Status Quo Bias under Uncertainty: An Experimental Study
Yüklə 392,99 Kb. Pdf görüntüsü
|
chooses as follows: c(A, f ) =
{f } if D Π,u
(f ) ∩ A = ∅ arg max
g∈A∩D Π,u
(f ) s∈S
ρ(s)E g(s)
(u) otherwise Such agent has a prior ρ which he trusts absent an endowment and is a standard Savagean agent in these contexts. However, when he is endowed with some act f he becomes uncer- tainty averse. That is, he considers as potential alternatives only those acts that generate a higher expected payoff than his endowment according to every prior he has in mind. If no feasible act satisfies this condition, he keeps his status quo option. If there are any acts which satisfy this requirement, i.e., belong to A ∩ D Π,u (f ), the agent chooses among them in the final stage using the prior ρ. The status quo act f , thus, affects choice in two ways: (1) The agent will choose it if no other act dominates it for all priors (that is, when D Π,u
(f ) does not contain any feasible act); and (2) by restricting the set of acts which the agent considers for choice. Predictions of the model within the experimental set up We now apply the model within our experimental set up to derive the prediction of interest. The two ambiguous bags used in our experiment, the Dow-Jones (DJ) and Standard & Poor’s (SP), contain 100 poker chips with an unspecified amount of whites and blacks. A state of the world is any possible joint composition of the two bags. States are indicated by s
i,j , i, j ∈ {0, 1, ..., 99} where i is the number of white chips in bag DJ and j is the number of white chips in bag SP . We denote by R the set of all constant acts, i.e., acts that deliver the same lottery in every state of the world. Let F DJ denote the set of all acts whose outcomes only depend on the realization of the Dow Jones Index. Similarly let F SP be the set of acts whose outcomes 30 only depend on the Standard & Poor’s realization. We are now ready to show that if a decision maker exhibits status quo biased behavior in choices between an act f ∈ F DJ (in
the experiment - the (10, 4) act played on the DJ bag) and some constant act r ∈ R (in the experiment - a risky lottery on the known bag) then, according to the above representation, he must also be biased in choices between f and another act g ∈ F SP . In other words, the presence of status quo bias in treatment A-R implies status quo bias in treatment A-A. We start by formally defining status quo biased behavior in our set up. Definition. We say that the choice between two acts, f and g, exhibits status quo bias in favor of f , if absent an endowment the DM chooses g but not f from the set {f, g}, but chooses f over g from the same set when f serves as the endowment. 26 Next, we introduce an assumption regarding the set of priors that the individual holds. This assumption roughly states that his subjective distributions over the composition of one ambiguous bag are independent of the subjective distributions over the possible compositions of the other bag. 27 We start with some simple notation. For any joint probability π ∈ Π denote by π DJ and π
SP the marginal probability distributions in- duced by π. Formally, for every i, j ∈ {0, ...., 99} , define π i DJ ≡ j π(s ij ) ,
π j SP ≡ i π(s ij ). Now, define Π DJ ≡ {p ∈ R
100 |p = π
DJ for some π ∈ Π} and Π
≡ {q ∈ R
100 |q = π
SP for some π ∈ Π}. Finally, for every π ∈ Π, define π DJ
SP ≡ ˜ π ∈ R 100
× R 100
|˜ π = π
T DJ · ¯ π SP for some ¯ π SP ∈ Π SP (where · is matrix multiplication, and v T
DJ ⊗ π
SP ≡ ˜ π ∈ R 100
× R 100
|˜ π = ¯
π T DJ · π SP for some ¯ π DJ ∈ Π DJ 26 This definition is a strict version of the weak status quo bias (WSQB) axiom introduced in Masatlioglu and Ok (2014). 27 This assumption is very natural in our set up given the construction of the ambiguous bags according to the decimals of two different stock market indices. 31
We are now ready to state our assumption. Assumption 1: For any π ∈ Π we have π DJ ⊗ Π
SP ⊆ Π and Π DJ ⊗ π
SP ⊆ Π.
Proposition 1 According to the representation given in Ortoleva (2010) and under As- sumption 1, if there exist acts f ∈ F DJ and r ∈ R such that the choice between f and r exhibits status quo bias in favor of f , then there exists an act g ∈ F SP such that the choice between f and g also exhibits status quo bias in favor of f . Proof. Let f and r be as in the Proposition. If the decision maker exhibits status quo bias in favor of f then the following must be true: s∈S
ρ(s)E f (s)
(u) < E r (u) (1) s∈S
˜ π(s)E
f (s) (u) > E
r (u), for some ˜ π ∈ Π (2)
According to inequality (1), act f evaluated at the prior ρ, i.e., without an endowment, delivers lower expected utility than r. However there is one prior in Π according to which f carries strictly higher expected utility than r; f is therefore chosen when it serves as the status quo. Now let g ∈ F SP be such that: s∈S ρ(s)E
g(s) (u) = E
r (u).
(3) Continuity of the utility function alongside monotonicity guarantee that such a g exists. 28 From (1) and (3) it follows that s∈S ρ(s)E
g(s) (u) >
s∈S ρ(s)E
f (s) (u), hence g is chosen 28 The act g is not only a theoretical construct. In our experiment, we find implicit evidence for the existence of such g as in all treatments we observe switching points in part 1 for the vast majority of subjects. 32
over f absent a status quo. We are left to show that when f serves as the endowment, it is chosen over g, that is, it carries higher expected utility than g according to some prior in Π. Recall ˜
π DJ ≡ (˜ π 1 DJ , ˜ π 2 DJ , ..., ˜
π 100
DJ ) and ρ
SP ≡ (ρ
1 SP , ρ 2 SP , ..., ρ 100 SP ). Define ˆ π ≡ ˜ π T DJ ·ρ SP . Assumption 1 ensures that ˆ π ∈ Π. By construction, ˆ π evaluates gambles on the DJ bag using ˜ π, and evaluates gambles on the SP bag using ρ. It follows that s∈S ˆ π(s)E f (s) (u) =
s∈S ˜ π(s)E f (s) (u) and
s∈S ˆ π(s)E g(s) (u) = E
r (u) which together with (2) completes the proof. Appendix C Corners Due to the limited number of questions asked in an experimental session, some subjects did not exhibit an interior switching point from the endowment to the alternative set, a situation we dub as a “corner”. A corner is the case where a subject always chooses the (10, 4) gamble or always chooses the alternative gambles in one or both subsets of ordered alternatives. To understand the role of corners, the reader could imagine extending the range of alternative options. It is conceivable that if we keep decreasing the expected payoff of the alternative gambles all subjects would eventually prefer the (10, 4) gamble. Similarly, all subjects would eventually switch to the alternative gamble if the expected payoffs of the latter are made large enough. Hence we interpret corners as cases in which a subject’s switching point is not captured by the selected range of alternative gambles. Corners have the potential to generate a bias in our estimate of the status quo effect. Luckily we can exploit the pattern of choice exhibited in proximity to the end of the range of questions to assess whether a certain corner is more likely to induce an overestimation or underestimation of status quo bias. 33
In this section we give an account of the emergence of corners across treatments. We show that in each treatment the bias is affected to a similar extent by instances of potential overestimation and underestimation. If anything, underestimation is slightly more likely in the two asymmetric treatments, which are the two cases where we do observe a statistically significant status quo bias. Thus, it seems that adjusting for corners would only strengthen the pattern of our findings. Table 6 TYPES OF CORNERS (a) Right corner in part 2 a 1 a 2 a 3 a N Part 1 0 0 1 ...
1 Part 2
0 0 0 ... 0 (b) Right corner in part 1 a 1 a 2 a 3 a N Part 1 0 0 0 ... 0 Part 2 0 0 1 ... 1 (c) Left corner in part 1 a 1 a 2 a 3 a N Part 1 1 1 1 ... 1 Part 2 0 0 1 ... 1 (d) Left corner in part 2 a 1 a 2 a 3 a N Part 1 0 0 1 ... 1 Part 2 1 1 1 ... 1 Table 6 provides examples of corners. The 0 entry represents a choice of the (10, 4) gamble in the question comparing the (10, 4) gamble to alternative a j . An entry of 1 represents the choice of the alternative gamble. The alternatives are monotonically ordered, that is, a j first order stochastically dominates a j−1 . As mentioned earlier, most subjects exhibit an interior switching point. That is, if they choose gamble a j over the (10, 4) gamble they also prefer gamble a j+1
over the (10, 4) gamble. This means that for most subjects all the 0 entries (if any) are located to the left of the 1 entries. This observation allows us to group corners into two categories, right and left corners. A right corner is represented by a full line of 0 entries. In this case the subject would eventually switch to the alternative gamble if offered an attractive enough alternative gamble. However, our 34
finite grid of alternatives does not capture the switching point. Similarly a left corner occurs when, under a certain frame, a subject always chooses gambles from the alternative set. In this case the switching point is located to the left of the grid of alternatives and all entries are 1. Right and left corners may lead to overestimation or underestimation of the bias depending on the frame under which they occur. To see this, consider the case described in table 6(a). The subject exhibits an interior switching point in part 1 and a right corner in part 2. Under a monotonicity assumption, if we were to add an additional gamble to the right of the grid it would turn either into an additional choice exhibiting status quo bias, or into a switch to the alternative gamble in part 2. In the former case, our findings would underestimate the true level of the bias while in the latter it would be accurate. For this reason we say that, in example 6(a), the truncation of the grid is more likely to have produced an underestimation of the bias, rather than an overestimation. Using a similar reasoning the truncation represented in Table 6(b) led more likely to an overestimation of the bias. The list below outlines all possible cases of underestimation and overestimation due to corners. • Left corner in part 1 and/or right corner in part 2: Potential underestimation. • Right corner in part 1 and/or left corner in part 2: Potential overestimation. • Left corner hit in both part 1 and part 2: Neutral. • Right corner hit in both part 1 and part 2: Neutral. Table 7 categorizes corners into instances of overestimation and underestimation and reports the occurrence of corners in all treatments. Notice that percentages of potential overestimation and underestimation are fairly balanced in each treatment. Underestima- tion appears slightly more prominent than overestimation in the two asymmetric treat- 35
Table 7 CORNERS BY TREATMENT R-R R-A
A-R A-A
Underestimation 4.55 (3)
19.7 (17) 12.9 (8)
6.9 (5) Overestimation 3 (2) 16.2 (14) 4.8 (3) 6.9 (5)
Difference 1.55
3.5 8.1
0 No. obs.
33 43 31 36 Percentages of occurences of subjects hitting corners which lead to overestima- tion or underestimation in each treatment (the number of occurences is reported in parenthesis). ments (especially in treatment A-R). This implies that if we were to correct our analysis to account for corners, our main result would only be strengthened. References Bewley, T. F. (1986): “Knightian Decision Theory. Part I,” Cowles Foundation discus- sion paper, 807. Dean, M. (2008): “Status Quo Bias in Large and Small Choice Sets,” Working Paper. Dean, M., O. Kibris, and Y. Masatlioglu (2017): “Limited Attention and Status Quo Bias,” Journal of Economic Theory, Forthcoming. Eisenberger, R. and M. Weber (1995): “Willingness-to-pay and willingness-to-accept for risky and ambiguous lotteries,” Journal of Risk and Uncertainty, 10(3), 223–233. Ericson, K. (2012): “Market Design When Firms Interact with Inertial Consumers: Ev- idence from Medicare Part D,” Working Paper. 36
Gill, D. and V. Prowse (2012): “A Structural Analysis of Disappointment Aversion in a Real Effort Competition,” American Economic Review, 102(1), 469–503. Hartman, R. S., M. J. Doane, and C.-K. Woo (1991): “Consumer Rationality and the Status Quo Bias,” Quarterly Journal of Economics, 106(1), 141–162. Horowitz, J. K. and K. E. McConnell (2002): “A Review of WTA/WTP Studies,” Journal of Environmental Economics and Management, 44(3), 426–447. Johnson, E. J. and D. G. Goldstein (2003): “Do Defaults Save Lives?” Science, 302, 1338–1339. Kahneman, D., J. L. Knetsch, and R. H. Thaler (1991): “Anomalies: The Endow- ment Effect, Loss Aversion, and Status Quo Bias,” Journal of Economic Perspectives, 5(1), 193–206. Kahneman, D. and A. Tversky (1979): “Prospect Theory: An Analysis of Decision Under Risk,” Econometrica, 47(2), 263–292. ——— (1992): “Advances in prospect theory: Cumulative representation of uncertainty,” Journal of Risk and Uncertainty, 5(4), 297–323. Kempf, A. and S. Ruenzi (2006): “Status Quo Bias and the Number of Alternatives: An Empirical Illustration from the Mutual Fund Industry,” Journal of Behavioral Finance, 7(4), 204–213. Knetsch, J. L. (1989): “The Endowment Effect and Evidence of Nonreversible Indiffer- ence Curves,” American Economic Review, 79(5), 1277–1284. Knetsch, J. L. and J. Sinden (1984): “Willingness to Pay and Compensation De- 37
manded: Experimental Evidence of an Unexpected Disparity in Measures of Value,” Quarterly Journal of Economics, 99(3), 507–521. Koszegi, B. and M. Rabin (2006): “A Model of Reference-Dependent Preferences,” The Quarterly Journal of Economics, 121(4), 1133–1165. ——— (2007): “Reference-Dependent Risk Attitudes,” American Economic Review, 97(4), 1047–1073. List, J. A. (2003): “Does Market Experience Eliminate Market Anomalies?” Quarterly Journal of Economics, 118(1), 47–71. ——— (2004): “Neoclassical Theory Versus Prospect Theory: Evidence From the Market- place,” Econometrica, 72(2), 615–625. Madrian, B. C. and D. F. Shea (2001): “The Power of Suggestion: Inertia in 401(k) Participation and Savings Behavior,” Quarterly Journal of Economics, 116(4), 1149– 1187. Maltz, A. (2016): “Exogenous Endowment - Endogenous Reference Point,” Working Paper. Masatlioglu, Y. and E. A. Ok (2014): “A Canonical Model of Choice with Initial Endowments,” The Review of Economic Studies, 81(2), 851–883. Mihm, M. (2016): “Reference dependent ambiguity,” Journal of Economic Theory, 163, 495–524. Munro, A. and R. Sugden (2003): “On the Theory of Reference-Dependent Prefer- ences,” Journal of Economic Behavior and Organization, 50(4), 407–428. 38
Ortoleva, P. (2010): “Status quo bias, multiple priors and uncertainty aversion,” Games and Economic Behavior, 69(2), 411–424. Pope, D. G. and M. E. Schweitzer (2011): “Is Tiger Woods Loss Averse? Persistent Bias in the Face of Experience, Competition, and High Stakes,” American Economic Review, 101(1), 129–157. Redelmeier, D. A. and E. Shafir (1995): “Medical Decision Making in Situations that Offer Multiple Alternatives,” The Journal of the American Medical Association, 273(4), 302–305.
Roca, M., R. M. Hogarth, and A. J. Maule (2006): “Ambiguity seeking as a result of the status quo bias,” Journal of Risk and Uncertainty, 32(3), 175–194. Sagi, J. S. (2006): “Anchored preference relations,” Journal of Economic Theory, 130(1), 283–295.
Samuelson, W. and R. Zeckhauser (1988): “Status Quo Bias in Decision Making,” Journal of Risk and Uncertainty, 1(1), 7–59. Schwartz, B. (2000): “Self-determination: The tyranny of freedom,” The American psychologist, 55(1), 79–88. Sprenger, C. (2015): “An Endowment Effect for Risk: Experimental Tests of Stochastic Reference Points,” Journal of Political Economy, 123(6), 1456–1499. Thaler, R. (1980): “Toward a positive theory of consumer choice,” Journal of Economic Behavior and Organization, 1(1), 39–60. 39 Yüklə 392,99 Kb. Dostları ilə paylaş:
|