Traffic Game Model with the Contract Model

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Introduction
e target of an Intelligent Transportation System is to configure the actual traffic demand rationally in time and space by a certain control means and the induction strategies, so that the distribution equilibrium of traffic flow can be reached in the road network, and the overall resources and the traffic capacity of the road network can be fully utilized. e effectively induced information should consider the traveler's response to the information and choice preferences, which can estimate the expected benefits of both the traveler and transport system. Furthermore, the expected benefits are optimized and integrated.
en, an intelligent transport system will give the inducing advice to accord with the transport system and the driver's choices. At this point, the behavior between managers and travelers is a game. e first application of game theory in transportation is in the form of Wardrop equilibrium in 1952, where Wardrop analyzed the traffic phenomena comprehensively and then proposed two basic principles of the user optimal equilibrium and the system optimal equilibrium about traffic flow, which was similar to Nash equilibrium of noncooperative game about N agents. After that, the game theory is increasingly applied to the field of transportation.
Game theory is the study of mathematical models of no cooperation and cooperation between rational agents. In these two kinds of game modes, players have two choices: to cooperate or to defect. In terms of traffic control at an intersection, some control strategy using a game-theoretic approach, the basic theory of independent and interdependent decision-making, has been introduced for intelligent traffic light control problem. Dong et al. [1] applied a two-person static game model in multi-intersection coordinate control problem. Technically, they proposed some concepts of game theory such as pure strategy Nash equilibrium, mixed strategy Nash equilibrium, Pareto efficiency solution, and Pareto improvement solution of Nash equilibrium for solving traffic control at multi-intersection coordination. Qi [2] establishes the congestion model of intersection between drivers and traffic administration based on the benefit-tending characteristics of the drivers and traffic administrations, an incorrect inducement for traffic administration to take the strategy that prevents vehicles from crossing the intersection during the amber light, where each intersection is regarded as a noncooperative game.
Regarding the noncooperative game, Dai et al. [3] proposed an algorithm to solve the multi-intersection coordinated control problem, which combines the maximal flow theory with the game theory and considers both the individual interests of one intersection and the interests of the whole traffic network. After that, Bel and Cassir [4], Zhou et al. [5], and Li [6] discussed the game equilibrium about ticket price between the transport operators and passengers. Transport operators made decisions on the ticket price, which is responded by passengers from changing the travel means. It is concluded that transport operators can get a competitive advantage by improving the service quality, and transport operators' strategy may benefit local targets. But it may be harmful to the whole road network target.
en Miyagi et al. [7] and Zhen [8] formulated this traffic game as a stochastic congestion game and proposed a naive user algorithm for finding a pure Nash equilibrium. An analysis of the convergence is based on the Markov chain. Finally, using a single origin-destination network connected by some overlapping paths, the validity of the proposed algorithm is tested.
Recently, in terms of the development of Internet of ings, intersections coordinate controls are regarded as smart agents which can communicate and coordinate with each other. e control of several signalized intersections in a network is decomposed into multiple subnetworks and each subnetwork is considered as a region agent. ese studies explored the benefits of reducing problem complexity and improving system performance and learning efficiency when cooperation between agents is enabled. Some scholars have studied the integrated management system on the traffic management system and the traveler information system based on the agent technology, and the information communication platform between them is constructed (Ezzedine et al. [9,10]; Liu and Wang [11]; Zhang et al. [12]; Zhen [13]; Yuan et al. [14]; Zhou et al. [15]; Zhang and Gao [16]). According to the noncooperative game between two agents, the reliability of transportation network performance was measured by Bel [17] in 2000. Miyagi et al. [18] showed a pure user Nash equilibrium which describes the route-choice behavior of a user in a traffic network comprising several discrete, interactive decision-makers, and the agents use only the utility information of the previous action to get a congestion game. It is a multiagent distributed traffic routing problem with both linear and nonlinear link cost functions. Bui et al. [19] applied a cooperative game-theoretic approach among agents to improve traffic flow with a large network. ereby, a distributed merge and split algorithm for coalition formation is presented. However, both the timely reflection on the traffic conditions and the actions for adapting to the environment are not considered in these systems.
If traffic management can take proactive actions to adapt to the environment according to the drivers' behavior, the traffic condition will be improved significantly. us, we can use agent technology to make signal control system in the intersection abstract and ideological, which is conducive to the implementation of intelligent traffic management, and the analysis of multiagent behavior is more easily transformed into traffic network management.
Regarding the intersection of the road network as an agent, where the agent is the signal control system of an intersection, we adopt bid and bargain between every two adjacent agents and then obtain the equilibrium strategy price for the traffic flow in the road network. According to the latest equilibrium strategy price information provided by the intelligent traffic information system, the drivers further optimize their path selection strategy and get the most favorable path for themselves. At the same time, better system goals are achieved, and the coordination between system optimality and user optimality is retained.

Price Game between Adjacent Agents. Let a city traffic network be
where V is the set of intersections in the traffic network and E is the set of directed roads between two adjacent intersections. Assuming that one intersection corresponds to a signal control system, the set of signal control system in the intersection can be also denoted by V. e control system of each intersection in the traffic network represents an agent. For any agent j ∈ V, V j is the set of the adjacent agents with j.
Denote the phase set of signal control system of intersection j by where s is the phase number of a signal control system of intersection j. Each signal phase of signal control system of intersection j gets the green light time alternately. For any i ∈ V j , let ij be the directed road from agent i to j, and the lane set of road ij controlled by phase k of intersection j can be expressed as where R ij is the lane set of the directed road ij and R k j is the import lane set controlled by phase k of intersection j. Let be the real-time traffic flow of road ij, where S ji is the phase composition set of S j that controls lanes of road ij, that is, S ji ⊆ S j , k is the phase of intersection j, and f k ij is the realtime traffic flow on lane set R k ij . Assume that any agent i can get real-time traffic information on road ij by using the Intelligent Transportation System. Each agent i is both a traffic flow sender and a receiver except for it is the origin and destination in the network and hopes to send traffic volume to lane set R k ij at the moment of t. According to the real-time traffic flow f k ij of lane set R k ij , free flow speed, traffic volume and length of 2 Scientific Programming road ij, signal cycle T j of intersection j, and corresponding phase green time of lane set R k ij , agent i will calculate the real-time traffic impedance of road ij and intersection j and is willing to quote the receiving end j for traffic impedance.
Since, on the one hand, f k ij , f ij , and the green time of the phase that corresponded to lane set R k ij are changing over time and, on the other hand, free flow speed, traffic capacity and length of road ij, and signal cycle T j of intersection j all have fixed values, we can consider the offering price of agent i to be the function of f k ij , f ij , and the green time of phase k of lane set R k ij . Hence, the acceptable traffic impedance price that agent i prefers to give to phase k of receiving end agent j can be expressed as where λ jk is green time of the k − th phase in intersection j and p is the function formula of offering price p k ij from i to j. Each agent i hopes to send certain traffic flow to lane set R k ij at the moment of t. If road ij accepts the traffic flow from i, then the traffic volume of road ij changes to where f k 0 is the traffic volume that agent i sends to lane set R k ij at the moment of t. According to the traffic flow f k1 ij , where after road ij accepts traffic flow f k 0 , green time λ jk , and static property of road ij, the traffic impedance of road ij and the delay of intersection j are calculated by the receiving end agent j, furthermore, the impedance bargain q k ji of intersection j which is willing to accept traffic flow f k 0 from i can be obtained as where q is the function of that j bargaining q k ji to i. When agent i makes an offer p k ij above or equal to bargaining q k ji of agent j, the two sides will reach an agreement which leads to the successful negotiations.
As set S ji may contain several elements, it means the traffic flow of road ij can be controlled by multiple phases of j intersection signal control system. For any k ∈ S ji , there exists a corresponding bid p k ij . Hence, there exists one set of bids from i to j, which will form a bid fracture surface, and it is denoted by where (p k ij ) k∈S ji is the set of bids from i to j. For example, if S ji � 1, 2, 3 { }, then p ij can be got as For any k ∈ S ji , p k ij is a function of f k ij , f ij , and λ jk , where f ij is given by and then a bid fracture surface p ij from i to j is indeed a function of f k ij and λ jk ; that is, where (f k ij ) k∈S ji is the traffic flow group of road ij controlled by different phases before road ij accepts f k 0 , and (λ jk ) k∈S ji is the green time group of traffic flow phase on the control road ij. Similarly, the bargain fracture surface and the function of bargain fracture surface from j to i can be deduced as follows: where (q k ji ) k∈S ji is the bargain group from j to i and (f k1 ji ) k∈S ji is the traffic flow group by different phases control on road ij after road ij accepts f k 0 . If the traffic volume of road ij is f ij , the traffic impedance of road ij is thus where l ij is the length of road ij and v(f ij ) is the vehicle speed on road ij. Let where u k ji is the traffic impedance of vehicle going through road ij and d k ij is the red light delay of the import lanes of road ij by the k − th phase control, and the traffic impedance u k ji can also be referred to as the status quo point from agent j to i, which is on road ij by the k − th phase control. Agent i quotes impedance p k ij by the k − th phase control lane of signal control system of intersection j, which should be higher than or equal to the status quo point u k ji from agent j to i; that is, Otherwise, agent j will refuse to continue negotiations with agent i on the k − th phase.
After j receives traffic flow f k 0 from i, we denote M k ij to be the status quo point from agent i to agent j on road ij controlled by the k − th phase, and it holds where M k ij is the traffic impedance that vehicles pass the directed road ij after j received traffic flow f k 0 from i, t 1 ij is the traffic impedance of road ij after j receives traffic flow f k 0 from i, and d k1 ij is the red light delay on the import lanes of road ij by the k − th phase control after j receives traffic flow f k 0 from i. Agent j counteroffers q k ji to agent i which should not be higher than the status quo point M k ij from agent i to j, and agent i will refuse negotiation with j if not; that is, Scientific Programming As S ji may have several elements and k ∈ S ji , the status quo points from j to i and from i to j will form an individual fracture surface, respectively, which are expressed as where (u k ji ) k∈S ji is the status quo point group from j to i, (M k ij ) k∈S ji is the status quo point group from i to j, u ji is the status quo point fracture surface from j to i, and M ij is the status quo point fracture surface from i to j. Quotes fracture surface p ij should be preferable to u ji ; that is to say, where ≻ means it holds that, for any k ∈ S ji , Furthermore, the bargain fracture surface from j to i should meet the demand of

Nash Equilibrium of the Negotiation between Agents.
Assume that all the agents in V are individually rational; that is to say their acquisition is as good as before after knowing the final results of the negotiation. Agent i quotes each fracture surface p ij which will compare own status quo point fracture surface M ij to j, and agent j counteroffers each fracture surface q ji which will compare own status quo point fracture surface u ji to i as well. For any k ∈ S ji , the utilities obtained by i and j are M k ij − p k ij and q k ji − u k ji , respectively, under each bid and bargain. If the negotiation reaches an agreement, the arithmetic product of utility with agents i and j in phase k can be got as where it must meet us, we have where U ij can be regarded as the measurement of negotiation price agreements effect between i and j. e negotiators are all individually rational, and they hope the final price agreement is preferable to that of their status quo points, and the parties have the will to reach an agreement. Now, let us talk about the final agreement. e bid price strategy between every two adjacent agents in the traffic network forms a fracture surface, which can be expressed as where (p ij , q ji ) j∈V,i∈V j is a bid and bargain strategy for any two adjacent agents in the traffic network. We denote R to be the set of all strategy fracture surfaces R and B + to be the set of all possible traffic distributions. For any traffic distribution B ∈ B + and R ∈ R, if there does not exist B / ∈ B + and B / ≻ B is made, we conclude that traffic distribution B is valid to strategy fracture R. Let P(R) be the valid set (Pareto set) of all traffic distributions to R.
Definition 1 (validity). e validity means all agents' strategy fracture surface R, of which the results of traffic distribution are B ∈ P(R); that is to say, all the agents always choose the most advantageous strategy. Theorem 1. If any two adjacent agents in the network are individually rational and follow the validity, the following hold: ① the negotiation between them can reach an agreement; ② the final agreement price reaches Nash equilibrium; that is, Proof. ① According to individual rationality, let the initial bid price of agent i be the status quo point u k ji from j to i for any k ∈ S ji . en, agent i increases his bid price according to the bargain price from agent j. On the other hand, the initial bargain price of agent j is the status quo point M k ij from i to j, and agent j then decreases his bargain price according to the increasing value size of bid price of agent i. Since both sides' bid prices are preferable to their status quo points, respectively, they both wish to reach an agreement. In order to do this, the bid price of i will be higher as long as the bargain price of j will be lower. When the n-th bid and bargain price of them is they finally achieve an agreement with the price: us, their negotiation can achieve an agreement. ② Assume that both sides achieve an agreement at the l-th bid p kl ij and bargain q kl ji ; that is, If according to formula (26), the effect of the price agreement of their negotiation is As we can get that If the agreed price of both sides is the arithmetic product of effectiveness for their negotiation is Since that is, the Nash equilibrium price of both sides is thus Hence, for any k ∈ S ji , if it holds that reaches maximum, and then p k * ij is the Nash equilibrium price of both agents i and j. According to the validity, bid price of both sides will be selected in the Pareto set, which yields that the final agreement price must be the Nash equilibrium price.
For agent j at any receiving side, it will receive many adjacent agents' quotation. Whose quotation will be prioritized is directly related to the utility obtained by j. So the priority will be given to the specific traffic route, which obtains the largest system utility. us, agent j has to balance all agents' transportation requests in V j and use the available resources (such as adjusting the green signal ratio of imported lanes, etc.) to meet the prior request. □ Theorem 2. If the traffic flow receiver agent j and the traffic flow sender agent i in V j are all individually rational and follow the validity, we can get the following: ① each agent in V j can reach an agreement with agent j, respectively; ② the Nash equilibrium of all agents utility arithmetic product, is the final agreement price, which is expressed as Proof.
(1) According to eorem 1, for any agent i and agent j, they can achieve a final agreement. Hence, each agent in V j can reach an agreement with agent j, respectively.
(2) According to eorem 1, for any agent i ∈ V j , k ∈ S ji , if the final agreement price between two agents is Scientific Programming we can obtain where that is, is leads to that is, us, the Nash equilibrium of is where k ∈ S ji and i ∈ V j . According to the validity, the bid prices of all agents in V j and agent j will be selected in the Pareto set, and the final agreement price between j and his adjacent agents must be the Nash equilibrium price: Agent j can change the price by adjusting the split ratio to achieve the purpose of regulating road traffic and control the traffic volume of the up-down stream. For example, according to the existing number of vehicles in the upstream road and the traffic volume that prepare to enter the road, the agent decreases the traffic volume that is prepared to enter the road by increasing the bargain price to the adjacent agent in the upper stream road and increase outflow by improving the quotation to the downstream adjacent agents.
is may guarantee the entire road network traffic flow to be balanced. □ Theorem 3. If every two adjacent agents in V have a price negotiation with each other and they are all individually rational and follow the validity, we can get the following: ① all adjacent agents of both sides will reach an agreement; ② the entire road network has Nash equilibrium price, and the final price agreement is the Nash equilibrium solution to which is expressed as Proof.
(1) According to eorem 2, for any j ∈ V and all the agents in V j , they can reach an agreement. By the arbitrariness of j, all adjacent negotiations of both sides can reach an agreement in the road network.
(2) According to eorem 2, for any j ∈ V, i ∈ V j , and k ∈ S ji , if it holds that and we thus get that that is, is yields that the Nash equilibrium solution of U is According to the validity, every two adjacent agents select their quotation price in the Pareto set, and the final agreement price must be Nash equilibrium solution in the road network. at is, the entire road network forms the Nash equilibrium price.

Confirming the Green Time of the Signal Control System in the Intersection.
It follows from eorem 3 that j could obtain the largest utility by adjusting the green time λ jk in the k-th phase when facing any agent i in V j . Due to the individual rationality, for any i ∈ V j and k ∈ S ji , j will increase the green time λ jk in the k-th phase if j can obtain larger utility from i and reduce λ jk conversely if not. Meanwhile, to prevent conflict of green light, we assume that when any phase in an intersection is a green light or yellow one, the others are all red light. Notice that the vehicle may get through the intersection on the yellow time; we fix the green time λ jk here including the yellow time in the k-th phase for discussing conveniently. erefore, it holds that where T j is the signal circle of the signal control system in intersection j. On the grounds of eorem 1, together with the status quo point from i to j, we thus obtain the equilibrium price as follows: where the traffic impedance of the intersection part is Taking the maximum of intersection stop delay controlled by the k-th phase in different roads as an example, that is, , the green time of the k-th phase in intersection j can then be denoted by where λ 0 is the minimum green time to make sure vehicles and pedestrians get through the intersection safely and λ 1 is the maximum green time of signal phase for intersection j.

The Driver Path Selection Rules Based on the Price Information
Let K be the origin set of the OD and let S be the destination set of the OD; it is obvious that K ⊂ V and S ⊂ V, and all the path set between origin site and the destination one is where κ ∈ K, ℓ ∈ S, and z m is the m-th path from origin κ to destination ℓ. For any path z λ ∈ Z κℓ , it can be expressed as where k 0 is the first intersection on path z λ from κ to ℓ and k m λ is the last intersection on path z λ . e driver hopes to pay the least expenditure for starting from the origin site κ to the destination site; that is to say, he will choose the path with the lowest equilibrium strategy price in the network. eorem 3 tells that all the adjacent agents in the network can reach an agreement and get an equilibrium price, and thus the intelligent traffic system can continuously provide the driver with the real-time equilibrium price between every two adjacent agents. With the equilibrium strategy price at hand, the driver calculates all path prices from κ to ℓ, which is indeed a price function that can be expressed as follows: where p k * ij is the Nash equilibrium price from i to j and U κℓ z λ is the price of path z λ .
Let the driver set with traffic demand between origin κ and destination ℓ be where n is the total number of the drivers with traffic demand between origin κ and destination ℓ. e price information in the Intelligent Transportation System is updated with a certain period of time which is assumed to be T 0 . Assume that the equilibrium price information of Intelligent Transportation Systems is updated once at the moment of t 1 . Now, during the period from moment t 1 to t 1 + T 0 , the equilibrium price of each path between origin κ and destination ℓ has changed with vehicles entering the road network, which is in sharp contrast to the price information in the Intelligent Transportation System that did not update in time. In this case, if the driver selects the path with the least sum of the price which is provided by Intelligent Transportation Systems, the actual price of the path may not be the lowest. As the received price information is not exactly coincided with the actual information, the different path strategies may be selected by different drivers according to their age, gender, education level, and other factors. Assume that the driver has two strategies to choose: the first one is to choose the path with the lowest price offered by the Intelligent Transportation Systems at the moment of t 1 ; the second one is based on the path price of the Intelligent Transportation System offered at the moment of t 1 , and choose the path in Z κℓ by using fuzzy-comparison strategy. Let where 1 ≤ e ≤ m. Set Step 6: If all the OD meet the condition we thus stop calculating; otherwise, we get back to Step 2.
is algorithm cycles on the basis of cycle time T 0 . ere is a traffic distribution to the entire network in each cycle and to test the distribution of each OD demand in the network. If the distribution of an OD demand is finished, the allocation of the next cycle will skip the OD demand, until all OD demand allocations come to an end. e price information in the algorithm is constantly updated over time, and it also reflects the human behavior factors in the distribution. Hence, it is much closer to the actual traffic distribution.  Table 1.

Example
ere are two cases in finding the solution of average delay on the lanes of import road ij of the k-th phase control in a time cycle. On the one hand, if the arrival rate is less than signal traffic capacity, it can be calculated by the steady-state delay model. On the other hand, if the arrival rate is higher than signal traffic capacity, it can be calculated by fixed number delay model. e steady-state delay model and fixed-number delay model (Wang and Yan) [20] can be, respectively, expressed as where v m ij is the free flow speed on road ij. e traffic impedance of each road is obtained by the numerical correspondence relationship between (traffic volume/traffic capacity) and (average speed/free flow speed), which is shown in Table 2 (Markose et al.) [21].
In Table 3, the traffic assignment result has been given based on the contract model and the shortest path algorithm. On the one hand, the assignment results based on the shortest path algorithm are listed as follows: there is 2988 veh·h − 1 traffic flow choosing path a 2 among the traffic demand volume of 3000 veh·h − 1 of OD pair (1, 7); there is 3988 veh·h − 1 traffic flow choosing path a 10 among the traffic demand volume of 4000 veh·h − 1 of OD pair (7, 1); there is 1016 veh·h − 1 traffic flow choosing path a 7 among the traffic demand volume of 1020 veh·h − 1 of OD pair (4,7). On the other hand, the traffic assignment results based on the contract model algorithm are stated as follows: the traffic demands of OD pair (1, 7) distributing the traffic flow to paths a 2 , a 3 , a 4 , and a 5 are 864 veh·h − 1 , 732 veh·h − 1 , 736 veh·h − 1 , and 664 veh·h − 1 , respectively; the traffic demands of OD pair (7, 1) distributing the traffic flow to paths a 10 , a 11 , a 12 , and a 13 are 1028 veh·h − 1 , 1000 veh·h − 1 , 1040 veh·h − 1 , and 932 veh·h − 1 , respectively; the traffic demands of OD pair (4, 7) distributing the traffic flow to paths a 7 and a 8 are 688 veh·h − 1 and 332 veh·h − 1 , respectively. Compared with the contract model algorithm, the traffic flow is overconcentrated on the shortest path algorithm and it does not respond to the human behavior factor in transportation route choice. For the contract model algorithm, according to the characteristic of traffic flow, every two adjacent intersections have a game, and the equilibrium price of the road network is obtained. It adjusts the intersection signal phase green time and meets the condition of the equilibrium price for the road network. Finally, it induces by an Intelligent Transportation Information System and considers behavior factors of the driver, which succeed in distributing the traffic flow to the road network equally. erefore, the contract model algorithm promotes traffic to be distributed more evenly to the road network and makes a more rational distribution of traffic flow in the network and thus avoids the phenomenon of overconcentration of the traffic flow.
It is shown from Table 4 that the red light delays of intersections 3 and 6 for the traffic flow assigned by shortest path method are far below the one assigned by the contract model algorithm. Indeed, since OD traffic demand distribution is overconcentrated on certain paths by the shortest path method, there is little traffic flow passing intersections 3 and 6, and the red light delays of intersections 3 and 6 by using the shortest path algorithm are fewer correspondingly. From the total red light delay of all the intersections in the network, we find that the total red light delay of all the intersections in the network by using the contract model algorithm is 2.53819 × 10 8 s, which is significantly lower than the total delay of 5.7141 × 10 8 s by using the shortest path algorithm. Hence, it has improved the total utility of the network signal control by 55.58%. us, the contract model algorithm makes the signal control of each intersection coordinated and unified by agent technology. It is a full initiative of traffic management and control and thus improves greatly the system utility in the road network. e contract model algorithm obtains the least price path by coordinating and unifying the control system of the road network. Furthermore, this combined with the positive inducement by the Intelligent Transportation System, and the uncertainty of driver's choosing the path, the system    utility in the network thus plays its role effectively. By the simulation example, we can find that the dynamic traffic game model based on contract considers fully the system optimization of the road network and also the utility maximization of the driver's choosing the path which achieves coordination between system optimization and user optimization and thus improves greatly the effectiveness of the system in the road network.

Conclusions
is paper combines the manager's initiatives with the driver's personal rational behavior closely by using agent technology and thus studies intensively the allocation of each intersection traffic right in the road networks. Based on the fact that the Intelligent Transportation Systems provides continually the real-time traffic information of road network, it is much easier for agents to achieve equilibrium strategy price. Next, the equilibrium strategy price will be given to the driver through the information system. en, the driver takes the corresponding path selection strategy by the latest price information of the road network, which succeeds in making the traffic flow assignment more reasonable in the road network. e example shows that the network utility will be greatly increased by using a dynamic traffic game model based on the contract model. e model gives full consideration to the system optimality and takes into account the user optimality, which conforms to the actual situation. e next step is based on the traffic conditions of the road network, to which agents (intersection) price control strategy makes the appropriate dynamic response, and pricing strategies influence each other between the various agents and so forth.

Data Availability
e data used to support the study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.