BENEFIT PACKAGES AND INDIVIDUAL BEHAVIOR: CHOICES OVER
DISCRETE GOODS WITH MULTIPLE ATTRIBUTES*
Mark Van Boening**
University of Mississippi
Tanja F. Blackstone***
Navy Personnel Research, Studies and Technology
Michael McKee****
University of Tennessee
Elisabet Rutstrom*****
University of Central Florida
August 2006 – Revised
Abstract: Managers and employers use an array of rewards to attract and retain quality
employees. An increasingly significant component of the overall compensation is the
employee’s benefits package. Flexible packages offer more choice but also incur higher decision
costs. We conduct an experiment on choices over stylized benefits packages where discrete
“goods” have multiple attributes affecting the payoff function. We investigate the degree to
which these complications affect choices. Eighty subjects play an individual-choice decisioncost game where they are implicitly asked to solve a complex programming problem. Our main
results are that (a) individual subjects respond to the relative tradeoff between the attributes, (b)
some combinations of the attributes (apparently) entice subjects to search more and thus earn
more, and (c) most subjects appear to adopt a heuristic that approximates the optimal solution.
Further, subjects appear to value the right to make choices, as they rarely choose a fixed payoff
option with a known payoff and low decision cost, even when the fixed payoff is 80% of the
maximum possible under the decision-making task.
* This research was supported by an Independent Laboratory Independent Research (ILIR) grant
from the Office of Naval Research. The opinions expressed are those of the authors and do not
necessarily reflect the views of the Department of the Navy. We thank participants at the
Economics Science Association 2001 North American Regional Meetings, John Conlon, and an
anonymous referee for comments on earlier drafts. All remaining errors are ours.
**
Associate Professor, Department of Economics, University of Mississippi, Holman Hall 373, University, MS
38677
***
Economist, Navy Personnel Research, Studies and Technology, 5720 Integrity Drive, Millington, TN, 380545026
****
Holly Professor of Economics, College of Business, University of Tennessee, Knoxville, TN 37996
*****
Professor, Department of Economics, College of Business, University of Central Florida, Orlando, FL 328161400
Blank page for printing duplex.
I.
Introduction
Managers and employers use an array of rewards to attract and retain quality employees.
An increasingly significant component of the overall compensation is the employee’s benefits
package. Flexible benefit plans have the potential of increasing job satisfaction (White, 1983),
thereby increasing the likelihood of retaining existing employees. Additionally, the labor supply
pool from which the employer can draw may increase if prospective employees place a greater
relative value on a flexible compensation package.1 Nevertheless, in order for the flexibility to
result in greater job satisfaction, employees must perceive the new plans as more valuable than
the existing ones and this requires that employees are able to make optimal choices from the
menu of benefits. Increasing the number of options and the flexibility of combinations increases
the subjective value of the compensation package in so far as the employee is able to effectively
decide on the optimal combination of benefits.
If the choice problem facing the employee becomes too complex, it is possible that the
subjective value of the compensation package actually decreases with the number of options.
Selecting an optimal benefit plan can be difficult because by nature the attributes of such plans
are both multiple and discrete. These two factors complicate the employee’s choice and may
push the limits of his or her cognitive abilities.2
The outcome may be a high degree of
dissatisfaction with the plan, leading to reduced job satisfaction and job tenure; the benefit
package has not been successful. In this paper we report an experimental investigation of
individual decision-making over complex (multiple and discrete) options broadly designed on
1
While a flexible compensation scheme may decrease the employer’s labor cost, it could increase other costs.
Offering a large number of benefit combinations would likely impose substantial monitoring and administrative
costs on the employer. Further, employees may attribute very little added value beyond the kth option (Bucci and
Grant 1995). The employer, therefore, has an incentive to offer just enough options and associated attributes such
that the marginal cost of offering an additional option is just equal to its marginal benefit.
2
These two factors also complicate the employer’s decision on what combinations to offer to the employee. Here,
we focus on the employee’s decision-making task.
-1-
the choice of an employment benefits compensation packages. Our research is motivated by
recruitment and retention issues facing managers as they compete in an increasingly
sophisticated labor market.3
Many decision scientists argue that individuals have computational or cognitive limits,
and that these limits become more critical when decisions involve difficult choices. Individuals
may cope by adopting simplifying techniques, known as heuristics. Following Simon’s (1955)
seminal work, different schools of thought have emerged ranging from the belief that human
decision making is intrinsically prone to errors to the belief that it is fundamentally efficient.4
Three strands of research address decision-making in complex settings (see von Winterfeldt and
Edwards, 1986). The first investigates individual assignment of values for commodities with
multiple attributes. This strand is largely normative, as the researcher constructs a decision aid
to facilitate the “correct” decision. The second strand studies whether individuals are capable of
making payoff-maximizing decisions in the presence of multiple attributes. The third looks at
whether or not individuals are able to consistently make payoff-maximizing decisions with
multidimensional decisions and meeting multiple constraints. The second and third strands are
more positive as they assess the ability of individuals from observations of unaided behavior.
3
These are especially significant issues for the United States Department of Defense (DoD) and the U.S. Navy in
particular. The Navy now utilizes numerous pecuniary incentives to increase enlistment and reenlistment, but the
existing benefits package provided to military personnel by the DoD is standardized and relatively inflexible.
Introducing flexible benefit plans may help solve staffing issues. Under such a scheme, the employer (Navy)
chooses the level of compensation and the employee (active duty personnel) chooses the mix of salary and benefit
options in the compensation package according to his/her preferences. Currently DoD is prohibited from offering
non-standard packages to specific groups.
4
The literature expressing the view that human decision makers are prone to errors has received recent attention,
with reviews in Camerer (1995) and Starmer (2000). Prospect Theory, introduced by Kahneman and Tversky
(1979), and theory of Framing Effects (Kahneman and Tversky, 1996), are particularly popular among behavioral
economists and psychologists as alternatives to the neo-classical optimization approach. An alternative decision
theory view, that human heuristic decision-making is fundamentally efficient, is based on a concept of ecological
rationality, which is promoted by Gigerenzer (1996) and Gigerenzer et al. (1999). According to this view, many
decision heuristics have evolved through time, affected by the successes and failures of individual decision makers.
-2-
Our research focuses on the second issue. The experimental literature in economics and
psychology has paid little attention to problems of complexity arising from choices over discrete,
multi-attribute objects (McKinney and van Huyck, 2006). To make optimal choices over multiattribute and discrete alternatives, an individual must, in effect, solve a complex programming
problem. Using data generated through laboratory experiments, we examine the choices of
individuals when faced with sets of discrete multi-attribute goods. We vary the relationship
between the relative values of the attributes, and the value of an outside (simple) option, in effect
varying both the complexity and the potential value of the options package. Our main results are
that (a) individual subjects respond to the relative tradeoff between the attributes, (b) some
combinations of the attributes (apparently) entice subjects to search more and thus earn more,
and (c) most subjects appear to adopt a heuristic that approximates the optimal solution to the
programming problem. Further, individuals appear to value the right to make choices; our
subjects rarely choose the fixed payoff option, with known payoff and low decision cost, even
when the fixed payoff is up to 80% of the maximum possible under the complex decision task.
II.
The Decision Problem, the Interface and Relation to Benefits Packages.
II.A. Decision problem.
Our experimental subject earns money either by playing a “cell selection” game or by
accepting a known fixed payoff in lieu of playing the game. In the cell selection game, the
subject’s choice set is discrete and represented by an n × n matrix. Each cell of the matrix has
three attributes: a cell payoff, a cell weight and a cell value. When a subject selects a cell, her
marginal reward is the cell payoff plus the product of the cell weight and the cell value:
Reward from
=
selecting cell i
Cell Payoffi + (Cell Weight i × Cell Valuei )
-3-
(1)
The subject’s total reward from selecting k cells is the sum of the rewards across those cells. As
part of our experimental design, we fix the cell weight and cell payoff at the same level for each
cell in the matrix, and we vary the cell value across cells.
In this case, the summation of
equation (1) can be expressed as:
k
Reward from
= k × Cell Payoff + [Cell Weight × ∑ Cell Value i ] .
selecting k cells
i =1
(2)
The subject has a value limit that constrains her choices. She can continue to select cells
as long as the sum of the cell values from the cells she selects does not exceed her value limit.
Formally, the k cells selected in equation (2) must satisfy:
k
∑ Cell Value
i =1
i
≤ Value Limit
(3)
In essence this decision task is a version of the “knapsack problem” (Greenberg, 1971)
where the payoff-maximizing subject maximizes the objective function (2) subject to the side
constraint (3). While this is analogous to the constrained optimization problem of neo-classical
consumer theory, here the goods are very discrete, so optimization at the margin is not strictly
available. This type of problem may appear intuitively simple, but the process of identifying the
optimal solution can be extraordinarily complex, typically requiring a programming algorithm
such as the branch-and-bound method.5
II.B. Computerized interface.
Our experimental parameters and procedures are presented in the next section. Here, we
describe the computerized interface that (a) provides the subject with the initial information she
needs to undertake the decision task as described by equations (2) and (3) above, (b) elicits and
5
For example, see Garfinkel and Nemhauser (1972) or Parker and Rardin (1988). The experimenter interface of our
computer program includes a choice of algorithms for calculating the optimal solution; we utilized the branch-and
bound option. We calibrated the program’s calculations with a separate exhaustive search algorithm.
-4-
records her decisions, and (c) reports the current status of her earnings and resource use in the
cell selection game, i.e., the values of equation (2) and the left-hand side of equation (3),
respectively, for her current cell choices. Figure 1 provides an illustration of the interface. The
upper half of the computer screen displays information, while the lower half presents the cell
selection game matrix. All decisions are entered via mouse input (clicks).
*** Figure 1. Subject Screen Display –– about here ***
The top of Figure 1 displays, from left to right, that the subject is in round #1 with a
conversion rate of 0.0001, a cell payoff of 20.000, a cell value weight or cell weight of 1.200,
and moves that are “Revocable.” (The round number, conversion rate and revocability are
discussed in the next section.) Directly below that information is a version of equation (1). The
middle-left side of the screen displays the fixed payoff that the subject can choose as an
alternative to the cell selection game. In Figure 1, the fixed payoff is shown as 2042. The
subject can select this option by clicking the “Decline Play – Accept Fixed Payoff” box directly
below the fixed payoff box; the screen displays a reminder that she can choose the fixed payoff
at any time.
To the right of the fixed payoff is a box labeled “You could earn this much or more” that
shows the maximum game payoff the subject can earn from playing the given cell selection
game.
Our software calculates this payoff when it identifies the optimal solution using a
specified integer programming algorithm. In Figure 1, this payoff is shown as 2552.8. As all of
the cell selection games in this experiment have exact solutions, the subject is explicitly
informed during the instructions that the number shown in the box is in fact the maximum she
can earn (in experimental dollars) for the given matrix.6 Among other things, this information
6
Our software has available several algorithms to identify optimal solutions; here we use an exhaustive search. The
“or more” designation covers those cases where an algorithm only provides approximate solutions.
-5-
allows the subject to compare the fixed payoff with the potential payoff to the game (2042 v.
2552.8 in Figure 1).
Continuing to the right, the screen displays the subject’s value limit or resource constraint
from the right-hand side of equation (3). In Figure 1, the value limit is shown as 2000. The
subject may continue to select cells as long as the sum of the cell values that she has selected
does not exceed this limit – if she tries to exceed it, the software informs her that she cannot
include the cell she is attempting to select because the value limit would be exceeded. The
screen also displays the subject’s current earnings as calculated by equation (2), and her value of
selected cells as calculated by the summation on the left-hand side of equation (3). In Figure 1,
the subject’s current earnings are 412 and her value of selected cells is 310; see the next
paragraph for calculations. This information allows the subject to compare her current earnings
with the maximum possible (e.g., 412 with 2552.8 in Figure 1), and to compare her current
consumption with her resource constraint (e.g., in Figure 1, her value of selected cells at 310 v.
her value limit of 2000). The subject clicks the “End Round” button on the middle right-hand
side of the screen when she is done selecting cells. Each round or game has a four-minute time
limit (discussed below), so a “Time Remaining for Playing Round” clock is provided in the
middle of the screen. Figure 1 shows 3:26 remaining in round #1.
The lower portion of the screen shows the n× n cell selection game matrix. The cell
value is shown in each of the individual cells. The subject chooses a particular cell by clicking
the box directly to the left of the corresponding number. Once a cell is selected, its color
changes from green to red; if a cell is subsequently deselected (see next section below), the color
reverts back to green. In the black-and-white version reproduced in Figure 1, the green or “not
selected” cells appear as grey, and the selected cells appear as black. In Figure 1, the subject has
-6-
selected two cells, with cell values of 196 and 114. Thus her value of selected cells is 196 + 114
= 310. In experimental dollars, her earnings from the first (196) cell are, as calculated by
equation 1 or by the expression in the upper portion of her screen, 20 + (1.2 × 196) = 255.2
while her earnings from the second (114) cell are 20 + (1.2 × 114) = 156.8. Her current earnings
are 255.2 + 156.8 = 412, or alternatively by equation (2), 2 × 20 + 1.2 × (196 + 114).
II.C. Relation to Benefits Package Choices.
The subject’s decision task is analogous to the problem of selecting components in a
flexible or cafeteria plan benefits package. The cells in the choice set (i.e., matrix) represent the
benefits options offered by the employer where each cell constitutes a component of the package
such as medical, dental and childcare policies. The combination of cells selected by the subject
represents the compensation package chosen by an employee. The employee’s valuation of an
individual component is represented by equation (1) above, and the employee’s valuation of a
given package is represented by equation (2). The value limit in equation (3) represents the
maximum amount of coverage that the employer is willing to offer the employee; it is the
employee’s benefit “budget.”
An individual cell contributes to the overall value by virtue of the benefits from the
component (cell value) and from the inclusion of a component (cell payoff).
The cell value
represents the dollar value of coverage for each component while the cell weight represents the
economic discount (or premium) at which the employee purchases that coverage. For example,
if a cell is chosen that has a cell weight of 1.2 and a cell value of 83, then the product of the cell
weight and the cell value contributes approximately 100 towards the total reward (see equations
(1) and (2) above). Alternatively, $83 of employer expenditure translates into $100 of benefits
coverage for the employee. In the experiment, we systematically alter the relative magnitude of
-7-
the cell payoff. Equation (1) reveals that ceteris paribus the higher the cell payoff the less the
relative contribution of the cell value to the reward from selecting a given cell. This corresponds
to an employee’s choice between having a particular benefit in the package (a cell’s reward is
determined primarily by the cell payoff) versus the amount of the coverage (a cell’s reward is
determined primarily by the cell value).
The fixed payment option represents the monetary value the employee places on an
alternative non-flexible benefits package. If the employee values choice, ceteris paribus the
flexible package will be more highly valued than a non-flexible one. But a flexible package
requires that the beneficiary incur the subjective cost of identifying the preferred package, so an
employee might opt for a non-flexible plan over a flexible one if the difference between total
benefits is relatively small. In the experiment, we systematically vary the relative difference
between the rewards to the flexible package (the cell selection game) and the non-flexible
package (the fixed payoff option).
III. Design and Procedure
III.A. Design.
Our experiment has a 2× 2 design with the cell payoff and fixed payoff option as the
treatment variables. These treatment variables allow us to examine two important aspects of the
decision-making task, which are elaborated below. So as to avoid confusion with the cells of the
game matrix, we refer to the four cells of the experimental design as “sessions” S1, S2, S3 and
S4.7
Our design and interface requires that we identify a total of twelve experimental
parameters, and we conducted several pre-test sessions to help determine appropriate values. We
chose values so that (a) the variety of choices within a given matrix and the variety of optimal
7
In the experiment, subjects are told the session labels are S17, S18, S19 and S20, respectively.
-8-
solutions across matrices are sufficiently rich for data analysis, and (b) the computational
difficulty facing the subject is significant but not overwhelming, i.e., the decision cost is
nontrivial but the problem tractable enough that the subject can (possibly) identify a useful
decision rule or heuristic to aid in the cell selection task.
Table 1 shows our parameters. Panel I reports the four parameters that vary across
sessions, including our cell payoff and fixed payoff treatment variables, and panel II shows the
eight parameters that are constant across sessions.
An individual session consists of a series of
decision-making “rounds” where the given cell payoff and fixed payoff applies to all rounds of
the session, but the subject sees a different cell selection game matrix each round. For example,
session S1 has nine rounds, so the subject sequentially views nine different cell selection game
matrices, each with a cell payoff of 20 and a fixed payoff option that is 80% of the maximum
possible earnings from selecting cells in the matrix.
*** Table 1. Experimental Parameters –– about here ***
The first treatment variable, the cell payoff, is shown in panel I as either 20 (in S1 and
S3) or 100 (in S2 and S4). By varying the cell payoff, we can examine how the subject’s choice
of cells is affected by the relative importance of the cell payoff versus the cell value. As we
describe above, this corresponds to an employee’s choice between having a particular benefit in
the package (a cell’s reward is determined primarily by the cell payoff) versus the amount of the
coverage (a cell’s reward is determined primarily by the cell value). In our data analysis, we
compare S1 v. S2 and S3 v. S4 to analyze the effect of this treatment variable.
The second treatment variable, the fixed payoff, is shown in panel I as either 80% (S1,
S2) or 50% (S3, S4) of the maximum possible reward from playing the cell selection game. By
varying the fixed payoff percentage, we can examine how the relative difference between the
-9-
fixed and game payoffs affects the subject’s decision on how to earn money. The game imposes
a subjective decision cost of deciding which cells maximize earnings while the fixed payoff
option does not, but the fixed payoff is less than the maximum possible game payoff. The
subject can (potentially) use the difference between the maximum possible game payoff and the
fixed payoff to approximate a “decision benefit” or return to playing the game. If this potential
benefit is less than her subjective decision cost, she may opt for the fixed payoff. As we describe
above, our cell selection game corresponds to a flexible benefits package and the fixed payoff
corresponds to a non-flexible package. We analyze the case where the fixed payoff is less than
the game payoff, i.e., ceteris paribus the employee values choice. Below, we compare S1 v. S3
and S2 v. S4 to analyze this treatment variable.
Two other parameters vary across sessions. The subject plays a total of thirty-eight
rounds across all four sessions, but as shown in panel I, no two sessions have the same number of
rounds. This is designed to control for “end of experiment” and/or “end of sequence” effects
(Davis and Holt, 1993). As a control for order effects, we also use four randomly determined
sequences for the four sessions. A total of eighty subjects participated in the experiment: twentyone subjects were assigned sequence S4-S1-S2-S3, twenty were assigned S1-S3-S4-S2, nineteen
were assigned S3-S2-S1-S4 and twenty were assigned S2-S4-S3-S1.8 The fixed deductions
shown in panel I also vary across sessions. At the end of each session, the corresponding
deduction is subtracted from the subject’s earnings. This deduction is explained to the subject as
part of the instructions, and the amount of the deduction is revealed to her as she begins each
8
Our design is not strictly balanced as one subject was inadvertently assigned sequence S4-S1-S2-S3 instead of S3S2-S1-S4. We became aware of this error only after the experiment was completed.
- 10 -
session. The deduction serves to maintain salient incentives at the margin while at the same time
keeping overall earnings within the research budget.9
Panel II of Table 1 shows the parameters that are constant across sessions. All cell
selection games are presented as 5× 5 matrices. 10 Each of the twenty-five cells of a given matrix
is randomly populated with a cell value from the range 100 to 1000 (inclusive). The cell weight
is fixed at 1.2 and the value limit is constant at 2000 in all rounds of all sessions. The subject is
allowed to make revocable choices when selecting cells, i.e., if she selects a cell by clicking on it
with her mouse, and she can later “deselect” the cell by clicking it a second time. She can select
and deselect cells at any time and as many times as she wishes (provided time has not expired).
We allow revocable moves to give the subject her best shot at achieving the maximum possible
game payoff.
Additionally, employees (and perhaps their spouses) might consider various
combinations of flexible benefits before making a final decision, so we thought it appropriate to
allow revocable moves in this initial experiment.
Each round lasts a maximum of four minutes or 240 seconds, and the round clock is
shown in the middle of the subject’s screen display (see Figure 1). We implement a time limit to
mimic the field setting in which enrollment periods typically have a (relatively short) fixed
horizon, e.g., thirty days; our choice of four minutes is based on our pre-tests that indicated most
subjects could complete decisions within that time limit. When a subject is finished making
decisions in a round, she mouse-clicks either the “Decline Play” box to take the fixed payoff, or
9
Without the fixed deduction, a subject would earn around $60 simply by choosing the fixed payoff each round.
Although she could earn an additional $5-$10 selecting cells, she might very well opt for fixed payoff because that
option has both a substantially lower time cost and substantially lower decision cost. Other methods of achieving
the desired earnings levels would have involved modifying parameter values in ways that would have reduced
incentives For example, small cell weights and cell payoffs may lead a subject to interpret equation (2) as
essentially zero, irrespective of the cell values.
Our pre-testing revealed that 4 × 4 or smaller matrices usually have trivial solutions, while 6 × 6 or larger matrices
have excessively burdensome solutions so that subjects almost always opt for the fixed payoff.
10
- 11 -
an “End Round” button to take the payoff she has earned from selecting cells. If she does not
click one of these two buttons before time expires, her reward for that round is zero.11 Finally,
during a round, the subject’s rewards are expressed in “experimental dollars” or E$, which
convert into U.S dollars or US$ at the rate of E$1 = US$0.001. At the end of each session, the
computerized interface converts the subject’s experimental rewards into U.S. currency and
displays a table showing her E$ and US$ earnings for each round and the total for the session.
III.B. Procedure.
We conducted the experiment at the Mississippi Experimental Research Laboratory
(MERLab) at the University of Mississippi and at the Business, Economics, Accounting and
Marketing Laboratory (BeamLab) at the University of South Carolina. The subjects are student
volunteers recruited from undergraduate business courses at the respective universities.
A
subject makes two visits, called Part I and Part II, to the laboratory. Part I is a training period
that lasts approximately one and one-half hours. Part II is the data-collection period in which
each subject completes the four sessions (thirty-eight total rounds) shown in Table 1, and it lasts
approximately two hours. Subjects are paid earnings from both parts upon completion of Part
II.12 Our reasons for choosing this two-part process include the fact many employees make
annual decisions about (or updates of) their cafeteria plans, i.e., it is a task with which they have
some degree of experience or familiarity.
Part I trains the subject as to the computerized interface, record keeping, and the
sequence of events (multiple sessions, fixed deductions, etc.). Upon arrival, the subject is
11
The “zero payoff if no decision is made” is reiterated during the subject’s instruction. In our results section, we
report that the zero payoff occurs in only 20 (0.7%) of the 3040 total rounds.
12
Our decision to divide the experiment into a training phase and a data collection phase is based on the same pretests mentioned in connection with Table 1 above. Subjects are recruited for participation in two parts, with each
part on a separate day, and the end-of-Part II payment schedule is explicitly explained during recruitment. The
purpose of Part I is to train the subject on the procedure and the interface; those parameters yield fairly trivial
decision tasks, and the data are not sufficiently rich to provide insight into the subject’s choices or decision-making.
- 12 -
assigned a subject number, given a blank earnings record sheet, and seated at a private computer
carrel. She completes the computerized instructions and two practice rounds with 2 × 2 matrices.
An experimenter then visits her carrel, and she is prompted for questions and/or clarification
regarding the task. After any discussion, the subject is presented with some brief printed
instructions and a consent form, and given a few moments to review that material. After she
signs the consent form, an experimenter writes the first session number and corresponding fixed
deduction on her printed record sheet. The experimenter then instructs the subject to press the
appropriate key to begin the first round of the first session.
The first round starts when the subject is presented with a screen similar to Figure 1
above and the round clock begins. Once the subject completes the round (she clicks “Decline
Play – Accept Fixed Payoff,” plays the game and then clicks “End Round,” or time expires), the
interface automatically proceeds to the next round. After the subject completes all rounds of the
given session, the interface pauses and displays her earnings. The subject records her US$
earnings from the computer display onto a paper record sheet, and she subtracts the fixed
deduction. She then raises her hand to signal that she is ready to begin the next session.
Upon seeing the subject’s raised hand, the experimenter approaches the subject’s carrel
and provides her with a new session number and fixed deduction. The experimenter watches
while the subject restarts the program and enters her subject number and new session number.
The subject then proceeds through all rounds of this new session. After completing the second
session, she again records her earnings on the paper record sheet and raises her hand. The
experimenter again provides the subject with a new session number and fixed deduction, and
watches while the subject restarts the program. This process continues until all Part I sessions
- 13 -
are completed. After the final Part I session, the subject totals her earnings on the record sheet,
signs up for Part II (a least one day later, and no more than seven), and is excused.
Upon returning for Part II, the subject is again seated at a private computer carrel and she
is given her record sheet from Part I (she uses the same subject number in both parts). Part I and
II procedures are identical13, but the Part II parameters and matrices differ from those used in
Part I. Upon completion of the final session of Part II, the subject is privately paid her cash
earnings and excused. In our experiment, cash earnings in Part I average $14.35, with a low of
about $4 and a high of about $25. In Part II, the average is $20.67, with a low of about $3 and a
high of about $28, exclusive of a $5 participation fee paid for timely arrival.
IV.
Results
IV.A. Summary.
Our Part II data comprise a panel with eighty subjects recording thirty-eight different
round-level decisions each, for a total of 3040 observations. The subjects complete 720 rounds
in session S1, 640 in S2, 800 in S3 and 880 in S4. Before reporting our statistical analysis, we
make four observations based on summary data provided in Table 2.
*** Table 2. Summary of Round-Level Outcomes – about here ***
First, subjects predominantly choose to play the cell selection game. The fixed payoff
option is chosen in only four percent (110) of the 3040 total rounds.14 However, it is chosen
significantly more often when the fixed payoff is high than when it is low. In sessions S1 and
13
The consent form is signed prior to participation in Part I. Prior to participation in Part II, each subject is
reminded that she signed the consent form during her previous visit, and offered an opportunity to review the form.
14
The fixed payoff choice is confined to less than a third (24/80) of the subjects. By session, the fixed payoff was
chosen 47 times in S1 by ten different subjects, 46 times in S2 by eighteen different subjects, 10 times in S3 by six
different subjects, and 4 times in S4 by three different subjects.
- 14 -
S2, where the fixed payoff is 80% of the maximum possible game payoff, the fixed payoff is
chosen in seven percent of the rounds (47 out of 720 and 46 out of 640, respectively). In
contrast, it is chosen only one percent of the rounds in S3 and S4 (10 out of 800 and 7 out of 880,
respectively), where the fixed payoff is 50% of the possible game maximum. This is consistent
with a decision-cost model of behavior, although the fact that it occurs in only seven percent of
the rounds with the high fixed payoff suggests that the typical subject’s decision cost is
substantially less than 20% of the maximum possible earnings.
Second, subjects are able to complete the task in the allotted time. A zero payoff occurs
if the subject fails to click either the “Decline Play – Accept Fixed Payoff” or the “End Round”
button before the round clock expires. The zero payoff occurs in less than one percent (20) of
the 3040 total rounds. Thus in over ninety-nine percent of the rounds, subjects are able to earn a
positive payoff within the four-minute time limit.
Third, subjects typically do quite well when they play the cell selection game. Rarely do
they earn less than 80% of the maximum possible, and in eighty-four percent of the rounds (855
+ 1715 out of 3040) they earn 90% or more or the maximum possible. Nonetheless, there is a
noticeable difference in the dispersion of earnings when the cell payoff is low versus when it is
high. In sessions S1 and S3, where the cell payoff is 20, earnings are concentrated in the 97–
100% range and rarely below 90% of the maximum possible. In contrast, when the cell payoff is
100 (sessions S2 and S4), less than half the observations are in the 97–100% range and roughly
fifteen to twenty percent of the observations are in the 80–89% range. We elaborate on these
differences across cell payoff treatments in our statistical analysis below.
Fourth, across all sessions, the average per-round earnings (including those rounds where
the fixed payoff is chosen or the subject times out) are 94.7% of the maximum possible, and the
- 15 -
per-session averages are at least 93% of the maximum possible. Given that we utilize a two-part
training and data collection sequence, the high earnings we observe here are consistent with
those reported elsewhere in the experimental literature for experienced subjects. We note,
however, that an important feature of our experimental design is the revocable-moves option,
i.e., subjects are able to search across alternative combinations of cells before making a final
decision. Had we invoked the non-revocable moves option in our software, mean per-round
earnings would almost certainly be less than those shown in Table 2.
IV.B. Statistical model.
We estimate a subject fixed effects model to statistically analyze our data and conduct the
hypothesis tests prescribed by our experimental design. We utilize three different dependent
variables, two that are performance measures and one that is an activity or search measure.15
Letting Y generically represent the three dependent variables, our model is:
Yitj =
4
∑ β SjSj + β F FixedPay itj + β T Timeout itj +
j =1
79
∑ μ i Sub i +ε itj
(4)
i =1
where
Yitj
= the Earnings Ratio, Cell Ratio or Search Ratio (defined below) for
subject i in round t of session Sj.
Sj
= 1 if the Yitj observation is from session Sj, = 0 otherwise.
FixedPayitj
= 1 if subject i chooses the fixed payoff option in round t of session Sj, = 0
otherwise.
Timeoutitj
= 1 if time expires before subject i makes a final decision (see Table 2
above) in round t of session Sj, = 0 otherwise.
15
We estimate the model with StataCorp (2003). The model presented here is the most parsimonious of several we
examined. For example, we included dummy variables for round specific effects, but found them statistically
insignificant using an F test; we also estimated a random effects model, but rejected it based on the Hausman (1978)
test statistic calculated by Stata. See Botelho et al. (2005) for a discussion of the choice of models in the presence of
unobserved heterogeneity.
- 16 -
Subi
= 1 if the Yitj observation is from subject i, = 0 otherwise.
εitj
= random error term.
With this model, we analyze our cell payoff treatment by comparing S1 v. S2 via a test of the
null hypothesis H0: βS1 = βS2, and by comparing S3 v. S4 via the test H0: βS3 = βS4. Similarly, we
analyze our fixed payoff treatment by comparing S1 v. S3 via the test H0: βS1 = βS3 and by
comparing S2 v. S4 via the test H0: βS2 = βS4.
The first two dependent variables, Earnings Ratio and Cell Ratio, measure the subject’s
performance relative to the optimal solution. The Earnings Ratio measures the subject’s perround earnings, as a percentage of the maximum possible earnings in the given round of the
given session. The Cell Ratio measures the number of cells in the subject’s final choice each round
(i.e., when she clicks the “End Round” button), as a percentage of the number of cells in the
optimal solution for the given round and session. The third dependent variable, Search Ratio,
uses the subject’s selection and de-selection of cells each round to measure her search activity.
This ratio is a count of all cells that the subject selects during the course of a round, including
those that the she subsequently deselects (as moves are revocable), divided by the number of
cells in her final choice. This measure normalizes search activity so that it (partially) accounts
for any heuristic that the subject might use.16
On the right-hand side of equation (4), the expected sign of both βF and βT is negative
The FixedPay and Timeout variables control for those 130 rounds (see Table 2 above) where,
16
Consider the following example. Two subjects both have three cells in their final selection. Suppose that the
second subject selects and then deselects two additional cells in the process of making her final choice, i.e., she
searches a total of five cells while the first subject only searches three. The first subject has a Search Ratio of 3/3 =
1.0, and the second has a ratio of 5/3 = 1.67, thus the subject who searches more has a higher Search Ratio. Now
consider a third subject who has five cells in her final choice but does not deselect any cells during the round. This
subject has a Search Ratio of 5/5 = 1.0. Thus according to the this measure of search activity, the first and third
subjects search an equal amount, after accounting for their final choice, and the second subject searches more.
- 17 -
respectively, the subject either chooses the fixed payoff option or she “times out.” If FixedPay =
1, then the Earnings Ratio equals the amount of the fixed payoff (divided by the maximum
possible game earnings for the round) and the Cell Ratio and Search Ratio equal zero. When
Timeout = 1, all three ratios have a value of zero. Also, the seventy-nine subject dummy variables
control for unobservable subject-specific effects, and the joint test H0: all μi = 0 tests for homogeneity
across subjects.17 We expect a substantial amount of heterogeneity across individual subjects, i.e., we
expect to reject H0, but we do not otherwise model their preferences with a specific functional form.
IV.B. Estimation and hypothesis tests.
Table 3 reports the estimated fixed effects equations and the hypothesis tests. The upper
portion of the table reports the estimated coefficients and standard errors, the adjusted R2 and
model F-statistic, and F-statistics for the subject-specific effects. The estimated models have
reasonably good explanatory power. The adjusted R2 values are 0.858, 0.777 and 0.483 for the
Earnings Ratio, Cell Ratio and Search Ratio, respectively, and the F-statistics indicate that the
models account for a statistically significant amount of the variation in the respective dependent
variable. We note, however, that over half of the variation in the Search Ratio is unaccounted for
by our specification. In each model, all four of the βSj coefficients are statistically significant.
These intercept terms estimate mean treatment effects; the Earnings Ratio estimates correspond
closely to the uncontrolled means in Table 2. The estimated βF and βT coefficients are, as
predicted, negative and statistically significant. Not surprisingly, all three F-statistics for the
subject specific μi coefficients reject H0, indicating there is significant variation across
experimental subjects.
17
Of course, the model cannot be estimated if all eighty subject dummy variables are included, and the choice of
which particular subject to omit is arbitrary. In all three regressions reported in Table 3, the StataCorp (2003)
default chose to drop subject 68. If a different subject is omitted, the βSj point estimates adjust slightly to reflect the
particular omitted subject, but all test statistics and R2 values remain unchanged. Thus in our analysis we do not
interpret the point estimates shown in Table 3, but focus on the hypothesis tests specified by our design..
- 18 -
*** Table 3. Fixed Effects Estimation and Hypothesis Tests –– about here ***
The lower portion of Table 3 presents the hypothesis tests specified by our experimental
design. We concentrate first on the Earnings Ratio and Search Ratio tests, as there is some
uniformity across those results. We then discuss the Cell Ratio, which tells a less consistent tale.
Our cell payoff treatment compares S1 v. S2 and S3 v. S4. Recall that S1 and S3 have
the lower cell payoff (20), and S2 and S4 have the higher cell payoff (100). The Earnings Ratio
results clearly indicate that the lower the cell payoff, the more the subject earns. The model
estimates βS1 > βS2 and βS3 > βS4, and the corresponding hypothesis tests both have p = .0000.
This is consistent with the data in Table 2, where earnings are higher and less dispersed when the
cell payoff is low. The Search Ratio results provide some evidence that the lower the cell
payoff, the more the subject searches. That model also estimates βS1 > βS2 and βS3 > βS4, but only
one of the corresponding hypothesis tests rejects H0 (p = .0198 and .7212, respectively). A
relatively low cell payoff implies that the cell value is more important in determining the reward
from selecting a particular cell; see equation (1). Perhaps this makes the subject concentrate
relatively more on the cell value, which leads her to try more combinations and thus more
closely approximate the optimal solution in her overall choice decision. This in turn increases
her earnings.
In our fixed payoff treatment, S1 and S2 have the higher fixed payoff (80% of the
maximum game payoff), and S3 and S4 have the lower fixed payoff (50%). Thus the relevant
comparisons are S1 v. S3 and S2 v. S4. In general, the Earnings Ratio and Search Ratio results
suggest that the higher the fixed payoff, the more the subject earns and searches. Both models
estimate βS1 > βS3 and βS2 > βS4, but under each model only one of the two tests rejects H0. The
Earnings Ratio tests have p = .2971 for S1 v. S3 and p = .0000 for S2 v. S4, while the Search
- 19 -
Ratio tests have p = .0296 and p = .9500, respectively. Prior to the experiment, we expected that
with a high fixed payoff, the subject might be less likely to incur the decision cost associated
with the cell selection game. But the high fixed payoff apparently entices the subject to pay even
more attention to her choices (e.g., search more) and thus earn more, though the effect is
stronger in some cases than in others. Importantly, we chose to make the fixed payoff option
available at any time during a round; our software can restrict the fixed payoff as an all-ornothing choice preceding the cell selection game.18 The results would almost certainly be
different if the subject’s decision to play the game precluded her ability to take the fixed payoff.
If, as the Earnings and Search Ratio data suggest, the low cell payoff and/or high fixed
payoff does in fact lead the subject to more closely approximate the optimal solution, then one
might expect the combined effect to be most potent.19 That is, the effect should be greatest in S1
(20 cell payoff, 80% fixed payoff) and least in S4 (100 cell payoff, 50% fixed payoff). Our
results are consistent with this conjecture.
In both the Earnings Ratio and Search Ratio
regressions, βS1 is indeed the highest of the four βSj estimates while βS4 is the smallest.
Furthermore, in both regressions βS3 is the second highest estimate and βS2 is the third highest.
As S3 has the 20 cell payoff and the 50% fixed payoff, and S2 has the 100 cell payoff and 80%
fixed payoff, this is marginal evidence that the cell payoff effect may be the stronger of the two.
Our Cell Ratio results are less consistent. In the cell payoff treatment, the Cell Ratio
model estimates βS1 < βS2 and βS3 > βS4, and the respective hypothesis tests have p = .0113 and p
= .0312. So apparently lowering the cell payoff reduces the Cell Ratio if the fixed payoff is high
(S1 v. S2), but increases it if the fixed payoff is low (S3 v. S4). In the fixed payoff treatment, the
18
If the subject selects any cell from the game matrix, the “Decline Play – Accept Fixed Payoff” box immediately
becomes inoperative.
19
Adding interaction terms to equation (4), and thus to Table 2, greatly complicates our presentation without adding
substantial explanatory power to the model. In the interest of brevity, we opt for this informal discussion.
- 20 -
model estimates βS1 < βS3 and βS2 > βS4, and the respective hypothesis tests have p = .0001 and p
= .4468. So apparently increasing the fixed payoff lowers the Cell Ratio if the cell payoff is low
(S1 v. S3), and marginally increases it or has no effect if the cell payoff is high (S2 v. S4).
None of these Cell Ratio results fit easily with our “low cell payoff and/or high fixed
payoff make the subject focus on the essence of the problem” story inspired by our combined
Earnings Ratio and Search Ratio results. Below we present evidence that the typical subject may
use a heuristic that approximates the optimal solution, but does not mimic it per se. Perhaps
what are crucial are which cells the subject finally selects rather than the number of cells she
selects. As she searches more, she earns more because she makes better choices, even if in some
cases (particularly when cell payoff is low) she decides on fewer cells than she initially selects.
IV.D. Heuristics.
We identify three simple (albeit ad hoc) heuristics designed to analyze the subject’s final
cell selections per round.
Thirty-six of the thirty-eight matrices we use in this experiment have
optimal solutions comprised of six or more cells, and the corresponding cell values typically
cover the entire 100–1000 range (see Table 1). One possibility is that the subject simplifies our
version of the complex “knapsack problem” by focusing on a subset of cell values within the
overall range, and picks primarily from that subset until she reaches her value limit. Focusing on
high cell values will lead her to select relatively fewer cells, and focusing on low cell values will
lead her to select relatively more. In terms of the number of cells in the final selection, the latter
strategy will more closely approximate the optimal solution (given our matrices), and thus yield
higher earnings. But to the extent that selecting fewer cells involves fewer decisions, the former
strategy has a lower decision cost. Presumably the subject would choose between strategies
based on her subjective earnings/cost tradeoff.
- 21 -
We conduct simulations on our thirty-eight matrices that utilize this “focus on a subset of
cell values from the overall range” strategy.20 Our simulations yield three viable alternatives:
Heuristic H focuses on “high” cell values in the 700–1000 range; it selects two or three
cells per round. H has the lowest decision cost and lowest payoff of the three.
Heuristic M focuses on “medium” cell values in the 350–700 range; it selects four to five
cells per round. M is between L and H in terms of both decision cost and payoff.
Heuristic L focuses on “low” cell values in the 100–350 range; it selects six or more
(most often seven or eight) cells per round. L has the highest decision cost and highest
payoff of the three.
In Table 4, we categorize ex post each subject according to these heuristics via a threestep process. In the first step (not shown in Table 4), the subject is given a round-level
designation of H, M, L or F/T in each of her thirty-eight rounds. If her final choice for the round
includes three or less cells then she is designated as H, if it includes four or five cells she is
designated as M, and if it includes six or more cells she is designated as L. If she chooses the
fixed payoff option or “times out” she is designated as F/T. In the second step, the subject is
given a session-level designation of H, M, L or Indeterminate for each of the four sessions, based
on her round-level designations in the respective session. Table 4 shows the session-level
summary data. If a majority of a subject’s round-level designations are of the same type (e.g.,
seven out of twelve rounds she is L), she is designated that type at the session-level. There are
cases where the session-level designation is admittedly subjective, i.e., they do not easily fit our
majority rule and we make a judgment call. Our conservative approach is to designate any
ambiguous case as Indeterminate. But generally individual subjects exhibit a great deal of
20
Our simulations use a spreadsheet with cell values ranked in one column from high to low (versus the randomly
populated matrix 5x5 matrix viewed by the subject) for each of the thirty-eight decision matrices. This alternative
format makes it easier to identify the most profitable choice(s) under any of the three heuristics, and our simulations
did not face a time constraint, as did the subject.
- 22 -
consistency in their choices across the rounds of a given session. In Table 4, only five or 2% of
our 320 session-level designations (80 subjects × 4 sessions) are Indeterminate.21
In the third step, the subject is assigned an overall-level designation of H, M, L or Mixed
based on her session-level categorizations. If the subject is designated as H in all four sessions,
then she is categorized as H at the overall level; an analogous “uniformity” rule applies to
categories M and L. If any one of the subject’s four session-level designations differs from
another (e.g., L in S1, S2, S3 but H in S4), she is designated as Mixed at the overall level. The
far right column of Table 3 shows overall-level summary data.
*** Table 4. Ex Post Categorization of Subject Heuristics –– about here ***
Because of the ad hoc nature of our heuristics, and the subjective nature of our
categorizations, we do not conduct formal statistical tests on the data summarized in Table 4.
Instead, we make two observations about subjects’ responses to the tradeoff between earnings
and decision cost. For simplicity, we use terminology like “the subject uses heuristic X” when
the more accurate terminology is “the subject’s choices are consistent with heuristic X.”
One, most of the subjects opt to incur the subjective decision cost in order to obtain the
higher payoff, while very few opt for the least onerous rule. Heuristic L is used by at least 60%
of the subjects in each of the sessions, and by 50% of the subjects at the overall level. In
contrast, heuristic H is used infrequently, roughly 10% in each of the sessions and 5% overall.
The regularity with which subjects use a heuristic that approximates the optimal solution is
consistent with the high average earnings reported in Table 2.
21
No individual subject is designated as Indeterminate in more than one round. Three subjects chose the fixed
payoff in all or nearly all of the rounds of a session (in S1 subjects 68 and 79, and in S2 subject 47), so there are
insufficient data to categorize them as H, M or L. In S3 subject 50 and in S4 subject 49 both appear to use all three
heuristics equally often.
- 23 -
Two, despite the apparent tendency of subjects to incur the decision cost, a significant
minority undertakes a strategy designed to avoid at least some of the cost. In Table 4, four
subjects always use heuristic H and eleven always use M at the session level (hence they receive
this designation at the overall level). Out of the twenty-five subjects designated as Mixed at the
overall level, eight (not shown in Table 4) never use L at the session level. Thus at the overall
level, 29% of the subjects (4 + 11 + 8 = 23 of 80) never use heuristic L at the session level, i.e.,
they use a heuristic that does not approximate the optimal solution. Presumably they find the
effort too arduous relative to the potential gain.
VI.
Conclusion
Although wage and salary compensation is an important factor in recruitment and
retention of employees, the scope and mix of the fringe benefits is also an oft-cited factor. For
managers and employers, the effectiveness of the benefit package as a recruitment and retention
tool depends on the ability of the workers to make optimal choices from the available options.
There is considerable evidence from the naturally occurring world that discrete, multi-attribute
goods complicate the decision-making task. Here, we use experimental methods to examine
individual decision-making in a stylized discrete multi-attribute goods setting. Two main results
emerge.
First, the relative tradeoff between the attributes of the discrete good is a significant
treatment variable. The discrete good has two attributes; one that is constant across choices and
one that varies across choices. When the variable attribute has relatively more weight in the
overall reward function, our typical subject earns a higher reward on average, apparently because
she engages in more search activity. One hypothesis is that with greater weight on the variable
- 24 -
attribute, the subject focuses on the “essence of the problem” and searches until she finds
(perhaps unwittingly, but in a response to incentives) an approximation to the optimal solution.
Overall, our typical subject does quite well, averaging over 90% of the maximum possible
reward. Further analysis reveals the majority of our subjects adopt heuristics that mimic the
optimal solution to the complex linear programming problem. From the perspective of a benefits
package, this suggests the employee gets greater satisfaction from a flexible plan that places
relatively more emphasis on the individual coverage amounts of benefits that are included, and
relatively less emphasis on its inclusion. Importantly, our subjects are allowed to search via a
revocable-moves option. Removing this option would almost certainly affect their ability to do
well (have high average earnings) in this game. We leave this potential treatment to future
experiments.
Second, our subjects rarely choose a fixed payoff option with a known payoff and low
decision cost, even when the fixed payoff is 80% of the maximum possible under the decisionmaking task. Thus the typical subject places a high implicit valuation on the flexibility in
making choices, as she is apparently confident in her ability to exceed the fixed payoff. This
indicates that a flexible benefits package may be strongly preferred to a pre-defined benefits
package. Additionally, even though the fixed option is rarely chosen, our typical subject is
(apparently) enticed to search more and subsequently earn more when the fixed option is highest.
This somewhat surprising result suggests that if an employer offers a fixed package as an option
to a cafeteria plan, the employee will get greater satisfaction if the fixed package has an overall
value comparable (but less) than the flexible one even if the employee ultimately chooses the
flexible plan. According to our results, the availability of a valuable fixed plan leads (or helps to
lead) the employee to choose a flexible plan that yields her greater satisfaction. An important
- 25 -
caveat is that the fixed plan be available while the employee is making choices in the cafeteria
plan. Our current experimental design does not address the case where shopping in the flexible
plan eliminates the fixed plan as an option. We leave this as an area of future work.
Collectively, our results suggest that an individual (at least, a financially motivated
experimental subject) can indeed handle difficult decision-making tasks like those involving
discrete multi-attribute goods. Based on our analysis, the typical subject applies a heuristic that
approximates the marginal setting of neo-classical consumer theory.
This suggests that
individuals prefer settings that allow for marginal decisions but it may also suggest that
individuals prefer, ceteris paribus, consumption bundles with larger numbers of goods. Further,
our experimental subjects are fairly versed at the decision making task. It remains to be seen
what more naïve decision makers might do. For example, choosing components of a flexible
benefits package might be an imposing task to new entrants into the labor force. This ambiguity
suggests that unanswered questions concerning individual behavior remain for further
investigation.
- 26 -
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- 28 -
Figure 1. Subject Screen Display
- 29 -
Table 1. Experimental Parameters
I. Parameters varied across sessions
Cell Payoff per round
Fixed Payoff per round a
Number of Rounds
Fixed Deduction (US$)
II. Parameters constant across sessions
Cell
Value
Cell
Matrix
Range
Weight
Size
5 ×5
100–1000
1.2
a
Session
S1
Session
S2
Session
S3
Session
S4
20
100
20
100
80%
80%
50%
50%
9
8
10
11
$17.00
$18.00
$20.00
$27.00
Value
Limit
2000
Revocable
Moves?
Yes
Seconds
per
Round
240
Conversion
Rate
(E$ to US$)
0.001
As a percentage of maximum possible earnings from playing the cell selection game.
- 30 -
Table 2. Summary of Round-Level Outcomes
Percent
(number)
who chose
fixed
payoff
Percent (number) who chose cell selection game
and earned:
0%
of max.
1–
80%
of max.
80 –
89%
of max.
90 –
96%
of max.
97 –
100%
of max.
Mean
per-round
earnings as
% of max.
possible
Overall
4%
(110)
1%
(20)
1%
(41)
10%
(299)
28%
(855)
56%
(1715)
94.7%
n = 3040
S1
7%
(47)
1%
(5)
1%
(4)
1%
(5)
25%
(181)
66%
(478)
95.9%
n = 720
S2
7%
(46)
1%
(7)
2%
(11)
16%
(99)
34%
(216)
41%
(261)
92.9%
n = 640
S3
1%
(10)
1%
(4)
0%
(0)
2%
(12)
23%
(180)
74%
(594)
96.9%
n = 800
S4
1%
(7)
1%
(4)
3%
(26)
21%
(183)
32%
(278)
43%
(382)
93.0%
n = 880
- 31 -
Table 3. Fixed Effects Estimation and Hypothesis Tests
Estimated
Coefficient
Dependent Variable (n = 3040)
Earnings Ratio
Cell Ratio
Search Ratio
βS1
(std. err.)
0.952 a
(0.007)
0.638 a
(0.022)
1.707 a
(0.062)
βS2
(std. err.)
0.926 a
(0.007)
0.656 a
(0.022)
1.660 a
(0.063)
βS3
(std. err.)
0.950 a
(0.002)
0.665 a
(0.022)
1.665 a
(0.062)
βS4
(std. err.)
0.912 a
(0.007)
0.651 a
(0.022)
1.659 a
(0.062)
βF
(std. err.)
–0.186 a
(0.004)
–0.737 a
(0.014)
–1.289 a
(0.040)
βT
(std. err.)
–0.955 a
(0.008)
–0.744 a
(0.028)
–1.468 a
(0.078)
Adj. R2 b
F5,2955
0.858 b
F = 220.01
p = .0000
0.777 b
F = 122.55
p = .0000
0.483 b
F = 32.88
p = .0000
H0: all μi = 0
F79,2955
F = 28.11
p = .0000
F = 70.09
p = .0000
F = 13.53
p = .0000
t = 11.8
p = .0000
t = –2.54
p = .0113
t = 2.33
p = .0198
t = 19.4
p = .0000
t = 2.15
p = .0312
t = 0.36
p =.7212
Fixed Payoff Treatment
S1 v. S3
t = 1.04
H0: βS1 = βS3
p = .2971
t = –3.93
p = .0001
t = 2.18
p = .0296
t = 0.76
p = .4468
t = 0.00
p = .9500
Cell Payoff Treatment
S1 v. S2
H0: βS1 = βS2
S3 v. S4
H0: βS3 = βS4
S2 v. S4
H0: βS2 = βS4
a
t = 6.94
p = .0000
p < .01 for two-tailed test H0: coefficient = 0.
b
Adj. R2 and model F statistic obtained by dropping dummy variable S1 and estimating the
model with a constant.
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Table 4. Ex Post Categorization of Subject Heuristics
Heuristic
H
M
L
Indeterminate b
Mixed c
Session-Level
S1
10%
(8)
26%
(21)
61%
(49)
3%
(2)
S2
10%
(8)
28%
(22)
61%
(49)
1%
(1)
S3
14%
(11)
25%
(20)
60%
(48)
1%
(1)
S4
9%
(7)
28%
(22)
63%
(50)
1%
(1)
---
---
---
---
a
Overall
Level
5% a
(4)
14% a
(11)
50% a
(40)
--31%
(25)
An individual subject has this row’s categorizations in all four sessions, e.g., the
four subjects designated as H at the overall level are assigned H at the session-level
in S1, S2 S3 and S4.
b
Unable to categorize as either H, M or L at the session level.
c
At least two session-level categorizations differ.
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