Strategic behavior and social outcomes in a bottleneck queue: experimental evidence

Abstract

We theoretically and experimentally study the differential incentive effects of three well known queue disciplines in a strategic environment in which a bottleneck facility opens and impatient players decide when to arrive. For a class of three-player games, we derive equilibrium arrivals under the first-in-first-out (FIFO), last-in-first-out (LIFO), and service-in-random-order (SIRO) queue disciplines and compare these predictions to outcomes from a laboratory experiment. In line with our theoretical predictions, we find that people arrive with greater dispersion when participating under the LIFO discipline, whereas they tend to arrive immediately under FIFO and SIRO. As a consequence, shorter waiting times are obtained under LIFO as compared to FIFO and SIRO. However, while our theoretical predictions admit higher welfare under LIFO, this is not recovered experimentally as the queue disciplines provide similar welfare outcomes.

Appendix

For the class of three-player games defined in Sect. 3, this appendix establishes the pure and mixed strategy equilibrium arrivals under each of the classical queue disciplines and compares their corresponding equilibrium equilibrium welfare. For arbitrary values of the relative queuing costs $\theta$, section “Pure strategy equilibrium analysis” presents the pure strategy equilibrium profiles and their corresponding equilibrium welfare. Section “Mixed strategy equilibrium analysis” presents symmetric mixed strategy equilibrium profiles and their corresponding expected equilibrium welfare. Section “Equilibrium comparison” compares the existing equilibrium profiles, their corresponding equilibrium welfare, and the conditions under which they are unique.

1.1 Pure strategy equilibrium analysis

We establish some auxiliary results for the pure strategy equilibrium profile that apply for the classical queue disciplines, and subsequently we analyze each queue discipline separately. We start by providing some useful notation.

Let $\alpha = (e_i^{a_i})_{i\in N}$ be a pure strategy profile where $a_i\in T$ denotes the period for which player i arrives with certainty. For any $\alpha$ and $i\in N$, let $a_{-i}$ denote the list $(a_j)_{j\in N\setminus \{i\}}$ of periods $a_j\in T$ for which all players, except i, arrive with certainty. We refer to the list $(a_i,a_{-i})$ as an arrival list. Moreover, let n(t) denote the number of players that arrive with certainty at period t. With this notation, we state the first auxiliary result.

Lemma 1

For any three-player game, let $\alpha ^*$ be a pure strategy equilibrium profile under either the FIFO, LIFO or SIRO discipline. Then $\alpha ^*$ has all players served at period 2.

Proof

We prove by contradiction. Suppose at some period $x\in \left\{ 0,1,2\right\}$ that the profile $a^*$ is characterized by $\sum _{t=0}^{x} n(t) < x+1$. That is, no player is served at period x despite available capacity at the facility. This implies that at least one of the three players is served at some later period $y>2$. With available service capacity at period x, however, the player served at period y will be better of by arriving at x instead and be served immediately without queuing time. A pure strategy equilibrium profile must therefore have $\sum _{t=0}^2 n(t)\ge t+1$, such that the last player is served at period 2. $\square$

Next, we define $C=\sum _{i\in N}S(e_i^{a_i},\alpha _{-i})-a_i$ as the total expected queuing time which measures the sum of expected queuing times under a pure strategy profile where each player $i\in N$ arrives at period $a_i\in T$. With this definition, we state the second auxiliary result.

Lemma 2

For any three-player game, let $\alpha ^*$ be a pure strategy equilibrium profile inducing the total queuing time $C^*$ and let $\alpha '$ be another pure strategy equilibrium profile inducing the total queuing time $C'$. Then any difference between the equilibrium welfare $\Phi _{\alpha ^*}$ and $\Phi _{\alpha '}$ is only caused by differences in $C^*$ and $C'$. Specifically, if $C^*=C'$ then $\Phi _{\alpha ^*}=\Phi _{\alpha '}$, whereas if $C^*<C'$ then $\Phi _{\alpha ^*}>\Phi _{\alpha '}$ and vice versa.

Proof

The expected welfare $\Phi _{\alpha }$ of a pure strategy equilibrium profile $\alpha$ may be reduced to

$$\begin{aligned} \Phi _{a} = \sum _{i\in N} \gamma - \omega \sum _{i\in N} S(e_i^{a_i},\alpha _{-i}) - \delta \sum _{i\in N} S(e_i^{a_i},\alpha _{-i})-a_i \end{aligned}$$

where $\sum _{i\in N} S(e_i^{a_i},\alpha _{-i})$ denotes the total expected service time, and $\sum _{i\in N} S(e_i^{a_i},\alpha _{-i})-a_i$ the total expected queuing time represented by C. We show that only C can cause differences in expected welfare between two pure strategy equilibrium profiles, since $\sum _{i\in N} S$ is constant under all pure strategy equilibrium profiles. To see this, consider the following observation: By Lemma 1, every pure strategy equilibrium profile must have the last player served at period 2, hence the facility uses its full service capacity within the periods 0, 1 and 2. That is, in equilibrium, the probability that one player is served at each period $s\le 2$ equals 1, such that $\sum _{i\in N} p_i(s\mid e_i^{a_i},\alpha _{-i})=1$ for each $s\in \{0,1,2\}$ and all $a_i\le 2$. Given that $S(e_i^{a_i},\alpha _{-i})=\sum _{s\in T} sp_i(s\mid e_i^{a_i},\alpha _{-i})$, we may write the total service time as

$$\begin{aligned} \sum _{i\in N} S(e_i^{a_i},\alpha _{-i})= & {} \sum _{i\in N} \sum _{s=0}^2 sp_i\left( s\mid e_i^{a_i},\alpha _{-i}\right) = \sum _{s=0}^2 \left[ s\sum _{i\in N} p_i\left( s\mid e_i^{a_i},\alpha _{-i}\right) \right] \\= & {} \sum _{s=0}^2 s = 3 \end{aligned}$$

The total expected service time is thus equal to the constant 3 under every pure strategy equilibrium profile, and only C can cause differences in expected welfare between two equilibrium profiles. Lemma 2 now follows immediately. $\square$

In the following, we characterize the pure strategy equilibrium profiles for each of the classical queue disciplines.

1.1.1 First-in first-out (FIFO)

We establish the following properties of a pure strategy equilibrium profile under FIFO.

Lemma 3

For any three-player game, let $\alpha ^*$ be a pure strategy equilibrium profile under FIFO, then

• (i) The number of players arriving for service at any period $t>0$ never exceeds the facility’s service capacity of one player per period.

• (ii) If all players do not arrive at period 0, then the last player arrives at period 2 and is served immediately with certainty.

Proof

Part (i) and (ii) are proven separately.

First, we prove part (i) by contradiction: Suppose profile $\alpha ^*$ admits $n(t)>1$ for some period $t>0$, i.e. the number of arrived players at period t exceeds the service capacity. However, this leads to a contradiction as one of the players arriving at period t could increase her expected utility by arriving at period $t-1$ and obtain an earlier expected service time for the same (or lower) expected queuing time. Thus, a pure strategy equilibrium profile under FIFO must have $n(t)\le 1$ for all $t>0$.

Next, we prove part (ii): Assume $\alpha ^*$ has $n(0)<3$, and let $\beta = \min \left\{ t\mid \sum _{t=0}^2 n(t)=3\right\}$ denote the earliest period for which all players have arrived under $\alpha ^*$. We note that $\alpha ^*$ cannot be an equilibrium profile for $\beta >2$ by Lemma 1. Conversely, if $0<\beta <2$, then [2, 1] is the only arrival distributions that corresponds to $\alpha ^*$ by Lemma 3, part (i). However, [2, 1] cannot be an equilibrium arrival distribution since the player arriving at period 1 could increase her expected utility by instead arriving at period 2 and obtain lower expected queuing time for the same expected service time. Therefore, $\alpha ^*$ must have $\beta =2$, and furthermore, have the last player arriving served immediately with certainty. This follows immediately by Lemma 1 and 3, part (i). $\square$

The only pure strategy profiles that satisfy Lemmas 1 and 3 are those with the arrival distribution [3], [1, 1, 1], or [2, 0, 1]. We next examine for which relative queuing costs, $\theta$, these are equilibrium profiles.

Proposition 1

For a three-player game with relative queuing cost $\theta$, under the FIFO queue discipline, the pure strategy equilibrium profiles and corresponding equilibrium welfare are given as follows:

$\theta <1$ :

There is a unique pure strategy equilibrium profile $\alpha ^*$ that induce the arrival distribution [3] and provides the equilibrium welfare $\Phi _{[3]}$ with $C=3$.

$\theta = 1$ :

Multiple pure strategy equilibrium profiles exist that induce the arrival distributions [2, 0, 1] and [3], respectively. Since [2, 0, 1] provides $C=1$, it follows that $\Phi _{[2,0,1]}>\Phi _{[3]}$.

$\theta \ge 3$ :

Multiple pure strategy equilibrium profiles exist that induce the same arrival distribution [1, 1, 1] and provide the maximum welfare $\Phi _{[1,1,1]}$ with $C=0$.

There exist no pure strategy equilibrium profiles for $1<\theta < 3$.

Proof

We start by introducing some useful notation: Recall that $U_i\left( \alpha \right)$ describes the expected utility for player i given the profile $\alpha$. Now, violating notation slightly, we write $U_i\left( a_i,a_{-i}\right)$ to describe the expected utility for player i given a pure strategy profile $\alpha$ that is induced by an arrival list $(a_i,a_{-i})$ where player arrives with certainty at period $a_i$ and the other players arrives according to $a_{-i}$.

$\mathbf {[3]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 0,(0,0)\right) \ge U_i\left( 2,(0,0)\right)$ which is true for all $\theta \le 1$, and (b) $U_i\left( 0,(0,0)\right) \ge U_i\left( 1,(0,0)\right)$ which is always true. Therefore, [3] is an equilibrium distribution for all $\theta \le 1$. The arrival distribution induce the total expected queuing time of $C=3$ and provides welfare $\Phi _{[3]}$.

$\mathbf {[2,0,1]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 0,(0,2)\right) \ge U_i\left( 1,(0,2)\right)$ which is true for $\theta \le 1$, and (b) $U_i\left( 2,(0,0)\right) \ge U_i\left( 0,(0,0)\right)$ which is true for $\theta \ge 1$. Since $\theta =1$ is the only binding condition for both (a) and (b), we conclude that [2, 0, 1] is an equilibrium arrival distribution for $\theta =1$ with $C=1$ and welfare $\Phi _{[2,0,1]}$.

$\mathbf {[1,1,1]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 2,(0,1)\right) \ge U_i\left( 0,(0,1)\right)$ which is true for $\theta \ge 3$; and (b) $U_i\left( 1,(0,2)\right) \ge U_i\left( 0,(0,2)\right)$ which is true for $\theta \ge 1$. Since $\theta \ge 3$ is the binding condition, we conclude that [1, 1, 1] is an equilibrium profile for $\theta \ge 3$ with $C=0$ and thus provides the maximum welfare $\Phi _{[1,1,1]}$.

Proposition 1 thus follows immediately as $\Phi _{[3]}<\Phi _{[2,0,1]}<\Phi _{[1,1,1]}$. $\square$

1.1.2 Last-in first-out (LIFO)

We establish the following properties of a pure strategy equilibrium profile under LIFO.

Lemma 4

For any three-player game, let $\alpha ^*$ be a pure strategy equilibrium under LIFO, then

• (i) For all $x<2$ where $\sum _{t=0}^x n(t)<3$, then period $x+1$ must have $n(x+1)>0$.

• (ii) All players do not arrive immediately, i.e. $n(0)<3$.

Proof

Part (i) and (ii) are proven separately.

First, we prove part (i) by contradiction: Suppose that $\alpha ^*$ for some period $x\in \{0,1\}$ has $\sum _{t=0}^{x} n(t)<3$, and moreover, the subsequent period $x+1$ has $n(x+1)=0$. However, this leads to a contradiction since the player arriving later than $x+1$ could increase her expected utility by arriving at $x+1$ instead and be serviced immediately with certainty as LIFO prioritizes later arrivals. Lemma 4, part (i), thus follows immediately.

We also prove part (ii) by contradiction: Suppose $\alpha ^*$ has $n(0)=3$. As simultaneous arrivals are served randomly with equal probability, then $p_i(s\mid e_i^0,\alpha ^*_{-i})=\frac{1}{3}$ for each $s\le 2$. Player i’s expected service and queuing time thus equal 1. However, this leads to a contradiction as player i could increase her expected utility by arriving at period 1 instead and be served immediately with certainty due to LIFO’s priority of latest arrivals. Thus, $\alpha ^*$ must in equilibrium have $n(0)<3$. $\square$

The only pure strategy profiles that satisfies Lemmas 1 and 4 are those with the arrival distributions [1, 1, 1], [1, 2], and [2, 1]. Similarly to FIFO, we examine for which values of $\theta$ these are equilibrium profiles.

Proposition 2

For a three-player game with relative queuing cost $\theta$, under the LIFO queue discipline, the pure strategy equilibrium profiles and corresponding equilibrium welfare are given as follows:

$\theta <1$ :

Multiple pure strategy profiles exist that induce the same arrival distribution [2, 1] and provide equilibrium welfare $\Phi _{[2,1]}$ with $C=2$.

$\theta =1$ :

Multiple pure strategy equilibrium profiles exist that induce the arrival distributions [1, 1, 1], [1, 2], and [2, 1], respectively. Since [1, 2] provides $C=1$, it follows that $\Phi _{[1,1,1]}>\Phi _{[1,2]}>\Phi _{[2,1]}$.

$\theta > 1$ :

Multiple pure strategy equilibrium profiles exist that induce the same arrival distribution [1, 1, 1] and provide the equilibrium welfare $\Phi _{[1,1,1]}$ with $C=0$.

Proof

We apply the slightly violated notation of $U_i$ as in the proof of Proposition 1.

$\varvec{[1,1,1]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 2,(0,1)\right) \ge U_i\left( 0,(0,1)\right)$ which is true for all $\theta \ge 1$, (b) $U_i\left( 1,(0,2)\right) \ge U_i\left( 0,(0,2)\right)$ which is true for all $\theta \ge 1$; and (c) $U_i\left( 2,(0,1)\right) \ge U_i\left( 1,(0,1)\right)$ which is true for all $\theta \ge 1$. Therefore, [1, 1, 1] is an equilibrium distribution for all $\theta \ge 1$. The arrival distribution provides $C=0$ and thus the maximum welfare $\Phi _{[1,1,1]}$.

$\varvec{[1,2]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 1,(0,1)\right) \ge U_i\left( 0,(0,1)\right)$ which is true for all $\theta \ge 1$, and (b) $U_i\left( 1,(0,1)\right) \ge U_i\left( 2,(0,1)\right)$ which is true for all $\theta \le 1$. Since $\theta =1$ is the only binding condition, we conclude that [1, 2] is an equilibrium distribution for $\theta =1$. The arrival distribution provides welfare $\Phi _{[1,2]}$ with $C=1$.

$\varvec{[2,1]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 1,(0,0)\right) \ge U_i\left( 0,(0,0)\right)$ which is always true, and (b) $U_i\left( 0,(0,1)\right) \ge U_i\left( 2,(0,1)\right)$ which is true for all $\theta \le 1$. Thus, [2, 1] is an equilibrium distribution for all $\theta \le 1$. The arrival distribution provides welfare $\Phi _{[2,1]}$ with $C=2$.

Proposition 2 thus follows immediately as $\Phi _{[2,1]}<\Phi _{[1,2]}<\Phi _{[1,1,1]}$. $\square$

1.1.3 Service in random order (SIRO)

Unlike under FIFO and LIFO, there are no further additions to Lemma 1 for which we can exclude other profiles to be pure strategy equilibria under SIRO. We thus turn to examining for which values of $\theta$ these are equilibrium profiles.

Proposition 3

For a three-player game with relative queuing cost $\theta$, under the SIRO queue discipline, the pure strategy equilibrium profiles and corresponding equilibrium welfare are given as follows:

$\theta <1$ :

There is a unique pure strategy equilibrium profile $\alpha ^*$ that induce the arrival distribution [3] and provides the equilibrium welfare $\Phi _{[3]}$ with $C=3$.

$\theta = 1$ :

Multiple pure strategy equilibrium profiles exist that induce the arrival distributions [2, 0, 1], [2, 1], and [3], respectively. Since [2, 0, 1] provides $C=1$ and [2, 1] provides $C=2$, it follows that $\Phi _{[2,0,1]}>\Phi _{[2,1]}>\Phi _{[3]}$.

$\theta > \frac{5}{3}$ :

Multiple pure strategy equilibrium profiles exist that induce the same arrival distribution [1, 1, 1] and provide the equilibrium welfare $\Phi _{[1,1,1]}$ with $C=0$.

There exist no pure strategy equilibrium profiles for $1<\theta < \frac{5}{3}$.

Proof

We apply the slightly violated notation of $U_i$ as in the proof of Proposition 1.

$\varvec{[1,1,1]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 2,(0,1)\right) \ge U_i\left( 0,(0,1)\right)$ which is true for all $\theta \ge \frac{5}{3}$, (b) $U_i\left( 1,(0,2)\right) \ge U_i\left( 0,(0,2)\right)$ which is true for $\theta \ge 1$, and (c) $U_i\left( 2,(0,1)\right) \ge U_i\left( 1,(0,1)\right)$ which is true for $\theta \ge 1$. As $\theta \ge \frac{5}{3}$ is binding, [1, 1, 1] is an equilibrium distribution under SIRO for all $\theta \ge \frac{5}{3}$ with the maximum welfare $\Phi _{[1,1,1]}$.

$\varvec{[1,2]}$ is an equilibrium arrival distribution if: (a) $U_i\left( 1,(0,1)\right) \ge U_i\left( 0,(0,1)\right)$ which is true for all $\theta \ge 3$, and (b) $U_i\left( 1,(0,1)\right) \ge U_i\left( 2,(0,1)\right)$ which is true for all $\theta \le 1$. No value of $\theta$ satisfy (a) and (a), thus [1, 2] is not an equilibrium arrival distribution under SIRO.

$\varvec{[2,1]}$ is an equilibrium distribution if: (a) $U_i\left( 1,(0,0)\right) \ge U_i\left( 0,(0,0)\right)$ which is true for all $\theta \ge 1$, (b) $U_i\left( 0,(0,1)\right) \ge U_i\left( 2,(0,1)\right)$ which is true for all $\theta \le \frac{5}{3}$, and (c) $U_i\left( 1,(0,0)\right) \ge U_i\left( 2,(0,0)\right)$ which is true for all $\theta \le 1$. As $\theta =1$ is the binding condition, [2, 1] is an equilibrium distribution under SIRO for $\theta =1$ with welfare $\Phi _{[2,1]}$.

$\varvec{[2,0,1]}$ is an equilibrium distribution if: (a) $U_i\left( 2,(0,0)\right) \ge U_i\left( 0,(0,0)\right)$ which is true for $\theta \ge 1$, (b) $U_i\left( 0,(0,2)\right) \ge U_i\left( 1,(0,2)\right)$ which is true for $\theta \le 1$, and (c) $U_i\left( 2,(0,0)\right) \ge U_i\left( 1,(0,0)\right)$ which is true for $\theta \ge 1$. As $\theta =1$ is binding, [2, 0, 1] is an equilibrium arrival distribution under SIRO for $\theta =1$ with welfare $\Phi _{[2,0,1]}$.

$\varvec{[3]}$ is an equilibrium distribution if: (a) $U_i\left( 0,(0,0)\right) \ge U_i\left( 1,(0,0)\right)$ which is true for $\theta \le 1$, and (b) $U_i\left( 0,(0,0)\right) \ge U_i\left( 2,(0,0)\right)$ which is true for $\theta \le 1$. Thus, [3] is an equilibrium distribution under SIRO for $\theta \le 1$ and provide the minimum welfare $\Phi _{[3]}$.

Proposition 3 thus follows immediately as $\Phi _{[3]}<\Phi _{[2,1]}<\Phi _{[2,0,1]}=\Phi _{[1,2]}<\Phi _{[1,1,1]}$. $\square$

1.2 Mixed strategy equilibrium analysis

This section presents the symmetric mixed strategy equilibrium (SMNE) profiles under the classical queue disciplines and establishes their corresponding equilibrium welfare for all values of relative queuing cost $\theta$. We restrict the analysis only to consider symmetric profiles where the players at most randomize over the periods 0, 1, and 2. As shown below (and highlighted in footnote 4, p. 9), this restriction does not affect the equilibrium predictions under our choice of experimental parameters.

We start by providing some useful notation. Let $\bar{T}=\{0,1,2\}$ denote the restricted set of admission periods. For a symmetric profile $\alpha =(\alpha _i)_{i\in N}$, let $\mathcal {S}(\alpha )\subseteq \bar{T}$ denote the support of $\alpha$ such that every strategy $\alpha _i$ assigns to each period $t\in \mathcal {S}(\alpha )$ the probability $\alpha _i(t)>0$ of arrival. For any symmetric $\alpha$ and any $i\in N$, let $\mathbf {P}(\alpha _{-i})$ be a vector that contains the probability of each possible combination of realized arrivals that two players can form when they arrive according to profile $\alpha _{-i}$. For example, if $\alpha$ has $\mathcal {S}(\alpha )=\{0,1\}$ then $\mathbf {P}(\alpha _{-i})$ contains the probabilities of the two players either arriving both at period 0, both at period 1, or one at period 0 and the other at period 1, i.e. $\mathbf {P}(\alpha _{-i})=\begin{pmatrix}\alpha _j(0)^2&\alpha _j(1)^2&2\alpha _j(0)\alpha _j(1)\end{pmatrix}$ for $j\in N{\setminus }\{i\}$. In other words, $\mathbf {P}(\alpha _{-i})$ is a vectorization of the terms generated by a binomial expansion with the number of variables equal to the cardinality of support $\mathcal {S}(\alpha )$.¹²

Fig. 6

Symmetric Mixed Strategy Equilibria and Expected Equilibrium Welfare under FIFO, LIFO, and SIRO

Next, recall that $e_i^x$ refers to a pure strategy where player i arrives at period x with certainty and the list $(e_i^x)$ refers to a pure strategy profile. For any pure strategy profile, let $e_{-i}^{(y,z)}=(e_j^y,e_k^z)$ denote the list of pure strategies for $j,k\in N{\setminus }\{i\}$ where $j\ne k$ such that $(e_i^x,e_{-i}^{(y,z)})$ for all $x,y,z\in \mathcal {S}(\alpha )$ is a pure strategy profile. Following this notation, for any profile $\alpha$, we define a corresponding vector $\mathbf {U}_{t}(\alpha _{-i})$ that contains player i’s expected utility by arriving at period $t\in \mathcal {S}(\alpha )$ with certainty under each of the possible combinations of the other players’ realized arrivals contained in $\mathbf {P}(\alpha _{-i})$. For example, if $\alpha$ has the support $\mathcal {S}(\alpha )=\{0,1\}$ then $\mathbf {U}_{t}(\alpha _{-i})=\begin{pmatrix}U_i\left( e_i^t,e_{-i}^{(0,0)}\right)&U_i\left( e_i^t,e_{-i}^{(1,1)}\right)&U_i\left( e_i^t,e_{-i}^{(0,1)}\right) \end{pmatrix}$ for each $t\in \mathcal {S}(\alpha )$ (here shown as the transpose of $\mathbf {U}_{t}$). Note that the position of the expected utilities in vector $\mathbf {U}_{t}(\alpha _{-i})$ is matched with the position of its probability of occurrence in $\mathbf {P}(\alpha _{-i})$.

For profile $\alpha ^*$ being a SMNE, player i obtains exactly the same expected utility by arriving at any period $t \in \mathcal {S}(\alpha ^*)$ with certainty, whereas she obtains at most the same expected utility by arriving at any period $m\in \bar{T}{\setminus }{\mathcal {S}(\alpha ^*)}$ outside the support domain. We thus say that $\alpha ^*$ is a SMNE if

$$\begin{aligned} \mathbf {P}(\alpha _{-i}^*)\mathbf {U}_{t}(\alpha _{-i}^*)&= \mathbf {P}(\alpha _{-i}^*)\mathbf {U}_{t'}(\alpha _{-i}^*) \quad \text { for all } t,t'\in \mathcal {S}(\alpha ^*)\end{aligned}$$

(3)

$$\begin{aligned} \mathbf {P}(\alpha _{-i}^*)\mathbf {U}_{t}(\alpha _{-i}^*)&\ge \mathbf {P}(\alpha _{-i}^*)\mathbf {U}_{m}(\alpha _{-i}^*) \quad \text { for all } t\in \mathcal {S}(\alpha ) \text { and } m\in \bar{T}{\setminus }{\mathcal {S}(\alpha )}\text {.} \end{aligned}$$

(4)

To establish every SMNE, we solve the system of equations in (3) wrt. $\alpha$ for a given $\theta$ and for all possible support domains using the following procedure:

• 1. Fix a support domain $\mathcal {S}\subseteq \bar{T}$.

• 2. Solve the system of equations $\mathbf {P}(\alpha _{-i})\mathbf {U}_{t}(\alpha _{-i}) = \mathbf {P}(\alpha _{-i})\mathbf {U}_{t'}(\alpha _{-i})$ for all $t,t'\in \mathcal {S}$ wrt. $\alpha _j(t)$ and given that $\alpha _j(t)=\alpha _k(t)$ for $j,k\in N{\setminus }\{i\}$ where $j\ne k$.¹³

• 3. Check condition (4) for any arrival $m\in \bar{T}{\setminus }{\mathcal {S}}$.

• 4. For all values of $\theta$ and for each $t\in \mathcal {S}$, compute $\alpha _j(t)$ solution derived in step 2.

• 5. Pick another support $\mathcal {S}'\subseteq \bar{T}$ where $\mathcal {S}'\ne \mathcal {S}$ and repeat the procedure.

Figure 6 shows the SMNE solutions under the classical queue disciplines for every $\theta \in [0,6]$. The graphs on the left show the probabilities $\alpha _i(t)$ for each $t\in \bar{T}$, while the graphs on the right show the corresponding equilibrium welfare. The solutions for each queue discipline are presented in separate figures. There exists a (non-degenerate) symmetric mixed strategy equilibrium for any $\theta > 0$ under LIFO, while for $\theta > 1$ under FIFO and SIRO due to strategic dominance of period 0 arrivals for $\theta < 1$. Depending on $\theta$ and the queue discipline, the SMNE profiles exists for different support sets $\mathcal {S}(\alpha )$. Under LIFO, the SMNE profile is $\alpha ^{\{0,1\}}$ for $0<\theta \le \sqrt{2}$, and $\alpha ^{\{0,1,2\}}$ is the SMNE profile for $\theta >\sqrt{2}$. Under FIFO, $\alpha ^{\{0,2\}}$ for $0<\theta \le 1+2\sqrt{2}$, and $\alpha ^{\{0,1,2\}}$ is the SMNE profile for $\theta >1+2\sqrt{2}$. Lastly under SIRO, $\alpha ^{\{0,1,2\}}$ for all $\theta >1$. In terms of SMNE welfare, LIFO provides the strictly highest equilibrium welfare for any relative queuing cost, while FIFO provides the lowest.

Table 2 Equilibrium profiles under FIFO, LIFO, and SIRO

1.3 Equilibrium comparison

This section outline the existence and uniqueness of pure and symmetric mixed strategy equilibrium profiles and the welfare properties across the queue disciplines.

Existence: Table 2 reports the equilibrium profiles under each queue discipline given the relative queuing cost. The equilibrium profiles are illustrated by rows and intervals of $\theta$ by columns. A queue discipline is stated at the intersection in which the equilibrium profile exists. All pure strategy equilibrium profiles are represented by arrival distributions (rows 1-5) while the symmetric mixed strategy profiles by their supported periods (rows 6-8). Table 2 shows that all queue disciplines provide at least one pure or symmetric mixed equilibrium profile for any $\theta$. LIFO is the only discipline that provides both a pure and a mixed equilibrium profile for all $\theta$. There exist no non-degenerated symmetric mixed equilibrium under FIFO and SIRO for $\theta <1$ due to strategic dominance of the pure strategy equilibrium profile [3]. Likewise, FIFO and SIRO provide no pure strategy equilibrium for $1<\theta < 3$ and $1<\theta < \frac{5}{3}$, respectively.

Uniqueness: Comparing the coexistence of pure and symmetric strategy equilibrium profiles, Table 2 shows that LIFO do not provide a unique equilibrium profile since there, at least, exists both a pure and symmetric mixed equilibrium for any $\theta >0$. Conversely, FIFO and SIRO provides the unique pure strategy equilibrium profile [3] for any $\theta \in [0,1)$ while it provides multiple equilibria for $\theta \ge 1$.

Welfare properties: With the existence of multiple equilibrium profiles, we cannot establish a coherent welfare ordering of the queue disciplines for all values of $\theta$. Nevertheless, we obtain insights into the welfare ordering by analyzing the best and worst case equilibrium welfares of the existing equilibrium profiles. That is, comparing the equilibrium profiles under each discipline that provide the highest and the lowest expected welfare, respectively. Figure 7 depicts the best and worst case equilibrium welfare under each queue discipline given the relative queuing cost $\theta$. The equilibrium welfare $\Phi _{\alpha }$ is indexed and thus illustrates the rank-orders of the queue disciplines for any set of parameter values.

Fig. 7

Best and worst case equilibrium welfare across queue disciplines

The best case equilibrium welfare under LIFO is induced by [2, 1] for $\theta \in [0,1)$ and [1, 1, 1] for $\theta \ge 1$, whereas the worst case is induced by $\alpha ^{\{0,1\}}$ for $\theta \in [0,\sqrt{2}]$ and $\alpha ^{\{0,1,2\}}$ for $\theta >\sqrt{2}$. Under FIFO, the best case is induced by [3] for $\theta \in [0,1)$, [2, 0, 1] for $\theta =1$, $\alpha ^{\{0,2\}}$ for $\theta \in (1,3)$, and [1, 1, 1] for $\theta \ge 3$, whereas the worst case is induced by [3] for $\theta \in [0,1)$, $\alpha ^{\{0,2\}}$ for $\theta \in [1,1+2\sqrt{2}]$, and $\alpha ^{\{0,1,2\}}$ for $\theta >1+2\sqrt{2}$. Lastly under SIRO, the best case is induced by [3] for $\theta \in [0,1)$, [2, 0, 1] for $\theta =1$, $\alpha ^{\{0,1,2\}}$ for $\theta \in (1,\frac{5}{3})$, and [1, 1, 1] for $\theta \ge \frac{5}{3}$, whereas the worst case is induced by [3] for $\theta \in [0,1)$ and $\alpha ^{\{0,1,2\}}$ for $\theta \ge 1$.

Figure 7 shows that both LIFO’s best and worst case equilibrium welfare are strictly higher than FIFO and SIRO’s best case equilibrium welfare for $\theta \in [0,1)\cup (1,\frac{5}{3})$, whereas they are strictly higher than FIFO’s for $\theta \in [0,1)\cup (1,3)$. Moreover, SIRO’s best and worst case equilibrium welfare is strictly higher than FIFO’s for $\theta \in (1,3)$. Lastly, for all $\theta$, LIFO’s worst case equilibrium is strictly higher than FIFO and SIRO’s worst case equilibrium, whereas LIFO’s best case equilibrium is at least as high as FIFO and SIRO’s best case equilibrium.

1.4 Statistics

See Table 3.

Table 3 Test statistics for experimental results