoint independence of time periods, which is why additional assumptions ofseparability are needed. Two further Debreu-type independence conditionsare required for exponential discounting: • independence of stochastic outcomes in periods { , } from determin-istic outcomes in { , , . . . } , • independence of stochastic outcomes in periods { , , . . . } from deter-ministic outcomes in period { } .To obtain quasi-hyperbolic discounting two additional Debreu-type inde-pendence conditions should be satisfied: • independence of stochastic outcomes in periods { , } from determin-istic outcomes in periods { } and { , . . . } , • independence of stochastic outcomes in periods { , , . . . } from deter-ministic outcomes in periods { , } .It is easy to see that these axioms applied to the non-stochastic consump-tion streams are analogous to the Debreu-type independence conditions usedin Bleichrodt et al. (2008) and Olea and Strzalecki (2014).In summary, to get a discounted utility representation with the discountfunction in either exponential and quasi-hyperbolic form separability mustbe assumed. The mixture independence axiom appears to be a strong as-sumption, however, it gives the desired separability without the need foradditional Debreu-type independence conditions.21 Appendix: Proof of Theorem 2
Proof.
Necessity of the axioms is straightforward to verify. Therefore we willfocus on the proof of sufficiency.
Step 1.
Applying Theorem 1 of Fishburn (1982) to the mixture set X ,it follows from Axioms I1, I3, I4 that there exists a linear utility function u preserving the order on X (unique up to positive affine transformations).Normalize u so that u ( x ) = 0. Note that by non-triviality u ( x ) is in theinterior of the non-degenerate interval u ( X ).Convert streams into their utility vectors by replacing the outcomes ineach period by their utility values. Define the following order: ( v , v , . . . ) < ∗ ( u , u , . . . ) ⇔ there exist x , y ∈ X ∞ such that x < y and u ( x t ) = v t and u ( y t ) = u t for every t . This order is unambiguously defined becauseof the monotonicity assumption, i.e., if x i ∼ x ′ i then ( x , . . . , x i , . . . ) ∼ ( x , . . . , x ′ i , . . . ).The preference order < ∗ inherits the properties of weak order, mixtureindependence and mixture continuity from < . Note that u ( X ) ∞ is a mixtureset under the standard operation of taking convex combinations: if v , u ∈ u ( X ) ∞ then v λ u = λ v + (1 − λ ) u for every λ ∈ (0 , . Therefore, by Theorem 1 of Fishburn (1982) we obtain a linear representation U : u ( X ) ∞ → R , where U is unique up to positive affine transformations.Hence v < ∗ u if and only if U ( v ) ≥ U ( u ). Step 2.
Normalize U so that U (0 , , . . . ) = U ( ) = 0. Since 0 is in theinterior of u ( X ), and since U ( v λ ) = λU ( v ) for any v ∈ R ∞ and for every λ ∈ (0 , U is defined on R ∞ .Mixture linearity of U implies standard linearity of U on R ∞ . To provethis, we need to show that U ( k v ) = kU ( v ) for any k and U ( v + u ) = U ( v ) + U ( u ) for any u , v ∈ R ∞ .As u ( X ) ∞ is a mixture set under the operation of taking convex combi-nations, U ( v k ) = U ( k v + (1 − k ) ) = U ( k v ) = kU ( v ) for any k ∈ (0 , k > U ( v ) = U ( kk v ) = k U ( k v ). Multiplying both parts of this equa-tion by k , we obtain U ( k v ) = kU ( v ) for all k >
1. Therefore, U ( k v ) = kU ( v )for any k >
0. 22o prove that U ( v + u ) = U ( v ) + U ( u ), consider the mixture v u . Bymixture linearity of U we have: U ( v u ) = 12 U ( v ) + 12 U ( u ) = 12 ( U ( v ) + U ( u )) . (6)On the other hand, v u = v + u = ( v + u ). Therefore, U ( v u ) = U (cid:18)
12 ( v + u ) (cid:19) = 12 U ( v + u ) (7)Comparing (6) and (7) we conclude that U ( v + u ) = U ( v ) + U ( u ).Finally, note that U ( ) = U ( v + ( − v )) = U ( v ) + U ( − v ) = 0 , hence U ( − v ) = − U ( v ). Therefore, if k <
0, then U ( k v ) = − kU ( − v ) = kU ( v ).For each T , consider the function f : R T → R defined as follows: f ( v , . . . , v T ) = U ( v , . . . , v T , , , . . . ) . This function is linear on R T and it satisfies f ( ) = 0, therefore, f ( v , . . . , v T ) = T X t =1 w Tt v t , where w T = ( w T , . . . , w TT ). By monotonicity w Tt ≥ t ≤ T .Note that w Tt = U ([1] t ), where [1] t is the vector with 1 in period t and 0elsewhere. It follows that w Tt = w T ′ t for any T and T ′ . Hence there is a vector w ∈ R ∞ such that U ( v , . . . , v T , , , . . . ) = ∞ X t =1 w t v t for any ( v , . . . , v T ) ∈ R T .Recalling that v t = u ( x t ) we obtain U ( u ( x ) , . . . , u ( x T ) , , , . . . ) = T X t =1 w t u ( x t ) for all x ∈ X ∗ . Therefore, for every x , y ∈ X ∗ we have x < y if and only if T X t =1 w t u ( x t ) ≥ T X t =1 w t u ( y t ) .
23y slightly abusing the notation, re-define U so that: U ( x , . . . , x T , x , x , . . . ) = T X t =1 w t u ( x t ) for all x ∈ X ∗ . Hence U ( x ) = ∞ X t =1 w t u ( x t ) represents preferences on X ∗ . Step 3.
Next, we show that U ( x , x , . . . ) converges for any ( x , x , . . . ).Define U T : X ∞ → R as follows: U T ( x ) = T X t =1 u t ( x t ), where u t ( x t ) = w t u ( x t ).Consider the sequence of functions U , U , . . . , U T , . . . According to theCauchy Criterion, a sequence of functions U T ( x ) defined on X ∞ convergeson X ∞ if and only if for any ε > x ∈ X ∞ there exists T ∈ N suchthat | U N ( x ) − U M ( x ) | < ε for any N, M ≥ T .Fix some x ∈ X ∞ and ε >
0. Suppose that for some k it is possibleto choose x + , x − such that [ x + ] k ≻ [ x k ] k ≻ [ x − ] k . By Step 2 the preference[ x + ] k ≻ [ x k ] k ≻ [ x − ] k implies that w k >
0. Therefore, as u is a continuousfunction, it is without loss of generality to assume that u k ( x + ) − u k ( x k ) < ε/ u k ( x k ) − u k ( x − ) < ε/ . It follows that u k ( x + ) − u k ( x − ) < ε, or u k ( x − ) − u k ( x + ) > − ε. By Axiom I6there exist T + and T − satisfying k ≤ min { T − , T + } such that x + k,N < x < x − k,M , for all N ≥ T + , M ≥ T − . Let T ∗ = max { T − , T + } . It is necessary to demonstrate that | U N ( x ) − U M ( x ) | < ε for any N, M ≥ T ∗ . If N = M the result is obviously true.If N = M then it is without loss of generality to assume that N > M . Bythe additive representation: U ( x + k,N ) ≥ U ( x − k,M ) . Expanding u k ( x + ) + N X t =1 ,t = k u t ( x t ) ≥ u k ( x − ) + M X t =1 ,t = k u t ( x t ) .
24y rearranging this inequality N X t = M +1 u t ( x t ) ≥ u k ( x − ) − u k ( x + ) > − ε. As N > M ≥ T ∗ it is also true that U ( x + k,M ) ≥ U ( x − k,N ), hence N X t = M +1 u t ( x t ) ≤ u k ( x + ) − u k ( x − ) < ε. Note that N X t = M +1 u t ( x t ) = U N ( x ) − U M ( x ) . Hence, | U N ( x ) − U M ( x ) | < ε and it follows that U ( x ) converges by the Cauchycriterion.Suppose now that it is not possible to find such k that[ x + ] k ≻ [ x k ] k ≻ [ x − ] k for some x + , x − ∈ X . If w t = 0 for all t then theresult is trivial. Suppose that w t > t . Then for every period t forwhich w t > x t ∈ X e ≡ { z ∈ X : z < z ′ for all z ′ ∈ X or z ′ < z for all z ′ ∈ X } . For some λ ∈ (0 ,
1) replace x t with the mixture x t λx for each t . Call theresulting stream x ∗ . Then U T ( x ) − U T ( x ∗ ) = T X t =1 u t ( x t ) − T X t =1 u t ( x t λx ) = (1 − λ ) T X t =1 u t ( x t ) = (1 − λ ) U T ( x ) . By rearranging this equation it follows that U T ( x ∗ ) = λU T ( x ). By the pre-vious argument U T ( x ∗ ) converges, therefore, U T ( x ) converges. Step 4.
Show that U ( x ) represents the order on X ∞ . Suppose that x < y ,where x , y ∈ X ∞ . If for some k, j it is possible to find x + , y − such that[ x + ] k ≻ [ x k ] k and [ y − ] j ≺ [ y j ] j , then [ x + λx k ] k ≻ [ x k ] k for every λ ∈ (0 , y − µy j ] j ≺ [ y j ] j for every µ ∈ (0 , x ∗ = x + λx k and y ∗ = y − µy j forsome λ, µ ∈ (0 , x ∗ k,N = ( x , . . . , x k − , x ∗ , x k +1 , . . . , x N , x , x , . . . ) , y ∗ j,M = ( y , . . . , y j − , y ∗ , y j +1 , . . . , y M , x , x , . . . ) . Then by Axiom I6, there exist T − , T + such that x ∗ k,N < x < y < y ∗ j,M for all N ≥ T + and for all M ≥ T − . Since x ∗ k,N < y ∗ j,M and U represents < on X ∗ we have: U ( x ∗ k,N ) ≥ U ( y ∗ j,M ) . By Step 3 we know that U ( x , . . . , x k − , x ∗ , x k +1 . . . ) and U ( y , . . . , y j − , y ∗ , y j +1 , . . . ) converge, so U ( x , . . . , x k − , x ∗ , x k +1 , . . . ) ≥ U ( y , . . . , y j − , y ∗ , y j +1 , . . . ) . Recall that x ∗ = x + λx k and y ∗ = y − µy j for some λ ∈ (0 ,
1) and some µ ∈ (0 , λ and µ are arbitrary, it follows that U ( x ) ≥ U ( y ).If it is not possible to find x + , y − such that [ x + ] k ≻ [ x k ] k and [ y − ] j ≺ [ y j ] j ,then either w t = 0 for all t , in which case U ( x ) = U ( y ); or x t < z ′ for all z ′ ∈ X and all t with w t >
0, in which case U ( x ) ≥ U ( y ); or z ′ < y t for all z ′ ∈ X and all t with w t > U ( x ) ≥ U ( y ).It is worth noting that as x < y implies U ( x ) ≥ U ( y ), then, by Axiom I2it follows that w t > t . Therefore, P ∞ t =1 w t >
0. Normalizingby 1 / P ∞ t =1 w t , we can assume that P ∞ t =1 w t = 1.Next, assume that U ( x ) ≥ U ( y ). Suppose that it is possible to find k and x + , x − ∈ X such that x + k,N < x < x − k,N for some fixed N . By mixture con-tinuity, the set { α : x + k,N α x − k,N < x } is closed. By assumption x + k,N < x ,so it follows that α = 1 is included into the set. Analogously, the set { β : x < x + k,N β x − k,N } is closed. In fact, β = 0 belongs to the set, as x < x − k,N .Therefore, as both sets are closed, nonempty and form the unit interval, theirintersection is nonempty. Hence, there exists λ such that x ∼ x + k,N λ x − k,N .Note that x + k,N λ x − k,N = ( x , . . . , x k − , x + λx − , x k +1 . . . , x N , x , x , . . . ). Let x + λx − = x ∗ . Define x ∗ k,N = ( x , . . . , x k − , x ∗ , x k +1 , . . . , x N , x , x , . . . ). There-fore, if there exist periods k, j and outcomes x + , x − , y + , y − ∈ X such that x + k,N < x < x − k,N and y + j,M < y < y − j,M for some N and some M , we can find λ, µ ∈ [0 ,
1] such that x ∼ x ∗ k,N and y ∼ y ∗ j,M = ( y , . . . , y j − , y ∗ , y j +1 , . . . , y M , x , x , . . . ) , y ∗ = y + µy − . We have already shown that if x < y then U ( x ) ≥ U ( y ).From x ∼ x ∗ k,N and y ∼ y ∗ j,M it therefore follows that: U ( x ∗ k,N ) = U ( x ) and U ( y ∗ j,M ) = U ( y ) . Hence, from the assumption U ( x ) ≥ U ( y ) we obtain: U ( x ∗ k,N ) ≥ U ( y ∗ j,M ) . Recall that U is an order-preserving function on X ∗ . Thus, x ∗ k,N < y ∗ j,M .Since x ∼ x ∗ k,N and y ∼ y ∗ j,M , we obtain x < y .Suppose now that there is no such k, j or outcomes x + , x − , y + , y − suchthat x + k,N < x < x − k,N and y + j,M < y < y − j,M for some N and some M .Then, using Axiom I6, we can conclude that either x t ∈ X e for every t with w t > y t ∈ X e for every t with w t >
0. Assume that there is only anupper bound to preferences; i.e., X e ≡ { z ∈ X : z < z ′ for every z ′ ∈ X } .Then U ( x ) ≥ U ( y ) means that x t ∈ X e whenever w t >
0. Therefore, U ( x ) = U ( x ), where x = ( x, x, . . . ) and x ∈ X e . Hence, it follows bymonotonicity that x < y . In the case when there is only a lower bound, i.e., x ∈ X e ≡ { z ∈ X : z ′ < z for every z ′ ∈ X } , the argument is similar.Next, suppose that X is preference bounded above and below, i.e., thereexist x, x ∈ X e with x < x < x for every x ∈ X . Assume that U ( x ) ≥ U ( y ).We need to demonstrate that x < y . By monotonicity and continuity thereexist λ, µ ∈ [0 ,
1] such that x ∼ x λ x and y ∼ x µ x . Since by assumption U ( x ) ≥ U ( y ) and U represents the preference order on constant streams, wehave U ( x λ x ) ≥ U ( x µ x ). By rearranging this inequality ( λ − µ )( U ( x ) − U ( x )),and using U ( x ) > U ( x ) it follows that λ ≥ µ . Therefore, as x ∼ x λ x and y ∼ x µ x and λ ≥ µ , we conclude that x < y .Thus ( w , u ) is an AA representation for < . Step 5.
Uniqueness of w t . Assume that ( w ′ , u ′ ) is another AA representa-tion. Then, for any t we have w t > w ′ t >
0. Consider theset of all constant programs { x ∈ X ∞ : x = ( a, a, . . . ) , where a ∈ X } , whichis a mixture set. Applying ( w ′ , u ′ ) and ( w , u ) to this set we conclude that u ( a ) > u ( b ) if and only if u ′ ( a ) > u ′ ( b ) for every a, b ∈ X . By Theorem 1Fishburn (1982) it implies that u = Au ′ + B for some A > B .Hence, ∞ X t =1 w t u ( x t ) ≥ ∞ X t =1 w t u ( y t ) if and only if ∞ X t =1 w ′ t u ( x t ) ≥ ∞ X t =1 w ′ t u ( y t ) . t, s with t = s and any x ′ , x ′′ ∈ X , let [ x ′ , x ′′ ] t,s denote the streamwith x ′ in the t th position, x ′′ in the s th position and x elsewhere. Fix t, s with w t > w s >
0. Using non-triviality, choose some x + , x − ∈ X suchthat x + ≻ x − . Define x = [ x + , x + ] t,s , y = [ x + , x − ] t,s , z = [ x − , x − ] t,s . Fromthe AA representation it follows that x ≻ y ≻ z . By continuity of the AArepresentation there exists λ ∈ (0 ,
1) such that y ∼ x λ z . Applying the AArepresentation to y ∼ x λ z we obtain w t u ( x + ) + w s u ( x − ) = λ ( w t + w s ) u ( x + ) + (1 − λ )( w t + w s ) u ( x − ) . It follows that (1 − λ ) w t = λw s . Similarly, (1 − λ ) w ′ t = λw ′ s . Therefore, w t /w s = w ′ t /w ′ s . As this is true for any t, s , we obtain that w = C w ′ forsome C > Acknowledgments
I would like to thank my co-supervisor Matthew Ryan for thoughtful advice andsupport. I am grateful to Arkadii Slinko and Simon Grant for helpful discus-sions. I also would like to thank Australian National University for hospitalitywhile working on the paper. Helpful comments from seminar participants at Aus-tralian National University, The University of Auckland and participants at Auck-land University of Technology Mathematical Sciences Symposium (2014), Centrefor Mathematical Social Sciences Summer Workshop (2014), the joint conferences”Logic, Game theory, and Social Choice 8” and ”The 8th Pan-Pacific Conferenceon Game Theory” (2015) Academia Sinica are also gratefully acknowledged. Fi-nancial support from the University of Auckland is gratefully acknowledged.
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