Journal of Mathematical Psychology 47 (2003) 66–74 Theoretical note Technical note on the joint receipt of quantities of a single good Thierry Marchanta,
and R. Duncan Luceb a Ghent University, H. Dunantlaan 1, 9000 Ghent, Belgium b University of California, Irvine, CA, USA Received 28 June 2000; revised 15 March 2002 Abstract The joint receipt of x and y is the fact of receiving them both. If x and y are objects that are valued, their joint receipt is valued as well. Assumming joint receipt is a binary operation that satisfies the conditions of extensive measurement, there is a numerical representation that is additive over joint receipt. We consider the case where x and y are quantities of the same infinitely divisible good. Different sets of assumptions are explored. Invariance with respect to multiplication proves to be interesting. Invariance with respect to addition yields a linear form. A relaxation of the latter yields an approximately linear form. Finally, we consider a noncommutative but bisymmetric joint-receipt operation with a representation arising from preferences over gambles. r 2003 Elsevier Science (USA). All rights reserved. 1. Introduction The joint receipt of x and y is the fact of having both x and y and is written as x"y: Tversky and Kahneman (1992) assumed that, if x; yARþ (the non-negative real numbers) are sums of money, then x"y; the joint receipt of x and y; is equal to x þ y: It can easily be shown (Luce, 2000, Section 4.3.3) that this, together with some other assumptions (the axioms of extensive measurement), implies that there exists a measure V ðxÞ ¼ ax over Rþ and it has strong consequences for the utility of binary gambles. Luce (2000, Section 4.5.2) suggests that it may be wise to consider ‘‘addition’’ rules different from x"y ¼ x þ y: In this paper, we explore different sets of assumptions about "; trying to obtain for V ðxÞ a more flexible form than the linear one. First, we introduce some notations and recall a classical result of extensive measurement. Then, we turn to two different kinds of invariance conditions: displaced multiplicative and translation invariance. The former generalizes a condition that has already been investigated in Luce (2000) and proved to be quite interesting. The latter motivates us to assume, in Section 4, that x"y ¼ x þ y þ r: In Section 5, we relax the assumption that x"y ¼ x þ y þ r by letting the equality be approximately true. In Sections 2–5, " is commutative. In Section 6, we consider a non-commutative but bisymmetric joint-receipt operation with a representation U such that Uðx"yÞ ¼ rUðxÞ þ UðyÞ: This representation arises from preferences over gambles (Aczél, Luce, & Ng, 2003 submitted). The last section is devoted to the proofs. 2. Additive value function Let /D; h; "; eS denote a structure of valued objects, where D is the set of objects, eAD; h is an ordering and " is a binary operation defined for all x and y in D: The interpretations are, respectively, h is a preference order of objects, x"y denotes a composite object consisting of x and y together, and e denotes no change in the status quo.1 The set of gains is defined as Dþ e ¼ fx: xAD; xheg: Similarly, we define the set of the losses as D e ¼ fx: xAD; x"eg: In some cases, D will be assumed to be 1
Corresponding author. Fax: +32-9-264-6487. E-mail addresses: thierry.marchant@rug.ac.be (T. Marchant), rdluce@uci.edu (R.D. Luce). Luce (2002) has reinterpreted the mathematics of Luce (2000) and of Aczél et al. (2003) (submitted) in terms of the psychophysics of signal intensity where the threshold plays the role of the status quo. The latter paper explores the case of non-commutative joint receipt. In the former paper, Luce makes use of Theorem 4. 0022-2496/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-2496(02)00020-2 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 the set of the real numbers (denoted by R) or the set of the rational numbers (denoted by Q). To ensure  the existence of an additive measure over Dþ e ; De or D; some axioms are needed. Let S denote either D; Dþ e or D e : A 1. Closure: The operation " is closed over S: A 2. Weak order: h is a weak order (transitive and connected). For each x; y; z in S; A 3. Weak commutativity: x"yBy"x: A 4. Weak associativity: ðx"yÞ"zBx"ðy"zÞ: A 5. Left weak monotonicity: xhy3x"zhy"z: A 6. Left weak identity: e"xBx: To introduce the next axiom, we need a new notation. Let n be a positive integer and let xðnÞ denote the joint receipt of n copies of x: Formally, xð1Þ ¼ x and xðnÞ ¼ xðn  1Þ"x: A 7. Archimedeanness: For all x; y; w; z; with xgy; there exists an integer n such that 67 When a representation satisfies the conditions (1) and (2) of Proposition 1, we call it an additive representation. Additive representations will be used throughout the paper. 3. Invariance Let the elements of D be quantities of the same infinitely divisible good (for example, idealized money), i.e. DDR: Then addition and multiplication are defined on D: In many places, we will consider h to be the same as X: One exception is Proposition 3. We now turn to two different kinds of invariance conditions: the first one is a very general form of invariance vis-à-vis multiplication, which is itself a generalization of homogeneity; the second one, invariance vis-à-vis addition, is called translation invariance. xðnÞ"whyðnÞ"z: Note that weak commutativity together with left weak monotonicity or left weak identity imply, respectively, the following: A 8. Right weak monotonicity: xhy3z"xhz"y: A 9. Right weak identity: x"eBx:  Definition 1. Let S denote either D; Dþ e or De : Then /S; e; h; "S is a joint-receipt preference structure iff the following conditions hold for all elements in S: closure, weak order, weak commutativity, weak associativity, left (or right) weak monotonicity, left (or right) weak identity. If, in addition, Archimedeanness holds, then we have an Archimedean joint-receipt preference structure. 3.1. Displaced multiplicative invariance A 10. Displaced multiplicative invariance: There exists a function f from Rþ to Rþ such that, for some real numbers r and s and for all x; y; zAD; with z40; ½zðx  rÞ þ r "½zðy  rÞ þ r ¼ f ðzÞ½ðx"yÞ  s þ s: In order to motivate this condition, let us consider several special cases. If r ¼ s ¼ 0; we obtain the classical multiplicative invariance. A 11. Multiplicative invariance: There exists a function f from Rþ to Rþ such that, for all x; y; zAD; with z40; zx"zy ¼ f ðzÞðx"yÞ: In the sequel, Rþ 0 denotes the non-negative real numbers and Rþ ; the positive real numbers. Proposition 1 (Roberts & Luce, 1968). The following are equivalent: (i) /Dþ e ; e; h; "S is an Archimedean joint-receipt preference structure. þ (ii) There is a representation V : Dþ e -R0 such that xhy3V ðxÞXV ðyÞ; ð1Þ V ðx"yÞ ¼ V ðxÞ þ V ðyÞ; ð2Þ V ðeÞ ¼ 0 ð3Þ This representation is unique up to a multiplication by a positive constant. The same holds for D e ; with  V : D -R and for D; with V : D-R: e 0 If r ¼ s ¼ 0 and f ðzÞ ¼ zk ; then " is said to be homogeneous of degree k: When " is homogeneous of degree 1, we are free to change all of the variables by any positive factor. Such numerical factors can either reflect systematic changes of the structure that are captured as its automorphisms or as changes of unit such as going from US$ to Belgian francs. This case has been worked out, with D ¼ R; in Luce (2000, Section 4.5 3). One more condition is needed in order to present our first theorem. A 12. Continuity: "; considered as a function of two variables, is continuous in each variable. Theorem 1. Suppose that /D; X; "S; DDR; satisfies the following conditions: weak order, weak associativity, weak monotonicity, continuity, and displaced multiplicative invariance. If D ¼ R or r; N½; then s ¼ r and f ðzÞ ¼ zk with k ¼ 1 or 2. 68 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 (i) If D ¼ R; then k ¼ 1 and, for some constants b40 and go0; x"y is given by 8 > ½ðx  rÞb þ ðy  rÞb 1=b þ r; > > > > > xXr; yXr; > > > > > b b 1=b > þ r; > ½ðx  rÞ þ 1gðr  yÞ > > > < xXr4y; x"yXr; ð4Þ x"y ¼ b b 1=b > > r  ½gðx  rÞ þ ðr  yÞ ; > > > > > xXr4y; x"yor; > > > > > > r  ½ðr  xÞb þ ðr  yÞb 1=b ; > > > : xpr; ypr: (ii) If D ¼ r; N½; then either k ¼ 1 in which case, for some b40; x"y ¼ ½ðx  rÞb þ ðy  rÞb 1=b þ r; ð5Þ ð6Þ (iii) If (4) or (5) hold, then there is an additive value function defined by ( aðx  rÞb ; xXr; V ðxÞ ¼ ð7Þ a0 ðr  xÞb ; xor; where a40 and a0 ¼ a=g; whereas, if (6) holds, then there is an additive value function defined by V ðxÞ ¼ c ln½Zðx  rÞ ; with c40: The reader will find the proofs of this theorem and all subsequent theorems in the last section. Corollary 1. Suppose that the structure /D; e; X; "S is such that /D; X; "S satisfies the conditions of Theorem 1, and e is a weak identity. If k ¼ 1; then r ¼ e; and if k ¼ 2; then Z ¼ 1=ðe  rÞ: Note that, under the conditions of Theorem 1, r"r ¼ r: If k ¼ 1; this is not surprising because r is then a weak identity but, if k ¼ 2; this is perhaps not what we intuitively expect. It is possible to arrive at a special case of (4) without imposing either continuity or displaced multiplicative invariance but we then need two additional conditions, namely solvability and weak positivity. A 13. Solvability: For all x; y; there exists z such that z"xBy: A 14. Weak positivity: x"xgx; for all xge: 3.2. Translation invariance An alternative assumption to A 10 is A 15. Translation invariance: There exists a function l : D-R such that, for all x; y; z in D; ðz þ xÞ"ðz þ yÞ ¼ lðzÞ þ ðx"yÞ: Theorem 2. Suppose that /D; X; "S satisfies the following conditions: weak order, weak associativity, weak monotonicity, continuity and translation invariance, with D ¼ R or Rþ : Then (i) lðzÞ ¼ bz with b ¼ 1 or 2. (ii) For b ¼ 2; or k ¼ 2; in which case, for some Z40; x"y ¼ Zðx  rÞðy  rÞ þ r: Proposition 2. Suppose that /D; e; X; "S is a weakly positive joint-receipt preference structure (with D ¼ R or Q). If " is solvable and homogeneous of degree 1, then (4) holds and there is an additive representation defined by (7), with r ¼ 0: x"y ¼ x þ y þ r ð8Þ and for b ¼ 1; 1 x"y ¼ ln½ekx þ eky ; ð9Þ k with k40: (iii) If (8) holds, then there is an additive representation given by V ðxÞ ¼ aðx þ rÞ; where a is strictly positive. If (9) holds, then there is an additive representation given by V ðxÞ ¼ lekx ; where l is strictly positive. Corollary 2. Suppose that the structure /D; e; X; "S is such that /D; X; "S satisfies the conditions of Theorem 2 and e is a weak identity. Then, b ¼ 2; r ¼ e and, so, x"y ¼ x þ y  e: ð10Þ It is possible to arrive at (10) without imposing continuity but with a stronger condition than translation invariance, as stated in the next proposition. Here, we do not assume that h is the same as X: Proposition 3. Suppose that /D; e; h; "S satisfies weak identity and D ¼ R or Q or the set of the integers. If translation invariance holds with lðzÞ ¼ 2z; then (10) holds. 4. The case of x"y ¼ x þ y þ r It may be assumed that receiving x"y (x and y) is valued differently from receiving the single amount x þ 69 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 y because there is some intrinsic value attached to the fact of receiving. Receiving x þ y is the receipt of a single thing, whereas receiving x"y is the receipt of two distinct things. A possible formalization of this idea is x"y ¼ x þ y þ r; where r represents the extra value associated with receiving two rather than one formally equal objects. Note that this form for the joint-receipt operator is also motivated by Theorem 2 and Proposition 3, dealing with translation invariance. In this section, we examine the consequences of this hypothesis in various cases and it is assumed that D is the set of the real numbers or the set of the rationals. In the two first results, we apply the condition x"y ¼ x þ y þ r separately to gains and losses because people may process gains differently from losses. Some support for this can be found in Luce (2000). Proposition 4. Suppose that /Dþ e ; e; X; "S is an Archimedean joint-receipt preference structure (with D ¼ R or Q) and x"y ¼ x þ y þ r for all x; y4e: Then r þ eX0 and there is an additive representation V over Dþ e as in Proposition 1 such that V ðxÞ ¼ br þ bx; 8x4e; ð11Þ where b40: It is unique up to a multiplication by a positive constant. If /D e ; e; X; "S is also an Archimedean joint-receipt preference structure and x"y ¼ x þ y  s for all x; yoe; then s  eX0 and V is a representation over D with V ðxÞ ¼ ds þ dx; 8xoe; ð12Þ where d40: It is unique up to multiplication by a positive constant. The constants for transforming V for positive and negative domains are chosen independently. Note that this proposition does not tell us what the value of x"y is when one of them is a gain and the other one a loss. This is due to the fact that we did not require that D be closed under ": The joint receipt of a gain and a loss is not necessarily in D: Nonetheless, it seems natural to explore the consequences of assuming that D is closed. Theorem 3. Let the set D be closed under ": Suppose  that /Dþ e ; e; X; "S and /De ; e; X; "S are two Archimedean joint-receipt preference structures (with D ¼ R or Q). Suppose, in addition, that x"y ¼ x þ y þ r for all x; y4e and x"y ¼ x þ y  s for all x; yoe: Then r ¼ s ¼ e and there is an additive representation V over D as in Proposition 1 such that V ðxÞ ¼ bðx  eÞ; 8x4e; 8xoe; But what happens when we impose some very weak condition linking gains and losses? We explore this issue in the next corollary. Corollary 3. Suppose that /D; e; X; "S is an Archimedean joint-receipt preference structure (with D ¼ R or Q). Suppose, in addition, that x"y ¼ x þ y þ r for all x; y: Then r ¼ e and there is an additive representation V as in Proposition 1 such that V ðxÞ ¼ bðx  eÞ; 8x; ð15Þ where b40: It is unique up to a multiplication by a positive constant. 5. Approximately linear value function In Section 4, assuming x"y ¼ x þ y þ r; we obtained different forms for V but all of them were piecewise linear. A possible way to avoid this linearity might be to relax the assumption x"y ¼ x þ y þ r: We could impose the less demanding condition that, for some E40 (not to be confused with e) and all x; y; jðx"yÞ  ðx þ y þ rÞjpE: Proposition 5. Suppose that /D; e; X; "S is an Archimedean joint-receipt preference structure with D ¼ R and an additive representation V (Proposition 1). Suppose that, for some real r; some E40 and all x; y; jðx"yÞ  ðx þ y þ rÞjpE and " is continuous. Then, maxf0; ax þ s  dgpV ðxÞpax þ s þ d; x4e; ð16Þ ax þ s  dpV ðxÞpminf0; ax þ s þ dg; xoe; ð17Þ where * * * ð2EÞ s ¼ V ð2EÞþV ; 2 a and d ¼ V ð2EÞ  s are strictly positive, je þ rjpE and jae þ sjpd: If E ¼ supx;y jðx"yÞ  ðx þ y þ rÞj then, aEp3d þ s: As shown in Fig. 1, the representation obtained in Proposition 5 is in some sense approximately linear: it is restricted between two parallel straight lines. 6. A non-commutative case ð13Þ and V ðxÞ ¼ dðx  eÞ; where b; d40: It is unique up to multiplication by independent positive constants for the positive and negative domains. ð14Þ In the previous section, " was always commutative; it was one of our assumptions (Sections 2, 4 and 5) or it was a consequence of our assumptions (Section 3). In 70 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 as in (19). If D ¼ R or r; N½; then s ¼ r and f ðzÞ ¼ zk with k ¼ 1 or 1 þ r: Fig. 1. An approximately linear value function. most previous work on joint receipt in utility theory, it is also assumed that " is commutative. This assumption may well be unrealistic. For instance, when we receive mail, we open the envelopes in some order. Receiving a $1000 check from a lottery followed by a $1000 correction on one’s income tax versus the correction followed by the check might produce a different overall reaction. The first one is likely to seem a disappointment whereas the second will give a sense of relief. A theory of non-commutative joint receipts has been worked out in two papers (Aczél et al., 2003 submitted; Ng, Luce, & Aczél, 2002). Under some relatively plausible assumptions concerning not only " but also the utility of gambles composed of joint receipts, they show that the utility U of a joint receipt can have one of the three following forms. Either Uðx"yÞ ¼ UðxÞ þ UðyÞ  dUðxÞUðyÞ; ð18Þ Uðx"yÞ ¼ rUðxÞ þ UðyÞ; r41 ð19Þ r0 41: ð20Þ (i) If D ¼ R; then k ¼ 1 and, for some constants b40 and go0; x"y is given by 8 > ½rðx  rÞb þ ðy  rÞb 1=b þ r; > > > > > > > xXr; yXr; > > > > ½rðx  rÞb þ 1gðr  yÞb 1=b þ r; > > > > < xXr4y; x"yXr; ð21Þ x"y ¼ b b 1=b > > r  r½gðx  rÞ þ ðr  yÞ ; > > > > > xXr4y; x"yor; > > > > > > r  ½rðr  xÞb þ ðr  yÞb 1=b ; > > > : xpr; ypr: (ii) If D ¼ r; N½; then either k ¼ 1 in which case, for some b40; ½ðx"yÞ  r b ¼ rðx  rÞb þ ðy  rÞb ; ð22Þ or k ¼ 1 þ r; in which case, for some Z40; x"y ¼ Zðx  rÞr ðy  rÞ þ r: ð23Þ (iii) If (21) or((22) hold, then aðx  rÞb ; xXr; UðxÞ ¼ ð24Þ a0 ðr  xÞb ; xor; where a40 and a0 ¼ a=g; whereas, if (23) holds, then UðxÞ ¼ c ln½Zðx  rÞ ; with c40: or Uðx"yÞ ¼ UðxÞ þ r0 UðyÞ; Here we use U instead of V to make clear that U is derived from preferences on gambles while V ; in this paper, is derived from preferences on joint receipts. In a given situation, it might be possible to derive a V and a U that are not identical. The first of the three forms (18) is commutative, the other two (19 and 20) are not; they are not associative either but they are bisymmetric, in the following sense. A 16. Bisymmetry: ðx"yÞ"ðz"wÞBðx"zÞ" ðy"wÞ: The first non-commutative form (19) has a left weak identity e while the second one (20) has a right weak identity. In this section, we examine the effect of Displaced Multiplicative Invariance and Translation Invariance when combined with one of the two non-commutative forms (19 and 20). Theorem 4. Suppose that /D; X; "S; DDR; satisfies displaced multiplicative invariance and has a closed, noncommutative, continuous, order-preserving representation Note that the forms obtained for " in Theorem 4 are generalizations of the forms obtained in Theorem 1. But it would be misleading to think that this is due to a weakening of the assumptions. The assumptions in Theorem 1 do not imply those of Theorem 4. The assumptions of Theorem 4 are about " and the utility representation arising from preferences over gambles whereas the assumptions of Theorem 1 are just about " and have nothing to do with gambles. Theorem 5. Suppose that /D; X; "S satisfies translation invariance and has a closed, non-commutative, continuous, order-preserving representation as in (19), with D ¼ R or Rþ : Then (i) lðzÞ ¼ bz with b ¼ 1 or 2. (ii) For b ¼ 2; x"y ¼ rx þ y þ r ð25Þ and for b ¼ 1; 1 x"y ¼ ln½rekx þ eky ; k with k40: ð26Þ 71 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 So, equating these, we see that, (iii) If (8) holds, then UðxÞ ¼ aðx þ rÞ; where a is strictly positive. If (9) holds, then UðxÞ ¼ lekx ; where l is strictly positive. f ðzwÞ ¼ f ðzÞf ðwÞ; ð28Þ z; w40: As F is strictly increasing, so then is f ; and thus the functional equation (28) has as its only solution f ðzÞ ¼ zk ; with k40: Hence, F ½zðx  rÞ þ r; zðy  rÞ þ r The forms obtained for " in Theorem 5 are also generalizations of those obtained in Theorem 2. The comments about the relation of Theorems 1 and 4 to their assumptions apply equally well to the relation between Theorems 2 and 5. If, in Theorems 4 and 5, we use representation (20) instead of (19), the theorems still hold with the obvious needed changes. In Section 5, we combined this relaxation with an additive representation. It might be interesting to relax other conditions, such as (22), (23), (25) or (26). But, currently, we do not know how to solve the functional ‘‘inequations’’ corresponding to these cases. Proof of Theorem 1. Denote x"y ¼ F ðx; yÞ and so displaced multiplicative invariance becomes ð27Þ By (27), F ½zwðx  rÞ þ r; zwðy  rÞ þ r ¼ f ðzwÞ½F ðx; yÞ  s þ s: Now, observe that if we set u ¼ wðx  rÞ þ r; v ¼ w ðy  rÞ þ r; using (27) twice, ¼ f ðzÞf ðwÞ½F ðx; yÞ  s þ s: ð30Þ Eq. (29) becomes ð31Þ for z40: Keeping z fixed, with the notations HðxÞ ¼ V ðzðx  rÞ þ rÞ; JðxÞ ¼ V ðzk ðx  sÞ þ sÞ; a ¼ V ðxÞ and b ¼ V ðyÞ; we obtain J½V 1 ða þ bÞ ¼ H½V 1 ðaÞ þ H½V 1 ðbÞ : The general solution to this Pexider equation is well known to be H½V 1 ðuÞ ¼ Au þ B; J½V 1 ðuÞ ¼ Au þ 2B: H½V 1 ðaÞ ¼ V ðzðx  rÞ þ rÞ ¼ AðzÞV ðxÞ þ BðzÞ; ð32Þ z40 and J½V 1 ðaÞ ¼ V ðzk ðx  sÞ þ sÞ ¼ AðzÞV ðxÞ þ BðzÞ þ B0 ðzÞ; z40: ð33Þ There are now two cases: 8. Proofs ¼ f ðzÞ½F ðwðx  r þ r; wðy  rÞ þ rÞ  s þ s F ðx; yÞ ¼ V 1 ½V ðxÞ þ V ðyÞ : Letting z vary again, we have jðx"yÞ  ðx þ y þ rÞjpE: ¼ F ½zðu  rÞ þ r; zðv  rÞ þ r ¼ f ðzÞ½F ðu; vÞ  s þ s Because F is continuous, strictly increasing and associative, we can apply a well-known result (Aczél, 1966, p. 256) that asserts there exists a continuous and strictly monotonic function V such that ¼ V 1 ½V ðz½x  r þ rÞ þ V ðz½y  r þ rÞ ; We explored different sets of assumptions about " and, with most of them, we ended up with some restrictive forms for ": The case of displaced multiplicative invariance, combined with a commutative or non-commutative representation, leads to very interesting forms (see Theorem 1, Proposition 2 and Theorem 4). The property x"y ¼ x þ y þ r arose as a consequence of Theorem 2. One of its possible relaxations or generalization is F ½zwðx  rÞ þ r; zwðy  rÞ þ r ð29Þ z40: zk ðV 1 ½V ðxÞ þ V ðyÞ  sÞ þ s 7. Conclusion F ½zðx  rÞ þ r; zðy  rÞ þ r ¼ f ðzÞ½F ðx; yÞ  s þ s; z40: ¼ zk ½F ðx; yÞ  s þ s; 1. A is identically 1. Then, V ðzðx  rÞ þ rÞ ¼ V ðxÞ þ BðzÞ; for all z40: Let u ¼ x  r and W ðxÞ ¼ V ðx þ rÞ: We obtain W ðzuÞ ¼ W ðuÞ þ BðzÞ: For positive values of u; this Pexider equation has a unique solution W ðuÞ ¼ c ln u þ d; where c; d are constants with c40; and BðzÞ ¼ c ln z: Therefore, V ðxÞ ¼ c lnðx  rÞ þ d; for x4r: If we replace V in (31), we observe that k ¼ 2 and r ¼ s: Finally, setting Z ¼ ed=c ; F ðx; yÞ ¼ Zðx  rÞðy  rÞ þ r; x; y4r: It is clear that this solution cannot be extended to accommodate values of x and y smaller than r: 2. A is not identically 1. Then, V ðzðx  rÞ þ rÞ ¼ AðzÞV ðxÞ þ BðzÞ; for all z40: Consider any x larger than r and introduce the notations x  r ¼ ep ; z ¼ eq ; W
ðxÞ ¼ V ðex þ rÞ; A
ðxÞ ¼ Aðex Þ and B
ðxÞ ¼ 72 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 Bðex Þ: We obtain



W ðp þ qÞ ¼ A ðqÞW ðpÞ þ B ðqÞ; ð34Þ p; qAR:
Given its definition, A cannot be identically 1 because, by assumption, A is not identically 1. Therefore, the unique solution of (34) is given by W
ðpÞ ¼ aebp þ d; ð35Þ pAR; with a; ba0 (Aczél, 1966, p. 150). Hence, V ðxÞ ¼ aðx  rÞb þ d for all x4r: If we replace V in (31), we observe that d ¼ 0; k ¼ 1 and r ¼ s; i.e. V ðxÞ ¼ aðx  rÞb ; ð36Þ x4r: Similarly, BðzÞ ¼ 0: Let us now extend this solution to values of x and y smaller than r: V being continuous, it is obvious that V ðrÞ ¼ 0: Let us set x ¼ r þ 1 in V ðzðx  rÞ þ rÞ ¼ AðzÞV ðxÞ and we obtain V ðz þ rÞ ¼ aAðzÞ: Hence, V ðzðx  rÞ þ rÞ ¼ V ðz þ rÞV ðxÞ=a: Setting x ¼ r  1; yields V ðz þ rÞ ¼ V ðz þ rÞV ðr  1Þ=a ¼ V ðr  1Þzb ¼ a0 zb ; z40: This is equivalent to b V ðxÞ ¼ a0 ðr  xÞ ; ð37Þ xor; with a0 ¼ V ðr  1Þo0: If x and y are larger than r; then b F ðx; yÞ ¼ ½ðx  rÞ þ ðy  rÞ b 1=b þ r: Proof of Corollary 1. By part (iii) of Theorem 1, we know that if k ¼ 1; then V ðrÞ ¼ aðe  rÞb ¼ 0 ¼ V ðeÞ and so r ¼ e: And if k ¼ 2; then V ðeÞ ¼ 0 ¼ c ln ½Z ðe  rÞ and so 1 ¼ Zðe  rÞ: & Proof of Proposition 2. First, we prove that the structure is Archimedean. By homogeneity of degree 1, zx"zy ¼ zðx"yÞ: Setting z ¼ 1=y and fðzÞ ¼ z"1; we find   x x"y ¼ yf ; ya0: ð38Þ y Let us set fð1Þ ¼ a: Then, ð39Þ xa0: n n Using repetitively (39), we find that xð2 Þ ¼ xa ; xa0: By weak positivity, a41 and xð2n Þaxð2nþ1 Þ; for xa0: But, eð2n Þ ¼ eð2nþ1 Þ: Thus, 0 is the identity. Take any x; y; w; z such that x4y: By solvability, we know that there are b and c such that b"y ¼ x and c"w ¼ z: Because x4y; we see that b4e and so b40: Therefore, there is n such that an b4c: By monotonicity, an b"w4c"w ¼ z: an b"an y"w ¼ an ðb"yÞ"w ¼ an x"w4z"an y: Because, as was shown after (39), xð2n Þ ¼ xan ; we get xð2n Þ"w4yð2n Þ"z and Archimedeanness is proved. Since we have an Archimedean joint-receipt preference structure, we know that there is an additive representation V : By the assumption of homogeneity of degree 1, for z40; V ½zðx"yÞ ¼ V ðzx"zyÞ ¼ V ðzxÞ þ V ðzyÞ: Setting Vz ðxÞ ¼ V ðzxÞ; we see that Vz ðx"yÞ ¼ Vz ðxÞ þ Vz ðyÞ: So, both V and Vz are additive representations of "; whence, for some function y; Vz ðxÞ ¼ yðzÞV ðxÞ ¼ V ðzxÞ: ð40Þ So, for xX0; it is well known that for some a40 and b40; V ðxÞ ¼ axb : For xo0; we conclude from (40), V ðrxÞ ¼ zb V ðxÞ: Set x ¼ 1 The other cases are treated in the same fashion, taking into account the position of x; y and x"y with respect to r: & xð2Þ ¼ x"x ¼ xa; By monotonicity, associativity, commutativity and homogeneity, V ðrÞ ¼ zb V ð1Þ ¼ a0 zb ; where a0 ¼ V ð1Þ40: Note that the function V is equivalent to the function V in the proof or Theorem 1. To complete the proof, simply substitute the expressions for V ðxÞ into V ðx"yÞ ¼ V ðxÞ þ V ðyÞ; taking into account the sign of the three terms. & Proof of Theorem 2. Because the proof is very similar to the proof of Theorem 1 we only sketch it. Applying translation invariance three times, we obtain the Cauchy equation lðw þ zÞ ¼ lðwÞ þ lðzÞ whose solution is lðzÞ ¼ bz; with b40: Then we use expression (30) and we find V 1 ½V ðx þ zÞ þ V ðy þ zÞ ¼ bz þ V 1 ½V ðxÞ þ V ðyÞ : After some change of notations, we arrive at a Pexider equation admitting two solutions, namely (8) and (9). & Proof of Corollary 2. By Theorem 2, we know that one of (8) or (9) holds. Expression (9) is immediately eliminated by weak identity and r ¼ e follows readily. & 73 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 Proof of Proposition 3. Translation invariance, in this particular case, is written as ðz þ xÞ"ðz þ yÞ ¼ 2z þ ðx"yÞ: V ðx þ uÞ  V ðxÞpV ðu  r þ EÞ: ð42Þ Because jðx"yÞ  ðx þ y þ rÞjpE and V is strictly increasing, we know that Setting y ¼ z and f ðxÞ ¼ x"0; we obtain x"y ¼ f ðx  yÞ þ 2y: that is ð41Þ If we replace y by e; we observe that x"e ¼ x ¼ f ðx  eÞ þ 2e: Hence, f ðxÞ ¼ x  e and replacing in (41) yields (10). & Proof of Proposition 4. Consider any x; y4e: By Proposition 1, V ðx"yÞ ¼ V ðxÞ þ V ðyÞ ¼ V ðx þ y þ rÞ: We can rewrite this functional equation as V ½ðx þ rÞ þ ðy þ rÞ  r ¼ V ½ðx þ rÞ  r þ V ½ðy þ rÞ  r : Let W ðxÞ ¼ V ðx  rÞ; x þ r ¼ u and y þ r ¼ v: The functional equation becomes W ðu þ vÞ ¼ W ðuÞ þ W ðvÞ: Because V is strictly increasing, so is W ; and this wellknown Cauchy equation admits only one solution (up to a multiplication by a constant): W ðuÞ ¼ bu: Therefore, V ðx  rÞ ¼ bx and V ðxÞ ¼ bðx þ rÞ for any x4e: To ensure that V is order preserving, b must be strictly positive. Let us consider the restrictions on r: Because V is order-preserving, we know that 0 ¼ V ðeÞp lim sup V ðxÞ ¼ br þ be: x-e This is possible only if bðr þ eÞX0 which, because b40; is equivalent to r þ eX0: To prove the second part of this proposition, we just  need to apply the same reasoning to Dþ e and De ; separately. & Proof of Theorem 3. By Proposition 4, V ðxÞ ¼ ds þ dx for all losses and V ðxÞ ¼ br þ bx for all gains, with d and b positive, r þ eX0 and s  eX0: Suppose now that bðr þ eÞ40: Because V is linear except at e and D is dense, we can choose x and y such that 0 ¼ V ðeÞoV ðxÞ þ V ðyÞobðr þ eÞ: Because V ðx"yÞ ¼ V ðxÞ þ V ðyÞ; we see that eox"y: But also V ðx"yÞ ¼ br þ bðx"yÞobr þ be V ðx"yÞ  V ðx þ yÞpV ðx þ y þ r þ EÞ  V ðx þ yÞ which, using (42) with x þ y for x and r þ E for u; is equivalent to V ðx"yÞ  V ðx þ yÞpV ð2EÞ: Setting now y ¼ u  r  E in x"ypx þ y þ r þ E; we easily follow the same reasoning as above and we come to the conclusion that V ðx"yÞ  V ðx þ yÞXV ð2EÞ: ð2EÞ Let s ¼ V ð2EÞþV and d ¼ V ð2EÞ  s: Then, 2 jV ðx"yÞ  V ðx þ yÞ  sjpd: Let GðuÞ ¼ V ðuÞ  s: Then, jGðxÞ þ GðyÞ  Gðx þ yÞjpd and it can be proved (Hyers, Isac, & Rassias, 1998, p. 13) that jGðxÞ  axjpd or, equivalently, jV ðxÞ  ðax þ sÞjpd: As V is strictly increasing and V ðeÞ ¼ 0; we easily obtain (16) and (17) with the condition on e: Let us consider the following absolute value: je þ rj ¼ jx  x  e  rj ¼ jðx"eÞ  ðx þ e þ rÞjpE: Because jV ðx"yÞ  V ðx þ yÞ  sjpd; using (16) and (17), we find that jðx"yÞ  ðx þ yÞj is necessarily smaller than ð3d þ sÞ=a: If E ¼ supx;y jðx"yÞ  ðx þ y þ rÞj; then we find that Epð3d þ sÞ=a: & Proof of Theorem 4. The proof is very similar to that of Theorem 1. The main difference lies at the beginning. Instead of using continuity, monotonicity and associativity to derive the existence of an additive representation (30), we directly use the non-commutative representation. That is, combining (29) and (19), we obtain zk ðU 1 ½rUðxÞ þ UðyÞ  sÞ þ s ¼ U 1 ½rUðz½x  r þ rÞ þ Uðz½y  r þ rÞ and we go on as in Theorem 1. & Proof of Theorem 5. The proof is very similar to those of Theorems 2 and 4. & and so, because b40; x"yoe; which is a contradiction. So r ¼ e: The same reasoning applies to dðs  eÞ: Thus, r ¼ s ¼ e: & Acknowledgments Proof of Corollary 3. We just have to solve the functional equation V ðxÞ þ V ðyÞ ¼ V ðx þ y þ rÞ on D; as in Proposition 4. & Proof of Proposition 5. Setting y ¼ u  r þ E in x"yXx þ y þ r  E; we have x"ðu  r þ EÞXx þ u: Because V is additive, V ðxÞ þ V ðu  r þ EÞXV ðx þ uÞ; The first author started working on this topic when visiting at UCI, thanks to grants from Université Libre de Bruxelles and NATO. The second author received partial support from US National Science Foundation Grant SBR-9808057 to the University of California, Irvine. The authors wish to thank J. Aczél for several 74 T. Marchant, R.D. Luce / Journal of Mathematical Psychology 47 (2003) 66–74 helpful suggestions and appreciate th comments of A.A.J. Marley on an earlier version of the paper. We are also indebted to an anonymous referee for his or her constructive remarks. References Aczél, J. (1966). Lectures on functional equations and their applications. New York: Academic Press. Aczél, J., Luce, R. D., & Ng, C. T. (2003). Functional equations arising in a theory of rank dependence and homogeneous joint receipt. Journal of Mathematical Psychology, in press. 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