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Incentives in simple bargaining games
Journal of Economic North-Holland
Psychology
263
13 (1992) 263-276
Incentives in simple bargaining games Martin
Sefton
*
Unkersity
of Iowa, Iowa City, USA
Received
January
8, 1992; accepted
March
22, 1992
One method of compensating subjects in experiments is to randomly select a subset of subjects to be paid according to their decisions. The rationale for this is that. according to economic theory, subjects in such an experiment will make the same decisions as they would if they were to be paid with certainty. Thus, a ‘random pay’ experiment provides the same information as a ‘full pay’ experiment, but at a lower cost to the experimenter. This invariance hypothesis is tested by comparing the outcomes of full pay and random pay experiments. In experiments with $5 dictator games the hypothesis is rejected. Subjects are more generous when 25% of the subjects are paid their shares of the $5 than when all of them are paid their shares of the $5. This contrasts with the results of Belle (1990), who accepted the hypothesis for ultimatum games.
Introduction This paper compares different methods of compensating subjects in economics experiments. Specifically, it investigates the effect of paying a randomly selected subset of subjects, rather than paying all subjects, according to their decisions at the end of a simple bargaining game experiment. The randomized payment procedure has been used in a variety of experiments as a means of conducting low cost experiments. The rationale for its use is that, according to theory, subjects in such experiments will behave as they would if they were to be paid with certainty. While the cost advantages of only paying some subjects is obvious, the invariance of behavior is not. In this paper the invariance hypothesis is tested by comparing outcomes of experiments which employ the * I wish to thank Bob Forsythe, comments and advice. The support Correspondence to: M. Sefton, Iowa City, IA 52242, USA.
0167-4870/92/$05.00
Joel Horowitz, John Kennan and Gene Savin for helpful of NSF grant SES 89-22460 is gratefully acknowledged. Dept. of Economics, University of Iowa, 679 Phillips Hall,
0 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
264
M. Sefton / Incentic>esin bargaining games
randomised payment procedure with the outcomes of experiments in which all subjects are paid. Smith (1976) compared the outcomes of market experiments when some and when all subjects were paid their earnings. He found that transaction prices did not converge to predicted prices as quickly under the ‘weaker’ random pay reward structure. In contrast, Bolle (1990) tested and accepted the invariance hypothesis for ultimatumtype games with moderate stakes. He noted that if this result holds on a broader empirical base the implications are far-reaching. For instance, in order to collect observations on how each of 100 subjects perform a $1000 reward allocation task the experimenter could select one subject at the end of the experiment to be paid according to his or her decision, at a cost to the research budget of $1000. Under the invariance hypothesis this experiment yields the same information as one which pays all 100 subjects at a cost to the research budget of $100,000. Another reason for examining the effect of this procedure is to make more informed comparisons between experiments. Many experiments involve a variation on the design or parameters of an earlier experiment. However, if the new experiment employs the randomized pay procedure while the earlier one did not, any comparison of the two experiments may be confounded by the variation in the incentive structure offered to subjects. Different experimental outcomes may be due to variation in some parameter of interest, but they may also reflect the different incentives. For instance, Giith and Tietz (1990) speculate that differences between the outcomes of experiments by Giith et al. (1982) and Kahneman et al. (1986) are ‘mainly due to the way of performing experiments’ (e.g. only a random sample of subjects were paid in the Kahneman et al. experiment). Whether this is, in fact, the reason for the different outcomes requires information about the effect of different incentives. The experiments reported in this paper provide evidence against the invariance hypothesis. The experiments involve dictator games, which will now be described. Dictator games The experiments considered here involve dictator games as studied by Forsythe et al. (1988). A dictator game is a two-player game in
M. Sefton
/ Incentiues
in bargaining
games
265
which a sum of money (a ‘pie’) is to be divided and one of the players (the dictator) chooses the division. It differs from an ultimatum game in that the second player cannot reject the proposed division. In experiments with these games all dictators have the same set of possible actions - they propose how much they and their partner each receive. Observed actions reflect dictators’ preferences over outcomes of the experiment and beliefs about how actions translate into outcomes. If preferences and beliefs vary from subject to subject, the offers observed in an experiment can be regarded as arising from applying a set of procedures to a sample from the pool of potential subjects. These procedures would include the rules of the game, the size of the pie, the way subjects communicate, and so on. The procedure of interest in this paper is the way experimental earnings are related to actions - we wish to see how the distribution of outcomes depends on the way subjects are paid. In a ‘full pay experiment’ all subjects are paid according to their decisions. This pay procedure elicits a truthful response to the question ‘How would you divide the sum of money?’ by actually confronting each dictator with the task. In a ‘no pay experiment’ none of the subjects are paid according to their decisions. The subjects have no incentive to give a truthful response to the hypothetical question. In a ‘random pay experiment’ at the end of the experiment the experimenter randomly selects a subset of the subjects to be paid according to their decisions. Thus, a random pay experiment may be thought of as one in which some subjects take part in a full pay experiment and some no a no pay experiment, although all subjects must make their decisions before they find out which experiment they are participating in. If subjects maximize expected utility, then behavior in these games will be independent of whether subjects are paid their earnings or receive them with some (positive) probability. Bolle (1990) discusses the theoretical conditions under which this invariance property holds. These conditions are quite general; in particular we do not require utility functions to depend only on own earnings. Let a subject’s expected utility from offering x in a full pay experiment be u(x), and the expected utility from offering x in a no pay experiment be U(X). In a random pay experiment a subject will choose x to maximize P(X) + (1 -p)u(x) where I, is the probability that the subject is selected in the lottery. In the event that the subject is not selected the
266
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result of any action will be the same, that is u(x) is constant with respect to X. Thus a subject in a random pay experiment should choose x to maximize pu(x> + constant, or equivalently to maximize U(X). The key assumptions are: (11 A subject can costlessly evaluate the (subjective) payoff from any action. (2) In a no pay experiment a subject gets the same (subjective) payoff from any action. With regard to the first assumption, what really matters are the magnitude of decision costs relative to the expected utility gain from making better choices. By making the expected amount to be divided (the pie times the probability with which a subject is paid) sufficiently large relative to decision costs an experiment should provide expected utility gains sufficiently large to offset any decision costs. The expected amount to be divided is kept above a dollar in the random pay experiments investigated here, which is comparable to the expected amount to be divided in Bolle’s (1990) experiments. Bolle presumes that the decision costs in ultimatum games are small. Presumably the decision costs in dictator games are even smaller. With regard to the second assumption, Bolle argues that this will be satisfied if anonymity of those not selected to be paid is enforced. In the random pay experiments investigated here all subjects are anonymous with respect to their partner, although not with respect to the experimenter. According to the invariance hypothesis differences between samples of offers from random pay and full pay experiments should reflect only sampling variation. This paper tests this proposition by comparing the outcomes of a $5 random pay dictator game experiment with the outcomes of a $5 full pay dictator game experiment. The experimental procedures for each experiment were identical except that at the end of the random pay experiment a lottery was conducted to determine which pairs were paid their shares of the pie, whereas at the end of the full pay experiment all pairs were paid their shares of the pie. Review of previous results Random pay procedures were suggested by Giith et al. (1982) as a means of investigating bargaining games at low cost. A procedure
M. Sefion / Incentives in bargaining games
267
essentially the same as the one investigated here was used in Kahneman et al. (1986). However, there have been few investigations of the effect of these procedures on experimental outcomes. The effect of different pay procedures is investigated directly by Bolle (1990) and Forsythe et al. (1988). Bolle compared the outcomes of 20 DM ultimatum games (worth about $10) under random pay and full pay procedures. In the random pay experiment 2 out of 20 subjects were randomly chosen to be paid at the end of the experiment. In the full pay experiment all 24 subjects were paid. The hypothesis that the distributions of offers in experiments with full pay and random pay are the same was accepted. Forsythe et al. (1988) conducted 20 full pay and 24 no pay ultimatum games with $5 to be divided. The hypothesis that the distributions of offers in experiments with full pay and no pay are the same was accepted. As an empirical issue it might at first appear that there is no need to pay any subjects according to their decisions. However, when the experiments were repeated five months later the distribution of offers in no pay games was significantly different from that in April. (The full pay experiments did replicate.) Thus the replicability of ultimatum game experiments appears to be dependent upon the incentives of the experiment. Forsythe et al. also found significant differences between the outcomes of full pay and no pay experiments with $5 dictator games. 48% of the offers were for half the pie and 13% were for nothing in the no pay experiment, 18% of the offers were for half the pie and 36% were for nothing in the full pay experiment. This suggests that whether subjects are paid or not matters a great deal in dictator games.
Hypotheses
and statistical
power
All of the statistical tests used here are tests of the equality of two distributions of offers. Denoting the distribution of offers in one experiment by F, and the distribution in another experiment by F2, the null and alternative hypotheses are: H,: F,(x)
= F*(x)
for all X,
H,: F,(x)
#F*(x)
for some x.
268
M. Sefton
/ IncentiL>es in bargaining games
Note that this is a test of equality of the entire distributions, as opposed to just equality of the means. Focussing on means alone can be misleading, since two distributions may differ in important respects yet still have the same mean. The tests of H, considered were the Anderson-Darling (AD), Cramer-von Mises (CM), Kolmogorov-Smirnov (KS), Epps-Singleton characteristic function (CF), and Wilcoxon Rank-Sum (RS) tests. The AD and CM tests are described by Shorack and Wellner (19861, the KS test by Kim and Jennrich (19731, the CF test by Epps and Singleton (1986) and the RS test by Wilcoxon et al. (1973). Also, one-sided versions of the KS and RS tests (referred to as the KS+ and RS, tests) were considered. In the Forsythe et al. dictator games no pay offers tended to be larger than full pay offers. If random pay offers tend to be in between full pay and no pay offers then one would expect random pay offers to tend to be larger than full pay offers. The directional tests are appropriate for testing the null hypothesis of invariance against the alternative that the random pay distribution stochastically dominates the full pay distribution. These tests require that the data be ordered, to do this ties were randomly broken. Choice among these tests was based on a Monte Carlo investigation of their powers. I first assumed that the distribution of full pay offers is the same as the sample distribution of full pay offers from the Forsythe et al. experiments. Next I assumed that the distribution of random pay offers is a probability mixture of the distributions of full pay and no pay offers, where the distribution of no pay offers is the same as the sample distribution of no pay offers from the Forsythe et al. experiments. I used these distributions to generate 2500 samples of II random pay offers and m full pay offers. Since the random pay offers would be compared with the Forsythe et al. full pay offers m was chosen to be the same as the FHSS sample size, 45. Initially, II was chosen to be 20. The probability with which a random pay offer is sampled from the full pay distribution is denoted by h and initially was chosen to be one. For each pair of samples the statistics were computed and compared with critical values. For each test the proportion of rejections is an estimate of its power. The process was repeated with different values of A and for IZ = 40. When A = 1 random pay offers and full pay offers are drawn from the same distribution, so that the power estimates are estimates of the size of the test. When A < 1 the invariance hypothesis
M. Sefton / Incentives in bargaining games
269
is false: random pay offers are drawn from a distribution in between the full pay and no pay distributions. Tables 1 and 2 display the results of the Monte Carlo simulations. Table 1 shows that either the KS or CM test has highest power among the five tests against H,, therefore these test statistics will be used for the analysis of the experiments. Table 2 reports the powers of the KS+ and RS, tests. As might be expected the powers are higher than for the other tests since they exploit information about the direction of departure from the null hypothesis. The KS+ test appears to perform slightly better than the RS, test and so the KS+ test will be used for a directional test. The tables also show that the powers are sensitive to A. This is important for the following reason. Suppose the invariance hypothesis is false, and that behavior in a random pay experiment is closer to behavior in a full pay experiment the higher the probability with which subjects are paid, p. Then as p decreases subjects’ behavior may be represented more closely by the alternatives with higher values of A. Thus by reducing p the experiment is more likely to detect the falsity of the invariance hypothesis. With moderate stakes and p = 0.01 one would not be surprised if the invariance hypothesis were rejected. A rejection could be attributed to decision costs: although small they may be large compared to the expected utility gain from making better decisions. An informative experiment requires a value of p that represents an appreciable cost saving while maintaining a reward structure in which decision costs are small relative to the variation in expected utility following from making different decisions. In order to achieve the second objective a lower bound on p was chosen that maintained an ‘expected pie’ of $1. This is comparable to the expected pie of Bolle’s (1990) ultimatum game, and presumably decision costs are lower in dictator than in ultimatum games. Thus, paying one in five subjects their share of a $5 pie is suggested as an informative test of the invariance hypothesis. In fact, one in four subjects were paid.
Experimental
procedures
and results
An additional set of full pay dictator games was conducted in order to ensure comparability between the new and old experiments. The hypothesis that the new full pay offers were drawn from the same
n
20 20 20 20 20 20 20 20 20 20 20
40 40 40 40 40 40 40 40 40 40 40
A
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
45 45 45 45 45 45 45 45 45 45 45
45 45 45 45 45 45 45 45 45 45 45
m
0.0860 0.1060 0.1740 0.2572 0.3912 0.5332 0.6832 0.8064 0.8968 0.9512 0.9876
0.0472 0.0560 0.1028 0.1748 0.2868 0.4168 0.5672 0.7172 0.8360 0.9244 0.9676
0.1052 0.1184 0.1864 0.2784 0.4040 0.5452 0.6872 0.7996 0.8896 0.9464 0.9828
0.1084 0.1256 0.1592 0.2184 0.3008 0.4132 0.5308 0.6272 0.7528 0.8412 0.9264
0.10
0.0384 0.0568 0.0712 0.1188 0.1784 0.2748 0.3844 0.4844 0.6760 0.7512 0.8656
0.05
0.10
0.0968 0.1056 0.1516 0.2048 0.2972 0.4028 0.5184 0.6184 0.7500 0.8484 0.9316
Cramer-Von (CM)
Kolmogorov-Smirnov
(KS)
Table 1 Dictator game: Powers of tests of H, against H,.
0.0496 0.0652 0.1136 0.1748 0.2940 0.4124 0.5544 0.6904 0.8144 0.9056 0.9572
0.0504 0.0612 0.0868 0.1388 0.2072 0.2948 0.4116 0.4952 0.6372 0.7416 0.8528
0.05
Mises
0.1000 0.1116 0.1848 0.2592 0.3800 0.5100 0.6436 0.7616 0.8656 0.9336 0.9756
0.1052 0.1248 0.1540 0.2136 0.2784 0.3828 0.5056 0.5864 0.7172 0.8072 0.9068
0.10
0.0504 0.0648 0.1088 0.1688 0.2704 0.3860 0.5152 0.6544 0.7836 0.8828 0.9504
0.0612 0.0828 0.1304 0.1944 0.2728 0.3860 0.4604 0.5944 0.7072 0.8220
0.05
Anderson-Darling (AD)
0.0760 0.0788 0.1324 0.2184 0.3260 0.4708 0.6156 0.7708 0.8724 0.9316 0.9804
0.0812 0.0864 0.1160 0.1644 0.2292 0.3276 0.4464 0.5500 0.7032 0.8020 0.9040
0.10
(CF)
0.0352 0.0392 0.0740 0.1336 0.2228 0.3364 0.4836 0.6560 0.8032 0.8860 0.9576
0.0416 0.0484 0.0656 0.0944 0.1468 0.2248 0.3336 0.4340 0.5876 0.7092 0.8352
0.05
Epps-Singleton
Wilcoxon
0.0904 0.1220 0.1688 0.2636 0.3580 0.4916 0.6156 0.7484 0.8456 0.8996 0.9592
0.0976 0.0976 0.1432 0.1968 0.2920 0.3644 0.4672 0.5752 0.6684 0.7640 0.8624
0.10
(RS) 0.05
0.0448 0.0668 0.0960 0.1672 0.2488 0.3664 0.4944 0.6448 0.7484 0.8420 0.9208
0.0472 0.0468 0.0852 0.1244 0.1852 0.2528 0.3392 0.4556 0.5556 0.6568 0.7792
271
M. Sefton / Incentives in bargaining games Table 2 Dictator game: Powers of directional tests of H,. h
n
m
KS+
RS+
0.10
0.05
0.10
0.05
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
20 20 20 20 20 20 20 20 20 20 20
45 45 45 45 45 45 45 45 45 45 45
0.0992 0.1508 0.2064 0.3212 0.4196 0.5604 0.6548 0.7736 0.8340 0.9192 0.9568
0.0476 0.0776 0.1268 0.2128 0.2932 0.4188 0.5232 0.6488 0.7460 0.8548 0.9116
0.0880 0.1580 0.2376 0.3036 0.4140 0.5424 0.6144 0.7060 0.8060 0.8776 0.9284
0.0428 0.0876 0.1404 0.1916 0.2796 0.3860 0.4568 0.5676 0.6896 0.7648 0.8588
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
40 40 40 40 40 40 40 40 40 40 40
45 45 45 45 45 45 45 45 45 45 45
0.0912 0.1672 0.2752 0.4136 0.5344 0.6656 0.8064 0.8892 0.9580 0.9816 0.9936
0.0512 0.0980 0.1736 0.3004 0.4068 0.5584 0.7052 0.8216 0.9120 0.9612 0.9820
0.0932 0.1780 0.2640 0.3740 0.5176 0.6152 0.7564 0.8460 0.9132 0.9564 0.9776
0.0416 0.1060 0.1612 0.2548 0.3816 0.4884 0.6268 0.7408 0.8392 0.9184 0.9524
distribution as the full pay offers from Forsythe et al. (1988) was accepted (see the first row of table 3) and so the full pay data were pooled giving 69 full pay observations. The random pay experiments were conducted with 24 pairs of subjects. Except for the way subjects were paid, the same procedures were used as in the experiments of Forsythe et al. The subjects for the random pay experiments were recruited from a subject pool consisting of students from undergraduate business classes and MBA classes at the University of Iowa who had completed a form indicating interest in participating in experiments. When called and asked to participate, they were told that they ‘would earn between $3 and $8 for participating in an experiment that would last about 20 minutes’. We used two connecting rooms, each of which could accommodate 8 individuals. Subjects were assigned to rooms randomly when they were recruited. At the beginning of a session, subjects were given
272 Table 3 Tests of invariance
M. Sefton / Incentic~esin bargaining games
hypotheses.
a Sample
size
Statistic
(p-value)
CM
KS
45 (1988) 24 (1990)
0.0780 (0.70)
0.2827 (0.4731)
Full pay dictator game offers vs. random pay dictator game offers
69 (Full) 24 (Random)
0.8062 (0.01)
0.3786 (0.0085)
0.3786 (0.0042)
No pay dictator game offers vs. random pay dictator game offers
46 (No) 24 (Random)
0.0455 (0.90)
0.1250 (0.9292)
0.1214 (0.5703)
Forsythe et al. (1988) full pay dictator game offers vs. new full pay dictator game offers
KS+
a The limiting null distribution of the CM test as tabled in Shorack and Wellner (1986) was used to provide p-values for CM statistics. Exact p-values for the KS and KS+ statistics were computed following the methods of Kim and Jennrich (1973).
instruction sets. The experimenter read the instructions aloud and answered questions. All subjects in the same session in the same room faced the same task (that is, they were either all senders or all receivers of a proposal). Each person was randomly and anonymously paired with someone in the other room, and each pair was given a $5 pie to divide. Communication between members of a pair was by a written proposal form that was carried between rooms by the experimenter. Each subject in room A chose how to allocate the $5; the subjects in room B could benefit from the allocation but had no decision to make. At the end of the experiment each subject was paid $3 for participating and a lottery was conducted to select 2 of the 8 pairs who would also receive their shares of the pie. Each session was completed in less than 20 minutes. Figures 1 and 2 are histograms of the full pay and random pay offers. The outcomes of the random pay experiment appear to be very different from the outcomes of the full pay experiment. In particular, the offers appear to be more generous in the random pay experiment. The modal offer is $2.50 in the random pay experiment and nothing in the full pay experiment, the median offers are $2.50 and $1, and the mean offers are $1.83 and $1.09. Formal tests of the invariance hypothesis are reported in the second row of table 3. The invariance
273
M. Sefton / Incentives in bargaining games
.s
.4
.3
.2
.I
0
I
I
0 Fig. 1. Distribution
1
I
2
I
offer in $5
3
I
I
4
5
of offers made in $5 full-pay dictator games, using 45 observations Forsythe et al. (1988) and 24 new observations.
from
.5 -
.4 -
.3 -
.‘-O .2 o- -
.“. 0
Fig. 2. Distribution
b 1
cl , 2
offer in $s
3
of observations made in random pay dictator ity p equal to 0.25. 24 observations.
4 games, with payment
5 probabil-
274
M. Sefton
.5
.A
/ Incentives
in bargaining
games
1 I
cl Fig. 3. Distribution
0
cl
I
I
1
2 offer
I
I
I
3
A
5
in $5
of offers made in no pay dictator games reported observations.
by Forsythe
et al. (1988). 46
hypothesis is rejected by each test at conventional significance levels, indicating that these differences cannot be attributed to sampling variation. The directional test confirms that random pay offers tend to be significantly larger than full pay offers. The saving in cost is substantial (in the random pay experiment subject fees were $90 less than they would have been had all subjects been paid their earnings) but it brings with it considerable changes in behavior. For comparison, figure 3 is a histogram of the Forsythe et al. no pay offers. The outcomes of the no pay experiment appear to be very similar to the outcomes of the random pay experiment. The average offer in the no pay experiment was $1.91 compared with $1.83 in the random pay experiment, the median and modal offers in the no pay experiment were $2.50, the same as in the random pay experiment. Formal tests do not detect statistically significant differences between the no pay and random pay distributions (see the third row of table 3).
M. Sefton / Incenliues in bargaining games
275
Conclusion The procedure of paying a randomly selected subset of subjects was suggested by Giith et al. (1982) as a means of investigating high stakes bargaining game experiments at low cost. A number of experimenters have used the procedure in moderate stakes bargaining games for this reason. Whether subjects behave the same way in a random pay experiment as they would in an experiment in which they received their shares of the pie with certainty is an empirical question. This paper investigated the use of the random pay procedure in experiments with moderate stakes ($5 dictator games), and found that behavior is not invariant to the random pay procedure. There are two reasons why this should be of concern. First, experimenters may be primarily interested in the full pay game, but conduct a random pay experiment believing that the results would be the same if they ran the more costly full pay experiment. The results of the experiments reported here suggest this belief is mistaken. Second, there is a theoretical basis for the use of the random pay procedure and so the experiments reported here can be viewed as a test of this economic theory. The theory is based on a simple ‘independence of irrelevant alternatives’ argument, yet the theoretical prediction (invariance) fails. An interesting area for further research would be to identify the assumptions that are made but which do not hold. Reconciliation with the results of Bolle (19901, where the invariance hypothesis was accepted for ultimatum games, may be achieved in numerous ways. First, the invariance hypothesis may be game-specific, This is not very satisfactory; we would like to identify the features of an experimental game which make the theory inapplicable. Second, Bolle’s results may be due to low power. Powerful tests of the invariance hypothesis against the alternative that randomizing pay moves experimental outcomes toward no pay outcomes requires a larger number of ultimatum game observations, simply because no pay and full pay distributions are very similar in the ultimatum game. In fact, I conducted Monte Carlo simulations using the Forsythe et al. (1988) ultimatum game results and found that even in the extreme case in which subjects treated a random pay game as a no pay game, with 20 random pay and 40 full pay observations the most powerful test rejects the null hypothesis less than 30% of the time at the 5% significance level. Third, Belle’s procedures include a device that
276
M. Sefton / Incentives in bargaining games
allows the decision of an unselected respect to the experimenter, while the This would be an interesting source imply that experimenter effects are dictator games, and at the same time these effects.
subject to be anonymous with procedure used here does not. of discrepancy since it would important in moderate stakes it suggest a way of controlling
References Belle, F., 1990. High reward experiments without high expenditure for the experimenter? Journal of Economic Psychology 11, 157-167. Epps, T.W. and K.J. Singleton, 1986. An omnibus test for the two-sample problem using the empirical characteristic function. Journal of Statistical Computation and Simulation 26, 177-203. Forsythe, R., J.L. Horowitz, N.E. Savin and M. Sefton, 1988. Replicability, fairness and pay in experiments with simple bargaining games. Working Paper 88-30, University of Iowa, IA. G&h, W., R. Schmittberger and B. Schwarze, 1982. An experimental analysis of ultimatum bargaining. Journal of Economic Behavior and Organization 3, 367-388. Giith, W. and R. Tietz, 1990. Ultimatum bargaining behavior: A survey and comparison of experimental results. Journal of Economic Psychology 11, 417-449. Kahneman, D., J. Knetsch and R. Thaler, 1986. Fairness and the assumptions of economics. Journal of Business 59, S285-S300. Kim, P.J. and Jennrich, 1973. ‘Tables of the exact sampling distribution of the two-sample Kolmogorov-Smirnov criterion, D,,,,, m 5 n’. In: H.L. Hatter and D.B. Owen (Eds.), Selected tables in mathematical statistics I (pp. 97-170). Providence, RI: American Mathematical Society. Shorack, G.R., and J.A. Wellner, 1986. Empirical processes with applications to statistics. New York: Wiley. Smith, V., 1976. Experimental economics: Induced value theory. American Economic Review, Papers and Proceedings, 274-279. Wilcoxon, F., S.K. Katti and R. Wilcox, 1973. ‘Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test’. In: H.L. Harter and D.B. Owen (Eds.), Selected tables in mathematical statistics I (pp. 171-235). Providence, RI: American Mathematical Society.
Psychology
263
13 (1992) 263-276
Incentives in simple bargaining games Martin
Sefton
*
Unkersity
of Iowa, Iowa City, USA
Received
January
8, 1992; accepted
March
22, 1992
One method of compensating subjects in experiments is to randomly select a subset of subjects to be paid according to their decisions. The rationale for this is that. according to economic theory, subjects in such an experiment will make the same decisions as they would if they were to be paid with certainty. Thus, a ‘random pay’ experiment provides the same information as a ‘full pay’ experiment, but at a lower cost to the experimenter. This invariance hypothesis is tested by comparing the outcomes of full pay and random pay experiments. In experiments with $5 dictator games the hypothesis is rejected. Subjects are more generous when 25% of the subjects are paid their shares of the $5 than when all of them are paid their shares of the $5. This contrasts with the results of Belle (1990), who accepted the hypothesis for ultimatum games.
Introduction This paper compares different methods of compensating subjects in economics experiments. Specifically, it investigates the effect of paying a randomly selected subset of subjects, rather than paying all subjects, according to their decisions at the end of a simple bargaining game experiment. The randomized payment procedure has been used in a variety of experiments as a means of conducting low cost experiments. The rationale for its use is that, according to theory, subjects in such experiments will behave as they would if they were to be paid with certainty. While the cost advantages of only paying some subjects is obvious, the invariance of behavior is not. In this paper the invariance hypothesis is tested by comparing outcomes of experiments which employ the * I wish to thank Bob Forsythe, comments and advice. The support Correspondence to: M. Sefton, Iowa City, IA 52242, USA.
0167-4870/92/$05.00
Joel Horowitz, John Kennan and Gene Savin for helpful of NSF grant SES 89-22460 is gratefully acknowledged. Dept. of Economics, University of Iowa, 679 Phillips Hall,
0 1992 - Elsevier
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Publishers
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M. Sefton / Incentic>esin bargaining games
randomised payment procedure with the outcomes of experiments in which all subjects are paid. Smith (1976) compared the outcomes of market experiments when some and when all subjects were paid their earnings. He found that transaction prices did not converge to predicted prices as quickly under the ‘weaker’ random pay reward structure. In contrast, Bolle (1990) tested and accepted the invariance hypothesis for ultimatumtype games with moderate stakes. He noted that if this result holds on a broader empirical base the implications are far-reaching. For instance, in order to collect observations on how each of 100 subjects perform a $1000 reward allocation task the experimenter could select one subject at the end of the experiment to be paid according to his or her decision, at a cost to the research budget of $1000. Under the invariance hypothesis this experiment yields the same information as one which pays all 100 subjects at a cost to the research budget of $100,000. Another reason for examining the effect of this procedure is to make more informed comparisons between experiments. Many experiments involve a variation on the design or parameters of an earlier experiment. However, if the new experiment employs the randomized pay procedure while the earlier one did not, any comparison of the two experiments may be confounded by the variation in the incentive structure offered to subjects. Different experimental outcomes may be due to variation in some parameter of interest, but they may also reflect the different incentives. For instance, Giith and Tietz (1990) speculate that differences between the outcomes of experiments by Giith et al. (1982) and Kahneman et al. (1986) are ‘mainly due to the way of performing experiments’ (e.g. only a random sample of subjects were paid in the Kahneman et al. experiment). Whether this is, in fact, the reason for the different outcomes requires information about the effect of different incentives. The experiments reported in this paper provide evidence against the invariance hypothesis. The experiments involve dictator games, which will now be described. Dictator games The experiments considered here involve dictator games as studied by Forsythe et al. (1988). A dictator game is a two-player game in
M. Sefton
/ Incentiues
in bargaining
games
265
which a sum of money (a ‘pie’) is to be divided and one of the players (the dictator) chooses the division. It differs from an ultimatum game in that the second player cannot reject the proposed division. In experiments with these games all dictators have the same set of possible actions - they propose how much they and their partner each receive. Observed actions reflect dictators’ preferences over outcomes of the experiment and beliefs about how actions translate into outcomes. If preferences and beliefs vary from subject to subject, the offers observed in an experiment can be regarded as arising from applying a set of procedures to a sample from the pool of potential subjects. These procedures would include the rules of the game, the size of the pie, the way subjects communicate, and so on. The procedure of interest in this paper is the way experimental earnings are related to actions - we wish to see how the distribution of outcomes depends on the way subjects are paid. In a ‘full pay experiment’ all subjects are paid according to their decisions. This pay procedure elicits a truthful response to the question ‘How would you divide the sum of money?’ by actually confronting each dictator with the task. In a ‘no pay experiment’ none of the subjects are paid according to their decisions. The subjects have no incentive to give a truthful response to the hypothetical question. In a ‘random pay experiment’ at the end of the experiment the experimenter randomly selects a subset of the subjects to be paid according to their decisions. Thus, a random pay experiment may be thought of as one in which some subjects take part in a full pay experiment and some no a no pay experiment, although all subjects must make their decisions before they find out which experiment they are participating in. If subjects maximize expected utility, then behavior in these games will be independent of whether subjects are paid their earnings or receive them with some (positive) probability. Bolle (1990) discusses the theoretical conditions under which this invariance property holds. These conditions are quite general; in particular we do not require utility functions to depend only on own earnings. Let a subject’s expected utility from offering x in a full pay experiment be u(x), and the expected utility from offering x in a no pay experiment be U(X). In a random pay experiment a subject will choose x to maximize P(X) + (1 -p)u(x) where I, is the probability that the subject is selected in the lottery. In the event that the subject is not selected the
266
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result of any action will be the same, that is u(x) is constant with respect to X. Thus a subject in a random pay experiment should choose x to maximize pu(x> + constant, or equivalently to maximize U(X). The key assumptions are: (11 A subject can costlessly evaluate the (subjective) payoff from any action. (2) In a no pay experiment a subject gets the same (subjective) payoff from any action. With regard to the first assumption, what really matters are the magnitude of decision costs relative to the expected utility gain from making better choices. By making the expected amount to be divided (the pie times the probability with which a subject is paid) sufficiently large relative to decision costs an experiment should provide expected utility gains sufficiently large to offset any decision costs. The expected amount to be divided is kept above a dollar in the random pay experiments investigated here, which is comparable to the expected amount to be divided in Bolle’s (1990) experiments. Bolle presumes that the decision costs in ultimatum games are small. Presumably the decision costs in dictator games are even smaller. With regard to the second assumption, Bolle argues that this will be satisfied if anonymity of those not selected to be paid is enforced. In the random pay experiments investigated here all subjects are anonymous with respect to their partner, although not with respect to the experimenter. According to the invariance hypothesis differences between samples of offers from random pay and full pay experiments should reflect only sampling variation. This paper tests this proposition by comparing the outcomes of a $5 random pay dictator game experiment with the outcomes of a $5 full pay dictator game experiment. The experimental procedures for each experiment were identical except that at the end of the random pay experiment a lottery was conducted to determine which pairs were paid their shares of the pie, whereas at the end of the full pay experiment all pairs were paid their shares of the pie. Review of previous results Random pay procedures were suggested by Giith et al. (1982) as a means of investigating bargaining games at low cost. A procedure
M. Sefion / Incentives in bargaining games
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essentially the same as the one investigated here was used in Kahneman et al. (1986). However, there have been few investigations of the effect of these procedures on experimental outcomes. The effect of different pay procedures is investigated directly by Bolle (1990) and Forsythe et al. (1988). Bolle compared the outcomes of 20 DM ultimatum games (worth about $10) under random pay and full pay procedures. In the random pay experiment 2 out of 20 subjects were randomly chosen to be paid at the end of the experiment. In the full pay experiment all 24 subjects were paid. The hypothesis that the distributions of offers in experiments with full pay and random pay are the same was accepted. Forsythe et al. (1988) conducted 20 full pay and 24 no pay ultimatum games with $5 to be divided. The hypothesis that the distributions of offers in experiments with full pay and no pay are the same was accepted. As an empirical issue it might at first appear that there is no need to pay any subjects according to their decisions. However, when the experiments were repeated five months later the distribution of offers in no pay games was significantly different from that in April. (The full pay experiments did replicate.) Thus the replicability of ultimatum game experiments appears to be dependent upon the incentives of the experiment. Forsythe et al. also found significant differences between the outcomes of full pay and no pay experiments with $5 dictator games. 48% of the offers were for half the pie and 13% were for nothing in the no pay experiment, 18% of the offers were for half the pie and 36% were for nothing in the full pay experiment. This suggests that whether subjects are paid or not matters a great deal in dictator games.
Hypotheses
and statistical
power
All of the statistical tests used here are tests of the equality of two distributions of offers. Denoting the distribution of offers in one experiment by F, and the distribution in another experiment by F2, the null and alternative hypotheses are: H,: F,(x)
= F*(x)
for all X,
H,: F,(x)
#F*(x)
for some x.
268
M. Sefton
/ IncentiL>es in bargaining games
Note that this is a test of equality of the entire distributions, as opposed to just equality of the means. Focussing on means alone can be misleading, since two distributions may differ in important respects yet still have the same mean. The tests of H, considered were the Anderson-Darling (AD), Cramer-von Mises (CM), Kolmogorov-Smirnov (KS), Epps-Singleton characteristic function (CF), and Wilcoxon Rank-Sum (RS) tests. The AD and CM tests are described by Shorack and Wellner (19861, the KS test by Kim and Jennrich (19731, the CF test by Epps and Singleton (1986) and the RS test by Wilcoxon et al. (1973). Also, one-sided versions of the KS and RS tests (referred to as the KS+ and RS, tests) were considered. In the Forsythe et al. dictator games no pay offers tended to be larger than full pay offers. If random pay offers tend to be in between full pay and no pay offers then one would expect random pay offers to tend to be larger than full pay offers. The directional tests are appropriate for testing the null hypothesis of invariance against the alternative that the random pay distribution stochastically dominates the full pay distribution. These tests require that the data be ordered, to do this ties were randomly broken. Choice among these tests was based on a Monte Carlo investigation of their powers. I first assumed that the distribution of full pay offers is the same as the sample distribution of full pay offers from the Forsythe et al. experiments. Next I assumed that the distribution of random pay offers is a probability mixture of the distributions of full pay and no pay offers, where the distribution of no pay offers is the same as the sample distribution of no pay offers from the Forsythe et al. experiments. I used these distributions to generate 2500 samples of II random pay offers and m full pay offers. Since the random pay offers would be compared with the Forsythe et al. full pay offers m was chosen to be the same as the FHSS sample size, 45. Initially, II was chosen to be 20. The probability with which a random pay offer is sampled from the full pay distribution is denoted by h and initially was chosen to be one. For each pair of samples the statistics were computed and compared with critical values. For each test the proportion of rejections is an estimate of its power. The process was repeated with different values of A and for IZ = 40. When A = 1 random pay offers and full pay offers are drawn from the same distribution, so that the power estimates are estimates of the size of the test. When A < 1 the invariance hypothesis
M. Sefton / Incentives in bargaining games
269
is false: random pay offers are drawn from a distribution in between the full pay and no pay distributions. Tables 1 and 2 display the results of the Monte Carlo simulations. Table 1 shows that either the KS or CM test has highest power among the five tests against H,, therefore these test statistics will be used for the analysis of the experiments. Table 2 reports the powers of the KS+ and RS, tests. As might be expected the powers are higher than for the other tests since they exploit information about the direction of departure from the null hypothesis. The KS+ test appears to perform slightly better than the RS, test and so the KS+ test will be used for a directional test. The tables also show that the powers are sensitive to A. This is important for the following reason. Suppose the invariance hypothesis is false, and that behavior in a random pay experiment is closer to behavior in a full pay experiment the higher the probability with which subjects are paid, p. Then as p decreases subjects’ behavior may be represented more closely by the alternatives with higher values of A. Thus by reducing p the experiment is more likely to detect the falsity of the invariance hypothesis. With moderate stakes and p = 0.01 one would not be surprised if the invariance hypothesis were rejected. A rejection could be attributed to decision costs: although small they may be large compared to the expected utility gain from making better decisions. An informative experiment requires a value of p that represents an appreciable cost saving while maintaining a reward structure in which decision costs are small relative to the variation in expected utility following from making different decisions. In order to achieve the second objective a lower bound on p was chosen that maintained an ‘expected pie’ of $1. This is comparable to the expected pie of Bolle’s (1990) ultimatum game, and presumably decision costs are lower in dictator than in ultimatum games. Thus, paying one in five subjects their share of a $5 pie is suggested as an informative test of the invariance hypothesis. In fact, one in four subjects were paid.
Experimental
procedures
and results
An additional set of full pay dictator games was conducted in order to ensure comparability between the new and old experiments. The hypothesis that the new full pay offers were drawn from the same
n
20 20 20 20 20 20 20 20 20 20 20
40 40 40 40 40 40 40 40 40 40 40
A
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
45 45 45 45 45 45 45 45 45 45 45
45 45 45 45 45 45 45 45 45 45 45
m
0.0860 0.1060 0.1740 0.2572 0.3912 0.5332 0.6832 0.8064 0.8968 0.9512 0.9876
0.0472 0.0560 0.1028 0.1748 0.2868 0.4168 0.5672 0.7172 0.8360 0.9244 0.9676
0.1052 0.1184 0.1864 0.2784 0.4040 0.5452 0.6872 0.7996 0.8896 0.9464 0.9828
0.1084 0.1256 0.1592 0.2184 0.3008 0.4132 0.5308 0.6272 0.7528 0.8412 0.9264
0.10
0.0384 0.0568 0.0712 0.1188 0.1784 0.2748 0.3844 0.4844 0.6760 0.7512 0.8656
0.05
0.10
0.0968 0.1056 0.1516 0.2048 0.2972 0.4028 0.5184 0.6184 0.7500 0.8484 0.9316
Cramer-Von (CM)
Kolmogorov-Smirnov
(KS)
Table 1 Dictator game: Powers of tests of H, against H,.
0.0496 0.0652 0.1136 0.1748 0.2940 0.4124 0.5544 0.6904 0.8144 0.9056 0.9572
0.0504 0.0612 0.0868 0.1388 0.2072 0.2948 0.4116 0.4952 0.6372 0.7416 0.8528
0.05
Mises
0.1000 0.1116 0.1848 0.2592 0.3800 0.5100 0.6436 0.7616 0.8656 0.9336 0.9756
0.1052 0.1248 0.1540 0.2136 0.2784 0.3828 0.5056 0.5864 0.7172 0.8072 0.9068
0.10
0.0504 0.0648 0.1088 0.1688 0.2704 0.3860 0.5152 0.6544 0.7836 0.8828 0.9504
0.0612 0.0828 0.1304 0.1944 0.2728 0.3860 0.4604 0.5944 0.7072 0.8220
0.05
Anderson-Darling (AD)
0.0760 0.0788 0.1324 0.2184 0.3260 0.4708 0.6156 0.7708 0.8724 0.9316 0.9804
0.0812 0.0864 0.1160 0.1644 0.2292 0.3276 0.4464 0.5500 0.7032 0.8020 0.9040
0.10
(CF)
0.0352 0.0392 0.0740 0.1336 0.2228 0.3364 0.4836 0.6560 0.8032 0.8860 0.9576
0.0416 0.0484 0.0656 0.0944 0.1468 0.2248 0.3336 0.4340 0.5876 0.7092 0.8352
0.05
Epps-Singleton
Wilcoxon
0.0904 0.1220 0.1688 0.2636 0.3580 0.4916 0.6156 0.7484 0.8456 0.8996 0.9592
0.0976 0.0976 0.1432 0.1968 0.2920 0.3644 0.4672 0.5752 0.6684 0.7640 0.8624
0.10
(RS) 0.05
0.0448 0.0668 0.0960 0.1672 0.2488 0.3664 0.4944 0.6448 0.7484 0.8420 0.9208
0.0472 0.0468 0.0852 0.1244 0.1852 0.2528 0.3392 0.4556 0.5556 0.6568 0.7792
271
M. Sefton / Incentives in bargaining games Table 2 Dictator game: Powers of directional tests of H,. h
n
m
KS+
RS+
0.10
0.05
0.10
0.05
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
20 20 20 20 20 20 20 20 20 20 20
45 45 45 45 45 45 45 45 45 45 45
0.0992 0.1508 0.2064 0.3212 0.4196 0.5604 0.6548 0.7736 0.8340 0.9192 0.9568
0.0476 0.0776 0.1268 0.2128 0.2932 0.4188 0.5232 0.6488 0.7460 0.8548 0.9116
0.0880 0.1580 0.2376 0.3036 0.4140 0.5424 0.6144 0.7060 0.8060 0.8776 0.9284
0.0428 0.0876 0.1404 0.1916 0.2796 0.3860 0.4568 0.5676 0.6896 0.7648 0.8588
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
40 40 40 40 40 40 40 40 40 40 40
45 45 45 45 45 45 45 45 45 45 45
0.0912 0.1672 0.2752 0.4136 0.5344 0.6656 0.8064 0.8892 0.9580 0.9816 0.9936
0.0512 0.0980 0.1736 0.3004 0.4068 0.5584 0.7052 0.8216 0.9120 0.9612 0.9820
0.0932 0.1780 0.2640 0.3740 0.5176 0.6152 0.7564 0.8460 0.9132 0.9564 0.9776
0.0416 0.1060 0.1612 0.2548 0.3816 0.4884 0.6268 0.7408 0.8392 0.9184 0.9524
distribution as the full pay offers from Forsythe et al. (1988) was accepted (see the first row of table 3) and so the full pay data were pooled giving 69 full pay observations. The random pay experiments were conducted with 24 pairs of subjects. Except for the way subjects were paid, the same procedures were used as in the experiments of Forsythe et al. The subjects for the random pay experiments were recruited from a subject pool consisting of students from undergraduate business classes and MBA classes at the University of Iowa who had completed a form indicating interest in participating in experiments. When called and asked to participate, they were told that they ‘would earn between $3 and $8 for participating in an experiment that would last about 20 minutes’. We used two connecting rooms, each of which could accommodate 8 individuals. Subjects were assigned to rooms randomly when they were recruited. At the beginning of a session, subjects were given
272 Table 3 Tests of invariance
M. Sefton / Incentic~esin bargaining games
hypotheses.
a Sample
size
Statistic
(p-value)
CM
KS
45 (1988) 24 (1990)
0.0780 (0.70)
0.2827 (0.4731)
Full pay dictator game offers vs. random pay dictator game offers
69 (Full) 24 (Random)
0.8062 (0.01)
0.3786 (0.0085)
0.3786 (0.0042)
No pay dictator game offers vs. random pay dictator game offers
46 (No) 24 (Random)
0.0455 (0.90)
0.1250 (0.9292)
0.1214 (0.5703)
Forsythe et al. (1988) full pay dictator game offers vs. new full pay dictator game offers
KS+
a The limiting null distribution of the CM test as tabled in Shorack and Wellner (1986) was used to provide p-values for CM statistics. Exact p-values for the KS and KS+ statistics were computed following the methods of Kim and Jennrich (1973).
instruction sets. The experimenter read the instructions aloud and answered questions. All subjects in the same session in the same room faced the same task (that is, they were either all senders or all receivers of a proposal). Each person was randomly and anonymously paired with someone in the other room, and each pair was given a $5 pie to divide. Communication between members of a pair was by a written proposal form that was carried between rooms by the experimenter. Each subject in room A chose how to allocate the $5; the subjects in room B could benefit from the allocation but had no decision to make. At the end of the experiment each subject was paid $3 for participating and a lottery was conducted to select 2 of the 8 pairs who would also receive their shares of the pie. Each session was completed in less than 20 minutes. Figures 1 and 2 are histograms of the full pay and random pay offers. The outcomes of the random pay experiment appear to be very different from the outcomes of the full pay experiment. In particular, the offers appear to be more generous in the random pay experiment. The modal offer is $2.50 in the random pay experiment and nothing in the full pay experiment, the median offers are $2.50 and $1, and the mean offers are $1.83 and $1.09. Formal tests of the invariance hypothesis are reported in the second row of table 3. The invariance
273
M. Sefton / Incentives in bargaining games
.s
.4
.3
.2
.I
0
I
I
0 Fig. 1. Distribution
1
I
2
I
offer in $5
3
I
I
4
5
of offers made in $5 full-pay dictator games, using 45 observations Forsythe et al. (1988) and 24 new observations.
from
.5 -
.4 -
.3 -
.‘-O .2 o- -
.“. 0
Fig. 2. Distribution
b 1
cl , 2
offer in $s
3
of observations made in random pay dictator ity p equal to 0.25. 24 observations.
4 games, with payment
5 probabil-
274
M. Sefton
.5
.A
/ Incentives
in bargaining
games
1 I
cl Fig. 3. Distribution
0
cl
I
I
1
2 offer
I
I
I
3
A
5
in $5
of offers made in no pay dictator games reported observations.
by Forsythe
et al. (1988). 46
hypothesis is rejected by each test at conventional significance levels, indicating that these differences cannot be attributed to sampling variation. The directional test confirms that random pay offers tend to be significantly larger than full pay offers. The saving in cost is substantial (in the random pay experiment subject fees were $90 less than they would have been had all subjects been paid their earnings) but it brings with it considerable changes in behavior. For comparison, figure 3 is a histogram of the Forsythe et al. no pay offers. The outcomes of the no pay experiment appear to be very similar to the outcomes of the random pay experiment. The average offer in the no pay experiment was $1.91 compared with $1.83 in the random pay experiment, the median and modal offers in the no pay experiment were $2.50, the same as in the random pay experiment. Formal tests do not detect statistically significant differences between the no pay and random pay distributions (see the third row of table 3).
M. Sefton / Incenliues in bargaining games
275
Conclusion The procedure of paying a randomly selected subset of subjects was suggested by Giith et al. (1982) as a means of investigating high stakes bargaining game experiments at low cost. A number of experimenters have used the procedure in moderate stakes bargaining games for this reason. Whether subjects behave the same way in a random pay experiment as they would in an experiment in which they received their shares of the pie with certainty is an empirical question. This paper investigated the use of the random pay procedure in experiments with moderate stakes ($5 dictator games), and found that behavior is not invariant to the random pay procedure. There are two reasons why this should be of concern. First, experimenters may be primarily interested in the full pay game, but conduct a random pay experiment believing that the results would be the same if they ran the more costly full pay experiment. The results of the experiments reported here suggest this belief is mistaken. Second, there is a theoretical basis for the use of the random pay procedure and so the experiments reported here can be viewed as a test of this economic theory. The theory is based on a simple ‘independence of irrelevant alternatives’ argument, yet the theoretical prediction (invariance) fails. An interesting area for further research would be to identify the assumptions that are made but which do not hold. Reconciliation with the results of Bolle (19901, where the invariance hypothesis was accepted for ultimatum games, may be achieved in numerous ways. First, the invariance hypothesis may be game-specific, This is not very satisfactory; we would like to identify the features of an experimental game which make the theory inapplicable. Second, Bolle’s results may be due to low power. Powerful tests of the invariance hypothesis against the alternative that randomizing pay moves experimental outcomes toward no pay outcomes requires a larger number of ultimatum game observations, simply because no pay and full pay distributions are very similar in the ultimatum game. In fact, I conducted Monte Carlo simulations using the Forsythe et al. (1988) ultimatum game results and found that even in the extreme case in which subjects treated a random pay game as a no pay game, with 20 random pay and 40 full pay observations the most powerful test rejects the null hypothesis less than 30% of the time at the 5% significance level. Third, Belle’s procedures include a device that
276
M. Sefton / Incentives in bargaining games
allows the decision of an unselected respect to the experimenter, while the This would be an interesting source imply that experimenter effects are dictator games, and at the same time these effects.
subject to be anonymous with procedure used here does not. of discrepancy since it would important in moderate stakes it suggest a way of controlling
References Belle, F., 1990. High reward experiments without high expenditure for the experimenter? Journal of Economic Psychology 11, 157-167. Epps, T.W. and K.J. Singleton, 1986. An omnibus test for the two-sample problem using the empirical characteristic function. Journal of Statistical Computation and Simulation 26, 177-203. Forsythe, R., J.L. Horowitz, N.E. Savin and M. Sefton, 1988. Replicability, fairness and pay in experiments with simple bargaining games. Working Paper 88-30, University of Iowa, IA. G&h, W., R. Schmittberger and B. Schwarze, 1982. An experimental analysis of ultimatum bargaining. Journal of Economic Behavior and Organization 3, 367-388. Giith, W. and R. Tietz, 1990. Ultimatum bargaining behavior: A survey and comparison of experimental results. Journal of Economic Psychology 11, 417-449. Kahneman, D., J. Knetsch and R. Thaler, 1986. Fairness and the assumptions of economics. Journal of Business 59, S285-S300. Kim, P.J. and Jennrich, 1973. ‘Tables of the exact sampling distribution of the two-sample Kolmogorov-Smirnov criterion, D,,,,, m 5 n’. In: H.L. Hatter and D.B. Owen (Eds.), Selected tables in mathematical statistics I (pp. 97-170). Providence, RI: American Mathematical Society. Shorack, G.R., and J.A. Wellner, 1986. Empirical processes with applications to statistics. New York: Wiley. Smith, V., 1976. Experimental economics: Induced value theory. American Economic Review, Papers and Proceedings, 274-279. Wilcoxon, F., S.K. Katti and R. Wilcox, 1973. ‘Critical values and probability levels for the Wilcoxon rank sum test and the Wilcoxon signed rank test’. In: H.L. Harter and D.B. Owen (Eds.), Selected tables in mathematical statistics I (pp. 171-235). Providence, RI: American Mathematical Society.