Two-stage security screening strategies in the face of strategic applicants, congestions and screening errors

Abstract

In a security screening system, a tighter screening policy not only increases the security level, but also causes congestion for normal people, which may deter their use and decrease the approver’s payoff. Adapting to the screening policies, adversary and normal applicants choose whether to enter the screening system. Security managers could use screening policies to deter adversary applicants, but could also lose the benefits of admitting normal applicants when they are deterred, which generates a tradeoff. This paper analyzes the optimal screening policies in an imperfect two-stage screening system with potential screening errors at each stage, balancing security and congestion in the face of strategic normal and adversary applicants. We provide the optimal levels of screening strategies for the approver and the best-response application strategies for each type of applicant. This paper integrates game theory and queueing theory to study the optimal two-stage policies under discriminatory and non-discriminatory screening policies. We extend the basic model to the optimal allocation of total service rate to the assumed two types of applicants at the second stage and find that most of the total service rate are assigned to the service rate for the assumed “Bad” applicants. This paper provides some novel policy insights which may be useful for security screening practices.

Appendix

1.1 Appendix 1: Proof of Proposition 1

The adversary potential applicant’s application rate $P_{B}$ depends on his utility function $u_{B}$, once the utility payoff is less than equal to 0, adversary applicants would have no interest in applying this system. His utility is $u_{B}=P_{B}\Lambda _{B}\Big (r_{B}(1-\Phi ^{B})-c_{B}\Phi ^{B}\Big )$. The revised screening probability across the two stage is $\Phi ^{B}\ge s_b\equiv \frac{r_B}{r_B+c_B}$, which is derived from $u_{B} \le 0$, results in $P_B=0$. The adversary potential applicants’ best response $P_{B}$ has the opposite change direction with $\Phi ^{B}$, which decreases in $e_{2b}$, where $\frac{{\partial }\Phi ^B}{{\partial } e_{2b}}=-\Phi _{1}\Phi ^{B}_{2}\le 0$. Therefore, adversary potential applicants’ best response $P_{B}$ increases in $r_{B}$ and $e_{2b}$, and decreases in $c_{B}$.

1.2 Appendix 2: Proof of Proposition 2

We define $\mu '_{1} \equiv \mu _{1}-\Phi _{1}P_{B}\Lambda _{B}$, $\mu '_{2G} \equiv \mu _{2G}-\Phi _{1}\Phi _{2G}e_{1b}P_{B}\Lambda _{B}$ and $\mu '_{2G} \equiv \mu _{2B}-\Phi _{1}\Phi _{2B}(1-e_{1b})P_{B}\Lambda _{B}$. There are upper bounds for the normal application probability at the first stage $P_{G}^{+}$ and at the second stage $P_{GG}^{++}$ and $P_{GB}^{++}$, respectively, where $P_{G}^{+} \equiv \frac{\mu '_{1}}{\Phi _{1}\Lambda _{G}}$, and $P_{GG}^{++}\equiv \frac{\mu '_{2G}}{\Phi _{1}\Phi _{2G}(1-e_{1g})\Lambda _{G}}$ and $P_{GB}^{++}\equiv \frac{\mu '_{2B}}{\Phi _{1}\Phi _{2B}e_{1g}\Lambda _{G}}$. The normal application probability $P_{G}$ must be at least smaller than or equal to them to satisfy the service rates.

• (i) We substitute $\Phi _{1}=0$ into Eqs. (1) and (2), which results in $u_{B}=P_{B}r_{B}>0$ and $u_{G}=P_{G}r_{G}>0$. Thus potential applicants would apply with probability 1. Since there is no screening at the first stage, naturally there will be no further screening at the second stage, thus we have $\Phi _{2G}=\Phi _{2B}=\Phi _{2}^{G}=0$.

• (ii) If the approver screens at both stages; i.e., $\Phi _{1} \in (0,1]$, $\Phi _{2}^{G}\in (0,1]$, then

  • (a) Once the service rates at the first and the second stage respectively can not satisfy normal applicants, they will drop the application. Thus when $\mu '_{1}\le 0$ or $\mu '_{2G}\le 0$ or $\mu '_{2B}\le 0$, we have $P_{G}=0$.

  • (b) Once the normal application probability $P_{G}$ satisfies the upper bounds and also makes the normal applicants’ utility $\mu _{G}({\varvec{\Phi }}, P_{G}=1, P_{B})\ge 0$, the normal applicants would apply to the system $P_{G}=1$.

  • (c) Once the normal application probability $P_{G}$ makes the normal applicants’ utility $\mu _{G}({\varvec{\Phi }}, P_{G}=1, P_{B})< 0$ or makes $\mu _{G}({\varvec{\Phi }}, P_{G}=1, P_{B})\ge 0$ with any upper bounds in the range of [0, 1) that $P_{G}^{+} \in [0, 1)$ or $P_{GG}^{++} \in [0, 1)$ or $P_{GB}^{++} \in [0, 1)$, it needs to decrease the probability value to the range of [0, 1), considering $P_{G}$ must be at least lower than or equal to the upper bounds $P_{G}^{+}$, $P_{GG}^{++}$ and $P_{GB}^{++}$. For the screening policy, the maximum traffic of screened normal applicants $\hat{\Lambda }_{G}$ is derived from the normal applicant’s zero utility [Eq. (2)] at the equilibrium:

    $$\begin{aligned}&r_{G}-(r_{G}+c_{G})\Phi _{1}\Phi _{2}^{G}e_{2g}-c_{W}\Phi _{1}\Big (\frac{1}{\mu '_{1}-\hat{\Lambda }_{G}}+\frac{\Phi _{2G}(1-e_{1g})}{\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G}}\nonumber \\&\quad +\,\frac{\Phi _{2B}e_{1g}}{\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G}}\Big )=0 \end{aligned}$$

    (15)

Then the maximum traffic of screened normal applicants is:

$$ \begin{aligned} \hat{\Lambda }_{G}= & {} -\frac{b'}{3a'}+\root 3 \of {\frac{b'c'}{6a'^2}-\frac{b'^3}{27a'^3}-\frac{d'}{2a'}+\sqrt{\left( \frac{b'c'}{6a'^2}-\frac{b'^3}{27a'^3}-\frac{d'}{2a'}\right) ^2+\left( \frac{c'}{3a'}-\frac{b'^2}{9a'^2}\right) ^3}}\\&+\,\root 3 \of {\frac{b'c'}{6a'^2}-\frac{b'^3}{27a'^3}-\frac{d'}{2a'}-\sqrt{\left( \frac{b'c'}{6a'^2}-\frac{b'^3}{27a'^3}-\frac{d'}{2a'}\right) ^2+\left( \frac{c'}{3a'}-\frac{b'^2}{9a'^2}\right) ^3}}\ \ (a' \ne 0)\\ \text {or} \ \hat{\Lambda }_{G}= & {} \frac{-c'\pm \sqrt{c'^2-4b'd'}}{2b'}\ \ (a' = 0 \& b' \ne 0)\\ \text {or} \ \hat{\Lambda }_{G}= & {} -\frac{d'}{c'} \ \ (a' = 0 \& b' = 0 \& c' \ne 0). \end{aligned}$$

where $\Phi _{2}^{G} \equiv (1-e_{1g})\Phi _{2G}+e_{1g}\Phi _{2B}$ is introduced in Sect. 2.3, and we define $a \equiv \Phi _{2G}(1-e_{1g})$, $b \equiv \Phi _{2B}e_{1g}$, $c \equiv \frac{r_{G}-(r_{G}+c_{G})\Phi _{1}\Phi _{2}^{G}e_{2g}}{\Phi _{1}c_{W}}$, $a'=abc$, $b'=(3ab-\mu '_{1}abc-\mu '_{2G}bc-\mu '_{2B}ac$, $c'=\mu '_{1}\mu '_{2G}bc+\mu '_{1}\mu '_{2B}ac+\mu '_{2G}\mu '_{2B}c-2a\mu '_{2B}-2b\mu '_{2G}-2ab\mu '_{1}$, and $d'=\mu '_{2G}\mu '_{2B}+a\mu '_{1}\mu '_{2B}+b\mu '_{1}\mu '_{2G}-\mu '_{1}\mu '_{2G}\mu '_{2B}c$. Using Theorem 1 in Balachandran and Schaefer (1980), there exists a unique equilibrium aggregate traffic rate: $\lambda _{G}=\max \big (\min (\hat{\Lambda }_{G},\Phi _{1}\Phi _{2G}(1-e_{1g})\Lambda _{G}, \Phi _{1}\Phi _{2B}e_{1g}\Lambda _{G}),0\big )$.

Since there are three maximum traffic of screened normal applicants, thus we have $P_{G}^{+}$, $P_{GG}^{++}$ and $P_{GB}^{++}$. We choose the larger normal application probability that satisfies $P_{G} \in [0, 1]$. Thus, normal potential applicants’ best response strategies satisfy:

$$\begin{aligned} P_{G}=\max \Big (\min \Big (\frac{\hat{\Lambda }_{G}}{\Phi _{1}\Phi _{2G}(1-e_{1g})\Lambda _{G}}, \frac{\hat{\Lambda }_{G}}{\Phi _{1}\Phi _{2B}e_{1g}\Lambda _{G}},1,P_{G}^{+},P_{GG}^{++},P_{GB}^{++}\Big ),0\Big ) \end{aligned}$$

We see that normal potential applicants’ best response $P_G$ has the opposite change direction with parameter $\Lambda _{G}$ but same change direction with parameter $\hat{\Lambda }_{G}$. We have $\frac{{\partial }P_G}{{\partial }\Lambda _{G}}=-\frac{\hat{\Lambda }_{G}}{\Phi _{1}\Phi _{2G}(1-e_{1g})\Lambda _{G}^2}<0$ or $-\frac{\hat{\Lambda }_{G}}{\Phi _{1}\Phi _{2B}e_{1g}\Lambda _{G}^2}<0$, and $\frac{{\partial }P_G}{{\partial }\hat{\Lambda }_{G}}=\frac{1}{\Phi _{1}\Phi _{2G}(1-e_{1g})\Lambda _{G}}>0$ or $\frac{1}{\Phi _{1}\Phi _{2B}e_{1g}\Lambda _{G}}>0$. Equation (15) shows that parameter $\hat{\Lambda }_{G}$ has the opposite change direction with parameters $c_W$ and $e_{2g}$ but same change direction with parameters $\mu _{1}$, $\mu _{2G}$, $\mu _{2B}$, and $r_{G}$ since we have:

$$\begin{aligned} \frac{{\partial }c_W}{{\partial }\hat{\Lambda }_{G}}= & {} -\frac{\frac{1}{(\mu '_{1}-\hat{\Lambda }_{G})^2}+\frac{\Phi _{2G}^2(1-e_{1g})^2}{(\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G})^2} +\frac{\Phi _{2B}^2e_{1g}^2}{(\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G})^2}}{\frac{1}{\mu '_{1}-\hat{\Lambda }_{G}}+\frac{\Phi _{2G}(1-e_{1g})}{\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G}} +\frac{\Phi _{2B}e_{1g}}{\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G}}}<0\\ \frac{{\partial }e_{2g}}{{\partial }\hat{\Lambda }_{G}}= & {} -\frac{c_{W}\Phi _{1}}{\Phi _{1}\Phi _{2}^{G}r_G}\left( \frac{1}{(\mu '_{1}-\hat{\Lambda }_{G})^2}+\frac{\Phi _{2G}^2(1-e_{1g})^2}{(\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G})^2} +\frac{\Phi _{2B}^2e_{1g}^2}{(\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G})^2}\right) <0\\ \frac{{\partial }\mu _{1}}{{\partial }\hat{\Lambda }_{G}}= & {} (\mu '_{1}-\hat{\Lambda }_{G})^2\left( \frac{\Phi _{2G}^2(1-e_{1g})^2}{(\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G})^2} +\frac{\Phi _{2B}^2e_{1g}^2}{(\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G})^2}\right) +1>0\\ \frac{{\partial }\mu _{2G}}{{\partial }\hat{\Lambda }_{G}}= & {} \frac{\left( \frac{1}{(\mu '_{1}-\hat{\Lambda }_{G})^2} +\frac{\Phi _{2B}^2e_{1g}^2}{(\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G})^2}\right) (\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G})^2}{\Phi _{2G}(1-e_{1g})}+\Phi _{2G}(1-e_{1g})>0\\ \frac{{\partial }\mu _{2B}}{{\partial }\hat{\Lambda }_{G}}= & {} \frac{\left( \frac{1}{(\mu '_{1}-\hat{\Lambda }_{G})^2}+\frac{\Phi _{2G}^2(1-e_{1g})^2}{(\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G})^2} \right) (\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G})^2}{\Phi _{2B}^2e_{1g}^2}+\Phi _{2B}^2e_{1g}^2>0\\ \frac{{\partial }r_{G}}{{\partial }\hat{\Lambda }_{G}}= & {} \frac{c_{W}\Phi _{1}}{1-\Phi _{1}\Phi _{2}^{G}e_{2g}}\left( \frac{1}{(\mu '_{1}-\hat{\Lambda }_{G})^2}+\frac{\Phi _{2G}^2(1-e_{1g})^2}{(\mu '_{2G}-\Phi _{2G}(1-e_{1g})\hat{\Lambda }_{G})^2} +\frac{\Phi _{2B}^2e_{1g}^2}{(\mu '_{2B}-\Phi _{2B}e_{1g}\hat{\Lambda }_{G})^2}\right) >0 \end{aligned}$$

• (iii) If the approver screens at the first stage $\Phi _{1}\in (0,1]$ but does not at the second stage $\Phi _{2}^{G}=0$, then

  • (a) Once the service rates at the first can not satisfy normal applicants, they will drop the application. Thus when $\mu '_{1}\le 0$, we have $P_{G}=0$.

  • (b) Once the normal application probability $P_{G}$ satisfies the upper bound at the first stage $P_{G}^{+}$ and also makes the normal utility $\mu _{G}({\varvec{\Phi }}, P_{G}, P_{B})\ge 0$, the normal applicants would apply to the system.

  • (c) Once the normal application probability $P_{G}$ makes the normal utility $\mu _{G}({\varvec{\Phi }}, P_{G}=1, P_{B})< 0$ or makes $\mu _{G}({\varvec{\Phi }}, P_{G}=1, P_{B})\ge 0$ with upper bound $P_{G}^{+} \in [0, 1)$, it needs to decrease the probability value to the range of [0, 1), considering $P_{G}$ must be at least lower than or equal to the upper bound $P_{G}^{+}$. For the screening policy, the maximum traffic of screened normal applicants $\hat{\Lambda }_{G}$ is derived from the normal applicant’s zero utility as follow: $r_{G}-(r_{G}+c_{G})\Phi _{1}e_{1g}-c_{W}\Phi _{1}\Big (\frac{1}{\mu _{1}-\Phi _{1}P_{B}\Lambda _{B}-\hat{\Lambda }'_{G}}\Big )=0$, and $\hat{\Lambda }'_{G}=\mu _{1}-\Phi _{1}P_{B}\Lambda _{B}-\frac{c_{W}\Phi _{1}}{r_{G}-(r_{G}+c_{G})\Phi _{1}e_{1g}}$. There exists a unique equilibrium aggregate traffic rate: $\lambda _{G}=\max \big (\min (\hat{\Lambda }'_{G},\Phi _{1}\Lambda _{G}),0\big )$. Thus, normal potential applicants’ best response strategies satisfy: $P_{G}=\frac{\lambda _{G}}{\Phi _{1}\Lambda _{G}}=\max \big (\min (\frac{\hat{\Lambda }'_{G}}{\Phi _{1}\Lambda _{G}},1,P_{G}^{+}),0\big )$. We see that the normal potential applicants’ best response $P_G$ has the opposite change direction with parameter $\Lambda _{G}$, but same change direction with parameter $\hat{\Lambda '}_{G}$. We have $\frac{{\partial }P_G}{{\partial }\Lambda _{G}}=-\frac{\hat{\Lambda }'_{G}}{\Phi _{1}\Lambda _{G}^2}<0$, and $\frac{{\partial }P_G}{{\partial }\hat{\Lambda }'_{G}}=\frac{1}{\Phi _{1}\Lambda _{G}}>0$. The parameter $\hat{\Lambda '}_{G}$ has the opposite change direction with parameter $c_W$ but same change direction with parameters $\mu _{1}$ and $r_{G}$. We have $\frac{{\partial }\hat{\Lambda }'_{G}}{{\partial }c_W}=-\frac{\Phi _{1}}{r_{G}(1-e_{1g}\Phi _{1})}<0$, $\frac{{\partial }\hat{\Lambda }'_{G}}{{\partial }\mu _{1}}=1>0$, and $\frac{{\partial }\hat{\Lambda }'_{G}}{{\partial }r_{G}}=\frac{c_{W}\Phi _{1}(1-e_{1g}\Phi _{1})}{(r_{G}-(r_{G}+c_{G})\Phi _{1}e_{1g})^2}>0$.

Therefore, normal potential applicants’ best response $P_G$ decreases in parameters $c_W$ and $e_{2g}$ and increases in parameters $\mu _{1}$, $\mu _{2G}$, $\mu _{2B}$, and $r_{G}$.

1.3 Appendix 3: Proof of Proposition 3 in discriminatory policy

According to the Proposition 1, when the total effective screening probability $\Phi _{1}\Phi _{2}^{B}$ is larger than or equal to the threshold $s_{b}$, none of the adversary potential applicants submit their applications. We assume that when the approver is indifferent between different levels of screening probabilities, the lowest level will be chosen.

• 1. When $\Phi ^{B}=s_{b}$, then $P_{B}=0$, there are no bad applicants. Then the approver’s objective value becomes: $J_{1}({\varvec{\Phi }})=\Lambda _{G}P_{G}R(1-\Phi _{1}\Phi ^G_{2}e_{2g})=R(1-\Phi _{1}\Phi _{2G}e_{2g})P_{G}({\varvec{\Phi }}, P_{B}=0)$.

• 2. When $\Phi ^{B} \in [0,s_{b})$, then $P_{B}=1$, all adversary potential applicants submit their applications. Then the approver’s objective value becomes: $J_{2}({\varvec{\Phi }})=R(1-\Phi _{1}\Phi _{2}^{G}e_{2g})P_{G}({\varvec{\Phi }}, P_{B}=1)-C(1-\Phi ^{B})$.

  $$\begin{aligned} J({\varvec{\Phi }})= \left\{ \begin{array}{ll} J_{1}({\varvec{\Phi }}), &{} \text {if} \ \Phi ^{B}=s_{b}\\ J_{2}({\varvec{\Phi }}), &{} \text {if} \ \Phi ^{B} \in [0,s_{b}).\\ \end{array} \right. \end{aligned}$$

  (16)

Thus, the optimal best strategy for the approver is to solve: $J^{*}=\underset{0 \le \Phi _{1}\Phi _{2}^{B} \le s_{b}}{{\max }}\ J({\varvec{\Phi }})$.

1.4 Appendix 4: Numerical sensitivity analyses under discriminatory policy

Figure 9 illustrates the numerical sensitivity analysis under a discriminatory policy, which implies that the screening probabilities for ‘Good’ and ‘Bad’ applicants at stage 2, $\Phi _{2G}$ and $\Phi _{2B}$, respectively, could be different. The screening probability $\Phi _{1}$ increases in C, $\Lambda _B$, $r_G$, $\mu _{1}$, $\mu _{2B}$, and $c_G$ (Fig. 9c, f, j, k, m, n), decreases in $\Lambda _{G}$, and R (Fig. 9g, h), and first increases and then decreases in $e_{g}$, r, $c_B$, R and $r_B$ (Fig. 9a, b, d, h, i). The probability of screening ‘Good’ applicants at the second stage $\Phi _{2G}$ generally is low except when $e_{g}$ is high (Fig. 9a), or when r is intermediate (Fig. 9b), or when R is low (Fig. 9h). The probability of screening ‘Bad’ applicants at the second stage $\Phi _{2B}$ generally remains high except when $e_{g}$ and r are high (Fig. 9a, b), or when C, $r_B$, $r_G$, and $\mu _{1}$ are low (Fig. 9c, i–k). The adversary application probability $P_{B}$ stays at zero except when $e_{g}$, r, $\Lambda _{G}$, R and $r_B$ are high (Fig. 9a, b, g–i), or when C, $c_B$, $\Lambda _B$, $r_G$, $\mu _{1}$, and $\mu _{2B}$ are low (Fig. 9c, d, f, j, k, m). The normal application probability $P_G$ generally increases in $c_B$, R, $r_G$, $\mu _{1}$, and $\mu _{2B}$ (Fig. 9d, h, j, k, m), decreases in $e_{g}$, C, $c_W$, $\Lambda _B$, $\Lambda _{G}$, R, $r_B$, and $c_G$ (Fig. 9a, c, e, f, g, i, n). The normal application probability $P_G$ keeps at 1 due to low screening probability $\Phi _{1}$ when $e_{g}$, r, and R are high (Fig. 9a, b, h), or when C, $c_B$, $\Lambda _B$, $\Lambda _{G}$, $r_B$, $r_G$, and $\mu _{1}$ are low (Fig. 9c, d, f, g, i, j, k).

Fig. 9

Numerical sensitivity analysis under discriminatory policies

1.5 Appendix 5: Comparisons between discriminatory policy and non-discriminatory policy

This section compares the model results between the non-discriminatory policy and the discriminatory policy: adversary application probabilities ($P_{BN}$ and $P_{BD}$, respectively), normal application probabilities ($P_{GN}$ and $P_{GD}$, respectively), and approver’s payoffs ($J_N$ and $J_{D}$, respectively). In particular, Figure 10 shows the comparison of adversary application probabilities $P_{BN}$ and $P_{BD}$ between the non-discriminatory and the discriminatory policies. The adversary application probability under the non-discriminatory policy is significantly higher than that in the discriminatory policy when the adversary applicants’ reward if passed $r_B$ is high (Fig. 10i), or when the penalty for approver once admitting each adversary applicant C, and the approver’s reward for admitting each normal applicant R are intermediate (Fig. 10c, h), or when the cost to adversary applicants being caught $c_{B}$, the adversary applicant arrival rate $\Lambda _{B}$, and the reward for normal applicant to pass the system $r_G$ are low (Fig. 10d, f, j).

Fig. 10

Comparing adversary application rates $P_{BN}$ and $P_{BD}$ between the non-discriminatory and the discriminatory policies

Figure 11 compares the normal application probabilities $P_{GN}$ and $P_{GD}$ between the non-discriminatory and the discriminatory policies. The normal application probability in a discriminatory policy is significantly higher than the one in a non-discriminatory policy when the error probabilities that normal applicants are screened as ‘Bad’ at the first and second stage $e_{1g}=e_{2g}$, the unit time cost $c_W$, the service rate at the first stage $\mu _{1}$, and the loss once normal applicants are rejected $c_G$ are high (Fig. 11a, e, k, n), or when the the cost to adversary applicants being caught $c_{B}$, the adversary applicants’ reward if passed $r_B$, and the second stage screening/service rate for applicants screened as ‘Bad’ $\mu _{2B}$ are intermediate (Fig. 11d, i, m), or when the power function coefficient r, the benefit of the approver for passing each normal applicant R, the reward for normal applicant to pass the system $r_G$, and the second stage screening/service rate for applicants screened as ‘Good’ $\mu _{2G}$ are low (Fig. 11b, h, j, l).

Fig. 11

Comparing normal application probabilities $P_{GN}$ and $P_{GD}$ between the non-discriminatory and the discriminatory policies

Figure 12 shows the comparison of the approver’s payoffs $J_{N}$ and $J_{D}$ between the non-discriminatory and the discriminatory policies. The approver’s payoffs under discriminatory policy is significantly higher than the one in a non-discriminatory policy, especially when the error probabilities that normal applicants are screened as ‘Bad’ at the first and second stage $e_{1g}=e_{2g}$, the penalty for the approver once admitting each adversary applicant C, the unit time cost $c_W$, the service rate at first stage $\mu _{1}$, and the loss once normal applicants are rejected $c_G$ are high (Fig. 12a, c, e, k, n), or when the cost to adversary applicants being caught $c_{B}$, the adversary applicant arrival rate $\Lambda _{B}$, the normal applicant arrival rate $\Lambda _G$, the reward for adversary to pass the system $r_B$, and the second stage screening/service rate for applicants screened as ‘Bad’ $\mu _{2B}$ are intermediate (Fig. 11d, f, g, i, m), or when the power function coefficient r, the the reward for normal applicant to pass the system $r_G$, and the second stage screening/service rate for applicants screened as ‘Good’ $\mu _{2G}$ are low (Fig. 11b, j, l).

Fig. 12

Comparing approver’s payoffs $J_{N}$ and $J_{D}$ between the non-discriminatory and the discriminatory policies

1.6 Appendix 6: Comparisons for one versus two-stage screening systems

This section introduces and compares one versus two-stage screening systems to find the best screening policy for the approver under certain situations. The utility for a one-stage imperfect screening system is $J_{1NP}$. The utility for a two-stage imperfect screening system is $J_{2NP}$. Figure 13 shows the comparison of the approver’s utilities in one- and two-stage screening systems. It shows that the approver’s payoff in a two-stage screening system $J_{2NP}$ is significantly larger than the one in a one-stage system $J_{1NP}$ when the adversary applicants’ reward if passed $r_B$, the service rate at the first stage $\mu _{1}$, and the loss once normal applicants are rejected $c_G$ are high (Fig. 13i, k, m), or when the error probability that normal applicants are screened as ‘Bad’ at the first stage $e_{1g}$, the penalty for approver once admitting each adversary applicant C, the adversary applicant arrival rate $\Lambda _{B}$, and error probabilities that normal applicants are screened as ‘Bad’ at the first and second stage $e_{1g}=e_{2g}$ are intermediate (Fig. 13a, c, f, n), or when power function coefficient r, the cost to adversary applicants being caught $c_B$, the unit time cost $c_W$, and the error probability at the second stage $e_{2g}$ are low (Fig. 13b, d, e, l).

Fig. 13

Comparisons of approver’s utilities in one-stage and two-stage systems