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How far ahead do people plan?
Economics Letters 96 (2007) 8 – 13 www.elsevier.com/locate/econbase
How far ahead do people plan? ☆ John D. Hey a,b,⁎, Julia A. Knoll c a
LUISS, Italy Department of Economics, University of York, Heslington, York, YO10 5DD, United Kingdom c Lehrstuhl IV, Institut für Experimentelle, Psychologie, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
b
Received 17 March 2006; received in revised form 6 December 2006; accepted 12 December 2006 Available online 12 April 2007
Abstract We report on an experiment which enables us to infer how far people plan ahead when taking decisions in a dynamic risky context. Just over half of the subjects plan fully, while the rest do not plan ahead at all. © 2006 Elsevier B.V. All rights reserved. Keywords: Planning; Dominance; Myopia; Naivety; Sophistication JEL classification: D9; C9
1. Introduction Economic theory assumes that people, when taking dynamic decisions, plan right to the planning horizon. Evidence that this might not be so comes from previous experimental investigations (for example, Bone et al. (2003), Carbone and Hey (2001) and Hey (2002)), but the inferences from these have been clouded by doubts concerning the preference functions of the subjects in the experiments. We present here the results from a simple experiment which requires only the assumption that preferences satisfy monotonicity and therefore do not violate dominance. This requires a particular design, which we ☆
We would like to thank the Economic and Social Research Council of the United Kingdom for a Research Fellowship (RES000-27-0077) to John Hey which provided funds to carry out the research and perform the experiment. We would also like to thank an anonymous referee whose comments led to a significant improvement in the paper. ⁎ Corresponding author. Tel.: +44 1904 433789. E-mail addresses: [email protected] (J.D. Hey), [email protected] (J.A. Knoll). 0165-1765/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.12.015
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
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discuss in the next section. We then report on the results from the experiment which suggest that only slightly more than half of the subjects actually do plan to the planning horizon. On the contrary, a little less than a half appears not to plan at all, and very few seem to plan ahead just one period. We speculate on the implications in the concluding section. 2. The experimental design We represent dynamic decision problems in the form of a tree. We use in this experiment what we call 3 + 3 trees. These have three decision nodes interleaved with three chance nodes. The tree starts with a decision node, and subsequently decision nodes are followed by chance nodes and vice versa. After the third chance nodes, there are payoff nodes. In the payoff nodes there are amounts of money which the subject is paid if he or she reaches that node. An example is shown in Fig. 1, in which the subject starts at the bottom and works up through the tree to one of the payoff nodes at the top. At each decision node, there are just two possible decisions — Left or Right. At each chance node, there are just two possible moves – by ‘Nature’ – Left or Right. Subjects are told that Nature moves Left and Right with equal probability and that all moves are independent of each other and of the moves by the subject. Even in this simple 3 + 3 tree branches and payoff nodes proliferate — so that there are 64 of the latter. Each of these contains a payoff denominated in money. The structure of the payoffs in these nodes is crucial to the design of the experiment. We built the 3 + 3 trees used in the experiment from 2 + 2 trees with what we call the dominance property. A 2 + 2 tree is simply a tree with two decision nodes interleaved with two chance nodes; it has 16 payoff nodes. A 3 + 3 tree is obtained by
Fig. 1. A 3 + 3 decision tree.
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J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
Table 1 The probabilities of the various paths through the tree (Model 2 Treatment 1) N.
Probability
Outcome
Probability implied by Model 2
11 12 13 14 15 16 17 18
p11 p12 p13 p14 p15 p16 p17 p18
+++ −++ +−+ ++− −−+ −+− +−− −−−
p0 t1 t2 (1 − t3) + p1 t1 (1 − t2) (1 − t3) + p2(1 − t1) (1 − t2) (1 − t3) p0 (1 − t1) t2 (1 − t3) + p1(1 − t1) (1 − t2) (1 − t3) + p2 t1 (1 − t2) (1 − t3) p0 t1 (1 − t2) (1 − t3) + p1 t1 t2 (1 − t3) + p2(1 − t1) t2 (1 − t3) p0 t1t2t3 + p1 t1 (1 − t2) t3 + p2(1 − t1) (1 − t2) t3 p0(1 − t1) (1 − t2) (1 −t3) + p1(1 − t1) t2 (1 − t3) + p2 t1 t2 (1 − t3) p0(1 − t1) t2 t3 + p1(1 − t1) (1 − t2) t3 + p2 t1 (1 − t2) t3 p0 t1 (1 − t2) t3 + p1 t1t2t3 + p2(1 − t1) t2 t3 p0(1 − t1) (1 − t2) t3 + p1(1 − t1) t2 t3 + p2 t1t2t3
combining four 2 + 2 trees. The dominance property is as follows, where we denote the payoffs in a 2 + 2 tree by xi i = 1, ...,16. The payoffs satisfy the following conditions, where by (y1, y2, …,yn) N (z1, z2, …,zn) we mean that yi⁎ ≥ zi⁎ for i = 1, …,n and that there is at least one i for which yi⁎ N zi⁎, where the set (x1⁎,x2⁎, …, xn⁎) denotes the set (x1,x2, …,xn) arranged in order from the highest to the lowest. 1. 2. 3. 4. 5. 6.
(x1,x2) N (x3,x4). (x5,x6) N (x7,x8). (x9,x10) N (x11,x12). (x13,x14) N (x15,x16). (x1,x2,x5,x6) N (x9,x10,x13,x14). (x9,x10,x11,x12,x13,x14,x15,x16) N (x1,x2,x3,x4,x5,x6,x7,x8).
In a 2 + 2 tree there is one decision node at the first decision level and four at the second. Conditions 1 through 4 imply that Left is the best decision (for a subject whose preferences satisfy dominance) at each of the four second level decision nodes. Condition 5 implies that Left therefore is the best decision at the first decision node for a subject whose preferences satisfy dominance and who plans ahead — thereby eliminating from the tree those payoff nodes that he or she knows will not be reached because of the decisions that will be made at the second level. Condition 6 implies that, for a subject who does not eliminate these payoff nodes, and thus thinks myopically about the implications of the decision at the first level, will choose Right at the first decision node. So one route through the tree (always play Left) is best for a subject who plans ahead while another (Right at the first level and Left at any of the second) appears best for a subject who does not plan ahead. An example is given by the set (15,17,2,4,20,8,8,0,16,8,8,13,6,20,6,18). For this, we have (x1,x2) N (x3,x4) since (15,17) N (2,4), (x5,x6) N (x7,x8) since (20,8) N (8,0), (x9,x10) N (x11,x12) since (16,8) N (8,13) and (x13,x14) N (x15,x16) since (6,20) N (6,18) and hence Left dominates Right at each of the second Table 2 The probabilities of the various paths through the tree (Model 2 Treatment 2) N.
Probability
Outcome
Probability Implied by Model 2
21 22 23 24
p21 p22 p23 p24
⁎++ ⁎−+ ⁎+− ⁎−−
p0 t2 (1 − t3) + (p1 + p2)(1 − t2) (1 − t3) p0(1 − t2) (1 − t3) + (p1 + p2)t2 (1 − t3) p0t2t3 + (p1 + p2)(1 − t2) t3 p0(1 − t2) t3 + (p1 + p2)t2t3
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
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Table 3 The observed and predicted frequencies (Treatment 1) Outcome
+++ −++ +−+ ++− −−+ −+− +−− −−−
Observed frequency
Predicted frequency Model 1
Model 2
Model 3
65 26 27 3 44 3 8 8
63 23 20 11 51 4 3 9
63 25 30 8 47 3 4 6
63 24 20 9 44 5 5 13
decision nodes. Moreover, (x1,x2,x5,x6) N (x9,x10,x13,x14) since (15,17,20,8) N (16,8,6,20) so Left dominates Right at the first decision node for the player who plans ahead; while (x9,x10,x11,x12,x13,x14,x15,x16) N (x1,x2,x3, x4,x5,x6,x7,x8) since (16,8,8,13,6,20,6,18) N (15,17,2,4,20,8,8,0) and so Right appears to dominate Left at the first decision node for the player who does not plan ahead. We construct our 3 + 3 trees from four different 2 + 2 trees with this dominance property. Moreover, we do it in such a way that (1) one route through the tree – LLL – is best for subjects who think two moves ahead (right to the end of the tree), (2) another route through the tree – RLL – appears best for those subjects who plan just one move ahead; and (3) a third route through the tree – RRL – appears best for those subjects who do not think ahead at all. Such a 3 + 3 tree has once again what we call the dominance property.1 We then randomise so that the respective routes through the tree are not the same for all subjects.2 Because of this randomisation, we use from now on the notation +++ to indicate the route chosen by the fully planning ahead person, −++ the route chosen by a subject planning one move ahead and by −−+ the route chosen by a person not planning ahead. Ninety two subjects took part in the experiment equally split between two separate treatments: Treatment 1 in which the dominance at the first level was strong; and Treatment 2 where the dominance was weak, in that either decision at the first decision level was optimal for the person fully planning ahead.3 Usual experimental protocols were observed and the subjects paid accordingly.4 Subjects were allowed four different attempts at the same decision tree with the same payoffs. 3. Results We fitted various models to the data. In the discussion that follows we start by restricting attention to the model that we think best fits the data, but later report the results for other models. Our favoured model – which we call Model 2 – is the following. Let us define a Type 0 subject as one that does not plan ahead at all, a Type 1 subject one who plans ahead just one period, and a Type 2 subject one who plans ahead two periods. We assume that there is a proportion pi of Type i (i = 0,1,2). In order to account for other 1
We can provide more formal statements of this property on request. Obviously the property is retained that one route through the tree is best for those who plan two moves ahead, a second appears best for those who plan just one move ahead and a third appears best for those who do not plan ahead at all. 3 Note, of course, that they had to do the planning ahead to realise this. 4 The written Instructions and full details of the experimental implementation are available on request. 2
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J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
Table 4 The observed and predicted frequencies (Treatment 2) Outcome ⁎++ ⁎−+ ⁎+− ⁎−−
Observed frequency
Predicted frequency Model 1
Model 2
Model 3
86 78 10 10
86 71 15 12
88 76 11 9
88 78 9 11
responses, and to allow for the usual phenomenon that subjects make mistakes when performing the experiment, we allow for trembles at each decision level. Specifically, we assume tremble probabilities of ti at decision level i = 1,2,3. So at decision level i the subject implements the intended decision with probability (1 − ti) and implements the unintended decision with probability ti. Thus we get Table 1 for treatment 1 and Table 2 for treatment 2. Let us denote by Fij the frequency of outcome labelled ij in Tables 1 and 2. We estimate p0, p1, p2, t1, t2 and t3 to minimise the sumPof squared differences P between the observed frequencies and those implied by Tables 1 and 2 — that is 8j¼1 ðF1j −np1j Þ2 þ 4j¼1 ðF2j −np2j Þ2 . Here n denotes the total number of observations in each treatment (equal to 46 — the number of subjects – times 4 – the number of repetitions that each subject performed — giving a total of 184), and pij is the probability of the outcome labelled ij according to the model. We use GAUSS to calculate the estimated parameters. The estimated proportions are p0 = 0.414, p1 = 0.008 and p2 = 0.578. The estimated trembles are t1 = 0.078, t2 = 0.298 and t3 = 0.109. The actual and fitted frequencies of the various outcomes are reported in columns 2 and 4 of Tables 3 and 4. This model seems to fit the data rather well. We fitted other models to the data. Model 1 assumes, as above, that a proportion p0 of the subjects is Type 0, a proportion p1 are Type 1, and a proportion p2 are Type 2. However, in Model 1 we assume that the same tremble probability t applies at each decision node. The estimated parameters are p0 = 0.430, p1 = 0.029, p2 = 0.541 and t = 0.150. The fitted frequencies are reported in column 2 of Tables 3 and 4. It will be noted that the estimated proportions of the subjects of the various Types are similar to those in Model 2, but the fit, as reflected in the fitted frequencies and the implied sum of squares (reported in Table 5) is considerably worse. Model 3 makes the same assumption about the p's but assumes that the trembles vary across types: t0 for Type 0, t1 for Type 1 and t2 for Type 2. The estimated proportions are p0 = 0.516, p1 = 0.025 and p2 = 0.459. The estimated trembles are t0 = 0.232, t1 = 0.066 and t2 = 0.112. The fitted frequencies are reported in column 5 of Tables 3 and 4. It will be noted that the estimated proportions are close to those of both Models 1 and 2 and that the trembles are perhaps more ‘reasonable’, but the overall goodness of fit (reported in Table 5) is significantly worse than that of Model 2. Whichever model we take, the fitted proportions are around 50% for each of Types 0 and 2 and only just positive for Type 1. We prefer Model 2 because of its significantly better fit. Table 5 The goodness of fit of the three models Sum of Squared differences
Model 1
Model 2
Model 3
298.86
72.93
125.16
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
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If we conclude using the results from Model 2, they suggest that some 41.4% of the subjects do not plan ahead at all, some 57.8% plan ahead to the planning horizon (as they should according to economic theory), while almost no-one plans ahead just one period. The conclusion is that the subjects neatly bifurcate into two groups — the planners and the non-planners. The tremble probabilities are interesting — with very small trembles at the first and third decision nodes and a rather large tremble at the second decision nodes. The first node decision appears easy to both the planners and the non-planners, while at the second decision nodes there appear to be lapses of concentration on the part of a considerable number of subjects. Trembles are significantly reduced at the final decision node where the decision is relatively straightforward. In terms of the observed outcomes it will be noted that the majority of observations are in the rows +++, −++, +−+ and −−+ of Table 3 and in rows ⁎++ and ⁎−+ of Table 4. With regard to the former (ignoring the very small number of Type 1 subjects): those in row +++ are Type 0 who make two mistakes and Type 2 who do not make any mistakes; those in row −++ are Type 0 who make a mistake at the second decision node and Type 2 who make a mistake at the first; those in row +−+ are Type 0 who make a mistake at the first decision node and Type 2 who make a mistake at the second; and those in row −−+ are Type 0 who make no mistake and Type 2 who make two mistakes (at the first and the second decision nodes). The observed frequencies in Table 4 can be similarly interpreted. 4. Conclusion The conclusion is straightforward, though we had hoped for greater richness in the data. There are two types of people out there: those who plan completely ahead and those who do not plan. If the same bifurcation is generally the case, then we get the kind of story that is often told in the economics literature — two kinds of agent: naïve and sophisticated. The implications of this have already been explored. This paper gives some evidence that it is a realistic story. References Bone, J.D., Hey, J.D., Suckling, J.R., 2003. Do people plan ahead? Applied Economics Letters 10, 277–280. Carbone, E., Hey, J.D., 2001. A test of the principle of optimality. Theory and Decision 50, 263–281. Hey, J.D., 2002. Experimental economics and the theory of decision making under risk and uncertainty. Geneva Papers on Risk and Insurance Theory 27, 5-2.
How far ahead do people plan? ☆ John D. Hey a,b,⁎, Julia A. Knoll c a
LUISS, Italy Department of Economics, University of York, Heslington, York, YO10 5DD, United Kingdom c Lehrstuhl IV, Institut für Experimentelle, Psychologie, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
b
Received 17 March 2006; received in revised form 6 December 2006; accepted 12 December 2006 Available online 12 April 2007
Abstract We report on an experiment which enables us to infer how far people plan ahead when taking decisions in a dynamic risky context. Just over half of the subjects plan fully, while the rest do not plan ahead at all. © 2006 Elsevier B.V. All rights reserved. Keywords: Planning; Dominance; Myopia; Naivety; Sophistication JEL classification: D9; C9
1. Introduction Economic theory assumes that people, when taking dynamic decisions, plan right to the planning horizon. Evidence that this might not be so comes from previous experimental investigations (for example, Bone et al. (2003), Carbone and Hey (2001) and Hey (2002)), but the inferences from these have been clouded by doubts concerning the preference functions of the subjects in the experiments. We present here the results from a simple experiment which requires only the assumption that preferences satisfy monotonicity and therefore do not violate dominance. This requires a particular design, which we ☆
We would like to thank the Economic and Social Research Council of the United Kingdom for a Research Fellowship (RES000-27-0077) to John Hey which provided funds to carry out the research and perform the experiment. We would also like to thank an anonymous referee whose comments led to a significant improvement in the paper. ⁎ Corresponding author. Tel.: +44 1904 433789. E-mail addresses: [email protected] (J.D. Hey), [email protected] (J.A. Knoll). 0165-1765/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.12.015
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
9
discuss in the next section. We then report on the results from the experiment which suggest that only slightly more than half of the subjects actually do plan to the planning horizon. On the contrary, a little less than a half appears not to plan at all, and very few seem to plan ahead just one period. We speculate on the implications in the concluding section. 2. The experimental design We represent dynamic decision problems in the form of a tree. We use in this experiment what we call 3 + 3 trees. These have three decision nodes interleaved with three chance nodes. The tree starts with a decision node, and subsequently decision nodes are followed by chance nodes and vice versa. After the third chance nodes, there are payoff nodes. In the payoff nodes there are amounts of money which the subject is paid if he or she reaches that node. An example is shown in Fig. 1, in which the subject starts at the bottom and works up through the tree to one of the payoff nodes at the top. At each decision node, there are just two possible decisions — Left or Right. At each chance node, there are just two possible moves – by ‘Nature’ – Left or Right. Subjects are told that Nature moves Left and Right with equal probability and that all moves are independent of each other and of the moves by the subject. Even in this simple 3 + 3 tree branches and payoff nodes proliferate — so that there are 64 of the latter. Each of these contains a payoff denominated in money. The structure of the payoffs in these nodes is crucial to the design of the experiment. We built the 3 + 3 trees used in the experiment from 2 + 2 trees with what we call the dominance property. A 2 + 2 tree is simply a tree with two decision nodes interleaved with two chance nodes; it has 16 payoff nodes. A 3 + 3 tree is obtained by
Fig. 1. A 3 + 3 decision tree.
10
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
Table 1 The probabilities of the various paths through the tree (Model 2 Treatment 1) N.
Probability
Outcome
Probability implied by Model 2
11 12 13 14 15 16 17 18
p11 p12 p13 p14 p15 p16 p17 p18
+++ −++ +−+ ++− −−+ −+− +−− −−−
p0 t1 t2 (1 − t3) + p1 t1 (1 − t2) (1 − t3) + p2(1 − t1) (1 − t2) (1 − t3) p0 (1 − t1) t2 (1 − t3) + p1(1 − t1) (1 − t2) (1 − t3) + p2 t1 (1 − t2) (1 − t3) p0 t1 (1 − t2) (1 − t3) + p1 t1 t2 (1 − t3) + p2(1 − t1) t2 (1 − t3) p0 t1t2t3 + p1 t1 (1 − t2) t3 + p2(1 − t1) (1 − t2) t3 p0(1 − t1) (1 − t2) (1 −t3) + p1(1 − t1) t2 (1 − t3) + p2 t1 t2 (1 − t3) p0(1 − t1) t2 t3 + p1(1 − t1) (1 − t2) t3 + p2 t1 (1 − t2) t3 p0 t1 (1 − t2) t3 + p1 t1t2t3 + p2(1 − t1) t2 t3 p0(1 − t1) (1 − t2) t3 + p1(1 − t1) t2 t3 + p2 t1t2t3
combining four 2 + 2 trees. The dominance property is as follows, where we denote the payoffs in a 2 + 2 tree by xi i = 1, ...,16. The payoffs satisfy the following conditions, where by (y1, y2, …,yn) N (z1, z2, …,zn) we mean that yi⁎ ≥ zi⁎ for i = 1, …,n and that there is at least one i for which yi⁎ N zi⁎, where the set (x1⁎,x2⁎, …, xn⁎) denotes the set (x1,x2, …,xn) arranged in order from the highest to the lowest. 1. 2. 3. 4. 5. 6.
(x1,x2) N (x3,x4). (x5,x6) N (x7,x8). (x9,x10) N (x11,x12). (x13,x14) N (x15,x16). (x1,x2,x5,x6) N (x9,x10,x13,x14). (x9,x10,x11,x12,x13,x14,x15,x16) N (x1,x2,x3,x4,x5,x6,x7,x8).
In a 2 + 2 tree there is one decision node at the first decision level and four at the second. Conditions 1 through 4 imply that Left is the best decision (for a subject whose preferences satisfy dominance) at each of the four second level decision nodes. Condition 5 implies that Left therefore is the best decision at the first decision node for a subject whose preferences satisfy dominance and who plans ahead — thereby eliminating from the tree those payoff nodes that he or she knows will not be reached because of the decisions that will be made at the second level. Condition 6 implies that, for a subject who does not eliminate these payoff nodes, and thus thinks myopically about the implications of the decision at the first level, will choose Right at the first decision node. So one route through the tree (always play Left) is best for a subject who plans ahead while another (Right at the first level and Left at any of the second) appears best for a subject who does not plan ahead. An example is given by the set (15,17,2,4,20,8,8,0,16,8,8,13,6,20,6,18). For this, we have (x1,x2) N (x3,x4) since (15,17) N (2,4), (x5,x6) N (x7,x8) since (20,8) N (8,0), (x9,x10) N (x11,x12) since (16,8) N (8,13) and (x13,x14) N (x15,x16) since (6,20) N (6,18) and hence Left dominates Right at each of the second Table 2 The probabilities of the various paths through the tree (Model 2 Treatment 2) N.
Probability
Outcome
Probability Implied by Model 2
21 22 23 24
p21 p22 p23 p24
⁎++ ⁎−+ ⁎+− ⁎−−
p0 t2 (1 − t3) + (p1 + p2)(1 − t2) (1 − t3) p0(1 − t2) (1 − t3) + (p1 + p2)t2 (1 − t3) p0t2t3 + (p1 + p2)(1 − t2) t3 p0(1 − t2) t3 + (p1 + p2)t2t3
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
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Table 3 The observed and predicted frequencies (Treatment 1) Outcome
+++ −++ +−+ ++− −−+ −+− +−− −−−
Observed frequency
Predicted frequency Model 1
Model 2
Model 3
65 26 27 3 44 3 8 8
63 23 20 11 51 4 3 9
63 25 30 8 47 3 4 6
63 24 20 9 44 5 5 13
decision nodes. Moreover, (x1,x2,x5,x6) N (x9,x10,x13,x14) since (15,17,20,8) N (16,8,6,20) so Left dominates Right at the first decision node for the player who plans ahead; while (x9,x10,x11,x12,x13,x14,x15,x16) N (x1,x2,x3, x4,x5,x6,x7,x8) since (16,8,8,13,6,20,6,18) N (15,17,2,4,20,8,8,0) and so Right appears to dominate Left at the first decision node for the player who does not plan ahead. We construct our 3 + 3 trees from four different 2 + 2 trees with this dominance property. Moreover, we do it in such a way that (1) one route through the tree – LLL – is best for subjects who think two moves ahead (right to the end of the tree), (2) another route through the tree – RLL – appears best for those subjects who plan just one move ahead; and (3) a third route through the tree – RRL – appears best for those subjects who do not think ahead at all. Such a 3 + 3 tree has once again what we call the dominance property.1 We then randomise so that the respective routes through the tree are not the same for all subjects.2 Because of this randomisation, we use from now on the notation +++ to indicate the route chosen by the fully planning ahead person, −++ the route chosen by a subject planning one move ahead and by −−+ the route chosen by a person not planning ahead. Ninety two subjects took part in the experiment equally split between two separate treatments: Treatment 1 in which the dominance at the first level was strong; and Treatment 2 where the dominance was weak, in that either decision at the first decision level was optimal for the person fully planning ahead.3 Usual experimental protocols were observed and the subjects paid accordingly.4 Subjects were allowed four different attempts at the same decision tree with the same payoffs. 3. Results We fitted various models to the data. In the discussion that follows we start by restricting attention to the model that we think best fits the data, but later report the results for other models. Our favoured model – which we call Model 2 – is the following. Let us define a Type 0 subject as one that does not plan ahead at all, a Type 1 subject one who plans ahead just one period, and a Type 2 subject one who plans ahead two periods. We assume that there is a proportion pi of Type i (i = 0,1,2). In order to account for other 1
We can provide more formal statements of this property on request. Obviously the property is retained that one route through the tree is best for those who plan two moves ahead, a second appears best for those who plan just one move ahead and a third appears best for those who do not plan ahead at all. 3 Note, of course, that they had to do the planning ahead to realise this. 4 The written Instructions and full details of the experimental implementation are available on request. 2
12
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
Table 4 The observed and predicted frequencies (Treatment 2) Outcome ⁎++ ⁎−+ ⁎+− ⁎−−
Observed frequency
Predicted frequency Model 1
Model 2
Model 3
86 78 10 10
86 71 15 12
88 76 11 9
88 78 9 11
responses, and to allow for the usual phenomenon that subjects make mistakes when performing the experiment, we allow for trembles at each decision level. Specifically, we assume tremble probabilities of ti at decision level i = 1,2,3. So at decision level i the subject implements the intended decision with probability (1 − ti) and implements the unintended decision with probability ti. Thus we get Table 1 for treatment 1 and Table 2 for treatment 2. Let us denote by Fij the frequency of outcome labelled ij in Tables 1 and 2. We estimate p0, p1, p2, t1, t2 and t3 to minimise the sumPof squared differences P between the observed frequencies and those implied by Tables 1 and 2 — that is 8j¼1 ðF1j −np1j Þ2 þ 4j¼1 ðF2j −np2j Þ2 . Here n denotes the total number of observations in each treatment (equal to 46 — the number of subjects – times 4 – the number of repetitions that each subject performed — giving a total of 184), and pij is the probability of the outcome labelled ij according to the model. We use GAUSS to calculate the estimated parameters. The estimated proportions are p0 = 0.414, p1 = 0.008 and p2 = 0.578. The estimated trembles are t1 = 0.078, t2 = 0.298 and t3 = 0.109. The actual and fitted frequencies of the various outcomes are reported in columns 2 and 4 of Tables 3 and 4. This model seems to fit the data rather well. We fitted other models to the data. Model 1 assumes, as above, that a proportion p0 of the subjects is Type 0, a proportion p1 are Type 1, and a proportion p2 are Type 2. However, in Model 1 we assume that the same tremble probability t applies at each decision node. The estimated parameters are p0 = 0.430, p1 = 0.029, p2 = 0.541 and t = 0.150. The fitted frequencies are reported in column 2 of Tables 3 and 4. It will be noted that the estimated proportions of the subjects of the various Types are similar to those in Model 2, but the fit, as reflected in the fitted frequencies and the implied sum of squares (reported in Table 5) is considerably worse. Model 3 makes the same assumption about the p's but assumes that the trembles vary across types: t0 for Type 0, t1 for Type 1 and t2 for Type 2. The estimated proportions are p0 = 0.516, p1 = 0.025 and p2 = 0.459. The estimated trembles are t0 = 0.232, t1 = 0.066 and t2 = 0.112. The fitted frequencies are reported in column 5 of Tables 3 and 4. It will be noted that the estimated proportions are close to those of both Models 1 and 2 and that the trembles are perhaps more ‘reasonable’, but the overall goodness of fit (reported in Table 5) is significantly worse than that of Model 2. Whichever model we take, the fitted proportions are around 50% for each of Types 0 and 2 and only just positive for Type 1. We prefer Model 2 because of its significantly better fit. Table 5 The goodness of fit of the three models Sum of Squared differences
Model 1
Model 2
Model 3
298.86
72.93
125.16
J.D. Hey, J.A. Knoll / Economics Letters 96 (2007) 8–13
13
If we conclude using the results from Model 2, they suggest that some 41.4% of the subjects do not plan ahead at all, some 57.8% plan ahead to the planning horizon (as they should according to economic theory), while almost no-one plans ahead just one period. The conclusion is that the subjects neatly bifurcate into two groups — the planners and the non-planners. The tremble probabilities are interesting — with very small trembles at the first and third decision nodes and a rather large tremble at the second decision nodes. The first node decision appears easy to both the planners and the non-planners, while at the second decision nodes there appear to be lapses of concentration on the part of a considerable number of subjects. Trembles are significantly reduced at the final decision node where the decision is relatively straightforward. In terms of the observed outcomes it will be noted that the majority of observations are in the rows +++, −++, +−+ and −−+ of Table 3 and in rows ⁎++ and ⁎−+ of Table 4. With regard to the former (ignoring the very small number of Type 1 subjects): those in row +++ are Type 0 who make two mistakes and Type 2 who do not make any mistakes; those in row −++ are Type 0 who make a mistake at the second decision node and Type 2 who make a mistake at the first; those in row +−+ are Type 0 who make a mistake at the first decision node and Type 2 who make a mistake at the second; and those in row −−+ are Type 0 who make no mistake and Type 2 who make two mistakes (at the first and the second decision nodes). The observed frequencies in Table 4 can be similarly interpreted. 4. Conclusion The conclusion is straightforward, though we had hoped for greater richness in the data. There are two types of people out there: those who plan completely ahead and those who do not plan. If the same bifurcation is generally the case, then we get the kind of story that is often told in the economics literature — two kinds of agent: naïve and sophisticated. The implications of this have already been explored. This paper gives some evidence that it is a realistic story. References Bone, J.D., Hey, J.D., Suckling, J.R., 2003. Do people plan ahead? Applied Economics Letters 10, 277–280. Carbone, E., Hey, J.D., 2001. A test of the principle of optimality. Theory and Decision 50, 263–281. Hey, J.D., 2002. Experimental economics and the theory of decision making under risk and uncertainty. Geneva Papers on Risk and Insurance Theory 27, 5-2.