The Simple Theory of Temptation and Self-Control
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1 The Simple Theory of Temptation and Self-Control Faruk Gul and Wolfgang Pesendorfer Princeton University January 2006 Abstract We analyze a two period model of temptation in a finite choice setting. We formalize the idea that temptation depends only on the most tempting alternatives and provide two representations of such preferences. The representation is an ordinal analogue of the self-control preferences in Gul and Pesendorfer (2001). This research was supported by grants SES , SES , SES and SES from the National Science Foundation.
2 1. Introduction In this paper, we analyze a two period model of temptation. In contrast to earlier work which considers preferences over lotteries, we assume an arbitrary finite set of alternatives. Our goal is to formulate a simple model of temptation and self-control for deterministic choice settings. Following Kreps (1979), we analyze preferences over sets of alternatives. The agent must take an action in period 0 that constrains the feasible choices in period 1. action corresponds to a period 0 choice over sets of alternatives. Hence, behavior in period 0 is described by a preference relation over sets of alternatives while behavior in period 1 corresponds to a choice from the given set of alternatives. This We consider individuals who may benefit from commitment, that is, adding an alternative to their choice set may make agents strictly worse off. In our model, this occurs if the added alternative is a temptation that alters choice behavior or requires costly self-control. We say that x is more tempting than y if (and only if) there exists some A such that y A and A A {x}. This definition assumes that only the most tempting alternatives in a set can reduce the welfare of the agent. We assume that more tempting is an acyclic relation, that is, if x n is more tempting than x n+1 for n =1,...,N then x N+1 is not more tempting than x 1. Acyclicity is equivalent to the assumption that there are maximally tempting alternatives in every set. Hence, our model of temptation can be paraphrased as lesser temptations can t hurt. Suppose adding x to a set A makes the agent strictly better off. We interpret this to mean that x is the unique optimal choice (in period 1) from the set A {x}. This interpretation assumes that the agent has no preference for flexibility as analyzed in Kreps (1979) and Dekel, Lipman and Rustichini (2001). 1 We say that x is a better choice than y if A {x} A for some A that contains y and assume that better choice is an acyclic binary relation. Hence, our model of period 1 choice can be paraphrased as lesser options can t help. We refer to the assumption that better choice and more tempting are acyclic relations as Acyclic Choice and Temptation (ACT). 1 Kreps (1979) and Dekel, Lipman and Rustichini (2001) analyze models where agents are uncertain in period 0 about their period 1 preference. In that case, A {x} A does not imply that x is the only possible choice in period 1. 1
3 Let be a complete and transitive preference relation over the set of all nonempty subsets of some nonempty set X. ACT yields the following representation: there are utility functions w, v and an aggregator u such that the function W defined by W (A) =u(max w(x), max v(y)) x A y A represents the preference. The aggregator u is non-decreasing in the first argument and non-increasing in the second argument. We call this representation a Temptation and Self- Control Utility (TSU). The function v weakly represents temptation: if x is more tempting that y then v(x) >v(y). The function w weakly represents choice: if x is a better choice than y then w(x) >w(y). The preference has self-control at C if there are A, B with B C = A such that A A B B (SC) Note that (SC) implies A contains a better choice than any element in B and that B contains a more tempting alternative than any element in A. Hence, (SC) implies that the agent does not choose a most tempting alternative. Let be a TSU preference (u, v, w) with self-control at A. Then, x arg max z A w(z),y arg max z A v(z) implies w(x) > w(y) and v(x) >v(y). The triple (u, v, w) isastrict TSU if u is strictly increasing in its first argument and strictly decreasing in its second argument. To obtain a strict TSU representation we add a regularity assumption. It requires that when two alternatives are equally tempting they must also be equally good choices. Similarly, when two alternatives are equally good choices they must be equally tempting. In other words, ties (in terms of choice or temptation) can only occur if two alternatives are perfect substitutes. Let be a strict TSU preference (u, v, w). Then has self-control at A if and only if w(x) >w(y) and v(y) >v(x) for x arg max z A w(z),y arg max z A v(z). Hence, for a strict TSU preference, the functions w and v identify instances of self-control. By contrast, for a (weak) TSU representation w(x) >w(y),v(y) >v(x) is necessary (but not sufficient) for self control to occur at A. 2
4 For the setting with lotteries, Gul and Pesendorfer (2001) obtain the representation W (A) = max(u(x)+v(x)) max v(y) x A y A Letting w(x) =U(x) +V (x) and v(x) =V (x) makes it clear that the representation here generalizes the earlier representation. In particular, the representation derived in this paper allows for a non-additive aggregator u. In Gul and Pesendorfer (2001), we assume that the objects are lotteries and that the agent is an expected utility maximizer. The main assumption in that paper is Set Betweenness: A B implies A A B B (SB) The expected utility hypothesis together with Set Betweenness are shown to imply the representation above. Theorems 3 establishes that the assumption of acyclic choice and temptation is stronger than Set Betweenness. Hence, Set Betweenness does not yield a TSU representation without the expected utility hypothesis. Strotz (1956) analyzes a model of consistent planning where tastes are changing over time. In Strotz s model, the agent anticipates his future behavior and chooses his current action so as to maximize the utility of the choice he will make in the future. This utility is evaluated with his current utility function which may be different than the utility function governing his behavior in the future. In our two-period framework, Strotz s model of consistent planning amounts to the following representation: there exists functions W, U, V such that W represents the individual s period 0 preferences and W (A) = max x M V (A) U(x) where M V (A) denotes the set of maximizers of V in A. Hence, the agent in period 0 understands that in period 1 he will make his choice, x A, according to V. His utility for this choice is U(x), and therefore his utility of choosing the set A is W (A). In Gul and Pesendorfer (2005), we show the preference relation over sets has a Strotz representation if and only if satisfies the No Compromise axiom below. 3
5 A A B or B A B (NC) for all A, B. Theorem 4 establishes that No Compromise is stronger than acyclic choice and temptation and therefore, the model analyzed here includes Strotz model as a special case. Dekel, Lipman and Rustichini (2005) analyze a model of temptation that generalizes Gul and Pesendorfer (2001) to allow for the possibility that the agent is simultaneously tempted by more than one alternative. Like Gul and Pesendorfer (2001), they consider lotteries and assume that the agent is an expected utility maximizer. By contrast, in this paper we give up the expected utility hypothesis but strengthen the model of temptation to obtain an ordinal analogue of our earlier representation. 2. Revealed Choice and Temptation Let X denote the finite nonempty set of alternatives and K denote the set of all nonempty subsets of X. The individual is identified with a preference relation on K. That is, is complete and transitive. The preference represents the individual s ranking of choice problems (in period 0) with the understanding that (in period 1) one alternative from the set must be chosen for consumption. We model a decision-maker who must deal with temptation. This means that adding an alternative to a choice problem may make the agent strictly worse off. If A A {x} we conclude that x is more tempting than any y A. If adding an alternative makes the agent better off, that is, A {x} A, we conclude that x will be chosen from A {x} and that x is a better choice than any y A. Definition: (i) The element x X is a better choice than y X (x c y) if there exists A Ksuch that y A and A {x} A. (ii) The element x X is more tempting than y X (x t y) if there exists A Ksuch that y A and A A {x}. Axiom ACT below requires that the binary relations c and t be acyclic. A binary relation on X acyclic if for any x 1,...,x n, x i x i+1 for i =1,...,n 1 implies x n x 1. 4
6 Axiom ACT: (Acyclic Choice and Temptation) Both c and t are acyclic. We say that W : K IR is a temptation-self-control utility (TSU) if there exists (u, w, v) such that w : X IR, v : X IR and u : w(x) v(x) IR, where u is non-decreasing in its first argument, non-increasing in its second, and W (A) =u(max w(x), max v(y)) x A y A We say that is a TS preference if only if it can be represented by a TSU. When convenient, we identify a TSU with the corresponding (u, w, v) and write W =(u, w, v). We sometimes write max w(a), max v(a) rather than max x A w(x), max y A v(y). Theorem 1: The preference relation satisfies ACT if and only if it is a TS preference. Let (u, w, v) be a TSU. The representation implies that w(x) >w(y) whenever x c y. Hence, the utility function w can be interpreted as a possible period 1 objective function. In particular, any maximizer of w can be interpreted as a possible choice in period 1. Similarly, we can interpret v as a possible representation of the temptation ranking; since x t y implies v(x) >v(y), any maximizer of v must be a most tempting alternative. On the other hand, it may be the case that w(x) > w(y), even though x c y. Hence, the function w associated with a particular TSU representation does not capture all possible optimal choices but rather a selection of optimal choices. In the next section, we use the criterion lesser options cannot help, to identify all possible choices from each set. Similarly, we identify all of the candidates for most tempting alternative in each option set. We use these constructions to provide an alternative characterization of TS preferences and a stronger axiom that yields a strictly monotone TSU representation. The proof of Theorem 1 relies on the following two Lemmas. Lemma 1 is a standard result (see Adams (1964)). For completeness, we provide a proof of the Lemmas in the appendix. Lemma 1: (ii) If is an acyclic binary relation on X, then there exists a function f : X IR such that x y implies f(x) >f(y). 5
7 Lemma 2: Let N = {1,...,n},M = {1,...,m}. Let D M N and let f : D IR be strictly increasing (non-decreasing). Then, there exists a strictly increasing (nondecreasing) function F : M N IR with f(i, j) =F (i, j) for (i, j) D. Proof of Theorem 1: To prove that the axioms are necessary for the representation, assume that such u, w, v exist. First, we show that x c y implies w(x) >w(y). To see this, note that if w(y) w(x) then for all A such that y A, max w(a {x}) = max w(a) and max v(a {x}) max v(a). Therefore, since u is non-increasing in its second argument, we have u(max w(a {x}), max v(a {x})) u(max w(a), max v(a)) for all A such that y A. Hence, x c y. Now, suppose for some x 1,...,x n we have x i c x i+1 for all i =1,...,n 1. Then, the above argument ensures that w(x i ) >w(x i+1 ) for all i =1,...n 1 and therefore w(x 1 ) >w(x n ), which again by the above argument ensures that x n c x 1. The proof of the acyclicity of t follows from a symmetric argument. Next, we prove that the axioms imply the representation. Define w : X IR, v : X IR such that x c y implies w(x) >w(y) and x c y implies w(x) >w(y). Lemma 1 implies that such functions w, v exist. Without loss of generality, assume that w(x) = {1,...,n},v(X) ={ m,..., 1}. Claim 1: A {x} {y} if x, y A such that w(x) = max w(a) and v(y) = max v(a). To prove claim 1, let B = {x} {y} for x, y A such that w(x) = max w(a) and v(y) = max v(a). If B = A there is nothing to prove. Otherwise, note that by our choice of x and y, we have z c x and z t y for all z {z 1,...,z n } = A\B. Hence, B B {z 1 }. Continuing in this fashion we get B {z 1 } B {z 1,z 2 } etc. Therefore A B. Claim 2: A B if max w(a) max w(b) and max v(a) max v(b). To prove claim 2, let x A be such that w(x) = max w(a) and let y B be such that v(y) = max v(b). Since w(x) w(y) it follows that y c x and hence A A {y} and since v(x) v(y) it follows that x t y and hence B {x} B. By Claim 1, we have A {y} B {x} {x} {y} and therefore A A {y} B {x} B which proves Claim 2. 6
8 Let g : K M N be defined as g(a) = (max w(a), max v(a)) and let D := g(k) M N be the set of values attained by g. Let W : K IR represent the preference. Claim 2 implies that W (A) =W (B) if max w(a) = max w(b) and max v(a) = max v(b). Therefore, we can define f : D IR by f(max w(a), max v(a)) = W (A) Note that f is non-decreasing by Claim 2 above and therefore Lemma 2 implies that we can extend f to a non-decreasing function F on M N. Define u : M N IR by u(i, j) :=F (i, j) and note that W (A) =f(i, j) =u(i, j) for (max w(a), max v(a)) = (i, j). Hence, (u, w, v) is a TSU representation. Typically, there are multiple, ordinally not equivalent, representations (u, w, v) for a single preference. Theorem 2 illustrates the extent of the non-uniqueness: any pair of utility functions w, v such that w represents c and v represents t (in the sense of Lemma 1) is part of a representation for some aggregator u. Moreover, if (u, w, v) is a TSU representation of then x c y implies w(x) >w(y) and x t y implies v(x) >v(y). Theorem 2: Let a preference relation and w, v : X IR. There exists u : w(x) v(x) IR such that the TSU (u, w, v) represents if and only if x c y implies w(x) > w(y) and y t x implies v(y) >v(x). Proof: To see necessity, note that x c y implies W (A {x}) > W(A) for some A containing y. This in turn implies w(x) >w(y) for any TSU representing. Similarly, x t y implies W (A) >W(A {x}) for some A containing y and therefore v(x) >v(y). To prove sufficiency, note that the only role of ACT in the proof of the representation was to yield w, v such that x c y implies w(x) >w(y) and y t x implies v(y) >v(v). Hence, starting with any w, v that has these properties, the arguments in the proof of Theorem 1 yield the desired u and establish that (u, w, v) is a TSU that represents. 7
9 3. A Strict Representation A TSU representation (u, v, w) isstrict if u is strictly increasing in its first argument and strictly decreasing in its second argument. To obtain a strict representation, we will strengthen our axioms to rule out accidental ties. We say that x and y are substitutes if x and y have the same effect on some set A: let A X with A = or A A {x}. Then x and y are substitutes if {x} A {y} A. We say that {x}, {y} are perfect substitutes if {x} A {y} A for all A X. Definition: The preference relation is regular if x, y are substitutes implies x, y are perfect substitutes. In a standard setting, where the agent maximizes a complete and transitive preference R on X, regularity is always satisfied. In that case, x and y are substitutes if and only if xry and yrx. Obviously this implies that x and y are also perfect substitutes. In our setting, regularity may be violated if, for example, x and y are equally good choices but x is more tempting than y. (In that case, we may have {x, z} {y, z} if z is more tempting than x but {w, y} {w, x} if w is a better choice than x). Regularity can be interpreted as the following two requirements. (i) The alternatives x and y are equally good choices if and only if x and y are equally tempting. (ii) If the agent is indifferent between commitment to x and commitment to y then x and y are equally good choices (and equally tempting). In other words, indifference occurs only if two alternatives are the same in every relevant dimension. We call a TSU representation generic if it satisfies this property. Definition: The strict TSU representation (u, w, v) is generic if u is one to one and if [w(x) =w(y) if and only if v(x) =v(y)]. For a regular preference, we can strengthen ACT. Consider, for example, the situation A {x} A {x}. In this case, it is clear that x is more tempting than y for all y A. However, when is regular, we can make a stronger inference: since x is the most tempting alternative in A {x}, if it were not an optimal choice from A {x}, we would have 8
10 A {x} {x}. Hence, x A {x} implies that x is a possible optimal choice. Moreover, regularity and the fact that x is the unique most tempting alternative implies that x must also be the unique optimal choice. Similarly, consider the case {x} A {x} A. Since A {x} A, x is the unique optimal choice from A {x}. Then, {x} A {x} implies that x is also a most tempting alternative from that set. The, finally, regularity ensures that x is the unique most tempting and best choice in A {x}. The following definition modifies our definition of more tempting and better choice to take account of regularity. Definition: (i) For x X, let x rc y if [x c y or A A {x} {x} for some A K containing y]. (ii) For x X, let x rt y if [x t y or {x} A {x} A for some A K containing y]. Axiom ACT : Both rc and rt are acyclic. Theorem 3: The preference is regular and satisfies ACT if and only if it has a generic strict TSU representation. Proof: See Appendix. For a generic, strict TSU representation (u, v, w), the function v represents temptation and w represents choice. That is, x rc y if and only if w(x) >w(y) and x rt y if and only if v(x) >v(y). Theorem 4: Let (u, v, w) be a generic, strict TSU. Then, w represents rc and v represents rt. Proof: Let w(x) >w(y). If v(y) >v(x), then {x, y} {x} and therefore x rc y. If v(x) >v(y), then {x} {x, y} and since genericity implies {x} {y}, it follows that x rc y. Conversely, let x rc y. Then, {x} {y} by regularity. If {x, y} {y}, then w(x) >w(y). If {y} {x, y} {x}, then v(x) >v(y) and w(x) w(y). By genericity this implies w(x) >w(y). An implication of Theorem 4 is that rc and rt are strict parts of preference relations. Hence, the multiplicity found in the weak representation (Theorem 2) is not present in the strict representation. 9
11 4. ACT, Set Betweenness, and No Compromise In this section, we relate ACT to two alternative temptation axioms that exist in the literature. In Gul and Pesendorfer (2001) we analyze preferences over sets of lotteries and provide axioms that yield the following representation: W (A) = max U(x)+V (x) max V (y) x A y A We refer to preferences that have such a representation as self-control preferences. Note that self-control preferences are a special case of TS preferences. To see this, set w = U +V, v = V and u = w v. Hence, TS preferences generalizes our earlier model by allowing for a non-additive aggregator u of choice utility w, and temptation utility v. In Gul and Pesendorfer (2001), we showed that the above representation obtains if preferences satisfy continuity, a version of the independence axiom, and the Set Betweenness axiom below. Axiom SB: (Set Betweenness) The preference relation satisfies Set Betweenness if A B implies A A B B for all A, B K. Without the linearity provided by lotteries and the independence axiom, Set Betweenness does not yield a representation that can be interpreted as a temptation model. More precisely, Theorem 4 below shows that Set Betweenness is too weak to yield a TSU representation. Hence, preferences that satisfy Set Betweenness may yield cyclic c or t. Theorem 5: ACT implies but is not implied by Set Betweenness. Proof: By Theorem 1, to prove that ACT implies Set Betweennes it is enough to show that a TSU representation implies Set Betweenness. Let (u, w, v) be a TSU representation of and suppose A B B. Then, max w(a B) > max w(b) and max w(a) = max w(a B). Hence, A A B. Similarly, A A B implies max v(a B) > max v(a) and max v(b) = max v(a B) and hence A B B by the representation. The following example shows that Set Betweenness does not imply ACT: Let X = {x, y, z} and {x} {x, y} {y} {y, z} X {x, z} {z}. Verifying that satisfies Set Betweenness is straightforward. Note that x c y since {x, y} c {y}. But {x, y, z} {x, z} and therefore y c x. Hence, c has a cycle. 10
12 Recall that a Strotz representation obtains if there are functions U, V such that the function W defined by W (A) = max U(x) x M V (A) represents, where M V (A) ={x A V (x) V (y) y A}. The utility function W describes a decision-maker who chooses (in period 1) to maximize V but evaluates these choices according to U. Ties are broken in favor of U. The above describes Strotz model of consistent planning with changing utility functions. In Gul and Pesendorfer (2005), we show that has a Strotz representation if it satisfies the following axiom: Axiom NC: (No Compromise) A A B or B A B for all A, B K. Theorem 6 below shows ACT is weaker than No Compromise. Hence, TS preferences include the Strotz model. Theorem 6: No Compromise implies but is not implied by ACT. Proof: In Gul and Pesendorfer (2005), we show that No Compromise is equivalent to the existence of U,V such that the function W defined by W (A) = max U (x) x M V (A) represents, where M V (A) ={x A V (x) V (y) y A}. Define w = U + kv, v = V and W (A) = max w(x) max kv(x) x A x A for some k large enough so that v(x) v(y) > 0 implies k(v(x) v(y)) >w(y) w(x) for all x, y X. Then, it is clear that W (A) = W (A) for all A K. Hence, No Compromise implies the existence of a TSU representation. By Theorem 1, we conclude that No Compromise implies ACT. To see that the converse does not hold, consider X = {x, y} and such that {x} X {y}. 11
13 5. Conclusion In this paper, we provide a model of temptation and self-control for a simple class of finite choice problems. In contrast to earlier work (Gul and Pesendorfer (2001), Dekel, Lipman and Rustichini (2005)), we consider an ordinal setting and therefore do not impose the expected utility hypothesis. We show how choice and temptation can be identified from an agent s ranking of sets of alternatives. Our key axiom requires that choice and temptation are acyclic relations. This axiom strengthens Set-Betweenness the key axiom in the setting with lotteries but is weaker than No Compromise the axiom that identifies preferences that correspond to Strotz model of changing tastes. 6. Appendix Lemma 1: If is an acyclic binary relation on X, then there exists a function f : X IR such that x y implies f(x) >f(y). Proof: Let F (A) ={x A y A implies y x}. Since A is finite and nonempty, acyclicity of ensures that F (A) is nomempty. Then, set X 0 = X and Y 1 = F (X) and X 1 = X 0 \Y 1. Define inductively, Y i+1 = F (X i ) and X i+1 = X i \Y i+1 for all i such that X i. Let k = max{i X i }. It is easy to verify that Y 1,...,Y k+1 is a partition of X. Let f(x) = i for i such that x Y i. Clearly, if x y, then x X i implies y X i. Hence, x y implies f(x) >f(y) as desired. Lemma 2: Let M = {1,...,m} and N = {1,...,n}. Let D M N and let f : D IR be strictly increasing (non-decreasing). Then, there exists a strictly increasing (non-decreasing) function F : M N IR with f(i, j) =F (i, j) for (i, j) D. Proof: We first prove the Lemma for the strictly increasing case. If f takes on a single value, define ɛ = 1. Otherwise, let ɛ be the minimal non-zero difference between two values of f. Let f be the maximal value of f. For (i, j) M N, let D ij = {(i,j ) D i i, j j} and for ij such that D ij, let I ij = min{i (i,j ) D ij }, J ij = min{j (I ij,j ) D ij }, κ ij =(I ij,j ij ). Similarly, let Jij = min{j (i,j ) D ij }, Iij = min{i (i,jij ) D ij}, and τ ij =(Iij,J ij ). 12
14 For (i, j) M N define, f + ɛi/m G(i, j) = f(i, j) f(κ ij ) ɛ + ɛi/m if D ij = if (i, j) D otherwise and let Finally, define f + ɛj/n H(i, j) = f(i, j) f(τ ij ) ɛ + ɛj/n if D ij = if (i, j) D otherwise F (i, j) = 1 / 2 G(i, j)+ 1 / 2 H(i, j) We will show that G is strictly increasing in its first argument and weakly increasing in its second argument. A symmetric argument for H then proves the lemma. Let (i, j), (k, j) M N with k>i. Note that D kj = implies that G(k, j) = f + ɛk/m > f + ɛi/m G(i, j). Since D kj D ij it follows that we are done if either D kj or D ij are empty. Therefore, assume that D ij and D kj are non-empty. In that case, κ ij and κ kj are well defined. If κ kj >κ ij, then f(κ kj ) f(κ ij )+ɛ and therefore G(k, j) f(κ kj ) ɛ+ɛk/m > f(κ ij ) G(i, j). If κ kj = κ ij, then G(i, j) =f(κ ij ) ɛ+i/m and since G(k, j) f(κ ij ) ɛ + k/m, wehaveg(k, j) >G(i, j). To prove that F is non-decreasing in its second argument, let (i, j), (i, k) M N with k>j. Note that D ik D ij and if either D ik or D ij the result follows from the argument above. Otherwise, κ ik κ ij. If κ ij = κ ik, then G(i, k) =G(i, j). If κ ik >κ ij then G(i, k) f(κ ik ) ɛ +1/m > f(κ ij ) G(i, j). If f is non-decreasing then define f F = f(i, j) f(κ ij ) if D ij = if (i, j) D ij otherwise The function F is non-decreasing since D ij D i j for i i,j j. Proof of Theorem 3: First, we prove the only if part. By Lemma 1, we can define w, v so that x rc y implies w(x) >w(y) and x rt y implies v(x) >v(y). Note that the construction in the proof of Lemma 1 ensures that if x, y are perfect substitutes then 13
15 w(x) =w(y) and v(x) =v(y). By Theorem 2, there exists u such that (u, w, v) isatsu representation. Next, we argue that the restriction of u to the set D = {(max w(a), max v(a)) A K} is strictly increasing in its first argument and strictly decreasing in its second argument. That is, if max w(a) max w(b), max v(b) max v(a) and (max w(a), max v(a)) (max w(b), max v(b)), then u(max w(a), max v(a)) >u(max w(a), max v(a)). Since (u, w, v) is a TSU, u(max w(a), max v(a)) u(max w(a), max v(a)), and hence it is enough to show that u(max w(a), max v(a)) u(max w(a), max v(a)). Consider first the case where x arg max w(a) arg max v(a), z arg max w(b) arg max v(b). If {x} {z}, then regularity implies that x, z are perfect substitues. Hence w(x) = w(z), v(x) = v(z), contradicting (max w(a), max v(a)) (max w(b), max v(b)). Next, let arg max w(a) arg max v(a) = and z arg max w(b) arg max v(b). Hence, w(x) >w(y) and v(x) <v(y) for x arg arg max w(a) and y arg max v(a). The argument in the previous paragraph ensures that {x} {y}. Also, we have y rc x, x rc y and therefore {x} {x, y} {y}. If {x, y} {z}, then w(x) w(z) implies {x, z} {z} {x, y, z}. Hence y rt z contradicting the assumption that v(z) v(y). Next, let x arg max w(a) arg max v(a) and arg max w(b) arg max v(b) =. Arguing as in the previous paragraph, we get w(x ) >w(z) and v(z ) <v(y) for x = arg max w(b) and z = arg max v(a) and {x } {x,z} {z}. Ifwehave{x} {x,z}, then v(z) v(x) implies {x, x,z} {x} {x, z}. Hence x rc x contradicting the assumption that w(x) v(x ). Finally, let arg max w(a) arg max v(a) = = arg max w(b) arg max v(b). Hence, {x} {x, y} {y}, {x } {x,z} {z} for x = arg max w(b), y = arg max v(a), x = arg max w(b), and z = arg max v(b). Suppose {x, y} {x,z}. Then, since (u, w, v) is a TSU, we have W ({x,y,z}) =u(w(x ),v(z)) = W ({x,z}) =W ({x, y}) = u(w(x),v(y)) u(w(x ),v(z)) = W ({x,y,z}). Hence, {x, y, z} {x,y,z} which implies that x, x are perfect substitutes. A symmetric argument ensures that y, z are perfect substitutes. Hence, w(x) =w(x ), v(y) =v(z), a contradiction. Hence, we have established that u is strictly increasing in its first argument and strictly decreasing in its second on D. Without loss of generality, we can assume that 14
16 D N M where N = {1,...,n} and M = { m,..., 1}. Let D = {(α, β) (α, β) D } and define f : D IR as follows: f(α, β) = u(α, β). Hence, f is a strictly increasing function on D. By Lemma 2, we have a F : N M IR so that F is also a strictly increasing and agrees with f on D. Now define u (α, β) =F (α, β) to obtain (u,w,v) as the desired strict representation. To see that this representation is generic, suppose w(x) =w(x ). Then, we must have {x} {x } so that x, x are substitutes. By regularity, they must be perfect substitutes. Hence, v(x) =v(x ). That v(x) =v(x ) implies w(x) =w(x ) follows from a symmetric argument. The if direction is straightforward. 15
17 References 1. Adams, E. W. (1965): Elements of a Theory of Inexact Measurement, Philosophy of Sciences, 32, Bridges, D. S. (1983): Numerical Representation of Intransitive Preferences on a Countable Set, Journal of Economic Theory, 30, Dekel, E., B. Lipman and A. Rustichini (2001): A Unique Subjective State Space for Unforeseen Contingencies, Econometrica, 69, Dekel, E., B. Lipman and A. Rustichini (2005): Temptation-Driven Preferences, mimeo. 5. Gul, F. and W. Pesendorfer, (2001): Temptation and Self-Control, Econometrica, 69, Gul,F. and W. Pesendorfer (2005): The Revealed Preference Theory of Changing Tastes, Review of Economic Studies 72, Kreps, D. (1979): A Representation Theorem for Preference for Flexibility, Econometrica, 47, May 1979, Strotz, R. H. (1955): Myopia and Inconsistency in Dynamic Utility Maximization, Review of Economic Studies, 23,
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