Experimental Economics, 8:117–144 (2005) c 2005 Economic Science Association


Joining a Queue or Staying Out: Effects of Information Structure and Service Time on Arrival and Staying Out Decisions DARRYL A. SEALE Department of Management, University of Nevada Las Vegas, Las Vegas, NV 89154, USA email: [email protected] JAMES E. PARCO Department of Management, United States Air Force Academy, Academy, CO 80840-2944, USA email: [email protected] WILLIAM E. STEIN Department of Information & Operations Management, Mays Business School, Texas A&M University, College Station, TX 77843 USA email: [email protected] AMNON RAPOPORT Department of Management and Policy, University of Arizona, Tucson, AZ 85721 USA; Hong Kong University of Science and Technology email: [email protected] Received October 28, 2003; Revised January 31, 2005; Accepted January 31, 2005

Abstract We study a class of single-server queueing systems with a finite population size, FIFO queue discipline, and no balking or reneging. In contrast to the predominant assumptions of queueing theory of exogenously determined arrivals and steady state behavior, we investigate queueing systems with endogenously determined arrival times and focus on transient rather than steady state behavior. When arrival times are endogenous, the resulting interactive decision process is modeled as a non-cooperative n-person game with complete information. Assuming discrete strategy spaces, the mixed-strategy equilibrium solution for groups of n = 20 agents is computed using a Markov chain method. Using a 2 × 2 between-subject design (private vs. public information by short vs. long service time), arrival and staying out decisions are presented and compared to the equilibrium predictions. The results indicate that players generate replicable patterns of behavior that are accounted for remarkably well on the aggregate, but not individual, level by the mixed-strategy equilibrium solution unless congestion is unavoidable and information about group behavior is not provided. Keywords: batch queueing, equilibrium solution, experiment, coordination JEL Classification: C71, C92, D81

118 1.

SEALE ET AL.

Introduction

Whenever the demand for service exceeds the capacity to provide it, customers must wait in line to be served. People routinely wait in queues—and get impatient, annoyed, or even frustrated by long waits—to purchase movie and concert tickets, conduct bank transactions, mail packages, call airline offices, and purchase food in university cafeteria and popular restaurants. Waiting in line is not just a personally annoying and inconvenient experience. In the USSR, notoriously long queues regularly formed to purchase basic food products and acquire other necessities had serious economic as well as political consequences. In the US, it has been estimated (Hillier and Lieberman, 2001) that 37 billion hours per year are spent waiting in queues. If, instead, this time could be spent productively, it would amount to 20 million person-years of useful work per year. Queueing theory (e.g., Hall, 1991; Hillier and Lieberman, 2001) constructs formal models to represent various queueing systems that arise in practice, and derives results for each of these models indicating how the corresponding queueing system should perform. The ultimate purpose of the analysis is to determine how to operate a queueing system, provided that it is modeled appropriately, in the most effective way. Once the basic characteristics of the system are formally specified, the analysis typically proceeds by assuming that the system has reached a steady state, where the probability distribution of the state of the system remains the same over time. When the queueing facility has just begun to operate— the so-called transient state—the state of the system is greatly affected by the initial state and the time that has since elapsed. With a few exceptions, queueing theory has focused on the steady state case because the transient case is, in general, analytically intractable (Hall, 1991; Medhi, 1991). Customers in queues typically share the same objective, namely, to receive service and derive utility from doing so while minimizing the time they spend waiting in line. Queueing theory often leaves no room for the customer to make decisions, as the characteristics of the queueing system are typically assumed to be exogenously determined. In contrast, our interest in this paper is in determining how customers decide whether to join a queue and if so at what time to arrive. Our approach differs from the one underlying queueing theory in three major respects: 1. Our goal is descriptive not prescriptive. Rather than deriving equations to determine how a particular queueing system should perform effectively, we wish to identify and explain whatever behavioral regularities emerge when customers decide independently whether to join a queue, and if so, at what time to arrive. Our approach, however, is not empirically based on field observations. Rather, we create a queueing system in the laboratory, recruit subjects to participate in an interactive decision making experiment with payoff contingent on performance, and then compare their behavior to predictions derived from theory. 2. We focus on the transient case by considering queues with non-stationary elements where the calling population is finite and the opening and closing times of the queue are fixed and commonly known. Queueing systems with fixed opening and closing times are, in fact, quite common in practice. Banks open and close at commonly known times,

JOINING A QUEUE OR STAYING OUT

119

and so do Motor Vehicle Departments, post offices, restaurants, and automobile emission inspection stations. 3. Our third point of departure is in the way we model arrival time. Queueing models typically assume exogenously determined arrival times. Most assume a Poisson input process, although some models assume that arrivals are scheduled at regular intervals or follow an Erlang interarrival time distribution. In sharp contrast, we consider queueing systems where arrivals are endogenously determined. We study behavior in such systems by imposing a common payoff structure, and then formulate the interactive decision making process as a non-cooperative n-person game in strategic (normal) form and complete information, construct the equilibrium solutions, examine their implications, and then test these solutions with experimental data. Section 2 describes the queueing game. Section 3 outlines the computational method used to construct the equilibrium solution to the queueing game. Numerical solutions are presented to two different games that are later subjected to experimental investigation. Section 4 describes related research on queueing systems in which customers choose their time of arrival. In particular, it discusses a previous experimental study by Rapoport et al. (2004), hereafter RSPS, on queueing systems with endogenous arrivals that the present study extends in several major ways. The experimental method and results of four different experimental conditions are presented in Sections 5 and 6, respectively. Section 7 concludes. 2.

The queueing game

In all the conditions of our experiment, the queueing game is presented to the subjects as the choice of an arrival time to a service facility. The assumptions of the game are stated below. 1. Opening and closing times. The service facility opens at time T0 and closes at time Te . 2. Calling population. The calling population is finite of size n. 3. Arrival pattern. Arrivals are made in discrete time. At the beginning of the game, each player must decide whether to join the queue, and if yes, at what time to do so. Decisions are made simultaneously and anonymously. 4. Tie-breaking rule. Because the strategy space is discrete, batches of simultaneously arriving players are possible. If m players arrive at the same time (2 ≤ m ≤ n), their order of arrival is determined randomly with equal probabilities. 5. Balking and reneging. Balking (not entering the queue upon arrival) and reneging (departing the queue after arrival) are prohibited. In particular, service will not be started unless it can be completed by Te . Therefore, arriving players are not guaranteed to receive service (and cannot leave the queue even if they know that service will not be provided). 6. Early arrivals. A player may obtain a favorable position in the queue by arriving before T0 , but no service is provided until this time. 7. Service requirement. The service time of each player is fixed with the same parameter d.

120

SEALE ET AL.

8. 9. 10. 11. 12.

Queue discipline. The queue discipline is FIFO (first in first out). Number of servers. There is a single server. Number of service stages. There is a single service stage. Service capacity. There is no physical limitation to the queue length. Prior information. Players cannot observe the queue before making their irrevocable decision of whether and when to join it. 13. Payoff structure. The payoff function—the same for each player i (i = 1, 2, . . . , n)—is given by:  g, if player i stays out of the queue       if player i waits wi minutes and does not complete −c × wi , Hi = service       r − c × wi , if player i waits wi minutes and service is completed where wi is the time (in minutes) spent in the queue waiting for service to commence. There is no waiting cost for the time spent being served. In the payoff structure above, c is the waiting cost per minute, r is the reward for completing service, and g is the payoff for staying out. Whereas c and r are assumed to be positive, g can take either positive or negative values. The values of T0 , Te , and d, as well as the (endogenous) time of waiting w are measured in commensurable units. The parameters n, T0 , Te , d, r , c, and g are all common knowledge, rendering the queueing task a non-cooperative strategic form game of complete information. Several brief comments on these assumptions are in order. Assumptions 8–12 are standard in the literature. Although not common, Assumption 7 of fixed service time is also made quite often in the queueing literature (e.g., Hillier and Lieberman, 2001; Mehdi, 1991). Queueing systems with fixed opening and closing times (Assumption 1) are usually not the focus of classical queueing theory because they result in analytical complexity. Although queueing theory almost always assumes the strategy space to be continuous, this assumption cannot be realized in experimental investigations. Instead, we assume a discrete strategy space (Assumption 3) and render it sufficiently rich to provide the players a wide selection of arrival times. Given a discrete strategy space, the assumption of random tie breaking (Assumption 4) is natural. Assumption 6 is a major point of departure from the previous study of RSPS (see Section 4 below) who disallowed early arrivals. In most queueing systems with fixed opening and closing times (e.g., banks, post offices, popular concerts), early arrivals are often observed. Assumptions 2 and 5 typically do not hold in practice. In particular, customers do not know in general the size of the calling population with precision, nor do they stay in line if they know with certainty that they will not be served. Both of these assumptions were primarily made for experimental reasons and may be relaxed in future research. Finally, the assumption that the waiting cost is proportional to the time of waiting (Assumption 13) is also common (e.g., Naor, 1969).

JOINING A QUEUE OR STAYING OUT

3.

121

Equilibrium analysis

Section 6 reports the results of an experiment using a 2 × 2 between-subject design (game type by information structure). The two versions of queueing game, termed Game G30 and Game G45, share all the parameter values except one. In both games, n = 20, T0 = 8:00, Te = 18:00, arrivals are restricted to 5-minute intervals, r = 100, c = 1, and g = 15. The two games only differ from each other in the (fixed) service time: d = 30 in Game G30 vs. d = 45 in Game G45. This difference has major implications for the equilibrium solutions and the way subjects perceive these two games. In Game G30, there are n! pure-strategy equilibria where all the n players enter the queue, none has to wait at all, and the system idle time is zero. This would be the case if players were to arrive, one at a time, at exactly 8:00, 8:30, . . . , 17:30. Technically, this is a game of coordination where players have to choose one of n! pure-strategy equilibria that are not Pareto-rankable, each yielding every player the maximum payoff r . If n is large, as in the present experiment, and pre-play communication is prohibited, coordination on a specific pure strategy equilibrium is practically impossible even with multiple iterations of the stage game. However, players who realize the possibility of all receiving the maximum payoff may focus on arrival times that are multiples of d, starting at 8:00, and attempt to signal their intention to do so to other players. In contrast, if d = 45 there is no symmetric pure-strategy equilibrium to the game and congestion cannot be avoided. Games G30 and G45 have symmetric mixed-strategy equilibrium solutions. The computational method used to construct these equilibria will now be described. We note that in equilibrium players will not arrive more than h time units before T0 , if by doing so they are guaranteed to earn less than the sure payoff g they could have earned by staying out. Thus, h is the largest integer satisfying the inequality r − 5ch ≥ g. Therefore, arrival times can be effectively restricted to the set {−h, −h + 1, . . . , −1, 0, 1, . . . , Te }. Renumbering these elements from 0 to T , we use the notation p0 , p1 ,. . . , pT to describe a probability distribution T over all arrival times {0, 1, . . . , T } in the queueing game, and 1− t=0 pt the probability of staying out of the queue. Because of the symmetry of the n players, our search is restricted to a symmetric equilibrium where each player uses the same mixed strategy { pt }. 3.1.

The computational method

Consider any one of the n players as the “designated player,” and assume that each of the other n − 1 players uses the same but arbitrary mixed strategy p0 , . . . , pT . At time t < 0 all the players are outside the system. At the initial period t = 0 a number of players may join the system. This number is binomial with parameters n − 1 and p0 . As these players move through the system over time additional players join the queue. Therefore, we can describe the system as a stochastic process and by using states that precisely characterize the system it will be a Markov chain. Assume that the designated player arrives at time t. The other n − 1 players act independently according to the same (but yet to be determined) mixed strategy. The progression of the designated player through the system from period t onward (and, consequently, the

122

SEALE ET AL.

payoff he receives) can be determined using the Markov chain. In this manner, the expected payoff E t of the designated player (the same for all players) can be computed conditional upon arriving at period t. Because future arrivals cannot affect the payoffs of those players already in the system, E t is a function of the mixed strategy only through p0 , p1 , . . . , pt . The standard characterization of a Nash equilibrium { pt } with expected payoff E is: (i) if pt > 0, then E t = E, and (ii) if pt = 0, then E t ≤ E. We choose an initial { pt } and compute {E t } and then iteratively adjust { pt }: if E t is relatively high (low), increase (decrease) pt . Of course, E is also an unknown quantity. The Appendix to RSPS contains further details on this computational method.

3.2.

Solution of the present game

The computational method described above was used to construct the symmetric mixedstrategy equilibria for Games G30 and G45. In both games, the facility is open for 10 hours from 8:00 to 18:00, and both arrival and waiting times are measured in minutes. However, to limit the strategy space, in the experiment players were restricted to arrive at multiples of 5 minutes between 6:00 and 18:00 or stay out of the queue. This gives rise to 146 strategies per player (145 entry strategies plus the strategy of staying out) and a joint strategy space with 14620 elements. The resulting mixed-strategy equilibrium solution for Game G30 is exhibited in figure 1, and the one for Game G45 is displayed in figure 2. Entry strategy 0 corresponds to early arrival at 6:40, strategy 1 to early arrival at 6:45, strategy 16 to arrival

Figure 1. Mixed-strategy equilibrium solution for the “short” service time queueing game with d = 30 minutes and early arrivals.

JOINING A QUEUE OR STAYING OUT

123

Figure 2. Mixed-strategy equilibrium solution for the “long” service time queueing game with d = 45 minutes and early arrivals.

at 8:00, and so on. The latest arrival time with positive probability in Game G30 is 17:30 (strategy 130) and in Game G45 17:15 (strategy 127). Consider first the solution to Game G30 (with the “short” service time). In equilibrium, players should always join the queue. Their expected payoff is E = 18.35 (compare this result with the certain payoff of r = 100 under any of the n! pure-strategy equilibria and the fixed payment of g = 15 for staying out). Figure 1 displays the periodicity of the solution that has also been observed in solving similar queueing games with fixed service time and smaller values of n (see examples of 2-person games in RSPS). The periodicity is entirely due to the discreteness of the arrival time and the resulting possibility of ties. In equilibrium, players should enter the queue at 6:40, 6:50, . . . , 8:00 with probability 0.00579 and at 6:45, 6:55, . . . , 7:55 with probability 0.01175. After 8:00, the probabilities of every six consecutive arrival times are grouped in patterns that change over time in a systematic manner. If we average the entry probabilities in adjacent time intervals (not shown here), then the entry probabilities for Game G30 are seen to be distributed more or less evenly between 6:40 and 14:00, and then slowly decline for arrival times between 14:05 and 17:30. The equilibrium solution for Game G45 (with the “long” service time) displays a very different pattern. The probability of staying out of the queue—not displayed in figure 2—is 0.4096. This means that in our experiment with n = 20 players, on average 8.2 players should stay out of the queue in Game G45. The expected payoff is clearly E = g = 15. Figure 2 also shows the periodicity of the solution, which is particularly pronounced before 8:00. In particular, players should join the queue at 6:40, 6:50, . . . , 8:00 with probability

124

SEALE ET AL.

0.0117 and not enter at times 6:45, 6:55, . . . , 8:05. After 8:05, the probabilities of each 18 consecutive arrival times are also grouped together in patterns that maintain the periodicity of the solution but change over time. If we neglect again the discreteness of the arrival time and smooth over the adjacent arrival intervals, then the probabilities of entry decline slowly from about 0.035 in the 30-minute interval [6:35, 7:00] to 0.030 in the interval [12:05, 12:30] to 0.011 in the interval [16:35, 17:00].

4.

Previous research

Several models have focused on the equilibrium behavior of customers and servers in queueing systems (for a recent literature review see Hassin and Haviv, 2003). Whereas in most of the models that explicitly allow for the customer’s behavior to be affected by actions taken by the system manager or other customers there is an underlying Markov process whose transition probabilities are induced by the common strategy selected by all of the players, they all usually assume, unlike the present paper, steady-state conditions. Holt and Sherman (1982) considered a model for allocating a fixed number, m, of identical prizes to N players (N > m) who independently choose their arrival times. Under a FIFO queue discipline, individual players are motivated to arrive early, thereby increasing their chances of obtaining a prize. However, early arrivals are associated with cost of waiting and their usefulness depends on the arrival times of the other players. In the terminology of our model, the service facility opens and the first m people that are waiting in line are served instantaneously (d = 0) and then the facility shuts down. Holt and Sherman assumed that the ratio r/c varies within the population, and then constructed a pure-strategy equilibrium solution to their incomplete information game in which players with higher time valuation of the prize arrive earlier. In a more closely related paper, Glazer and Hassin (1983) considered the decision of a customer when to visit a facility that opens at time t = 0, closes at time t = T , and uses a FIFO queue discipline. Similarly to our study, each customer’s objective is to minimize the time of waiting in line (service is guaranteed to be completed if arrival is before T ). However, unlike our model, the total number of customer arrivals during any day is assumed to be a Poisson random variable with mean n whose value is exogenously determined. Thus, we never observe the players who do not join the queue. The model assumes that the service requirement of each customer is exponential with equal mean. Glazer and Hassin constructed an equilibrium solution to their game in which the arrival rate prior to the starting time is constant (time is continuous in their model) but that after the starting time it declines slowly. Most relevant to the present study is the previous experiment conducted by RSPS. Using the same experimental procedure, the queueing game in this earlier study is of the same type as Game G30 but with four major differences. First, the payoff for staying out was set at g = 0 rather than g = 15 in the current experiment. This resulted in a positive probability of staying out in the RSPS study. In the present study, we fixed the payoff of staying out at g = 15 so as to render the equilibrium probability of staying out equal to zero in Game G30 but positive and large in game G45. This was done in order to sharpen the difference between

JOINING A QUEUE OR STAYING OUT

125

Figure 3. Mixed-strategy equilibrium solution for the “short” service time queueing game with d = 30 minutes and no early arrivals (RSPS).

Games G30 and G45. Second, with arrival time in the RSPS study restricted to 1-minute intervals, the strategy space included 602 elements. Third, the cost per-minute (c) to reward (r ) ratio in the RSPS study was set at 1/60 (rather than 1/100 in Game G30.) Finally and most importantly, early arrivals before time T0 were explicitly forbidden by RSPS. As a result, the symmetric mixed-strategy equilibrium solution constructed by RSPS (reproduced here as figure 3) differs markedly from the one for Game G30 (figure 1). Figure 3 assigns a positive probability (0.06) to staying out compared to zero probability for this strategy in figure 1. Also, figure 3 assigns a high probability (0.211) to arriving at the time T0 (8:00) when service starts, whereas this arrival time is not particularly distinguished in figure 1. Given the large arrival probability at T0 , it is not surprising that figure 3 shows zero entry probabilities for arrival times between 8:01 and 9:03, whereas no such discontinuity in the arrival pattern appears in figure 1. 5. 5.1.

Experimental method Subjects

One hundred and sixty subjects, in roughly equal proportions of males and females, participated in the experiment. Recruited by class announcements and bulletin board advertisements, the subjects were primarily business administration undergraduate students who volunteered to participate in a group decision-making experiment with payoff contingent on performance. The subjects were divided into eight equal-size groups of n = 20 members

126

SEALE ET AL.

each (two groups in each of four conditions). Each group participated in a single session that lasted about 110 minutes and included 75 iterations (trials) of the stage game. 5.2.

Procedure

The experiment used a 2×2 game type (Game G30 vs. Game G45) by information structure (minimal or private vs. maximal or public information) between-subject design with two groups in each condition. Hereafter, we refer to these four treatments as Conditions MIN30, MAX30, MIN45, and MAX45. The experiment was conducted at the Economic Science laboratory (ESL) at the University of Arizona. The ESL includes forty networked PC terminals separated from one another by partitions. Upon arrival to the lab, each subject chose a poker chip out of a bag containing 20 chips to determine his or her seating. Any form of communication between subjects was strictly forbidden. Once randomly seated in their cubicles, the subjects proceeded to read the instructions (see Appendix for the instructions for Condition MAX45) at their own pace. Questions asked during this period (there were very few) were privately answered by the experiment supervisor. The parameters for Conditions MIN30 and MAX30 were n = 20, d = 30, c = 1, r = 100, and g = 15 (c, r , and g are in cents). Those for Conditions MIN45 and MAX45 were the same with the only exception that d = 45. Arrival times to the service facility were restricted to 5-minute intervals from 6:00 to 18:00 with service starting at 8:00. This restriction was imposed to reduce the number of strategies in the strategy space so that all could be displayed legibly on the results screen (see Appendix). At the beginning of the session, each subject was provided with an endowment of $10. At the end of the session, the cumulative earnings across all the 75 trials were added to the initial $10 endowment to determine the payoff for the session (thus, payoff = cumulative earnings plus endowment). All the subjects in Conditions MIN30, MAX30, and MAX45 completed the session with positive payoffs. Some of them (about 12%) ended up the session with a positive payoff less than $10. Most of the subjects in Condition MIN45 completed the session with negative payoffs, and none of them ended the session with a payoff exceeding the initial endowment of $10. Subjects who ended up with less than their initial endowment were paid a flat payment of $10.00 (thus, payment = max($10, payoff)). The subjects had not been told so in the instructions (see Appendix). Individual payments across the eight sessions, excluding subjects who ended up with negative payoffs (but including the $10 endowment), ranged between $3.20 and $34.00. Mean payoff for these subjects was $18.00. Each trial had the same structure. Once the trial number was displayed, each subject was asked to choose whether to join the queue. If choosing to stay out, the subject was asked to wait until all members of the group recorded their decisions. Otherwise, the subject was asked to choose an arrival time from a list of all possible arrival times that were displayed on the computer monitor. On each trial, a subject could review her previously chosen arrival times and associated payoffs. The decisions were made independently and anonymously with no time pressure. In this particular environment, in order not to provide any cues for coordinating behavior, the subject’s station identification number—a number between 1 and 20—was never revealed.

JOINING A QUEUE OR STAYING OUT

127

Once all the twenty group members typed in their decisions, a “Results” screen was displayed with information about the trial outcome. In all four conditions, each subject was informed of the following: – the subject’s arrival time; – number of players tied with the subject at the same arrival time, if any, and the outcome of the tie breaking rule; – the subject’s queue wait time (0, 5, . . . ); – the subject’s payoff for the trial; – the subject’s cumulative payoff. In addition, subjects in Conditions MAX30 and MAX45 were also informed of the arrival times, service starting time, and payoffs of all the twenty members in their group. This was accomplished by presenting a “Results” screen that, in addition to including the private information described above, included a 20 × 3 table with rows corresponding to the 20 players ordered in terms of their arrival times, and 3 columns designating the arrival time, starting time of service, and individual payoff for the trial (see Appendix for Condition MAX45). The instructions also included a detailed example with n = 20 players and hypothetical arrival times that illustrated the game, the random breaking of ties, and the computation of the individual payoffs. 6.

Results

The two groups (sessions) in each of the four conditions were compared to each other on four different measures: the individual payoff, individual number of entry decisions, mean individual waiting time, and individual number of switches. The individual payoff, number of entry decisions, and mean waiting time were computed separately for each subject across the 75 trials. If a subject’s decision on trial t + 1 differed from her decision on trial t, we scored this event as “switch.” The number of switches per subject could take any value between 0 and 74. We tested null hypotheses that the two groups in Condition MAX30 have the same mean response for each of the four measures. The null hypothesis could not be rejected for any of these four measures (t(38) = 0.94, 1.28, −1.56, and −1.29 for mean payoff, mean number of entries, mean waiting time, and mean number of switches, respectively, p > 0.01). Similar results were obtained in comparing to each other the two groups in Condition MAX45 (t(38) = 1.41, 0.33, −0.63, and −0.75, p > 0.01), and the two groups in Condition MIN45 (t(38) = 2.05, 0.73, −1.15, and −1.25, p > 0.05). However, of the four mean comparisons for the two groups in Condition MIN30, two resulted in significant differences. On the average, members of Group 1 in Condition MIN30 earned significantly less than members of Group 2 in the same condition (t(38) = −4.70, p < 0.01) and waited longer for service (t(38) = 4.27, p < 0.01). The two other differences between these two groups in mean number of entries (t(38) = 0.02, p > 0.05) and mean number of

128

SEALE ET AL.

Table 1.

Frequencies of arrival time on trial 1. Arrival time in one-hour intervals 6:35– 7:35– 8:35– 9:35– 10:35– 11:35– 12:35– 13:35– 14:35– 15:35– 16:35– 7:30 8:30 9:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 Total

d = 30

3

8

2

5

10

5

11

15

7

5

4

75∗

d = 45

4

7

0

4

10

5

14

15

9

3

3

74∗∗

*One subject arrived after 17:30 and four stayed out. **One subject arrived before 6:35 and five stayed out.

switches (t(38) = 1.31, p > 0.05) were not statistically significant. These results indicate that members of Group 1 in Condition MIN30, although behaving similarly to members of Group 2, waited longer (62.7 minutes for Group 1 vs. 50.0 minutes for Group 2) and, consequently, were paid less money ($22.50 vs. $32.55). Because we find no significant difference between the two groups in each of the other three conditions and only differences in waiting time and earnings in Condition MIN30, in subsequent analyses of aggregate data the results are collapsed across the two groups in each of the four conditions. 6.1.

Arrival time on trial 1

When making their decisions on trial 1 the subjects have had no opportunity to observe previous outcomes. Whatever differences are observed in their decisions on trial 1 must be attributed to individual differences in beliefs, personal history, or risk attitude. Table 1 presents frequency distributions of arrival time for d = 30 and d = 45. The frequencies are computed across the MIN and MAX conditions in intervals of one hour starting at 6:35, thereby yielding 11 class intervals. Our first observation is that the two frequency distributions are remarkably similar. The null hypothesis that the two waiting time conditions do not differ from each other in the distributions of arrival time on trial 1 could not be rejected (χ 2 (10) = 3.57, p > 0.05). In sharp contrast to the equilibrium solutions (figures 1 and 2), we observe no differences between Games G30 and G45 in the number of players staying out. Of the 80 subjects in the MIN30 and MAX30 conditions, one arrived erroneously after 17:30 and four stayed out. Of the 80 subjects in the MIN45 and MAX45 conditions, one arrived erroneously before 6:35 and five stayed out. These cases are not included in Table 1. Our second observation is the considerable and unexpected variability in arrival time already observed on trial 1 with a more or less uniform distribution across the 11 hours from 6:35 to 17:30. 6.2.

Frequency distributions of arrival time

With 40 subjects in each condition and 75 trials we have a total number of 3000 observations per condition. Because of the very large number of entry strategies, observed arrival times were grouped in 30-minute intervals (6:05–6;30, 6:35–7:00, . . . , 17:05–17:30) for inspection only. However, in comparing observed and equilibrium cumulative distributions of

JOINING A QUEUE OR STAYING OUT

129

arrival times, all the T +1 strategies were considered. Figure 4 (four separate panels) displays side by side as bar graphs the theoretical and observed results for each condition separately. In each of the four panels, the equilibrium probability of staying out of the queue and the corresponding relative frequency of staying out are exhibited on the right-hand side bar.

Figure 4.

Observed and predicted distributions of arrival time by experimental condition.

130

SEALE ET AL.

Consider first the arrival time distributions for the two d = 30 conditions (displayed in 30-minute intervals in the left-hand panel of figure 4). Inspection of these two figures suggests that across subjects the mixed-strategy equilibrium solution accounts for the observed arrival times when d = 30 quite well. In equilibrium, the probability of entry should equal 4% at 7:00, remain more or less constant at 5% from 7:30 until 2:30, and then decline gradually. The observed relative frequencies display this pattern. Using the player as the unit of analysis1 and considering all entry strategies, the one-sided Kolmogorov-Smirnov (K-S) test could not reject the null hypothesis of no difference between the observed and predicted cumulative distributions of arrival time (D = 0.059 and D = 0.069 for the MAX30 and MIN30 conditions, respectively, n = 40, p > 0.05). Figure 4 also suggests no difference between the two observed relative frequency distributions of arrival time with complete information (Condition MAX30) and no information (Condition MIN30) about the players’ decisions on the immediately preceding trial. The two-sided two-sample K-S test could not reject the null hypothesis of no difference between the two cumulative arrival time distributions (D = 0.026, p > 0.05). Notwithstanding the results of the one-sided K-S test, the two figures for conditions MAX30 and MIN30 show two minor discrepancies between the observed and predicted arrival time distributions. First, in both the MIN30 and MAX30 conditions subjects tended to enter the queue before 7:35 less often than predicted, and in the time interval [7:35– 8:00] more often than predicted. Second, approximately 4% of all the decisions were to stay out compared to 0% in equilibrium. A more detailed analysis that breaks down the 75 trials into three blocks of 25 trials each shows that the discrepancy between observed and predicted results decreased with experience of playing the game. Thus, the frequency of entering the queue between 7:35 and 8:00 decreased from 9% in block 1 (trials 1–25) to 5.5% in block 3 (trials 51–75) of Condition MAX30, and from 9.8% in block 1 to 6.6% in block 3 of Condition MIN30 (compared to 5.3% under equilibrium play). In contrast, we observe no effects of experience on the relative frequency of staying out.2 Turning next to the two d = 45 conditions, where the mixed-strategy equilibrium probability of staying out is 0.4096, the two figures on the right-hand panel of figure 4 indicate that the equilibrium solution continues to account remarkably well for the arrival time distribution in Condition MAX45 but not in Condition MIN45. As before, the one-sided K-S could not reject the null hypothesis of no difference between the observed and predicted cumulative distributions in Conditions MAX45 (D = 0.061, n = 40, p > 0.05) and MIN45 (D = 0.175, n = 40, p > 0.05). Nor did the two-sided two-sample K-S test reject the null hypothesis of no difference between the two observed arrival time distributions (D = 0.121, p > 0.05). However, comparison of the two displays on the right-hand side of figure 4 clearly shows that when no information about previous arrival times of all the n players was provided, subjects in Condition MIN45 did not stay out as frequently as predicted. This was the major reason for the poor performance of the subjects in this condition (in both groups) who not only earned no money in the experiment but, with very few exceptions, lost their $10 endowment. A more detailed analysis of the decisions in the two d = 45 conditions shows that the discrepancy between observed and predicted frequency of staying out decreased with

JOINING A QUEUE OR STAYING OUT

131

experience in Condition MAX45 but not in Condition MIN45. When the players in Condition MAX45 could observe and study the decisions and outcomes of all the group members at the end of each trial, the percentage of staying out decisions steadily increased from 30% in block 1 through 35.5% in block 2 to 40.5% in block 3 (compare with 40.96% in equilibrium). We observe no such trend in Condition MIN45. As the subjects in Condition MIN45 could not ascertain how many players opted to stay out on the previous trial, they had no way of determining whether their payoff for the trial, which was typically negative, was due to insufficient staying out decisions of the group members or to simply poor choice of an arrival time. This was not the case in Condition MAX45, where each subject was accurately informed of the number of players who stayed out on the previous trial and the arrival times of players who entered the queue.

6.3.

Payoffs

Even small deviations from the equilibrium solution have considerable effect on the individual payoffs. Figure 5 displays the ten-period moving average of the group payoff for each condition separately. Recall that in equilibrium the group payoff is 18.35×20 = 367 if d = 30 and 15 × 20 = 300 if d = 45. Figure 5 shows that, on the average, subjects in Conditions MIN30 and MAX30 earned about the same payoff. Indeed, after 25 trials or so the subjects’ payoffs in these two conditions exceeded the expected payoff under equilibrium play.3 The subjects in the two d = 45 conditions performed rather poorly at the beginning of the session. However, the mean payoffs in Condition MAX45 steadily increased across trials and converged to a value slightly above the equilibrium expected payoff of 300. In contrast, the mean group payoff in Condition MIN45 did not change across trials in any

Figure 5.

Ten-period moving average of group payoff.

132

SEALE ET AL.

systematic manner and stayed negative during the entire course of the session, considerably below the equilibrium expected payoff.

6.4.

Frequency of staying out

The previous analysis suggests that knowledge of the number of players staying out is critical for the difference between conditions MIN45 and MAX45. To explore this suggestion further, figure 6 displays the ten-period moving average of number of players who stayed out on any given trial. Similarly to figures 4 and 5, the results are exhibited separately for each condition. (Note change in scale between the left-hand and right-hand panels.) The figure explains the trends in the mean earnings shown in figure 5. In equilibrium, players in Conditions MIN30 and MAX30 should not have stayed out. In contrast, figure 6 shows a small but consistent tendency to stay out in these two conditions. The mean number

Figure 6.

Ten-period moving average of number of subjects staying out of the queue by experimental condition.

JOINING A QUEUE OR STAYING OUT

133

of players who stayed out in these two conditions is about 0.8 (4%). Small as it is, this percentage was sufficient to introduce slack into the system that resulted in players earning substantially more than the equilibrium expected payoff (figure 5). In equilibrium, about 8.2 of the 20 players in Conditions MIN45 and MAX45 should have stayed out. Figure 6 shows that the mean number of players who stayed out in Condition MIN45 was only slightly higher than 4 and hardly changed across trials. This resulted in considerable congestion that led to the heavy losses. In contrast, and in qualitative agreement with figure 5, the mean number of players in Condition MAX45 who stayed out steadily increased from about 5 in trials 1–10 to about 8.2 in the last twenty trials. This resulted in considerable increase in mean payoff that, in the last twenty trials converged to the equilibrium payoff (figure 5).

6.5.

Switches in decision

A switch was recorded if the subject’s decision on trial t + 1 differed from her decision on trial t. This was the case if a subject stayed out on trial t but entered the queue on trial t + 1, entered the queue on trial t and stayed out on round t + 1, or entered the queue on both trials t and t + 1 but at different times. Denote the total number of switches per trial by s where s = 0, 1, . . . , 20. Using a sample of 10,000 simulations of the 75-trial queueing game with 20 fictitious players whose arrival times on each trial were chosen randomly and independently from the mixed-strategy equilibrium distributions of arrival time for d = 30 (figure 1) and d = 45 (figure 2), we computed the expected number of switches per trial to be E(s) = 19.83 for Condition d = 30 and E(s) = 16.58 for Condition d = 45. In contrast to the equilibrium prediction, the mean number of switches in each of the four conditions was smaller than predicted. Figure 7 exhibits the ten-period moving average of the number of switches for each condition. Three observations are in order. First, in all four conditions the subjects exhibited an “inertia” effect switching their arrival times from trial to trial less than expected. The only exception is the first 30 trials or so in Condition MIN45. The agreement with the mixed-strategy equilibrium solution on the aggregate level reported above could not be due to individual subjects randomly and independently sampling their decisions from the equilibrium strategies. Second, in qualitative agreement with the equilibrium solution, subjects in Conditions MAX45 and MIN45 switched their decisions less often than subjects in Conditions MAX30 and MIN30. Thirdly, figure 7 indicates weak learning effects in the two d = 45 conditions. No such trends can be discerned in Conditions MAX30 and MIN30.

6.6.

Switching between entry and staying out decisions

To better understand the switching behavior, the next analysis considers only the decisions to enter (EN) or stay out (SO) irrespective of arrival time. Table 2 presents 2 × 2 tables, one for each condition, that display the joint frequencies of EN and SO on trials t and t + 1. For example, out of 2960 (74 × 40) joint decisions in Condition MAX30, in 2771 cases entry decisions were recorded on trials t and t + 1, and in 89 cases a decision to enter

134

SEALE ET AL.

Figure 7.

Ten-period moving average of observed number of switches between arrival times.

was followed by a decision to stay out. In all four conditions, p(SOt+1 | SOt ) > p(SOt+1 | ENt ). Moreover, the ratios p(SOt+1 | SOt )/ p(SOt+1 | ENt ) are large, ranging from 2.77 in Condition MAX45 to 13.7 in Condition MIN30 compared to 1 in equilibrium. Similarly, in all four conditions p(ENt+1 | ENt ) > p(SOt+1 | ENt ). These results indicate that players have a strong tendency to repeat their decisions—either enter or stay out—more often than expected.

6.7.

Individual differences

The hypothesis that the mixed-strategy equilibrium solution accounts for the individual behavior is flatly rejected. A process of adjustment that may differ between subjects generated the consistent and predicted patterns of aggregate behavior that we reported above. Table 1 shows that already on trial 1, before the subjects could obtain information about individual (MIN condition) or group (MAX condition) outcome, arrival times varied

135

JOINING A QUEUE OR STAYING OUT Table 2.

Joint decisions of entry (EN) and staying out (SO) on adjacent trials. Condition MAX30

Condition MAX45

Trial t + 1 Trial t

EN SO Sum

EN 2771 97% 91 91% 2862

SO 89 3% 9 9% 98

Trial t + 1 Sum 2860

Trial t

EN

100

SO

2960

Sum

Condition MIN30

EN SO Sum

EN 2758 97% 75 59% 2833

SO 75 3% 52 41% 127

SO 419 22% 624 61% 1073

Sum 1933 1027 2960

Condition MIN45

Trial t + 1 Trial t

EN 1514 78% 403 39% 1917

Trial t + 1 Sum 2833

Trial t

EN

127

SO

2960

Sum

EN 2035 89% 242 36% 2277

SO 253 11% 430 64% 683

Sum 2288 673 2960

considerably across subjects and were spread over the entire interval of possible arrival times [T0 , Te ]. Inspection of the 160 individual distributions of arrival time (not displayed here because of space limitation) further shows considerable heterogeneity in the adjustment process. The individual percentage of staying out in Conditions MAX45 and MIN45 ranged between 0 and 82.7. In Conditions MAX30 and MIN30, where overall only 4% of the decisions were to stay out, only very few subjects contributed most of these decisions. Of a total of 80 subjects in these two conditions, the majority entered the queue on all 75 trials of play in agreement with the equilibrium solution. Finally, individual differences in the “inertia” effect were also substantial. When entering, some subjects switched their arrival time on almost every occasion, whereas others repeated the same arrival time on successive trials presumably in an attempt to “scare off” other players from choosing the same arrival time. The joint explanation of aggregate behavior that, with the exception of Condition MIN45, is accounted for by the mixed-strategy equilibrium solution, and individual behavior that constantly changes across trials and exhibits patterns of arrival that defy simple classification is a topic that we undertake in a separate paper (Bearden et al., in press).

7.

Conclusions

As long waits in lines are costly in time and effort, there is theoretical and practical interest in determining the behavioral regularities, if any, that emerge when agents decide

136

SEALE ET AL.

independently and anonymously whether to join a queue, and if yes, at what time to arrive. There is an additional interest in finding out if, and under what conditions, the Nash equilibrium solution can account for these behavioral patterns. Evidence concerning the descriptive power of the Nash equilibrium is presently mixed. Continuing a previous study by Rapoport et al. (2004) that focused on a single queueing game with endogenous arrivals, private information about the game outcome, and no early arrivals, the present study investigated arrival and staying out decisions in two new queueing games each under two different information structures. The two games, G30 and G45, shared the same parameter values except for service time. The service time in Game G30 was set at 30 minutes so that with perfect coordination under pure-strategy equilibrium play all the n agents could be served without waiting and thereby maximize their payoff while the system incurs no idle time. In sharp contrast, service time in Game G45 was set at 45 minutes so that congestion was unavoidable unless about 40% of the agents opted to stay out. In both games, early arrivals were permitted. The major finding is that in both games inexperienced subjects generated replicable patterns of arrival and staying out decisions that were accounted for quite well by the mixed-strategy equilibrium solution. Moreover, and in complete agreement with previous results reported by Rapoport et al., these patterns of behavior were generated by subjects operating independently with no trial-to-trial information about the decisions and outcomes of the other members of their groups. The only exception occurred in Game G45 under the minimal information structure. Our results suggest that common information about the decisions and payoffs of the all the group members matters only when congestion is unavoidable. At the beginning of the session, subjects in both Conditions MAX45 and MIN45 performed equally poorly, entering the queue more often than predicted and thereby causing congestion that resulted in losses in earnings. When this type of information was provided, subjects in Condition MAX45 slowly recovered and their behavior and subsequent payoffs converged to equilibrium. Without public information about the decisions of the other group members, subjects in Condition MIN45 had no way of determining whether their poor performance was due to congestion or unfortunate choice of arrival time. Rather than staying out and earning a positive payoff (g) on each trial, they continued to enter the queue more often than predicted, causing congestion, and consequently incurred substantial losses. We do not interpret the findings of this study and the previous study by Rapoport et al. as support for equilibrium play. Random and independent drawing of arrival and staying out decisions from a given probability distribution is prescribed by the mixedstrategy equilibrium solution for individual agents, not groups. We find no support for mixed-strategy equilibrium play on the individual level. Rather, we observe considerable individual differences in arrival and staying out patterns as well as strong sequential dependencies. The theoretical challenge is to account simultaneously for the emergence of replicable patterns of aggregate behavior that are described quite well by the mixedstrategy equilibrium and individual patterns of behavior that defy simple characterization. A first step in meeting this challenge has been reported by Bearden et al. Analyzing aggregate and individual data reported both by the present study and the previous study of Rapoport et al., they proposed and subsequently tested a reinforcement-based learning

JOINING A QUEUE OR STAYING OUT

137

model that, with a few exceptions, accounted for many of the major findings reported in these two studies. As the performance of the learning model is mixed, more research to explain the dynamics of play in repeated interaction large group queueing games is warranted.

Appendix: Experiment instruction This experiment has been designed to study how individual members of a group decide if and when to join a queue (waiting line). The instructions are simple. If you follow them carefully and make good decisions, you may earn a considerable amount of money. At the end of this session, you will be paid in cash your earnings from the experiment. Therefore, it is important that you do your best. A research foundation has contributed the funds to support this line of research. In case you have any questions after reading the instructions, please raise your hand and the supervisor will come to answer them. General description of the task This experiment focuses on how individual members of a group decide if and when to join a queue when communication among them is not possible. The task that we study is of a vehicle inspection station (with a single service line) that supports a given population. The station’s opening (8:00 a.m.) and closing (6:00 p.m.) times are known by everyone. The population consists of 20 car owners who need to bring their cars for inspection. Each inspection lasts exactly 45 minutes. In the present task, a player must first decide whether to seek inspection on the next day.

If deciding not to seek inspection, he/she will receive a small payoff that does not depend on the decisions of the other 19 group members. If deciding to seek inspection, he/she must specify the time of arrival. Depending on the arrival times of the other players, he/she may have to wait in line to be served. His/her service may either be completed by the closing time (6:00 p.m.) or it may not. Because waiting is costly, each player who must wait in line will be charged a fixed amount for every minute waited before the service commences. Waiting costs will cease when service commences or the station closes for the day, whichever comes first. Once a player joins a queue, he/she cannot leave it. A player’s earnings for each trial are calculated by subtracting the cost of waiting from the reward of successfully completing the inspection (reward—cost). Because it is possible for waiting costs to exceed the value of the reward, a player who joins the queue may lose money.

138

SEALE ET AL.

To summarize, there are 20 group members. Each is given an initial endowment of $10.00. The station opens at 8:00 a.m., but early arrivals after 6:00 a.m. are possible. An individual inspection takes exactly 45 minutes. The reward for successfully completing the inspection is $1.00. Each player is charged $0.01 per minute waited in line. Communication between the players during the experiment is strictly forbidden.

Procedure There are 75 trials. At the beginning of each trial you will be asked to make at most two decisions. Decision 1: First, the computer will ask you whether you want to join the queue. If you decide to stay out, you will earn $0.15. The decision screen, which will be presented to you at the start of each trial, will look as follows:

To join the queue—If you wish to join the queue, simply click the YES button. To stay out—If you do not wish to join the queue, simply click the NO button. Decision 2: If you choose to join the queue, you must also state the time you want to arrive (in 5-minute increments). To do so, the computer will present you with another screen (see the example below). To choose your arrival time, simply “click” the button next to the time you wish to arrive. You may choose any time between 6:00 a.m. and 6:00 p.m. Please note that the station does not open until 8:00 a.m. Once you have chosen your arrival time, you must click on the Submit button. The computer will then ask you to verify your arrival time. If you decide at that stage to change your mind, you can always click on the Stay Out button. No player will know how many other players have chosen to join the queue during any given trial. Similarly, no player will be informed of the arrival times of the other entrants. Trial outcome. Once you have made your decision(s), you will be presented with a wait screen until all group members have made their decisions. Once everyone types in his/her decision, the computer will present you with a “Results” screen that looks as follows.

JOINING A QUEUE OR STAYING OUT

139

This screen summarizes all the information that will be provided to each player about his/her own payoff. For each player, it reiterates his/her chosen arrival time and identifies the time his/her inspection started. You will know whether or not you received an inspection by looking at the left-hand boxes in the middle of the screen. The Completed Inspection box will always show either “Yes” or “No,” and the Reward for Completion box will always show either “$1.00” or “$0.00”. If you do not receive an inspection, you will be notified. In this case, you will not receive the reward of $1.00, but will incur waiting costs from the time you arrived until 6:00 p.m. If you do receive an inspection, you will see the time in which you received it.

140

SEALE ET AL.

Service time is the most critical aspect of this experiment. Ideally, you would like to choose an arrival time where there is no waiting (and, therefore, no wait cost) but there is still ample time (at least 45 minutes) for an inspection. The longer you have to wait, the higher the cost you will incur and, consequently, the less money you will earn. At the bottom of the “Results” screen, you will see how much money you have earned (could be positive or negative) for the trial as well as how much money you have earned throughout the experiment. Breaking ties. It is possible for two or more players to choose the exact arrival time. If this happens to you, the computer will randomly break the tie and inform you of the results. You will know if you tied with one or more other players because the text in the top-right portion of the “Results” screen will be visible. The message will tell you how many people chose to arrive at the same time as you and what order you will join the queue in relation to the other player(s). The player who is, say, “1 of 3” will join the queue first, followed by “2 of 3,” and lastly followed by “3 of 3.” Options. Prior to making any decision, the computer will provide you with two choices to view your previous results (see the top screen on page 2), if you wish to do so. • Previous Trial Results: Selecting this button brings up to the “Results” screen (discussed above) from the previous trial. • All Trial Results: Selecting this button provides you with a log of all your past decisions.

141

JOINING A QUEUE OR STAYING OUT

You will be able to view your cumulative earnings by selecting either of the two buttons at any time. Example Assume that you decide to join the queue on this trial. You select your arrival time to be 3:00 p.m. Although you don’t know the arrival times of the other players, the table below identifies the actual results. Assume that you are Player 9. In this case, only 17 of the 20 players (including you) decided to enter the queue for this trial, whereas 3 other players (Players 4, 11, and 16) decided to stay out.

Player

Arrival time

Service time

Minutes waited

Wait cost

Reward

Earnings

13

7:05 a.m.

8:00 a.m.

55

$0.55

$1.00

$0.45

3

7:25 a.m.

8:45 a.m.

80

$0.80

$1.00

$0.20

18

7:30 a.m.

9:30 a.m.

120

$1.20

$1.00

−$0.20

15

8:00 a.m.

10:15 a.m.

135

$1.35

$1.00

−$0.35

1

8:00 a.m.

11:00 a.m.

180

$1.80

$1.00

−$0.80

5

8:45 a.m.

11:45 a.m.

180

$1.80

$1.00

−$0.80

20

10:00 a.m.

12:30 p.m.

150

$1.50

$1.00

−$0.50

10

12:10 p.m.

1:15 p.m.

65

$0.65

$1.00

$0.35 $0.85

8

1:45 p.m.

2:00 p.m.

15

$0.15

$1.00

14

2:55 p.m.

2:55 p.m.

0

$0.00

$1.00

$1.00

YOU

3:00 p.m.

3:40 p.m.

40

$0.40

$1.00

$0.60

12

3:00 p.m.

4:25 p.m.

85

$0.85

$1.00

$0.15

19

3:30 p.m.

5:10 p.m.

100

$1.00

$1.00

$0.00

2

4:20 p.m.

NA

100

$1.00

$0.00

−$1.00

17

4:35 p.m.

NA

85

$0.85

$0.00

−$0.85

7

5:00 p.m.

NA

60

$0.60

$0.00

−$0.60

6

5:00 p.m.

NA

60

$0.60

$0.00

−$0.60

Based on the above information, the computer determines the queue order and service times and informs you of the following: • • • • •

Your arrival time: 3:00 p.m. Your service time: 3:40 p.m. You earned: $0.60 Number of other players arriving at 12:00: 1 If you were tied, what order you were served in relation to the others tied with you: 1 of 2

142

SEALE ET AL. Results of the queue Arrival time

Service time

Earnings

7:05 a.m.

8:00 a.m.

$0.45

7:25 a.m.

8:45 a.m.

$0.20

7:30 a.m.

9:30 a.m.

−$0.20

8:00a.m.

10:15 a.m.

−$0.35

8:00 a.m.

11:00 a.m.

−$0.80

8:45 a.m.

11:45 a.m.

−$0.80

10:00 a.m.

12:30 p.m.

−$0.50

12:10 p.m.

1:15 p.m.

$0.35

1:45 p.m.

2:00 p.m.

$0.85

2:55 p.m.

2:55 p.m.

$1.00

3:00 p.m.

3:40 p.m.

$0.60

3:00 p.m.

4:25 p.m.

$0.15

3:30 p.m.

5:10 p.m.

$0.00

4:20 p.m.

NA

−$1.00

4:35 p.m.

NA

−$0.85

5:00 p.m.

NA

−$0.60

5:00 p.m.

NA

−$0.60

Did not arrive

NA

$0.15

Did not arrive

NA

$0.15

Did not arrive

NA

$0.15

Thus, you would have earned $0.60 for this trial ($1.00 reward—$0.40 wait cost), which would have been added to your cumulative total. (Recall that each player starts the experiment with $10.00.) Had you incurred a loss during a trial, the loss would have been subtracted from your cumulative earnings. Note the decisions of the other players in this example. • Players 13, 3, and 18 arrived before the station opened at 8:00 a.m. and had to wait in the order they arrived (55, 80, and 120 minutes, respectively). • Players 15 and 1 arrived exactly 8:00 a.m. when the station opened, but had to wait until those players who had arrived earlier (Players 13, 3, and 18) completed their inspections. Thus, service for Player 15 commenced after 135 minutes of waiting at exactly 10:15 a.m. • Player 14 (arrived at 2:55 p.m.) Player chose an arrival time when the station was idle that allowed him to be serviced immediately without incurring waiting costs and earning the full amount of the $1.00 reward. • Even though Player 5 (arrived at 8:45 a.m.) arrived at the inspection station 45 minutes after the previous player, (Player 1 who arrived at 8:00 a.m.), she still had to wait in line for three hours for Players 3, 18, 15, and 1 to complete their inspections.

JOINING A QUEUE OR STAYING OUT

143

• Both you and Player 12 arrived at the same time (3:00 p.m.). The computer randomly determined that you would be served first and Player 12 would be served second. Players 6 and 7 also arrived at the same time, namely, 5:30. Player 7 was randomly selected to be served first. However, because there was not sufficient time for either player to get an inspection, both players had to wait in line for 30 minutes until 6:00 p.m. when the station closed. (Recall that once a player joins the queue, he/she cannot leave it.) Thus, she did not get inspected ad did not receive the reward. • Approximately half of the players generated negative earnings for the trial. • Each of the three players who decided to stay out of the queue earned $0.15 for the trial.

Payment at the end of the session All 75 trials have the same structure. To reiterate, each player starts the experiment with $10.00, each inspection takes 45 minutes, the reward for a successful inspection is $1.00, and the cost for waiting in line is 1 penny per minute waited. If you choose not to join the queue, you will receive $0.15 for that trial. At the end of the experiment, your total earnings will be paid to in cash. Please look up to indicate to the supervisor that you have completed reading the instructions. The supervisor will start the experiment in just a few minutes. Thank you for your participation.

Acknowledgments We gratefully acknowledge financial support by NSF Grant No. SES-0135811 to D. A. Seale and A. Rapoport and by an AFOSR/MURI contract F49620-03-1-0377 to A. Rapoport. We also wish to thank an anonymous reviewer for very constructive comments.

Notes 1. We explicitly acknowledge the methodological problem that when the same n players interact repeatedly with one another, the statistical unit of analysis is the group, not the individual player. In the rest of the analysis we assume that players within a group (but not the repeated decisions of the same player) are statistically independent. In partial justification of this assumption, we note that as n become very large the effect of any particular player on the other group members becomes infinitely small (e.g., like the effect of a single investor in the stock market). With n = 20 interacting players, considering the group as a population is a reasonable approximation. 2. There are several possible reasons for staying out in Game G30. One of them is that a risk-averse player may prefer the sure payoff of g to the gamble associated with entering the queue. Note that the difference between the sure payoff of staying out of g = 15 and the expected payoff in equilibrium of E = 18.5 is less than 20% and the variance of the expected payoff is substantial. Yet another reason is to simply take some time off in order to re-consider one’s strategy. 3. This finding is due to a tendency to enter the queue at “round” times (i.e., 7:00, 7:30, . . . , if d = 30) and a small percentage of staying out decisions. This tendency was already reported by RSPS.

144

SEALE ET AL.

References Bearden, J.N., Rapoport, A., and Seale, D.A. (in press). “Entry Times in Queues with Endogenous Arrivals: Dynamics of Play on The Individual and Aggregate Levels. In R. Zwick and A. Rapoport (eds.), Experimental Business Research II. Norwell, MA: Kluwer Academic Publishers. Glazer, A. and Hassin, R. (1983). “?/M/1: On the Equilibrium Distribution of Customer Arrivals.” European Journal of Operational Research 13, 146–150. Hall, R.W. (1991). Queueing Methods. Upper Saddle River, NJ: Prentice-Hall. Hassin, R. and Haviv, M. (2003). To Queue or not to Queue: Equilibrium Behavior in Queueing Systems. Boston: Kluwer. Hillier, F.S. and Lieberman, G.J. (2001). Introduction to Operations Research 7th edition. Boston: McGraw-Hill. Holt, C.A. Jr. and Sherman, R. (1982). “Waiting-Line Auctions.” Journal of Political Economy. 90, 280–294. Medhi, J. (1991). Stochastic Models in Queueing Theory. Boston: Academic Press. Naor, P. (1969). “The Regulation of Queue Size by Levying Tolls.” Econometrica 37, 15–23. Rapoport, A., Stein, W.E., Parco, J.E., and Seale, D.A. (2004). “Strategic Play in Single-Server Queues with Endogenously Determined Arrival Times.” Journal of Economic Behavior and Organization 55, 67–91.