CN113282348A - Edge calculation task unloading system and method based on block chain - Google Patents
Edge calculation task unloading system and method based on block chain Download PDFInfo
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Abstract
The invention discloses a block chain-based system and a block chain-based method for unloading edge calculation tasks, wherein the system comprises: the wireless network equipment unit comprises a plurality of Internet of things equipment connected through a network, and the Internet of things equipment collects data information of an Internet of things application program from a physical environment and forwards the data information to the MEC server unit to execute task calculation; and the MEC server unit comprises a plurality of MEC servers and is used for realizing a consensus mechanism of the block chain and realizing signing registration and task unloading of the wireless network equipment according to an intelligent contract arranged in a block chain network on the MEC servers. By using a block chain technology, the traceability and visibility of the unloading process of the edge computing task are ensured; by the task unloading method based on the priority, the requirements of users with different time delay requirements on the current unloading task can be more reasonably met; by the aid of the unloading result verification method based on the game theory, correctness and verifiability of calculation results are guaranteed, and quality of task unloading is improved.
Description
Technical Field
The invention belongs to the technical field of mobile edge calculation and block chains, and particularly relates to a block chain-based high-reliability task unloading system and method in edge calculation.
Background
With the development of various mission critical applications of the internet of things, such as vehicle networks (including vehicle-to-vehicle, vehicle-to-infrastructure, augmented reality technology, and city sensing, etc.), it is necessary to ensure that sufficient computing power and low latency connections are provided for the devices used in these applications.
However, in the application of the internet of things, there is a contradiction between having enough computing resources and low delay. This is especially true in locations closer to the internet of things devices, as the network capacity of the first network transmission from the internet of things device to the cloud may be limited. To compensate for the cloud Computing deficiency, the european telecommunications standards institute has proposed a Mobile Edge Computing (MEC) framework. The mobile edge computing is a technology based on a 5G evolution architecture and integrates a mobile access network with internet service in depth. One aspect is to reduce the latency associated with forwarding data from a customer-owned internet of things device to a cloud platform. On the other hand, computing power and storage capacity are sunk to the mobile edge node, and third-party application integration and deployment are provided, so that more innovations of mobile edge computing are realized. This overcomes some potential limiting factors in current cloud services: the energy consumption of the terminal equipment is low, and the requirements for transmitting the security sensitive data to the cloud platform and the network capacity requirements between the Internet of things equipment and the cloud platform are reduced.
Some problems still exist in the existing unloading scheme of the edge computing task, and when the time delay requirements of users for different unloading tasks are different, some emergency tasks cannot be solved at the first time; the mobile edge computing server provider has limited audit trail capability for processing user data, and a user is difficult to trace back a task unloading process; the edge calculation is a service provided by the outside, and the correctness of the calculation result of the edge server is difficult to guarantee.
Disclosure of Invention
In order to solve the problem that the unloading result is unreliable in the task unloading in the edge calculation, the invention provides a high-reliability task unloading system and method based on a block chain in the edge calculation. The unloading technical scheme comprises a priority-based unloading method in a block chain and an unloading result verification method based on a game theory. The entire offload policy is deployed in blockchain intelligent contracts to ensure visibility and traceability of the entire offload process.
In order to achieve the purpose, the invention adopts the following technical scheme:
an edge computing task unloading system based on a block chain comprises a wireless network equipment unit and an MEC server unit. The wireless network equipment unit comprises a plurality of Internet of things equipment connected through a network.
The wireless network equipment unit comprises a plurality of Internet of things equipment, such as Internet of things sensors, and the Internet of things equipment collects data information of various Internet of things application programs from a physical environment and forwards task information to the MEC server unit to execute task calculation;
the MEC server unit comprises a plurality of MEC servers and is used for realizing a consensus mechanism of the block chain and realizing signing registration and task unloading of wireless network equipment according to an intelligent contract arranged in a block chain network on the MEC servers.
Further, the wireless network device unit completes the signing and registration of the internet of things device to the block chain network on the MEC server, and the registration authentication information is stored in the intelligent contract.
Further, the internet of things device in the wireless network device forwards data information collected from the physical device or uploaded by the user to an intelligent contract in the block chain network on the MEC server.
Furthermore, an intelligent contract in the MEC server unit selects a proper MEC server to unload the task according to the uploaded task information through a priority-based task unloading method.
Further, cross validation is carried out on the calculation result through an unloading result validation method based on the game theory in the intelligent contract. The intelligent closing date in the MEC server unit records the unloading process and the verification of the calculation result in the blockchain.
Further, the MEC server unit returns the calculation result to the internet of things device that sends the task and the data information.
The invention relates to a block chain-based edge computing task unloading method, which comprises the following steps:
s1, uploading unloading task information to an intelligent contract of a block chain through Internet of things equipment;
s2, selecting a proper edge server to unload the task according to the task information to be unloaded by using a priority-based unloading method;
preferably, assuming that the task set M uploaded by the wireless network device is {1, 2., M }, the task set may represent a set of different types of computing tasks, and is used to describe the set of different types of computing tasks
Attribute, u, describing the offload task iiIs the size of the data uploaded by task i, ciIndicating the number of CPU cycles required to complete task i, ziData size, t, representing output resultiRepresents the maximum delay tolerance of the computing task, which represents the maximum delay that the task can tolerate.
The tasks in the priority queue PQ are ordered from small to large according to the task delay tolerance, and the order of the tasks in the queue is dynamically adjusted according to the newly added task. And the tasks are put into a priority queue from the tail of the queue, and the first task of the queue is selected in the priority queue for unloading each time.
Preferably, the respective unloading time delay of each MEC server in the communication transmission range of the Internet of things equipment is calculated, the MEC server with the optimal time delay is selected for task unloading calculation, and the MEC server with the suboptimal time delay is selected for cross validation of the calculation result. And after calculating the task unloading result, the MEC server records all the results into an intelligent contract deployed in the block chain.
S3, cross-verifying the calculation result through an unloading result verification method based on a game theory in the intelligent contract;
preferably, reasonable trading rules are formulated by allowing two MEC servers to complete the same task, so that rational MEC servers cannot collude and cheat each other. In the absence of collusion, cross-validation can be performed by examining the results of two MECs. The method proves the game in the intelligent contract through the extended form of the game. The game algorithm is realized by deploying the game on an Etherhouse intelligent contract and combining transaction rules in the intelligent contract. Thus, the customer only needs to pay two MEC servers a fee and a small transaction fee to be able to use the smart contract to effect the calculation.
The game in the method represents the possible behaviors of the game party in the form of decision trees through the expansion form of the game so as to solve the Nash equilibrium. Here, the user is denoted as Client, and the uninstalling MEC server is denoted as U1Verify MEC server as U2The intelligent contract is specifically introduced as follows:
verifying that the intelligent contract initiator is a Client, and verifying the MEC server U2Sign up the contract, U2The task offload results need to be sent in the validation contract. When U is turned2And if dishonest behavior occurs, a third-party trusted authority is introduced to verify the unloading result. Suppose that when the task is completed, the user pays w for the MEC server, the computing cost required by the MEC server is c, the deposit for the task is d, and the verification cost of the third-party trusted authority is e.
Under normal reasonable circumstances, the following relationship is clearly true:
w ≧ c-the edge server gains revenue is typically greater than its computational cost.
e >2w the third party trusted authority computing task cost should be greater than the amount the user needs to pay to both MEC servers. Otherwise, the MEC server is not needed, and the client only uses the third-party trusted computer to calculate.
If an MEC server is dishonest, e is paid by the MEC server, and a honest customer need only hire two MEC servers plus the transaction cost of the intelligent contract. In addition, to satisfy the ideal nash balance, a constraint d > c + e is set in the smart contract.
Preferably, any node of the extended game tree represents each possible state in the game play. The game starts from the only initial node, the path to the terminal node is determined by the action taken by the MEC server, and the game is ended, and the MEC server obtains corresponding benefits. Each non-terminal node only belongs to one MEC server participating in the game; the participant selects his or her availability at the nodeActions that are possible, each possible action passing from node to another node through an edge. The game is divided into three stages in total: first stage, verifying MEC server U2Discovery of U1A dishonest behavior occurs, which may or may not be optionally reported; second stage, by uninstalling MEC server U1Selecting actions, uninstalling MEC server U1The correct calculation result or the wrong calculation result is returned; the third stage, by verifying the MEC server U2Selecting action, verifying MEC server U2Either the correct calculation or the incorrect calculation is selected to be returned. According to the transaction rule in the intelligent contract, the MEC servers of both parties obtain the highest income for the MEC servers, U2Will choose not to report and U1And U2The correct calculation results are returned, the final nash equilibrium point is reached, and the gains of w-c are all obtained.
And S4, verifying and recording the unloading process and the calculation result in a block chain, and transmitting the result of task unloading back to the wireless network equipment unit.
The technical scheme of the invention for high-reliability task unloading based on the block chain in the edge calculation has the following advantages: by using a block chain technology, the traceability and visibility of the unloading process of the edge computing task are ensured; by the task unloading method based on the priority, the requirements of users with different time delay requirements on the current unloading task can be more reasonably met; by the aid of the unloading result verification method based on the game theory, correctness and verifiability of calculation results are guaranteed, and accordingly quality of task unloading is improved.
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Fig. 1 is an edge computing task offloading system based on a block chain according to an embodiment.
Fig. 2 is a schematic flowchart of offloading an edge computation task based on a block chain according to an embodiment.
Fig. 3 is a schematic diagram of an offload result verification method based on an extended game tree according to an embodiment.
Detailed Description
The embodiments of the present invention are illustrated below by specific examples. Other advantages and effects of the present invention will be readily apparent to those skilled in the art from the disclosure herein. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It is to be noted that the features in the following embodiments and examples may be combined with each other without conflict.
The invention provides a high-reliability task unloading system and method based on a block chain in edge calculation, aiming at the defects of task unloading in the existing edge calculation.
Example one
The embodiment provides a block chain-based edge computing task offloading system, as shown in fig. 1, which includes a wireless network device unit and an MEC server unit. The wireless network equipment unit consists of a plurality of pieces of Internet of things equipment; the MEC server unit is composed of a plurality of MEC servers.
And the wireless network equipment unit is used for acquiring task data collected by the user and the wireless sensor and forwarding the data information to the MEC server unit with the intelligent contract.
The wireless network equipment is composed of internet of things equipment, comprises various sensors, cameras and the like, is usually deployed in various different environments such as intelligent automobiles, buildings, streets and the like, and the data of the wireless network equipment is mainly used in system applications such as intelligent transportation and intelligent buildings. Since the wireless network device itself has limited computing power and cannot process the collected data, the task needs to be offloaded to a device with computing power and a certain storage space.
And the MEC server unit is used for realizing a consensus mechanism of the block chain and realizing signing registration and task unloading of the wireless network equipment according to an intelligent contract arranged in the block chain network on the MEC server. The block chain network consists of a plurality of MEC servers, and block chain link points are deployed on a plurality of edge computing nodes.
Edge server nodes can be divided into proxy MEC server nodes and normal MEC server nodes. The proxy MEC server node is responsible for deploying intelligent contracts in the blockchain network and adding or changing unloading strategies. And the common MEC server unit is responsible for calculating and processing the task information and data uploaded by the wireless network equipment unit.
Each MEC server can interact with the intelligent contracts on the block chain, the tasks are unloaded to the MEC servers through the unloading strategies in the intelligent contracts, and the processes from the selection of the proper MEC server to the task unloading process and the verification of the calculation result are recorded in the block chain so as to facilitate tracing.
Fig. 2 shows a flow chart of highly reliable task offloading based on block chains according to this embodiment, which includes the following steps:
s1, uploading unloading task information to an intelligent contract of a block chain through Internet of things equipment.
And S2, selecting a proper edge server to unload the task according to the task information needing to be unloaded by using a priority-based unloading method.
And S3, cross-verifying the calculation result by an unloading result verification method based on a game theory in the intelligent contract.
And S4, verifying and recording the unloading process and the calculation result in a block chain, and transmitting the result of task unloading back to the wireless network equipment unit.
In step S1, the wireless network device unit uploads the task information to the intelligent contract in the block chain network, records the task upload information in the block chain, and selects the MEC server for task offloading according to the task offloading policy in the intelligent contract.
In step S2, in the unloading system, an unloading policy method plays a key role. Currently, the design of conventional task offloading strategies takes into account a number of factors, such as: the distance between the internet of things equipment and the MEC server, the rental cost of the MEC server, the computing capacity of the MEC server and the like. In this embodiment, a method for offloading based on priority in a blockchain is proposed, specifically, assuming that a task set M ═ 1, 2.. multidata, M } uploaded by a wireless network device, the task set may represent a set of different types of computing tasks,by using
Attribute, u, describing the offload task iiIs the size of the data uploaded by task i, ciIndicating the number of CPU cycles required to complete task i, ziData size, t, representing output resultiRepresents the maximum delay tolerance of the computing task, which represents the maximum delay that the task can tolerate.
In this model, Frequency Division Multiple Access (FDMA) is used for data communication between the wireless network device and the MEC server. FDMA techniques allow multiple users to transmit data over a single communication channel (e.g., coaxial cable or microwave beam) by dividing the bandwidth of the channel into separate non-overlapping frequency subchannels and assigning each subchannel to a separate user. A user may transmit data by modulating it on a carrier at the frequency of the subchannel. In FDMA technology, a channel is divided into a number of sub-channels, each of which has a radio propagation rate that is related to the channel bandwidth and channel fading. Therefore, the task offload transmission rate between network devices in each layer in the model can be expressed as:
wherein HiFor channel gain, Hi＝di r，diDistance from invalid network equipment unit to MEC server unit, r is channel fading factor, BiIs the bandwidth of the channel, PiIs the average power of the signal transmitted by the channel, and N is the power of gaussian noise inside the channel.
Depending on the task offload transfer rate, the task data transfer delay may be expressed as
Task computation latency for task offloading to MEC server can be expressed as
Wherein f isiThe frequency is calculated for the MEC server.
The MEC server returning the task computation results back to the delay may be expressed as
The total time delay for task offloading can be expressed as
t＝t1+t2+t3
The tasks in the priority queue PQ are ordered from small to large according to the task delay tolerance, and the order of the tasks in the queue is dynamically adjusted according to the newly added task. And the tasks are put into a priority queue from the tail of the queue, and the first task of the queue is selected in the priority queue for unloading each time.
Further, by calculating the respective unloading time delay of each MEC server in the communication transmission range of the Internet of things equipment, the MEC server with the optimal time delay is selected for task unloading calculation, and the MEC server with the suboptimal time delay is selected for cross validation of the calculation result. And after calculating the task unloading result, the MEC server records all the results into an intelligent contract deployed in the block chain.
In step S3, the calculation result is cross-verified by the unloading result verification method based on the game theory in the intelligent contract. Specifically, two MEC servers complete the same task, and reasonable transaction rules are formulated, so that the rational MEC servers cannot mutually collude and cheat. In the absence of collusion, cross-validation can be performed by examining the results of two MECs. The method proves the game in the intelligent contract through the extended form of the game. The game algorithm is realized by deploying the game on an Etherhouse intelligent contract and combining transaction rules in the intelligent contract. Thus, the customer only needs to pay two MEC servers a fee and a small transaction fee to be able to use the smart contract to effect the calculation.
The following are the currency variables that the present example will use in contracts, all of which are non-negative.
c, the cost of the computing task of the edge server.
And e, calling a third-party trusted authority to recalculate tasks and solve the dispute of the cost of the problem.
The margin calculates the deposit that needs to be paid to the customer in order to obtain the job.
w the amount the customer agrees to pay the edge server to complete the task.
Under normal reasonable circumstances, the following relationship is clearly true:
w ≧ c-the edge server gains revenue is typically greater than its computational cost.
e >2w the third party trusted authority computing task cost should be greater than the amount the user needs to pay to both MEC servers. Otherwise, the MEC server is not needed, and the client only uses the third-party trusted computer to calculate.
If dishonest behaviour occurs, e is paid by the MEC server and a honest customer only needs to hire two MEC servers plus the transaction cost of the intelligent contract. In addition, to satisfy the ideal nash balance, a constraint d > c + e is set in the smart contract.
The main difficulty in designing the game algorithm is how to avoid the counter-act. The client may make collusion less desirable by providing extra rewards to the honest MEC servers and altering the balance. However, once the dishonest MEC servers know what the customer has provided, they can create a reverse contract to check out that it is again true. This cycle can be continued endlessly.
To escape the cycle, the gaming method does not counter collusion, but instead incentivizes the MEC server to report collusion. If the intelligent contract is signed and collusion is reported, a third-party trusted authority intervenes to verify the calculation result and decide who cheats. Reverse contracts are meaningless because once a third party trusted authority is involved, the revenue of one MEC server depends only on whether it is dishonest, and not on the behavior of another MEC server.
According to the unloading result verification method based on the game theory, the game expresses the possible behaviors of game parties in the form of decision trees through the expansion form of the game so as to solve Nash equilibrium. The user is marked as Client, and the uninstalling MEC server is marked as U1Verify MEC server as U2.. The intelligent contract is specifically introduced as follows:
verifying that the intelligent contract initiator is a Client and the MEC server U needs to be verified2Sign up the contract, U2The task offload results need to be sent in the validation contract. When U is turned2And if dishonest behavior occurs, a third-party trusted authority is introduced to verify the unloading result.
Fig. 3 is a schematic diagram illustrating an unloading result verification method based on an extended game tree according to this embodiment.
Specifically, any node V of the extended game tree in the figure represents each possible state in the game play. Game is from a unique initial node V0Initially, the MEC server takes action to determine the path to the terminal node, and at the end of the game, the MEC server receives the corresponding benefit. Each non-terminal node only belongs to one MEC server participating in the game; the participant selects its possible actions at the node, each possible action passing from node to node through an edge. The game is divided into three stages in total: a first stage when the game is in V0State, validation MEC Server U2Discovery of U1A dishonest behavior occurs, which may or may not be optionally reported; second stage, by uninstalling MEC server U1Selecting actions, uninstalling MEC server U1The correct calculation result or the wrong calculation result is returned; the third stage, by verifying the MEC server U2Selecting action, verifying MEC server U2Either the correct calculation or the incorrect calculation is selected to be returned. According to the transaction rule in the intelligent contract, the MEC servers of both parties obtain the highest income for the MEC servers, U2Will choose not to report and U1And U2All canAnd returning a correct calculation result, reaching a final Nash equilibrium point and obtaining the benefits of w-c.
In summary, the invention provides a high-reliability task unloading system and method based on a block chain. The system architecture of the present invention supports mobility of networked devices and allows them to join or leave at any time; by using a block chain technology, the traceability and visibility of the unloading process of the edge computing task are ensured; by the task unloading method based on the priority, the requirements of users with different time delay requirements on the current unloading task can be more reasonably met; by the aid of the unloading result verification method based on the game theory, correctness and verifiability of calculation results and quality of unloading tasks are guaranteed.
The embodiments described above are merely preferred possible embodiments of the invention and the technical principles applied. Those skilled in the art may supplement or modify the embodiments described or substitute them in other ways. Therefore, although the objects and advantages of the present invention will be made clearer by the above embodiments, the present invention is not limited to the above embodiments only. The description is not to be construed as limiting the invention.
Claims (10)
1. An edge computing task unloading system based on a block chain is characterized by comprising the following units:
the wireless network equipment unit comprises a plurality of Internet of things equipment connected through a network, and the Internet of things equipment collects data information of an Internet of things application program from a physical environment and forwards the data information to the MEC server unit to execute task calculation;
and the MEC server unit comprises a plurality of MEC servers and is used for realizing a consensus mechanism of the block chain and realizing signing registration and task unloading of the wireless network equipment according to an intelligent contract arranged in a block chain network on the MEC servers.
2. The system of claim 1, wherein the wireless network device unit performs the signing of the IOT device to the blockchain network registered to the MEC server, and the registration authentication information is stored in the intelligent contract.
3. The blockchain-based edge computing task offloading system of claim 1 or 2, wherein the internet of things device in the wireless network device forwards data information collected from the physical device or uploaded by the user to the intelligent contract in the blockchain network on the MEC server.
4. The system of claim 1, wherein an intelligent contract in the MEC server unit selects an appropriate MEC server for task offloading through a priority-based task offloading method according to the uploaded task information.
5. The system for unloading edge calculation tasks based on the blockchain as claimed in claim 1, wherein the calculation results are cross-verified by an unloading result verification method based on a game theory in an intelligent contract; the intelligent contracts in the MEC server units record the offload process and the computation result validation in the blockchain.
6. The blockchain-based edge computing task offloading system of any one of claims 1,2, or 4-5 wherein the MEC server unit returns the results of the computation to the IOT device sending the task and data information.
7. An edge calculation task unloading method based on a block chain is characterized by comprising the following steps:
s1, uploading unloading task information to a block chain intelligent contract of an MEC server unit through Internet of things equipment of a wireless network equipment unit;
s2, selecting a proper edge server to unload the task according to the task information to be unloaded by using a priority-based unloading method;
s3, cross-verifying the calculation result through an unloading result verification method based on a game theory in the intelligent contract;
and S4, verifying and recording the unloading process and the calculation result in a block chain, and transmitting the result of task unloading back to the wireless network equipment unit.
8. The method for offloading an edge computation task based on a block chain as claimed in claim 7, wherein the step S2 is as follows: assume that a task set M ═ 1, 2.., M } uploaded by a wireless network device represents a set of different types of computing tasks, as used herein
Attribute, u, describing the offload task iiIs the size of the data uploaded by task i, ciIndicating the number of CPU cycles required to complete task i, ziData size, t, representing output resultiRepresenting a maximum delay tolerance of the computing task, the maximum delay tolerance representing a maximum delay that the task can tolerate;
the tasks in the priority queue PQ are sorted from small to large according to the task delay tolerance, and the sequence of the tasks in the queue can be dynamically adjusted according to the newly added tasks; and the tasks are put into a priority queue from the tail of the queue, and the first task of the queue is selected in the priority queue for unloading each time.
9. The method according to claim 8, wherein step S2 further comprises calculating respective unloading delays of each MEC server in the communication transmission range of the internet of things device, selecting an MEC server with the best delay for task unloading calculation, and selecting a MEC server with the suboptimal delay for cross-validation of the calculation results; and after calculating the task unloading result, the MEC server records all the results into an intelligent contract deployed in the block chain.
10. The method for offloading task of edge computation based on block chain according to any of claims 7-9, wherein step S3 is embodied as follows: by enabling the two MEC servers to complete the same task and formulating reasonable transaction rules, the rational MEC servers can not mutually collude and cheat; cross-validation was performed by examining the results of two MECs without collusion;
the game represents the possible behaviors of the game party in the form of a decision tree through the expansion form of the game so as to solve the Nash equilibrium; the user is marked as Client, and the uninstalling MEC server is marked as U1Verify MEC server as U2The method comprises the following steps:
verifying that the intelligent contract initiator is a Client, and verifying the MEC server U2Sign up the contract, U2A task offload result needs to be sent in the validation contract; when U is turned2Reporting the occurrence of dishonest behaviors, and introducing a third-party trusted institution to verify the unloading result; supposing that when the task is completed, the expenditure of the MEC server required to be paid by the user is w, the calculation cost required by the MEC server is c, the deposit of the task is obtained as d, and the verification cost of the third-party trusted authority is e;
under normal and reasonable circumstances, the following relationship holds:
w is more than or equal to c, the income obtained by the edge server is generally more than the calculation cost;
e >2w, the calculation task cost of the third-party trusted authority is more than the amount of money required by the user to pay the two MEC servers; otherwise, the client only uses the third-party trusted computer to calculate without using the MEC server;
if the MEC server has dishonest behavior, e is paid by the MEC server, and one honest customer only needs to hire two MEC servers and add the transaction cost of the intelligent contract; in addition, in order to meet the ideal Nash equilibrium, a limiting condition d > c + e is set in the intelligent contract;
any node of the expanded game tree represents each possible state in the game play; the game starts from the only initial node, the path reaching the terminal node is determined through the action taken by the MEC server, and the game is ended, and the MEC server obtains corresponding income; each non-terminal node only belongs to one MEC server participating in the game; participants select their possible actions at the node, each possible action passing from node to another through an edgeA node; the game is divided into three stages in total: first stage, verifying MEC server U2Discovery of U1A dishonest behavior occurs, and the behavior is selected to be reported or not reported; second stage, by uninstalling MEC server U1Selecting actions, uninstalling MEC server U1The correct calculation result or the wrong calculation result is returned; the third stage, by verifying the MEC server U2Selecting action, verifying MEC server U2The correct calculation result or the wrong calculation result is returned; according to the transaction rule in the intelligent contract, the MEC servers of both parties obtain the highest income for the MEC servers, U2Will choose not to report and U1And U2The correct calculation results are returned, the final nash equilibrium point is reached, and the gains of w-c are all obtained.
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