Agent Heterogeneity and Facility Congestion

Abstract

This paper extends Selten et al.’s (Games Econ Behav 58:394–406, 2007) laboratory experiment of an agent’s choice behavior by offering eight subjects a choice between one of two facilities that provide an identical service. We assume that the cost of using these facilities depends on their congestion level and that there are two types of agents with different congestion costs: high- and low-cost agents. High-cost agents are affected by congestion more severely than low-cost agents. The theory of congestion games predicts that cost heterogeneity will not affect the facility choices of agents. We obtain experimental results that support this theoretical prediction, namely that cost heterogeneity influences neither the facility choices of agents nor the congestion levels of facilities. By using data derived from the laboratory experiment, we then develop state-action tables and computationally simulate the facility choices of subjects. We find that a subject decides whether he or she remains in the same facility or moves to the other facility in the next period according to the current congestion level.

Appendices

Appendix 1: Instruction to Subjects

2009/2

1.1 About Today’s Experiment

We thank you for your participation in the PG Lab experiment. The purpose of this experiment is to examine how an individual changes his or her behavior by using information provided. There are a few things to note about this experiment. Please read this instruction first.

Examples imagine the situation in which you are waiting for your turn in line to obtain a service at a particular facility. For example, you are waiting for your turn in front of an ATM to withdraw money or for your turn at the postal office to send a parcel. In these situations, you obtain the same service regardless of the facility. Therefore, you choose the least crowded facility.

Instruction of this experiment

• In this experiment, we will ask you to choose one of two facilities repeatedly.

• If you choose the facility that a small number of people choose, then the waiting time becomes shorter. Consequently, your payoff becomes larger.

• Your cumulative payoff constantly changes with your facility choice.

• The cumulative payoff you obtain is shown on the PC screen in each period.

• The maximum payoff you can obtain through this experiment is 9,600 yen, while the minimum is 3,300.

• After each session, please fill in your payoff in the prescribed form.

• In addition to the payoff you obtain through the experiment, you will receive 2,000 yen remuneration.

Cautions

• (1) All instructions are provided on the PC screen. If you cannot understand them, please raise your hand and ask the instructor directly.

• (2) Please do not talk to other participants during this experiment.

• (3) Please do not look at the screens of other participants during this experiment.

• (4) Once this experiment begins, you cannot go to the bathroom for about 1 h. Please go to the bathroom now if you want to.

Thank you for your cooperation.

Appendix 2: The Computation of the Mixed Equilibrium

In this experiment, there are two types of agents, namely high-cost and low-cost. It is assumed that an agent knows his or her own type only. Suppose the H-type agent expects that other agents choose Facilities A and B with probabilities of $x_{H}^{A}$ and $x_{H}^{B} =1-x_{H}^{A},$ respectively. Suppose the L-type agent expects that other agents choose Facilities A and B with probabilities of $x_{L}^{A}$ and $x_{L}^{B} =1-x_{L}^{A},$ respectively. The conditional expected payoff is

$$\begin{aligned} \hbox {E}\left[ {U_{H}^{A} } \right] =V^{A}-c_{H} \left( {1+x_{H}^{A}({n-1})} \right) , \end{aligned}$$

if the H-type agent chooses Facility A. That is

$$\begin{aligned} \hbox {E}\left[ {U_{H}^{B} } \right] =V^{B}-c_{H} \left( {1+\left( {1-x_{H}^{A}}\right) ( {n-1})} \right) , \end{aligned}$$

if he or she chooses Facility B. Here, n is the total number of agents, $c_{H}$ is the waiting cost of the H-type agent, and $V^{A}$ and $V^{B}$ are facility-specific benefits. At the equilibrium, the conditional expected payoff must be equal. The solution becomes

$$\begin{aligned} x_{H}^{A} =\frac{V^{A}-V^{B}}{2c_{H} ( {n-1})}+\frac{1}{2}. \end{aligned}$$

Similarly, the solution for the L-type agent becomes

$$\begin{aligned} x_{L}^{A} =\frac{V^{A}-V^{B}}{2c_{L}( {n-1})}+\frac{1}{2}. \end{aligned}$$

Since it is assumed that the two facilities provide the same service, $V^{A}=V^{B}.$ Therefore, both high-cost and low-cost agents expect the other agents to choose the facility with a probability of 0.5.

The variance of a binomial distribution with a success probability of 0.5 is $V_{x} =0.25.$ The standard deviation for eight subjects in one period is

$$\begin{aligned} \sigma _{x} =\sqrt{0.25\times 8}=\sqrt{2}. \end{aligned}$$

In each session, the number of agents that uses either facility is 120 (=$30\times 8\times 0.05).$ The corresponding variance is

$$\begin{aligned} V=30\times 8\times \frac{1}{4}=60. \end{aligned}$$

By assuming the five sessions are independent of each other, the agent’s facility choice is approximated by a normal distribution:

$$\begin{aligned} N( {nx,\,nx( {1-x})})=N({120,\,60}). \end{aligned}$$

The variance for the mean of the five sessions is

$$\begin{aligned} \frac{V}{5}=\frac{60}{5}=12. \end{aligned}$$

The corresponding standard error is

$$\begin{aligned} \sigma =\sqrt{12}=3.464. \end{aligned}$$

The expected probability of facility changes is

$$\begin{aligned} y=2x( {1-x})=2\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{2}. \end{aligned}$$

Therefore, the expected number of facility changes is

$$\begin{aligned} R=( {30-1})\times 8\times \frac{1}{2}=116. \end{aligned}$$

The variance of this is given by

$$\begin{aligned} V_{y} =y( {1-y} )=\frac{1}{4}. \end{aligned}$$

The variance of R is

$$\begin{aligned} V_{R} =( {30-1})\times 8\times V_{y} =58. \end{aligned}$$

The variance for the mean of the five sessions is

$$\begin{aligned} \frac{V_{R} }{5}=\frac{58}{5}=11.6. \end{aligned}$$

The corresponding standard error is

$$\begin{aligned} \sigma _{R} =\sqrt{11.6}=3.406. \end{aligned}$$

Appendix 3: Sample Questions Used in the First Three Sessions

Appendix 4: Sample Questions Used in the Last Two Sessions