D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200 195
to place the responsibility for the loss of transaction on the buyer, and consequently leaves his ask function unaltered.
In summary, the buyer’s bids are accounted for by four parameters: b
t
— the single parameter of the exponential bid function, w
+ t
and w
− t
— the two impact parameters for successful and unsuccessful trades, and d
b
— a parameter discounting the effects of w
+
and w
−
. The seller’s asks are described by the same reinforcement-based learning model.
6. Dynamic analysis
In an attempt to account for the round-to-round decisions of the sellers and buyers in our experiments, the DSR learning model was estimated and tested on the individual rather than
aggregate level. For each of the twenty sellers and buyers in conditions SA and SLA, the four parameters of the learning model were estimated separately so as to best replicate
the individual decisions of that trader. The estimation was conducted in the following manner. For each trader separately, the data were divided into two blocks of 30 and 20
trials respectively. A quasi-Newton search procedure was used to find the set of values of the four parameters that minimize the sum of squared errors between observed and predicted
decisions over the first 30 trials. As is generally the case with such non-linear minimization procedures Hamilton, 1992, there is no guarantee of achieving absolute minimum; the
error function may have several local minima relative to the parameters of both the buyer’s and seller’s models. Using the best set of parameters thus found, the model was tested on
the second block of 20 trials.
Table 5 presents the estimated parameter values for all the 20 traders in condition SA. The top section of the table displays the results for the buyers and the lower part for the sellers.
In each case, the estimated values are based on the first 30 rounds of play. The performance of the fitted model over the last 20 rounds is shown in the two right-hand columns of the
table. Two statistics are used to measure goodness of fit: R
2
, which measures the linear relationship between observed and predicted decisions, and the root mean squared error
RMSE between observed and predicted decisions. Beginning with the sellers in condition SA, the lower panel of Table 5 shows that the
learning model describes the individual asks very well for all except seller 6 our ‘irrational’ subject. Omitting seller 6, the median R
2
is 0.94. The learning model is equally successful in accounting for the buyers’ bids. The top panel of Table 5 shows that the model provides
an excellent description of the last 20 offers in nearly all cases; the R
2
values are above 0.90 for buyers 1, 2, 4, 5, 6, 7, and 9.
Consider next the estimated parameter values in Table 5. For the sellers, the discount values, d
b,
vary between 0.00 and 0.21 with a mean of 0.05, and for the buyers they vary between 0.00 and 0.15 with a mean of 0.04. The difference in mean rate of learning between
the buyers and sellers is not significant t
10
1. Hence, for both buyers and sellers, the impact of profits and lost profits on future bids declines at approximately the same rate,
namely, about five percent per period. Our learning model contains two different weight parameters to represent the differential
effects on subsequent bids of gains from trades successfully completed and opportunity losses from trades missed. The large body of psychological literature on the effects of
196 D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200
Table 5 Individual parameters and fit of learning model for condition SA
Buyer Parameters
Trials 1–30 Trials 31–50
d
b
w
+
w
−
b R
2
RMSE R
2
RMSE 1
0.01 0.0001
0.0500 400
0.89 10.76
0.99 6.80
2 0.01
0.0001 0.0500
1000 0.39
23.90 1.00
6.29 3
0.06 0.0054
0.0500 700
0.93 7.05
0.88 11.08
4 0.02
0.0020 0.0500
700 0.92
7.15 0.96
5.53 5
0.09 0.0062
0.0000 2230
0.94 9.15
0.98 11.15
6 0.00
0.0000 0.0500
800 0.97
5.88 1.00
6.75 7
0.00 0.0000
0.0400 800
0.98 6.42
1.00 5.77
8 0.02
0.0016 0.0106
700 0.89
8.87 0.86
8.56 9
0.02 0.0016
0.0150 1000
0.98 7.75
0.94 6.46
10 0.15
0.0091 0.0000
1000 0.47
25.61 0.81
25.84 Mean
0.04 0.0026
0.0316 933
0.84 11.25
0.94 9.42
Seller d
s
w
+
w
−
s R
2
RMSE R
RMSE 1
0.10 0.0053
0.0000 400
0.94 8.81
0.91 8.53
2 0.00
0.0026 0.0232
788 0.97
16.17 0.81
17.72 3
0.04 0.0002
0.0010 500
0.91 16.84
0.70 49.31
4 0.02
0.0050 0.0500
284 0.89
13.39 0.87
15.22 5
0.04 0.0045
0.0500 250
0.94 7.74
0.98 7.32
6 0.00
0.0000 0.0000
1000 0.38
57.75 0.52
45.21 7
0.05 0.0070
0.0100 175
0.81 12.09
0.94 6.51
8 0.21
0.0079 0.0000
287 0.94
9.23 0.94
9.49 9
0.01 0.0010
0.0200 180
0.97 6.48
0.94 7.17
10 0.05
0.0034 0.0232
200 0.85
12.25 0.94
17.00 Mean
0.05 0.0037
0.0177 406
0.86 16.08
0.86 18.35
reference points suggests stronger effects associated with losses than with gains.
1
In terms of our learning model, the implication is that w
−
w
+
. Table 5 shows that this prediction holds for seven of nine sellers omitting seller 6 again and eight of the 10 buyers. The mean
value of w
−
is 0.0177 for the sellers whereas the mean w
+
is just 0.0037. This difference is significant F = 5.07, p ≤ 0.037. For the buyers, the means are even more significantly
different. The mean w
−
is 0.0316 compared to the mean w
+
at 0.0026 F = 16.5, p ≤ 0.0007.
Tested in condition SLA, the learning model was clearly less successful for the buyers; the model accounts for only a small portion of the variation in their bids. Buyers’ reservation
values were constrained to the narrow interval from 180 to 200, and most buyers appeared to change bids from trial to trial in a way that is largely independent of these values. For
the sellers, the performance of the model was considerably better, but still not as good as for condition SA median R
2
is 0.84. The general observations regarding the impact of the two parameters w
−
and w
+
continue to hold under condition SLA.
1
Post experiment conversation with subjects almost always confirmed this point. Most remarked that ‘losses hurt more’.
D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200 197
6.1. Comparison of information advantage Our study was designed to determine whether or not the information advantage conferred
on buyers in the DSR experiment carried over to situations where sellers were given a corresponding advantage. As we observed in previous sections of the paper, the profit
advantage buyers had in DSR translates into an equivalent advantage for sellers in conditions SA and SLA in the present study. One would then expect this symmetry to hold in the
comparison of the two sets of experiments with respect to the estimated parameter values and the measures of goodness of fit of the learning model. Indeed, we find no significant
differences between the two conditions. Table 6 shows that the learning model accounts equally well for the individual offers of the buyers in Experiment 1 of DSR as it did for
the sellers in condition SA. The R
2
values are above 0.86 for all 10 buyers in Table 6. Discarding seller 6 in condition SA, the difference between the mean R
2
values of the buyers in Experiment 1 of DSR and the sellers in condition SA is not significant t
17
1. Similarly, the mean RMSE scores do not differ significantly across the two sets of subjects.
With the exception of buyer 2 in Table 6, the discount parameter, d
b
, falls in a similar range for all buyers in Experiment 1 of DSR as it does for the sellers in condition SA, and
the mean discount value is again about five percent. On the average, the mean w
−
value
Table 6 Individual parameters and fit of learning model for experiment 1 of DSR
Buyer Parameters
Trials 1–30 Trials 31–50
d
b
w
+
w
−
b R
2
RMSE R
2
RMSE 1
0.04 0.0008
0.0063 250
0.91 13.45
0.95 17.98
2 0.30
0.0016 0.0500
173 0.82
14.36 0.92
9.37 3
0.02 0.0036
0.0294 170
0.94 10.00
0.94 8.10
4 0.03
0.0031 0.0100
600 0.89
15.23 0.93
9.29 5
0.00 0.0031
0.0651 400
0.85 16.57
0.94 10.58
6 0.11
0.0017 0.0299
170 0.94
11.57 0.95
9.60 7
0.02 0.0024
0.0209 115
0.91 9.37
0.96 7.51
8 0.06
0.0012 0.0001
275 0.89
16.64 0.88
9.82 9
0.00 0.0014
0.1000 800
0.89 23.19
0.86 27.74
10 0.03
0.0013 0.1000
500 0.90
22.55 0.95
16.31 Mean
0.06 0.0020
0.0412 345
0.89 15.29
0.93 12.63
Seller d
s
w
+
w
−
s R
2
RMSE R
RMSE 1
0.06 0.0022
0.0052 500
0.83 9.66
0.93 10.05
2 0.10
0.0070 0.0200
400 0.89
7.49 0.76
15.45 3
0.00 0.0026
0.0265 200
0.67 15.03
0.92 14.38
4 0.05
0.0015 0.0138
400 0.37
25.10 0.71
18.37 5
0.12 0.0100
0.1000 500
0.46 27.92
0.44 26.07
6 0.11
0.0100 0.0000
1500 0.24
29.36 0.28
51.34 7
0.11 0.0059
0.0359 349
0.77 8.93
0.73 11.16
8 0.01
0.0002 0.0500
700 0.97
6.44 1.00
3.59 9
0.02 0.0008
0.0700 300
0.97 7.36
0.98 6.17
10 0.03
0.0010 0.0700
300 0.97
7.26 0.98
5.94 Mean
0.06 0.0041
0.0391 515
0.71 14.46
0.77 16.25
198 D.A. Seale et al. J. of Economic Behavior Org. 44 2001 177–200
exceeds the mean w
+
by a factor of six for the sellers and 11 for the buyers. These results are in close agreement with the ones reported earlier in Table 5. We find no significant
differences between the mean w
−
of the sellers in condition SA and the buyers in the DSR study. The same holds for the comparison of these two sets of traders with respect to the
mean of the weight parameter w
+
. Statistical comparisons of the mean estimated values of w
−
and w
+
between the buyers in condition SA and sellers in the DSR study also yield non-significant differences.
7. Discussion and conclusions
