HUEI-CHUNG LU, MINGSHEN CHEN and JUIN-JEN CHANG
ARE MORE ALTERNATIVES BETTER FOR DECISION-MAKERS? A NOTE ON THE ROLE OF
DECISION COST
ABSTRACT. While the traditional economic wisdom believes that an individual will become better off by being given a larger opportunity set to choose from, in this paper we question this belief and build a formal the-oretical model that introduces decision costs into the rational decision process. We show, under some reasonable conditions, that a larger feasible set may actually lower an individual’s level of satisfaction. This provides a solid economic underpinning for the Simon prediction.
KEY WORDS: bounded rationality, considered subset, decision cost. JEL CLASSIFICATIONS: D11, D83.
1. INTRODUCTION
The traditional economic wisdom believes that an individual will become better off (or, at least not worse off) by being given a larger opportunity set to choose from. This notion implies that the agent can give up those incremental alternatives that are unwanted without incurring any cost. As pointed out by Conlisk (1988), agents are typically assumed to reason cost-lessly in regard to a decision about how much information to collect and, given that information, to reason costlessly in relation to an optimal final action. However, it is well recog-nized pragmatically (see, for instance, Baumol and Quandt, 1964; Williams and Findlay, 1981) that an agent will incur some decision costs when he makes a choice. The experimental economists, such as Smith (1989), Pingle (1992), and Wilcox (1993), use psychological experiments to prove that the decision
cost (measured in terms of decision-making time) is often a key factor affecting an agent’s choice among alternatives.
Simon (1955) has argued that people may not always follow the rational decision-making rule as described by economists, one of the main reasons for this being the exis-tence of decision costs. Accordingly, he proposes the concept of bounded rationality to challenge the traditional economics of the unbounded rational decision. In a survey paper, Conlisk (1996) also concludes that the deliberation cost is one of the major reasons why an individual does not act ‘‘rationally.’’ However, the decision costs are suppressed in most economic analyses.
The aim of this paper is to shed light on the importance of the decision cost in rational decision theory. While we do not set out to provide yet another proof of the existence of bounded rationality and decision costs, our intention is to set up a formal theoretical model and demonstrate how we should view things differently when having a larger feasible set to choose from, given the existence of decision costs. To be more specific, we will show that, under certain reasonable conditions, the net expected benefit of choosing from a larger feasible set might be smaller. Consequently, the decision-maker might not be better off by being given a larger number of choices. This will provide a solid economic underpinning for the Simon prediction.
This issue has become more important than ever during the present era of information overload. By using an Internet search engine, it is easy for an individual to obtain an enormous feasible set to choose from. People often embrace this increase in the size of the feasible set, which seems to be beneficial at first glance, while failing to realize the implicit costs involved in the decision-making process. When faced with such decision costs, we can observe that in reality some people would actually prefer to choose from a smaller, but more manageable, feasible set. In Sections 2 and 3, we will build a new model that runs counter to the usual model in rational decision theory and use it to express our viewpoint. Section 4 discusses the robustness of
our main results and provides numerical calibrations for these results. Section 5 concludes.
2. THE DECISION PROCEDURE AND OPTIMIZATION PROBLEM
Our model shares some concepts with the bounded rationality theory, such as in Simon (1955) and Lipman (1991), and introduces decision costs into the decision-making process. Naturally, there are N bundles in the feasible set that can be ranked according to the order of preferences x1 x2 xN. The same can be done for the corre-sponding benefits bðxÞ, that is, bðx1Þ > bðx2Þ > > bðxNÞ. Following Simon (1955), we consider
ASSUMPTION 1. A decision-maker (henceforth DM), facing non-trivial decision costs, follows a two-stage decision process as follows:
• Stage 1. (screening process): The DM pre-selects those alternative bundles x that fit into the profile of having the potential to become the ‘‘best bundle’’. Specifically, the DM will pick up n bundles from the feasible set that has N alternative bundles. The n selected bundles constitute the considered subset.1
• Stage 2. (evaluation process): The DM evaluates all of the selected bundles x, calculates their corresponding benefits bðxÞ, and chooses the best bundle xk (and hence derives benefit bðxkÞ) from the n considered bundles.2 In addition, the evaluation process entails some evaluation (or decision) costs C,3which could be thought of as either a pecuniary cost of evaluating or a psychological disutility stemming from deliberating over a decision, such as hesitation and uneasi-ness. For simplicity, the evaluation cost is specified as a linear increasing function with respect to n, i.e. CðnÞ ¼ c n. To be more specific, given N alternatives, fxigNi¼1, the DM knows there are N possible values of benefits, fbigNi¼1.
4
match the correct value b with an alternative x without further evaluation. In other words, the DM does not have exact knowledge regarding the true ranking for N bundles during the screening process; the evaluation process, however, can match the bundles x with their correct benefits b.
By means of both the screening process and the evaluation process, the DM maximizes his net expected benefit, V, as follows:
ASSUMPTION 2. V EBðx; n; NÞ c n.
Because the agent does not have exact knowledge regarding the ranking of x and bðxÞ for N bundles during the screening process, he can only ex-ante consider the following factor, namely, the expected benefit EBðx; n; NÞ. The expected benefit is related to the size of the considered subset n, the benefit arising from the best bundle bðxkÞ among the n considered bundles after the evaluation is performed, and the number of alternative bundles in the feasible set N. Notice that, even after evaluation, the DM still cannot be sure that xk is the best bundle out of all of the alternatives, i.e. x1, unless he chooses to evaluate all alternative bundles, i.e. n¼ N.
Backward induction is applied to solve this two-stage opti-mization decision problem. During the second stage, given that n is chosen during the first stage, the DM evaluates these n considered bundles, finds the best optimal bundle xk(and hence derives benefit bðxkÞ), and incurs the decision cost c n. By internalizing these possible results, the DM’s goal during the first stage is to choose an optimal n so as to maximize V. If the agent chooses a larger n, on the one hand, he has a higher probability of obtaining a higher level of bðxkÞ (i.e. a smaller k), but, on the other hand, he will also incur a higher decision cost. If n¼ N, then the agent can obtain the highest benefit bðx1Þ for sure, but he will also incur the highest decision cost as measured by c N.
Given the evaluated result bðxkÞ and the decision costs c n, the optimal size is:
n ¼ arg max n V EBðx; n; NÞ C ¼ X Nnþ1 k¼1 Pkðn; NÞ bðxkÞ c n; ð1Þ where Pkðn; NÞ ¼ N k n 1 N n ¼ ðN kÞ!=ðn 1Þ!ðN n k þ 1Þ! ðn! ðN nÞ!=N!Þ; 1 k N n þ 1:
To obtain the expected benefit EBðx; n; NÞ, we should first calculate the probability of obtaining the bundle xk, namely Pkðn; NÞ.5Given that the sizes of the considered subset and the feasible set are n and N, respectively, the total number of possible combinations of choosing n from N is Nn
. Further-more, the number of possible combinations of picking the best bundle xk out of n considered bundles is Nn 1 k
. Thus, the probability of obtaining the bundle xk, Pkðn; NÞ, is the ratio of
N k n 1
to Nn
as demonstrated in (1). In an extreme case where n ¼ N, we then have P1ðN; NÞ ¼ 1.
A numerical example might be helpful in understanding the inference of Pkðn; NÞ. We set N ¼ 10, n ¼ 3, and the considered bundles asðxi; xj; xkÞ, where i 6¼ j 6¼ k and 8i; j; k ¼ 1; . . . ; 10. Accordingly, the total number of possible combinations for the considered subset is 103
¼ 10!ð3! 7!Þ ¼ 120. If the considered subset consists of x1, i.e. k¼ 1 (of course, x1 is the best among the three considered bundles), then the considered bundles are ðx1; xi; xjÞ, where i 6¼ j and 8i; j ¼ 2; . . . ; 10. Given that the position of x1 is certain, obtaining the possible combinations of ðx1; xi; xjÞ involves choosing two bundles xi and xj from the nine bundles x2–x10. In other words, the number of possible combinations of picking up x1 from the three considered bundles is 9
2
obtaining the bundle x1 is P1ðn ¼ 3; N ¼ 10Þ ¼ 36
120¼ 310. If k¼ 2 and x2 (the second best among all bundles) is the best among three considered bundles, then the considered subset must rule out x1. Hence the number of possible combinations that would pick up x2 is 82
¼ 8!ð2! 6!Þ ¼ 28. Accordingly, we have P2ðn ¼ 3; N ¼ 10Þ ¼ 28
120¼ 730. We can infer the probability of obtaining xkas Pkðn ¼ 3; N ¼ 10Þ ¼
10 k 2
.
120 in the case where N¼ 10 and n ¼ 3. By the same logic, in the general case the probability of obtaining xk is Pkðn; NÞ ¼ N k n 1 , N n , as expressed in (1). Define a cumulative probability function as
Ukðn; NÞ PrðbðxÞPbðxkÞÞ ¼ Xk i¼1 Piðn; NÞ; and we have: LEMMA 1. Ukðn þ 1; NÞPUkðn; NÞ 8k. Proof. See Appendix A.
Furthermore, according to the concept of ‘‘stochastic domi-nance’’ defined by Hardar and Russell (1969), the following lemma is also obtained immediately:
LEMMA 2. Suppose that there are two probability density func-tions fðÞ and gðÞ with respect to a random variable x. Let us denote their corresponding cumulative probability density functions as FðxÞ and GðxÞ, respectively. If FðxÞ GðxÞ 8x and bðÞ is a decreasing function of x, thenPbðxÞ fðxÞPPbðxÞ gðxÞ must hold.
From the relationship bðx1Þ > bðx2Þ > > bðxNÞ and Lemmas 1 and 2, we establish:
PROPOSITION 1. Under Assumptions 1 and 2, then EBðx; n þ 1; NÞPEBðx; n; NÞ.
This indicates that the expected benefit EB increases with the size of the considered subset n.
According to Proposition 1, the marginal decision-making approach indicates that the optimal size of the considered subset n satisfies the condition in which the marginal benefit of changing n (MB) equals its marginal cost (MC). That is,
MBðn; NÞ EBðx; n; NÞ EBðx; n 1; NÞ ¼ c: ð2Þ One point should be noted. Since n is a discrete variable, (2) may not always be satisfied. If so, the optimal n is an integer such that , MB MC c MC c MC c MC MB c MC 0 1 * n N n (a) MB M ,B MC 0 n* N n MC A B c (b) Figure 1.
EBðx; n; NÞ EBðx; n 1; NÞ > c and
EBðx; nþ 1; NÞ EBðx; n; NÞ < c: ð20Þ However, in order to make our analysis more treatable and without significant loss of generality, we focus on the case where (2) is held.
To be sure that the optimal n is acceptable to the DM, the following condition.
ASSUMPTION 3. V ¼ EBðx; n; NÞ c nP0 is required, implying that, based on the optimal size of the considered subset, the net expected benefit is non-positive.
We now explore the role of evaluation costs in determining the optimal size of the considered subset in Figure 1(a) and (b). In Figure 1(a) and (b), the MC curve is horizontal due to the assumption of a fixed marginal decision cost. Since MB can be either negatively or positively related to n, we need to discuss these two possible cases. We start the discussion with the case where MB has negative relationship with n, as described in Figure 1(a). In the case of the traditional theory of rational choice, the decision cost is trivial and infinitesimal (hence, the marginal decision cost is close to zero, say c0). Thus, we yield a corner solution, which indicates that the optimal size of the considered subset is that of the feasible set, i.e. n ¼ N. How-ever, in most cases, the cost of evaluation is non-trivial, namely, the marginal decision cost is c. In such a case, there is an interior solution and the intersection of the curves MB and MC deter-mines the optimal size of the considered subset n. If the mar-ginal decision cost is increased from c to c00, as depicted in Figure 1(a), the optimal number of the considered bundles is unity, because MB intersects MC at 1. This potentially implies that the DM may randomly select his alternative and the ex post benefit of his decision will be completely dependent upon luck. An extreme case is that, if the marginal cost is extremely high, say c000, the optimal set of the considered bundles will be empty. In such a situation, the DM will give up all possible alternatives. These results show that the optimal size of the considered subset n may fall short of the total number of alternatives N.
We turn to the case where MB is positively related to n. Let c be a critical level of marginal cost such that the net expected benefit is zero, i.e. EB ¼ C, which implies that the area A is equal to the area B, as illustrated in Figure 1(b). If MC c, the optimal n will entail a positive net expected benefit V (due to the area A being larger than the area B). Because the net ex-pected benefit increases with n, the DM will increase the number of considered bundles until n ¼ N. On the contrary, if MC > c, the DM will decrease the size of n as much as he possibly can due to EB < C 8n. In this case, n ¼ 0 may be a possible solution. In other words, given that MB is upward sloping, the optimal number of considered bundles is a corner solution, i.e. either n ¼ 0 or n ¼ N. We summarize the above results as: PROPOSITION 2. Under Assumptions 1–3, in the presence of non-trivial decision costs, the DM will not necessarily consider all alternatives in the feasible set.
3. ARE MORE ALTERNATIVES BETTER FOR THE DM?
To shed light on our main point, we will henceforth place our focus on the case of interior solutions (hence the case where MB is decreasing with n). Now, we consider a larger feasible set, say, where the size of the feasible set increases from N to Nþ 1. Corresponding to this size of feasible set of Nþ 1, the prob-ability of picking xk from n considered bundles is denoted by Pkðn; N þ 1Þ, which is expressed as:
Pkðn; N þ 1Þ ¼ N k þ 1 n 1 , Nþ 1 n ¼ ðN k þ 1Þ! ðn 1Þ! ðN n k þ 2Þ! n!ðN n þ 1Þ! ðN þ 1Þ! ¼ ðNnþ1ÞðNkþ1Þ ðNþ1ÞðNnkþ2Þ Pkðn; NÞ if 1O kO N n þ 1; ðNnþ1Þ! ðNþ1Þ! if k ¼ N n þ 2; 0 if k > N n þ 2: 8 > < > :
According to (3), we have
LEMMA 3. Ukðn; NÞPUkðn; N þ 1Þ 8k. Proof. See Appendix B.
After the size of the feasible set is increased by 1, the rank and the corresponding benefits are specified as ~x1 ~x2
~
xN ~xNþ1 and as bð~x1Þ > bð~x2Þ > > bð~xNÞ > bð~xNþ1Þ, respectively. Accordingly, from Lemmas 2 and 3, it is easy to derive the difference between EBðx; n; N þ 1Þ and EBðx; n; NÞ as: EBðx; n; N þ 1Þ EBðx; n; NÞ ¼ X Nnþ2 k¼1 Pkðn; N þ 1Þ Pkðn; NÞ ½ bð~xkÞ þ X Nnþ2 k¼1 Pkðn; NÞ ½bð~xkÞ bðxkÞ: ð4Þ
In order to make our point more striking, we further assume that the DM does not care about the intrinsic benefit of the chosen bundle, but simply about the rank of the chosen bundle among the feasible set. Specifically, we propose:
ASSUMPTION 4. bðxkÞ ¼ /ðkÞ, where /0 <0.
The DM does not know the actual benefit arising from any alternative bundle before the evaluation process is undertaken, and therefore his goal may be to maximize satisfaction brought in by the optimal bundle according to its ‘‘rank’’. The benefit function may be specified more explicitly as bðxkÞ ¼ z kc, where z and c are coefficients. To examine the validity of the proposition, in Section 4, based on this specification, we will perform numerical simulations.
If the DM is only concerned with whether the selected bundle is the best one out of the considered subset based on its ‘‘rank’’ within the feasible set, and is not concerned with whether the best bundle is chosen from either N alternative bundles or ðN þ 1Þ bundles, the satisfaction (the feeling) from
obtaining the highest ranking bundle from a considered subset out of the feasible set ðN þ 1Þ will be the same as getting it out of the feasible set N.6 That is, bð~xkÞ ¼ bðxkÞ, k ¼ 1; 2; . . . ; N and bðxNÞ > bð~xNþ1Þ. Given this assumption, we modify (4) as: EBðx; n; N þ 1Þ EBðx; n; NÞ ¼ X Nnþ2 k¼1 Pkðn; N þ 1Þ Pkðn; NÞ ½ bðxkÞ: ð5Þ
Proposition 3 is immediately derived from (5).
PROPOSITION 3. Under Assumptions 1, 2 and 4, EBðx; n; NÞP EBðx; n; N þ 1Þ 8n and N.
Proof. See Appendix C.
Proposition 3 indicates that, a larger size of feasible set may result in a lower expected benefits EB given a particular size of the considered subset. However, more alternatives may lead the DM to change his decision concerning the size of the considered subset’s optimal n, and, as a result, alter his expected benefits EB and decision costs C. Therefore, in what follows, we will further explore the net change in welfare of the DM.
It easily follows from (2) that an increase in the number of alternatives from N to Nþ 1 has an ambiguous effect on the optimal n, depending on the relative curvature of EBðx; n; NÞ and EBðx; n; N þ 1Þ. Based on this and Proposition 3, we then have:
PROPOSITION 4. Under Assumptions 1–4, a larger feasible set may make the DM worse off.
Proof. See Appendix D.
A remark should be made here: To make our argument more striking, Proposition 4 is established under Assumption 4. However, we do not intend to claim that more alternatives should make the DM worse off, particularly when Assump-tion 4 is relaxed. Now we turn to the discussion concerning the robustness of Proposition 4.
4. DISCUSSION AND NUMERICAL SIMULATIONS
Propositions 3 and 4 are established under Assumption 4 in which the DM does not care about the intrinsic benefit of the chosen bundle, but simply about the rank of the chosen bundle among the feasible set. However, in some cases the DM does care about the intrinsic benefit of the alternative bundle, for instance, utility from consumption. One may thus inquire about the robustness of these propositions.
In what follows, we will show that Propositions 3 and 4 may still hold, even though the DM is concerned with not only the ‘‘rank’’ of the chosen bundle, but also the ‘‘level’’ of benefit derived from a chosen bundle. Suppose that the additional
650 700 750 800 850 900 950 1000 0 5 10 15 20 25 30n V N = 20 N = 25 N = 30 (a) 825 850 875 900 925 950 975 0 3 6 9 12 15 18 21n V original ˆ ( ) ˆ ( ) ˆ ( ) b x = 1002 b x = 890 b x = 580 (b) Figure 2.
alternative is ^x, when the size of the feasible set increases from N to Nþ 1. By letting x1 xm1 ^x xm xN, we have
bð~xkÞ ¼ bðxkÞ for k ¼ 1; . . . ; m 1 and bð~xkÞ > bðxkÞ for k ¼ m; . . . ; N:
Accordingly, the second item in (4) is positive and decreasing with m. For example, in extreme cases, if ^x x1, this will result in the largest value for the second item in (4) and if xN ^x, this will result in the smallest value for the second item in (4). Given that the sign of the first item in (4) must be negative (inferred by Lemma 3), Proposition 3 is more likely to be true when m is larger (or bð^xÞ is smaller). Provided that the negative effect of the first term in (4) is substantially strong, Propositions 3 and 4 may be valid.
We next perform two sets of numerical simulations in order to explicitly show the possibility of Proposition 4.
(1) The DM is only concerned with the rank of a chosen bundle Let bðxkÞ ¼ 1000 k2 and C¼ c n ¼ 5 n, where k ¼ 1; . . . ; N. Consider three possible sizes of the feasible set, say 20, 25, and 30, respectively. Figure 2(a) shows that, given that N is 20, 25 and 30, the associated optimal sizes of considered subset n are 5 (or 6), 7 and 8, respectively. Furthermore, we can evaluate their corresponding net expected benefits V as 956.5, 948.75 and 941.4. Obviously, the largest net expected benefit (956.5) is derived from the smallest feasible set, N ¼ 20, rather than the larger feasible sets, N¼ 25 or N ¼ 30.
(2) The DM cares not only about the rank of a chosen bundle but also its intrinsic value
Consider a benefit setfbðxkÞg20k¼1, namely F1, consisting of {999, 996, 991, 984, 975, 964, 951, 936, 919, 900, 879, 856, 831, 804, 775, 744, 711, 676, 639, 600}. The setting of the decision cost is the same as in (1). Obviously, the benefit pertaining to the best bundle is 999 while the benefit from the worst bundle is 600. Now, assume that there is an additional bundle ^x and that its
corresponding benefit is denoted by bð^xÞ. The benefit set with a larger number of alternatives, N ¼ 21 (including bð^xÞ), is de-fined as F2.
We consider three possible cases: (i) bð^xÞ is greater than all of elements of the set F1 and is specified as 1,002 (> 999); (ii) bð^xÞ is smaller than all of elements of the set F1 and is spec-ified as 580 (< 600); (iii) bð^xÞ is a middle value among the elements of the set F1 and is specified as 890. Given these specifications, Figure 2(b) shows that, under the feasible set F1, the optimal size of the considered subset is n ¼ 5 or 6, (and hence the maximum net expected benefit is V ¼ 956:5); however, in response to an increase in the feasible set (i.e. under the larger set F2), the optimal size of the considered subset may either increase (n ¼ 6 in case (ii) and (iii)) or decrease (n ¼ 5 in case (i)). The result is as predicted by our deductions in Section 3.
Of great importance, when we consider a larger feasible set F2, as indicated by Figure 2(b), is that the maximum net ex-pected benefits become 961.38 in case (i), 955.22 in case (ii), and 955.07 in case (iii), respectively. Given that the maximum net expected benefit is V ¼ 956:5 under F1 with N¼ 20, except in case (i), the net expected benefit V does not increase with the larger size N¼ 21 of the feasible set F2. In other words, more alternatives do not necessarily make the DM better off unless they can provide better choices for the DM.
5. CONCLUDING REMARKS
Conlisk (1996, p. 671) stressed that, for an individual with bounded rationality, heuristics often provide adequate solu-tions that are cheap, whereas more elaborate approaches would be unduly expensive. By taking decision costs into account, we set up a formal model and show that having more alternatives may not make a DM better off. The numerical simulations that we perform also support this argument. This result, that runs contrary to the traditional rational choice theory, has impor-tant implications for DM.
APPENDIX A
The Proof of Lemma 1. Given the n considered bundles, the definition of Uk immediately yields:
Ukðn; NÞ ¼X k i¼1 Piðn; NÞ ¼ Pk i¼1 ðNiÞ! ðn1Þ! ðNniþ1Þ! n!ðNnÞ! N! if k N n þ 1; 1 if k > N n þ 1: 8 < : ðA1Þ
In the case of a larger considered subset with ðn þ 1Þ bun-dles, the cumulative probability function will be changed into
Ukðn þ 1; NÞ ¼ Pk i¼1 ðNiÞ! n!ðNniÞ! ðnþ1Þ! ðNn1Þ! N! if k N n; 1 if k > N n: 8 < : ðA2Þ
By rearrangement, Ukðn þ 1; NÞ can be written as
Ukðn þ 1; NÞ ¼ nþ 1 n X k i¼1 Piðn; NÞ N n i þ 1 N n ¼ nþ 1 n X k i¼1 Piðn; NÞ 1 i 1 N n : ðA3Þ According to (A3), we further derive
Ukþ1ðn þ 1; NÞ ¼ nþ 1 n X kþ1 i¼1 Piðn; NÞ 1 i 1 N n ¼X kþ1 i¼1 Piðn; NÞ þ 1 n Xkþ1 i¼1 Piðn; NÞ nþ 1 n X kþ1 i¼1 Piðn; NÞ i 1 N n
¼ Ukþ1ðn; NÞ þ 1 n Xk i¼1 Piðn; NÞ nþ 1 n X k i¼1 Piðn; NÞ i 1 N n þ1 n Pkþ1ðn; NÞ 1 ðn þ 1Þk N n :
Let DnUk Ukðn þ 1; NÞ Ukðn; NÞ and Piðn; NÞ ¼ Pi, then the above equation can be rewritten as
Ukþ1ðn þ 1; NÞ ¼ Ukþ1ðn; NÞ þ DnUk þ1 n Pkþ1 1 ðn þ 1Þk N n ) DnUkþ1 ¼ DnUkþ Hkþ1; ðA4Þ where Hkþ1 1 n Pkþ1 1 ðn þ 1Þk N n :
Suppose that there exists a critical k such that k ¼ ðN nÞ=ðn þ 1Þ. When k < k, then ½1 ðn þ 1Þk= ðN nÞ > 0; otherwise, when k > k, then ½1 ðn þ 1Þk= ðN nÞ < 0. Since Pkþ1 and ½1 ðn þ 1Þk=ðN nÞ both de-crease with k, Hkþ1 is also a decreasing function of k. Accordingly, Figure A.1 follows equation (A4) and depicts the
1 n k 1 (1 )n P 0 k* 1 N n k Figure A.1.
relationship between k and DnUkþ1, which indicates that Hkþ1>0 (i.e. DnUkþ1>DnUk) if k < k and Hkþ1 <0 (i.e. DnUkþ1 <DnUk) if k > k. In Figure A.1, in order to sketch the relationship between k and DnUkþ1, we have utilized the following relationships U1ðn þ 1; NÞ ¼ ðn þ 1Þ=n P1 > P1 ¼ U1ðn; NÞ, DnU1 ¼ ð1=nÞ P1, and UNnþ1ðn þ 1; NÞ ¼ UNnþ1 ðn; NÞ, i.e. DnUNnþ1¼ 0. Obviously, Figure A.1 indicates that DnUkP0 8k, implying that Ukðn þ 1; NÞP Ukðn; NÞ 8k. h
APPENDIX B
The Proof of Lemma 3. According to (3), we have
Ukðn; N þ 1Þ ¼ Xk i¼1 Piðn; N þ 1Þ ¼X k i¼1 Piðn; NÞ N n þ 1 Nþ 1 N i þ 1 N n i þ 2 ¼ Ukðn; NÞ þ Xk i¼1 Piðn; NÞ N n þ 1 Nþ 1 N i þ 1 N n i þ 2 1 : Letting DNUk¼ Ukðn; N þ 1Þ Ukðn; NÞ, then 0 ** 1 k N n 1 k 1 1 (N 1) P 1 N k Figure A.2.
DNUk¼ Xk i¼1 Piðn; NÞ N n þ 1 Nþ 1 N i þ 1 N n i þ 2 1 and DNUkþ1 ¼ DNUkþ Pkþ1ðn; NÞ N n þ 1 Nþ 1 N k N n k þ 1 1 : ðA5Þ In Equation (A5) there exists a critical k ¼ ðN n þ 1Þ=n such that DNUkþ1¼ DNUk, and DNUkþ1<DNUk if k < k, while DNUkþ1 >DNUk if k > k. It follows from equation (3) that DNU1 ¼ P1ðn; N þ 1Þ P1ðn; NÞ < 0 if k ¼ 1, and DNUk¼ 0 if kPN n þ 2. Based on these inferences, we sketch the relationship between k and DnUkþ1 in Figure A.2. Figure A.2 indicates that DNUk 0 8k, i.e. Ukðn; NÞP
Ukðn; N þ 1Þ 8k. h
APPENDIX C
The Proof of Proposition 3. From Lemmas 2 and 3, it is clear that EBðx; n; N þ 1Þ will be less than EBðx; n; NÞ for all n if the DM is only concerned with the rank of a chosen bundle. h
APPENDIX D
The Proof of Proposition 4. Assume that the optimal size of the considered subset is n when the size of a feasible set is N, and that the optimal size of the considered subset is n when the size of a feasible set is Nþ 1. Their corresponding optimal net expected benefits are V and V. Accordingly, by defining Dn n n, we have
V V¼ EBðx; n½ ; NÞ c n
EBðx; n½ ; Nþ 1Þ c n ¼ EBðx; n½ ; NÞ EBðx; n; Nþ 1Þ
þ ½EBðx; n; NÞ EBðx; n Dn; NÞ c Dn
X1þ X2: ðA6Þ
In (A6) the term X1 is positive due to EBðx; n; NÞ > EBðx; n; Nþ 1Þ based on Proposition 3. Moreover, if n ¼ n, the term X2 will be reduced to zero. Thus, V< V is true and Proposition 4 holds.
If n6¼ n, more discussions are needed. If Dn¼ 1, X2is zero since the optimal condition (2) indicates that MBðn; NÞ EBðx; n; NÞ EBðx; n 1; NÞ ¼ c. Given Prop-osition 3, the result V < V is still valid in such a case.
If Dn6¼ 1, for example, Dnj j ¼ 2, then X2 ¼ EBðn; NÞ EBðn 2; NÞ c 2 ¼ EBðn½ ; NÞ EBðn 1; NÞ þ EBðn½ 1; NÞ EBðn 2; NÞ 2c MBðn; NÞ þ MBðn 1; NÞ 2c; ðA7Þ when Dn ¼ 2, and X2 ¼ EBðn; NÞ EBðnþ 2; NÞ c ð2Þ ¼ EBðn½ ; NÞ EBðnþ 1; NÞ þ EBðn½ þ 1; NÞ EBðnþ 2; NÞ þ 2c MBðn½ þ 1; NÞ þ MBðnþ 2; NÞ þ 2c; ðA70Þ when Dn¼ 2. With an interior solution, the optimal condi-tions (2) or (2¢) are satisfied, implying that
MBðn; NÞ EBðn; NÞ EBðn 1; NÞ> <c if n
< >n
: ðA8Þ
Therefore, we can conclude
X2 ¼ MBðn; NÞ þ MBðn 1; NÞ 2c > c þ c 2c ¼ 0 if Dn ¼ 2;
X2 ¼ MBðn½ þ 1; NÞ þ MBðnþ 2; NÞ þ 2c > c c þ 2c ¼ 0 if Dn ¼ 2:
That is to say, the term X2 of (A6) is always positive whenever Dn¼ 2 or 2. Based on the similar inference, the result can be applied to Dn being any integer number. Moreover, in the ex-treme case where n ¼ N and n¼ N þ 1, V< V is also true since bðx1Þ c N > bðx1Þ c ðN þ 1Þ. Thus, Proposition 4 is
proved. h
ACKNOWLEDGEMENTS
Financial support from the National Science Council, Taiwan, R.O.C (NSC90-2415-H-030-001) is gratefully acknowledged. The authors would also like to thank the editor of this journal and an anonymous referee for their helpful suggestions and insightful comments on an earlier version of this paper. Of course, the usual disclaimer applies.
NOTES
1. The term ‘‘considered subset’’ is taken from Simon (1955).
2. For example, if k¼ 3, this means that x3(the third best among all) is the
best among the n considered bundles that are picked up during the screening process.
3. Takahashi and Takayanagi (1985) propose two other approaches to search for the optimal bundle. One is the ‘‘fixed-size procedure’’ and the other is the ‘‘sequential procedure.’’ Under the fixed-size procedure, the decision-maker comprehensively surveys all possible alternatives before the evaluation process begins. On the other hand, when the ‘‘sequential procedure’’ is adopted, the DM considers one alternative at a time and accepts such an alternative when it reaches the acceptable level; otherwise, he rejects it and returns to the screening process. Obviously, these two different procedures give rise to different decision costs.
4. In this paper x can be regarded as either a consumption bundle for a consumer, or an investment project for an entrepreneur. Thus, b(Æ) is regarded as the utility function or return function, respectively. When making an investment decision, the entrepreneur’s return must involve some uncertainty. Therefore, the DM may only have a priori information that each alternative x has a benefit b drawn from some distribution, or,
more specifically, a priori benefit of x is i.i.d. such that b(x)~ F(b). In such a case, a Bayesian decision procedure will be applied. To keep matters simple and focus on our main point, we do not explicitly deal with this problem. Nevertheless, we can think of b(Æ) as the expected benefit of alternative x in order to simplify our model. We are grateful to an anonymous referee for bringing this point to our attention.
5. Notice that since the DM picks up the n considered bundles randomly from the feasible set, he cannot therefore know the exact value of xk.
6. For example, the DM has the same satisfaction level if he can obtain the first-best bundle (i.e. x1or ~x1) regardless of whether it is chosen from the
NorðN þ 1Þ bundles.
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Huei-Chung Lu, Department of Economics, Fu-Jen Catholic University, Taiwan.
Mingshen Chen, Department of Finance, National Taiwan University, Taiwan.
Address for correspondence: Juin-Jen Chang, Institute of Economics, Academia Sinica, and Department of Economics, Fu-Jen Catholic Uni-versity, Taiwan. E-mail: jjchang@econ.sinica.edu.tw
