The Method of Trajectory Selection Based on Bayesian Game Model

Abstract

To cope with the problem that most of the en-route spatial-temporal resource allocation in the collaborative trajectory options program (CTOP) only considers the air traffic control system command center (ATCSCC) while ignoring the needs of the airlines, which results in the loss of fairness, this study explores resource allocation methods oriented to airline trajectory preferences with optional trajectory and entry slots of flights over the flow constrained area (FCA) as the research object. Using game theory to analyze airline trajectory preference information and a Bayesian game model based on mixed strategies is constructed, the process of incomplete information game among airlines is studied. The equilibrium theory is used to solve the guarantee strategy of airline trajectory selection, which makes the airline trajectory selection strategy robust and provides a basis for the selection of schemes for ATCSCC to implement en-route network resource allocation under the CTOP. Experimental analysis was carried out to verify the feasibility of the method based on the actual operation data of high-altitude sectors of Shanghai. The results show that the solution obtained by the game can provide airlines with flight trajectory and entry slots over the FCA that are more in line with their actual operational needs and which provide data reference for the ATCSCC to select the final plan in multiple global Pareto optimal solutions in the subsequent process of the CTOP so as to better play the decision-making role of airlines in the CTOP while improving the fairness of en-route resource allocation.

1. Introduction

As the main operation mode of future air traffic, trajectory-based operations (TBO) [1] are based on the four-dimensional trajectory of aircraft safety operation cycle, and through collaborative decision making (CDM) between ATCSCC, airlines, airports, aircraft, and other related parties which assign more flexible trajectories for flights to better meet the needs of airspace users. The CTOP is a new type of traffic management initiative for TBO. Based on the CDM mechanism, CTOP focuses on the FCA and takes the en-route resources (that is, the set of optional trajectory and the set of entry slots when the flights fly by the FCA) as the deployment object and uses a combination of rerouting and slot allocation strategies to solve the imbalance of demand and capacity in en-route and terminal airspace [2]. It achieves the efficient and equitable utilization of en-route resources. In this process, en-route resource allocation is no longer limited to the traditional point of considering only the demand of ATCSCC but adopts a distributed decision-making approach between ATCSCC and multiple airlines, incorporating airline trajectory preferences into the decision-making process, which effectively enhances the decision-making status of airlines in en-route resource allocation and improves their enthusiasm to participate in CDM.

Due to the early emergence of the CDM concept, existing early research mainly focused on the application of game theory in CDM time slot trading mechanism: using prediction game theory to analyze the slot auction strategy for ground delay program (GDP) [3]; using the matching approach for two-sided markets of game theory to build a CDM based on deferred acceptance mechanism [4] and the top trading cycle CDM(TTC-CDM) algorithm for time slot allocation [5]; using cooperative game theory to improve the delay allocation algorithm [6]; and constructing a new satisficing CDM model using satisficing game theory [7]. Some scholars have also applied game theory to ground transportation to study the impact of parking preferences on the parking methods of shared autonomous vehicles [8], to analyze the causes of traffic accidents [9], and to use incomplete information games to study pricing strategies [10,11]. As the latest technical extension in the field of CDM, CTOP is oriented to optional en-route resources and time slot resources, which mainly involves two aspects: airlines submitting trajectory options set (TOS, it includes trajectory parameters such as route, altitude, speed, and cost of per trajectory) to express their trajectory preferences and ATCSCC finally selecting the en-route resource allocation scheme [12]. Therefore, some studies have integrated game theory with global deployment strategies of en-route resources dominated by ATCSCC, such as slot allocation and rerouting: applying cooperative game theory to CTOP slot allocation strategies, establishing single-stage delay allocation model [13], slot switching model [14], and cooperative slot allocation model [15] among multiple airlines to provide airlines with better time slot resource allocation schemes and to reduce its delay and increase their participation in the slot allocation process. The game equilibrium theory is applied to the CTOP rerouting strategy, which mainly takes the principle of efficiency of resource allocation and the principle of optimal interests of airlines as the conflict points of game participants: establishing the optimal trajectory selection model [16], the global information exchange model [17], rerouting trajectory allocation model [18], and so on. These models meet the ATCSCC’s en-route global goal of optimal resource allocation and the respective trajectory preferences of airlines. However, in the above-mentioned process of selecting the en-route resource allocation plan led by the ATCSCC, most of the research tends to use multiple objectives optimization and decision making for global allocation plan selection [19,20,21]. This approach tends to result in more final alternatives and uncertainties due to too many considerations, and fewer considerations of airline trajectory preferences, which, to some extent, loses fairness.

In balancing the global objective of CDM and the individual objectives of airlines, game theory can be used to fully consider the influence of airlines’ decisions, thus enhancing airlines’ participation. At the same time, calculating the airline’s preferred en-route resource allocation plan from the game theory perspective can provide a stronger quantitative reference for the ATCSCC to choose the final plan among multiple global Pareto optimal solutions. The final result is more in line with the operational reality. At present, the game theory still focuses on solving the time slot allocation problem of ground delayed departure time or rerouting in the CDM process. For the time slot allocation problem of entering FCA in CTOP, the relevant results mainly focus on solving the combination problem of game theory and the CDM framework on which CTOP is based [22,23]. The research on solving the conflict of interests among airlines completely for the process of CTOP is incomplete.

In this paper, we introduce advanced collaborative traffic management measures—CTOP, analysis of the CTOP operational process, integration of the airlines’ decision-making preferences, incorporation of the game process into collaborative en-route network resource management, and the establishment a Bayesian game model based on mixed strategies among the airlines under the combination the domestic operational reality. It satisfies the requirements of all parties as much as possible and achieves an equilibrium solution for each airline’s en-route selection in restricted airspace. In addition, it provides the upper limit of each airline’s guarantee strategy for the ATCSCC to carry out the final plan of optimal allocation of en-route resources, which improves the fairness of the decision-making process of multi-party participation in the CTOP and final allocation strategy.

2. Analysis of the Construction of the Trajectory Preference Selection Model

The process of CTOP implementation includes: (1) identifying the FCA [24]; (2) the ATCSCC initiating a CTOP for the FCA; (3) the airlines submitting preference information for the affected flights (that is, the TOS) and giving the set of trajectory within the FCA that they prefer and the set of time slots into the FCA; (4) the ATCSCC makes the trajectory and slot assignments based on the TOS submitted by each airline. The specific implementation process of CTOP is shown in Figure 1. This study is aimed at the step in the red box in Figure 1. After the airlines submit the TOS, considering the trajectory preference of other airlines, how can the airlines choose strategies to ensure their own benefits (i.e., the delay time is as small as possible). Based on this framework, this paper studies the game process design experiment between airlines to ensure that the subsequent distribution is more effective and fairer.

Airline trajectory preferences are usually expressed by TOS. In order to design the TOS scheme for airlines, it is necessary to build the corresponding game model for the behavioral strategies of airlines within the CTOP framework and calculate the guarantee strategies of the corresponding participants so that their payoff functions have a lower bound value regardless of the strategies chosen by the other participants. Assuming that there are two airlines, that is, two game participants in one game process, and there is no cooperative behavior among airlines emphasizing individual rationality, the type of game model is analyzed as follows:

• Although each airline submits its own TOS in sequence, the ATCSCC, as the decision maker of the trajectory and slot allocation, needs to collect the TOS of each airline before the next allocation, so it is considered a static game.
• Although CTOP requires TOS information sharing among airlines, it is considered as an incomplete information game due to the uncertainty of each aircraft’s expected entry time (EET) at the designated route point, which causes the arrival time to be a set rather than a definite time point.

In summary, the airline trajectory preference model is set up as a static game of incomplete information, which can also be referred to as a Bayesian game model. A two-person Bayesian game model typically contains seven elements, which are assumed to be $(K,L,A,B,p_0,q_0)$:

• K and L are non-empty finite sets, representing the type sets of participants 1 and 2, respectively.
• A and B are non-empty finite sets, representing the sets of strategies of participants 1 and 2, respectively.
• M: $K\times L\times A\times B\to R$ denotes the payoff function for participant 1. $M^{kl}$ denotes the payoff matrix given the type $k\in K$ for participant 1 and the type $l\in L$ for participant 2. $M_{ab}^{kl}$ is a sub-element of $M^{kl}$ and can also be described as $M(k,l,a,b)$, denoting the payoff for participant 1 in the state of $k\in K$ and $a\in A$ and for participant 2 in the state of $l\in L$ and $b\in B$.
• $p_0\in \mathsf{\Delta }(K)$ and $q_0\in \mathsf{\Delta }(L)$ are the initial probabilities of the type sets K and L, respectively. In general, we assume $p_0,q_0>0$ for any $k\in K$ and $l\in L$.

Before building a Bayesian game model of the trajectory preference selection process, the basic rules for selecting candidate tracks need to be set. The specific rules are as follows:

(1)

To reduce computational complexity, when there are multiple trajectories in a single FCA, only the trajectory that is expected to enter the FCA earliest is considered [25]; this is because even sending multiple trajectory options for each FCA does not reduce global delay [13].

(2)

In order to obtain a better allocation order in CTOP, for each flight, the TOS will include the option of the earliest EET at the FCA in the candidate trajectory database.

(3)

In order to maximize the number of allocated slots, the option to fly around the FCA trajectory will not be included in the TOS.

We follow the above rules to design TOS for affected flights of airline A and assume that airline B also follows these rules.

3. Analysis of the Elements of Trajectory Preference Selection Model

Typically, a single CTOP execution procedure affects multiple airlines, and all affected airlines are participants. However, from the perspective of airline A, airline A itself is a participant (participant 1) whose strategy set can be represented by A; all other competing airlines are other participants (e.g., participant 2), called airline B, whose strategy set is represented by B. Before building the game model, the elements involved in the model are first analyzed.

The first element is the participant’s type set. That is, the non-empty finite sets K and L mentioned in Section 2. In the CTOP, the airline’s type set is the EET of each affected flight to the FCA. Suppose a CTOP contains M FCAs and airline A has a total of N affected flights. When it crosses FCA_m ($m=1,2,\dots ,M$), flight n ($n=1,2,\dots ,N$) may have multiple trajectory options. And according to rule (1) described above, only the earliest EET will be considered by the CTOP. Therefore, defining the EET of airline A’s flight into FCA_m as ${\mathrm{EET}}_{n,m}^A$, the set $({\mathrm{EET}}_{1,1}^A,\cdots ,{\mathrm{EET}}_{1,M}^A,{\mathrm{EET}}_{2,1}^A,\cdots ,{\mathrm{EET}}_{2,M}^A,\cdots ,{\mathrm{EET}}_{N,1}^A,{\mathrm{EET}}_{N,M}^A)$ is denoted as airline A type set K. Similarly, the type set of airline B can be defined.

The second element is the participant’s strategy set, also known as the action set. That is, the non-empty finite sets A and B mentioned in Section 2. The airline’s strategy set is the option to provide each affected flight with an FCA and a corresponding certain entry slot. $S_n^A\in \left\{1,\cdots ,M\right\}$ denote a slot selected by flight n of airline A from the FCA slot set, then the set $(S_1^A,\cdots ,S_2^A,S_N^A)$ denotes a combination of strategies for airline A.

The third element is the payoff matrix under all possible types set of participants. That is, the matrix composed of the values of the payoff function $M(k,l,a,b)$ mentioned in Section 2. Once the type of airlines (that is, the EET of all affected flights) is determined, the elements in sets K and L are determined. Then the allocated slots to the affected flights of airline A are determined by its strategy set (sets A and B), i.e., the selected trajectory of the affected flights determines the allocated slots. Then this will produce a $(\left|K\right|\times \left|L\right|, \left|A\right|\times \left|B\right|)$ dimensional payoff matrix, which is the matrix M.

The fourth element is the initial set of probabilities for all types of all airlines. That is, the set of $p_0\in \mathsf{\Delta }(K)$ and $q_0\in \mathsf{\Delta }(L)$ mentioned in Section 2. To obtain the probabilities for each type of airlines, the historical flight trajectory data of the affected flights need to be analyzed and the probabilities of the affected flights on the EETs need to be further obtained.

In Bayesian games, mixed strategies are usually used. The mixed strategy for participant 1 is a mapping from K to $\mathsf{\Delta }(A)$ and the mixed strategy for participant 2 is a mapping from L to $\mathsf{\Delta }(B)$, and $\mathsf{\Delta }(A)$ and $\mathsf{\Delta }(B)$ denote the probabilities of participant 1 and participant 2 on the set of strategies, respectively. The mixed strategy is used because in the worst case, a mixed strategy can achieve better results than a pure strategy.

4. Construction of the Trajectory Preference Selection Model

$x={(x^k)}_{k\in K}$ is the mixed strategy for participant 1, where $x^k\in \mathsf{\Delta }(A)$ is the probability of participant 1 (airline A) on the set of strategies under the given type $k\in K$ for it. Similarly, let $y={(y^l)}_{l\in L}$ be the mixed strategy of participant 2, where $y^l\in \mathsf{\Delta }(B)$ is the probability of participant 2 (airline B) on the set of strategies under the given type $l\in L$ for it. Based on the Bayesian finite zero-sum game model, the aim is to find the guarantee strategy $x^{*}$ of participant 1 (airline A) to reach its security level $V(p_0,q_0)$:

$$V(p_0,q_0)=\underset{x\in \mathsf{\Delta }{(A)}^{\left|K\right|}}{\min }\underset{y\in \mathsf{\Delta }{(B)}^{\left|L\right|}}{\max }{E}_{p_0,q_0,x,y}M(k,l,a,b)$$

(1)

That is, the mixed strategy x of airline A is found by Formula (1) such that the security level of airline A $V(p_0,q_0)$ is guaranteed regardless of the strategy adopted by airline B.

The above model is a complex nonlinear model, and in order to simplify the computation, the Harsanyi transformation [26,27,28] is used to transform it into a simple linear model, and the process is shown in Figure 2. Every time the Harsanyi transformation is executed, a full information game matrix is formed, and the linear planning method is combined with the mixed strategy can transform Formula (1) into a linear programming model. The specific transformation process is as follows. Firstly, Formula (1) is deformed into:

$$\begin{array}{l}V(p_0,q_0)=\underset{x\in \mathsf{\Delta }{(A)}^{\left|K\right|}}{\min }\underset{y\in \mathsf{\Delta }{(B)}^{\left|L\right|}}{\max }{E}_{p_0,q_0,x,y}M(k,l,a,b)\\ =\underset{x\in \mathsf{\Delta }{(A)}^{\left|K\right|}}{\min }\underset{y\in \mathsf{\Delta }{(B)}^{\left|L\right|}}{\max }{p}_0{}^k{x}^{T}M(k,l,a,b)q_0{}^{l}y\\ =\underset{x\in \mathsf{\Delta }{(A)}^{\left|K\right|}}{\min }\underset{y\in \mathsf{\Delta }{(B)}^{\left|L\right|}}{\max }{p}_0{}^k{x}^{T}M(k,l,a,b)({\displaystyle \sum _{l\in L}{q}_0{}^l{y}^l{e}^l})\\ =\underset{x\in \mathsf{\Delta }{(A)}^{\left|K\right|}}{\min }\underset{y\in \mathsf{\Delta }{(B)}^{\left|L\right|}}{\max }{\displaystyle \sum _{l\in L}{A}_l{q}_0{}^l{y}^l}\end{array}$$

(2)

where $e^l$ is a unit vector with only the component l being 1 and the remaining components being zero, $A_l=p_0{}^k{x}^T{M}^{kl}{e}^l$.

We assume $u^l=A_k=\underset{l}{\max }{A}_l$, and since ${\displaystyle \sum _{l\in L}{y}^l}=1$, $\underset{Y}{\max }{\displaystyle \sum _{l\in L}{A}_l{y}^l}$ reaches its maximum at $y^t=1$ and $y^l=0(l\ne t)$$u^l$, thus the security level of participant 1 satisfies the following relationship.

$$V(p_0,q_0)=\underset{x\in \mathsf{\Delta }{(A)}^{\left|K\right|},u\in R^{\left|L\right|}}{\min }{\displaystyle \sum _{l\in L}{q}_0^l{u}^l}$$

(3)

$$s.t.{\displaystyle \sum _{k\in K}{p}_0^k{M}^{k{l}^T}{x}^k\le u^{l}1},\forall l\in L$$

(4)

$$1x^k=1,\forall k\in K$$

(5)

$$x^k\ge 0,\forall k\in K$$

(6)

where the airline’s guarantee strategies are $x^{*}$.

Then, based on the maximin equilibrium theory, using Formula (3), the guarantee strategy of airline A can be calculated for each type of occurrence, and the optimal guarantee strategy for the whole Bayesian game can be found on the final probability set of each type so that the airline gets a lower bound on its revenue and makes its submitted TOS proposal more reasonable and explanatory.

5. Experiment Analysis

The actual operation data and simulation data of a certain route sector in the Shanghai Flight Information Region in Eastern China in April 2019 are used for analysis. Within the restricted airspace, one FCA is set for each of the two routes, named FCA_ZSHA_001 and FCA_ZSHA_002, respectively, as shown in Figure 3. Based on the actual air traffic control experience and available airspace capacity conditions, the available time slots are set for both routes on average, as shown in

Table 1. FCA time slot information.

FCA_ZSHA_001	FCA_ZSHA_002
17:05	17:10
17:12	17:15
17:18	17:21
17:26	17:27
17:31	17:33
17:39	17:38
17:46	17:44
17:54	17:49
18:02	17:54
18:08	18:00
18:13	18:06
18:18	18:11
18:23	18:17
18:27	18:23
18:32	18:28
18:36	18:33
18:40	18:38

.

After determining the available time slot information for FCAs, airlines need to share the information related to the affected flights from their own company. In the game model, the possible set of time points at which each affected flight is expected to arrive at each of the two FCAs needs to be entered. Here, based on the history data of the flight, the probable time points when each flight is most likely to arrive at the affected area are determined. Considering the practical situation, it is necessary to make the assumptions: (1) the affected flight is not affected by other factors during the flight and only two route options need to be considered; (2) the time of arrival of the affected flight at the expected FCA is not deterministic and the three most likely arrival times at the two FCA regions are selected separately; and (3) the affected flight is equally likely to select both route options. The affected flight in the historical data may select only one trajectory, then the set of time to the other trajectory is known from the calculation of its flight plan with the average speed when flying the original trajectory.

Obtain the time of the flight flying by UGAGO (in Figure 3) from the automatic dependent surveillance-broadcast (ADS-B) data, then add the flight time of the flight from UGAGO to the two FCAs (the flight time is based on the route length and flight altitude provided by the flight plan and the high-altitude flight performance data provided by the Base of Aircraft Data (BADA) database, and the flight time obtained by 4D trajectory prediction). Then calculate a certain time forward/backward from this time and obtain the three time points at which the flight is most likely to arrive at the FCAs. Based on the above assumptions, the affected flight information for each airline is shown in

Table 2. Affected flight information.

Number	Flight Number	Airline	Expected Entry Time at FCA1			Expected Entry Time at FCA2
1	CSN3573	CZ	17:00:18	17:06:15	17:11:00	17:05:18	17:11:15	17:16:00
2	CES9521	MU	17:03:46	17:09:25	17:14:41	17:08:46	17:14:25	17:24:41
3	CBJ5508	JD	17:06:17	17:10:39	17:15:45	17:11:17	17:15:39	17:20:45
4	CSN3847	CZ	17:13:07	17:17:19	17:21:31	17:08:07	17:12:19	17:16:31
5	CES5931	MU	17:10:16	17:15:11	17:19:38	17:15:16	17:20:11	17:24:38
6	CBJ5690	JD	17:12:03	17:15:23	17:20:20	17:17:03	17:20:23	17:25:20
7	CSZ1748	ZH	17:19:13	17:25:44	17:28:11	17:14:13	17:20:44	17:23:11
8	CSZ9513	ZH	17:17:58	17:21:52	17:25:26	17:22:58	17:26:52	17:30:26
9	CBJ5128	JD	17:21:05	17:25:45	17:31:27	17:26:05	17:30:45	17:36:27
10	CSN393	CZ	17:28:28	17:33:16	17:37:10	17:23:28	17:28:16	17:32:10
11	CSN3251	CZ	17:30:34	17:35:26	17:39:47	17:25:34	17:30:26	17:34:47
12	CES5560	MU	17:33:03	17:36:17	17:38:25	17:28:03	17:31:17	17:33:25
13	CBJ5665	JD	17:36:58	17:39:43	17:43:10	17:31:58	17:34:43	17:38:10
14	CBJ7628	JD	17:32:01	17:36:49	17:40:58	17:37:01	17:41:49	17:45:58
15	CSN6875	CZ	17:38:53	17:42:23	17:47:18	17:33:53	17:37:23	17:42:18
16	CSZ9829	ZH	17:35:50	17:42:01	17:46:52	17:40:50	17:47:01	17:51:52
17	CSN6351	CZ	17:43:12	17:46:09	17:50:15	17:38:12	17:41:09	17:45:15
18	CSN3501	CZ	17:45:33	17:49:26	17:52:46	17:50:33	17:54:26	17:57:46
19	CSZ9809	ZH	17:47:28	17:53:18	17:57:43	17:52:28	17:58:18	18:02:43
20	CES5932	MU	17:49:15	17:55:35	17:58:45	17:54:15	18:00:35	18:03:45
21	CES8022	MU	17:51:40	17:56:30	17:59:42	17:56:40	18:01:30	18:04:42
22	CES5793	MU	17:58:01	18:05:55	18:12:22	17:53:01	18:00:55	18:07:22
23	CSN3757	CZ	17:57:51	18:06:37	18:10:11	18:02:51	18:11:37	18:15:11
24	CSZ8320	ZH	17:59:27	18:05:19	18:12:18	18:04:27	18:10:19	18:17:18

.

From the previous model analysis, it is clear that the complexity of the game model increases exponentially with the number of type sets as well as strategy sets. To reduce the model complexity, the affected flights are grouped modularly to reduce the number of type sets and action sets. The affected flights are grouped according to the average of the earliest time points at which they are expected to arrive at FCA1 and FCA2 at 15-min intervals, and the result of grouping the flights is shown in Figure 4.

As can be seen from Figure 4, the affected flights are divided into four groups: A, B, C, and D, with a total of six flights in each group. First, the game is modeled for group A flights. Since only two participants are involved in the game process, when dealing with games with more than two participants, the game is played sequentially from the perspective of multiple participants, respectively, and the remaining participants are considered as one participant. From the distribution of flights in group A, we can see that the flight of China Southern Airlines (CZ) enters the congestion period first (that is, the flights of China Southern Airlines will enter the FCA first in chronological order, and the congestion period refers to the period during which the flight flies over the FCA), so China Southern Airlines is considered as participant 1 first, and the game model is built from its perspective, and China Eastern Airlines (MU) and Beijing Capital Airlines (JD) are jointly considered as participant 2.

The game parameters of group A are determined as follows:

(1) Type set: two flights of China Southern Airlines in group A are CSN3573 and CSN3847. A Cartesian product operation is performed on the set of expected entry times of these two flights at two FCAs. Since all sets of expected entry times contain three elements, there are a total of $3\times 3\times 3\times 3=81$ possible type sets for China Southern Airlines. Similarly, for the affected flights of the other two airlines, which are automatically grouped into the flight set of participant 2, there are a total of $3^8=6561$ possibilities for the type set of participant 2 since there are four flights. It is assumed that the selection probabilities of the types in each participant’s type set are uniformly distributed.

(2) Strategy set: participant 1, i.e., China Southern Airlines, has a total of two flights, so four strategies exist, namely, flights CSN3573 and CSN3847 both choose FCA1 (11), flight CSN3573 chooses FCA1 and CSN3847 chooses FCA2 (12), flight CSN3573 chooses FCA2 and CSN3847 chooses FCA1 (21), and both flights choose FCA2 (22). Similarly, for participant 2 with a total of four flights, there are a total of $2^4=16$ possibilities for its strategy set.

After the basic game parameters are given, the game needs to be Harsanyi transformed into a full information game model, and each payoff matrix in the game model is calculated using the mixed strategy solving method. The specific transformation process is shown in Figure 5.

As shown in Figure 5, on the first level of the Figure, it is assumed that participant 1 has chosen type K1, and at this time participant 1 does not know the type of participant 2, but participant 2 knows it himself, so it is necessary to introduce the virtual participant “Nature”. On the second level of the Figure, it actually becomes a three-player game, and “Nature” decides that the type of participant 2 is L. From the above analysis, it can be seen that there are 6561 possible types of participant 2. For the choice of “Nature”, participant 1 does not know it, but participant 2 knows. On this basis, participant 1 needs to choose its own game strategy among 81 possible cases. For type K1, there are 6561 possibilities for participant 2 and so on, and the participant 1′s type set needs to face a total of $81\times 6561=\mathrm{531,441}$ possibilities for participant 2. So, participant 1 needs to find out the strategy that gives itself the advantage as much as possible inside the 531,441 possibilities, and at this point the linear programming method for mixed strategies can help the game model to be solved. Based on the above data, an example of the calculus for layer 3 is given here. When the type of layer 3 is K1 and L1, K1 $({\mathrm{EET}}_{1,1}^A,{\mathrm{EET}}_{1,2}^A,{\mathrm{EET}}_{2,1}^A,{\mathrm{EET}}_{2,2}^A)$ is (17:00:18, 17:05:18, 17:08:27, 17:13:27), L1 $({\mathrm{EET}}_{1,1}^B,{\mathrm{EET}}_{1,2}^B,{\mathrm{EET}}_{2,1}^B,{\mathrm{EET}}_{2,2}^B,{\mathrm{EET}}_{3,1}^B,{\mathrm{EET}}_{3,2}^B,{\mathrm{EET}}_{4,1}^B,{\mathrm{EET}}_{4,2}^B)$ is (17:03:26, 17:08:26, 17:06:17, 17:11:17, 17:10:16, 17:15:16, 17:12:03, 17:17: 03). When participant 1 and participant 2 select strategies “11” and “1111” respectively, i.e., when FCA1 is selected for all flights, the final flight time sequence is obtained (17:00:18, 17:03:26, 17:06:17, 17:08:27, 17:10:16, 17:12:03), based on this. The time sequence determines the order of slot assignments. The slots are assigned to six flights in turn, and the sum of delays obtained by participant 1 is calculated, from which the calculation of each element of the matrix $M_{ab}^{kl}$ is carried out, and so on, filling the 64 elements of the payoff matrix to form a complete payoff matrix; the payoff matrix under K1 and L1 types is shown in Figure 6.

As a result, all elements of the CTOP Bayesian game have been well defined and computed. According to Formula (2), the security level and the guarantee strategy of participant 1 can be calculated using the linear programming method for mixed strategies. Given the strategy of participant 2 as “1111” and the type as L1, there are 81 possible cases for participant 1 for the four strategies, and the probability of choosing a mixed strategy for all types of participant 1 is shown in Figure 7. The horizontal coordinate in Figure 7 indicates the Cartesian product number of the set of times when flight CSN3847 is expected to enter two FCAs of participant 1 (China Southern Airlines, Guangzhou, China), and the vertical coordinate indicates the Cartesian product number of the set of times when flight CSN3573 is expected to enter two FCAs, and the order of Cartesian product operations is shown in

Table 3. The relationship between the flight Cartesian product number and its expected entry time at each FCA.

Cartesian Product Number	Expected Entry Time at Each FCA	Cartesian Product Number	Expected Entry Time at Each FCA
0	(17:00:18, 17:05:18)	5	(17:06:15, 17:16:00)
1	(17:00:18, 17:11:15)	6	(17:11:00, 17:05:18)
2	(17:00:18, 17:16:00)	7	(17:11:00, 17:11:15)
3	(17:06:15, 17:05:18)	8	(17:11:00, 17:16:00)
4	(17:06:15, 17:11:15)

. Each flight has three expected times of arrival at each of the two FCAs, corresponding to numbering in the range 0–8. The data in Table 3 describes the correspondence between the Cartesian product number of flight CSN3573 and the expected times of arrival at each FCA. The specific type of all flights of participant 1 can be determined by its Cartesian product number.

It shows the probability distribution of choosing different strategies for all types of participant 1 in Figure 7. From Figure 7a, we can see that in most cases, the probability of choosing strategy “11” for participant 1 is close to 0. Since the strategy of participant 2 is fixed here as “1111”, it is obviously unwise to choose strategy “11” when the competitors all choose FCA1. Figure 7b,c shows the probability that each of participant 1’s two flights will receive slots from different FCAs, which is typically higher than choosing a single FCA. Even when the competitor’s strategy is “1111”, pparticipant 1’s flights can gain a greater competitive advantage due to its higher arrival order among the flights participating in the game. This also suggests that in order to achieve stable performance, in many cases airlines should choose to allocate flights with different FCAs. Figure 7d shows two flights of participant 1 acquiring slots from FCA2, which is similar to Figure 7a, but with a larger overall probability than Figure 7a, mainly because pparticipant 2’s “1111” strategy allows participant 1 to avoid fierce competition by favoring FCA2 in choosing its strategy to select slots, thus gaining greater slots selection advantage.

In order to verify the experimental results, the CTOP game process is completely simulated with a total of 531,441 possibilities, and since the probability distribution of its type set is assumed to be uniform in this paper, its average value needs to be calculated. participant 1 has five strategies, except for the above-mentioned “11”, “12”, “21” and “22” as well as the choice of the guarantee strategy solved by the Bayesian game model. Accordingly, participant 2 has 17 strategies, including the 16 strategies mentioned above as well as a random strategy, which is the average of the 16 strategies. The final computed average total delay payoff matrix and guarantee strategy for participant 1 is shown in Figure 8. The total delay value under its guarantee strategy is compared with the delay values of the other four strategies as shown in Figure 9.

Figure 9 shows that among the five strategies, the total delay value of strategy “12” and the guarantee strategy is at a lower level, and the average value of strategy “12” is even lower than that of the guarantee strategy. However, in terms of robustness performance, the guarantee strategy is more advantageous. As the guarantee strategy always provides a more robust delay level for participant 1 regardless of the strategy chosen by participant 2, and the upper bound of its guarantee strategy is the lowest among the five strategies. This ensures that participant 1 can obtain a more stable and smaller delay loss in the game. Thus the guarantee strategy of the two flights of China Southern Airlines in group A chooses a delay value that is the maximum value of the guarantee strategy, which is both the lowest delay loss and more robust.

In addition, the delay value obtained by the game is compared with that obtained by other traditional methods, and the results are shown in Figure 10. The several methods in the figure are first come first serve (FCFS), two-stage framework (T.F.) [29], multi-objective programming (M.O.P.) [30], ration-by-schedule (RBS) and the guarantee strategy obtained by the game. It can be seen from Figure 10 that the delay value obtained by this method is lower and more robust, which shows the effectiveness of this method.

Similarly, the delay values under each strategy for the other airlines in group A and each airline in the other three groups are calculated, and the final guarantee strategies for the three airlines in group A are calculated. Their guarantee strategy value is finally taken as the maximum value in the set of guarantee strategies, so as to ensure the upper limit of delay that each airline can afford. The specific data are shown in

Table 4. The guarantee strategy value of each participant in group A.

Airlines	CZ	MU	JD
Guarantee strategy	13.17	13.32	15.55

Delay time unit: minutes.

.

In order to avoid the excessive computational complexity of the game model, the affected flights are divided into four groups according to the time interval. However, the impact of flights between groups is not taken into account in the separate game process after the grouping, which can greatly affect the selection of the guarantee strategy value of each airline at the end. Therefore, to minimize the impact of grouping, it is necessary to consider the influence between groups after the game process of the previous group. After the game of the current group, each airline will receive a guarantee strategy upper bound, which can guarantee the gain of each airline in any case; therefore, when playing in the subsequent group, it is necessary to consider the impact of the guarantee strategy value of the affected airlines in the previous group on their own game. Since the guarantee strategy value is the level of delay that all airlines would be willing to accept, the actual game situation of each airline is selected as the set of strategies that satisfy its guarantee strategy, and the intersection of the set of optional strategies of each airline is selected to determine a strategy that best meets the criteria of the guarantee strategy value as its optional strategy. The slots that have been occupied by the previous group under the strategy are excluded from the set of FCA slots so as to provide a realistic reference for the later group game. Taking Group A as an example, the selection process is shown in Figure 11.

As shown in Figure 11, each airline in group A places their flights at the top when playing as a participant, resulting in inconsistent flight order in the three groups of games. It is necessary to first arrange the flight order of each airline according to the order in Table 2 and subsequently find the combination of strategies with delay values less than the guarantee strategy values of all airlines. When only one strategy exists, then that strategy is taken as the possible guarantee strategy of group A and occupies the slots in the FCA in turn, and when more than one strategy exists, the sum of the delay values of each airline under these strategies is calculated and the strategy corresponding to the maximum value is selected as the possible guarantee strategy to maximize the possible adjustment space of group A. Here, it can also be considered as the result of the non-cooperative game between group B and group A.

After analysis, it can be concluded that the total delay value of the strategy combination “111222” in group A is 40.89 min according to the order given in Table 2, The slots (17:05, 17:12, 17:18) in FCA1 and the slots (17:15, 17:21, 17:27) in FCA2 are occupied. Therefore, group B is unable to use these six slots when gaming. On this basis, the analysis of the group B game is performed.

The subsequent calculation of the delay values of each strategy for each airline is carried out based on the above rules, and the calculation for the remaining groups follows this method. The values of the guarantee strategies for each airline are shown in

Table 5. Guarantee strategy value of each participant in each group.

Airlines	ZH	JD	CZ	MU
Group A	None	15.55	13.17	13.32
Group B	16.28	10.94	20.31	12.82
Group C	14.28	19.78	35.49	None
Group D	24.09	None	12.83	38.77

None: it indicates that there is no flight of the airline in this group, so there are no corresponding calculation results.

.

After completing the calculation of all the cases in the four groups, the total upper bound value of the guarantee strategies for each airline is calculated to provide an indicator reference for the determination of the multi-objective allocation scheme for route resources. The final total upper bound value of the guarantee strategies for each airline is shown in

Table 6. The upper bound value of each airline’s guarantee strategy.

Airlines	ZH	JD	CZ	MU
Guarantee strategy	54.65	46.27	81.80	64.91

.

Thus, the upper bound value of the guarantee strategy of each airline is obtained, which provides data support and process basis for the subsequent process of CTOP in the deployment method of route resources by ATCSCC. This can be combined with the multi-objective decision-making method to select the solution that meets the guarantee strategy among the obtained strategies, which can ensure higher efficiency and fairness at the same time and increase the enthusiasm of airlines to participate in CDM.

6. Conclusions

Based on game theory, the uncertainty of the arrival time of the affected airline flights at the scheduled point is considered, and the actual situation of information exchange between each airline and ATCSCC in CTOP is combined to carry out the research on the airline trajectory preference selection method by using the method of the static game of incomplete information. By studying the combination of Bayesian game with Harsanyi transformation and mixed strategy, a linear programming model of guarantee strategy of the airline trajectory preference is established to find the guarantee strategy value of each airline, which provides a more robust solution. This method guarantees a security level of its delay value regardless of the strategy adopted by other airlines and provides a reference basis for the final selection of the en-route resource allocation scheme led by ATCSCC for the airline trajectory preferences. This method is conducive to airline participation in collaborative decision-making and promotes the improvement of fairness in the allocation of en-route spatial-temporal resources from the perspective of collaborative decision-making.

In the experiment analysis in this paper, only the en-route resource allocation of the affected flight to a single FCA is considered. However, there may be multiple FCAs on a single trajectory and the frequent delays caused by the connectivity of the en-route network. At this time, more factors leading to higher delay costs must be considered. For example, the air-holding cost of the affected flight in two consecutive FCAs, and the random influencing factors caused by the flight trajectory selected. In future research, resource allocation methods in the presence of multiple FCAs need to be considered to apply to more general reality.