Simulation method for asymmetric information multi-ship cooperative defense based on bayesian stackelberg game

Abstract

The application discloses a kind of asymmetric information multi-boat cooperative defense simulation methods based on bayesian Stackelberg game, for the asymmetric information scene formed by the private type information of observed boat in multi-unmanned boat cooperative defense, constructs the bayesian Stackelberg game model with observed boat as leader, observer boat as follower, define type space, strategy space and benefit function;Through online bayesian updating mechanism, the private type of observed boat is dynamically inferred, and the information disadvantage of observer boat is compensated;Based on posterior belief, solve bayesian Stackelberg equilibrium, generate the distributed cooperative defense strategy of observer boat.The application realizes the integrated intelligent process from intention recognition to cooperative decision, can effectively handle the information asymmetry problem in dynamic interaction, has good real-time, adaptability and expansibility, is suitable for multi-unmanned boat cooperative scene such as regional patrol, cooperative task execution.

Description

Technical Field

[0001] This invention relates to the field of dynamic game decision-making and unmanned system cooperative control technology, and in particular to an asymmetric information multi-vehicle cooperative defense simulation method based on Bayesian Stackelberg game. Background Technology

[0002] With the increasing prevalence of unmanned surface vessels (USVs), multi-USV cooperative systems are widely used in various dynamic tasks due to their flexibility, robustness, and high efficiency. Building USV systems with real-time perception, intent understanding, and collaborative decision-making capabilities is crucial for accomplishing complex dynamic tasks. Unlike research under conditions of complete information, in real-world multi-USV cooperative defense scenarios, the observed vessel typically possesses private information unknown to the observer vessel (such as true intent and action preferences), creating an information asymmetry advantage that makes cooperative defense by the observer vessel more challenging.

[0003] Game theory-based methods model both the observed and defending vessels as rational decision-makers, formally describing the strategic interaction process in collaborative defense scenarios and providing a theoretical framework for constructing adaptive interactive systems. Among numerous game models, Stackelberg games, with their "leader-follower" sequential decision-making structure, closely align with the logic of "observation vessels" making decisions based on observations. However, traditional Stackelberg game research suffers from two shortcomings: first, it often presupposes the observation vessel (defender) as the game leader, which contradicts the reality that the observed vessel typically has the first move in decision-making; second, it largely relies on the assumption of complete information, failing to characterize the uncertainty arising from the opponent's private information.

[0004] The aforementioned shortcomings make it difficult for existing methods to learn the camouflage type of the observed vessel online, and also prevent them from supporting collaborative decision-making by the observation vessel under conditions of incomplete information. Summary of the Invention

[0005] This invention discloses an asymmetric information multi-ship cooperative defense simulation method based on Bayesian Stackelberg game to overcome the above-mentioned technical problems.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows: A simulation method for multi-vessel cooperative defense based on Bayesian Stackelberg game with asymmetric information includes the following steps: S1: Construct a Bayesian Stackelberg game model, model the observed boat as the leader of the game with private type information, and model the observer boat as the follower in the game. At the same time, establish the private type space of the observed boat, the strategy space of the observed boat, the strategy space of the observer boat, the payoff function of the observed boat, the payoff function of the observer boat, and initialize the prior beliefs of the observed boat based on the Bayesian Stackelberg game model. S2: Based on the Bayesian Stackelberg game model, obtain the posterior concept of the private type of the observed vessel; S3: Based on the posterior concept of the private type of the observed vessel and the Bayesian Stackelberg game model, obtain the joint strategy that minimizes the expected cost of the observed vessel; S4: The observation vessel executes the joint strategy that minimizes the expected cost of the observation vessel. If the interception of the observed vessel is successful or the observed vessel leaves the protected area's warning range, the simulation of multi-vessel cooperative defense is completed; otherwise, S2-S3 are re-executed until the interception of the observed vessel is successful or the observed vessel leaves the protected area's warning range.

[0007] Furthermore, the payoff function of the observed vessel is expressed as follows: ; In the formula: Let be the payoff function of the observed vessel; Indicates the strategy of the observed vessel; This indicates a joint strategy involving all observation vessels; The private type of the observed vessel; An index for the private type of the observed vessel; To get closer to the profit; The cost of interception; For the cost of maneuver; in, ; In the formula: for The weights; The shortest distance that the observed vessel is expected to reach the mission area; This is the reference distance used for normalization; ; ; In the formula: for Risk aversion weight; For the purpose of indexing observation vessels; The total number of observation boats; Risk sensitivity coefficient; For an exponentially decaying function, when The smaller the value, the higher the risk and the easier it is to be intercepted; For the first The time difference between the arrival of the observation vessel and the observed vessel at the predicted rendezvous point; For the first The expected meeting point and distance between the observation vessel and the observed vessel; For the first The speed of the observation boat; The current speed of the observed vessel; ; In the formula: This is the mobility cost coefficient; For the observed vessel at all times Selected heading angle; For the observed vessel at all times Selected heading angle; It is an absolute value.

[0008] Furthermore, the observation vessel's profit function is as follows: ; In the formula: This indicates a measurement of the strategy adopted by the observed vessel. The observation vessels adopted a joint strategy. And the private type of the observed vessel is At that time, the total benefits paid by the observation vessel; Let be the payoff function of the observed vessel; To counteract the weighting coefficient; For the cost of collaboration; This comes at the cost of fuel / energy consumption. in, ; In the formula: The weighting coefficient for the cost of collaboration; Indicates standard deviation; For the first The predicted rendezvous distance between the observation vessel and the observed vessel; For the first The speed of the observation boat; The set of predicted interception times for all observation vessels; ; In the formula: This is the fuel cost weighting coefficient; For the first The square of the speed of the observation vessel.

[0009] Furthermore, the formula used to obtain the posterior concept of the private type of the observed vessel is as follows: ; In the formula: The posterior concept and posterior belief representing the private type of the observed vessel, i.e., the moment... The private type of the observed vessel is The probability of; It is the likelihood function; In order to be in Data observed at all times; For the private type of the observed vessel, the posterior concept and a prior belief, that is, at time... The private type of the observed vessel is The probability of; in, ; ; In the formula: This is a normalization constant; For sensitivity parameters, ; This is a difference measurement function; For at any time The strategy adopted by the observed vessel; The theoretically optimal strategy for the observed vessel; for The payoff function of the constantly observed vessel; The strategic space for the observed vessel; The parameter is the argument when the function reaches its maximum value.

[0010] Furthermore, the formula used to obtain the joint strategy that minimizes the expected cost of the observation vessel is as follows: ; ; In the formula: For at any time The joint strategy that minimizes the expected cost of the observation vessel; This provides strategic space for observation vessels; To and Calculate the expected cost of the observation vessel; This indicates a measurement of the strategy adopted by the observed vessel. The observation vessels adopted a joint strategy. And the private type of the observed vessel is At that time, the total benefits paid by the observation vessel; For private type The observed vessel adopts a joint strategy when facing the observation vessel. Theoretically optimal strategy at that time; This represents the expression that maximizes the value of the variable. .

[0011] Beneficial Effects: This invention provides a simulation method for multi-unmanned surface vessel (USV) collaborative defense based on Bayesian Stackelberg game theory. Addressing the asymmetric information scenario in multi-USV collaborative defense where observed USVs possess private information types (such as true intentions and action preferences), this method constructs a Bayesian Stackelberg game model with the observed USV as the leader. It establishes the payoff functions of the observed USV and the observer USV based on the Bayesian Stackelberg game, dynamically inferring the private information types of the observed USV through an online Bayesian update mechanism to compensate for the observer USV's information disadvantage. Based on posterior beliefs, it solves the Bayesian Stackelberg equilibrium to generate a distributed collaborative defense strategy for the observer USVs, achieving an integrated intelligent process from intention recognition to collaborative decision-making. This method effectively handles information asymmetry issues in dynamic interactions and possesses good real-time performance, adaptability, and scalability. Attached Figure Description

[0012] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0013] Figure 1 This is a flowchart of the asymmetric information multi-ship cooperative defense simulation method based on Bayesian Stackelberg game of the present invention; Figure 2 This is the initial situation diagram for the multi-ship cooperative defense simulation in this embodiment of the invention; Figure 3 This is a process situation diagram of multi-ship cooperative defense simulation in an embodiment of the present invention; Figure 4 This is the final situation diagram of the multi-ship cooperative defense simulation in this embodiment of the invention. Detailed Implementation

[0014] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0015] This embodiment introduces an asymmetric information multi-vessel cooperative defense simulation method based on Bayesian Stackelberg game, including the following steps: Figure 1 As shown: This embodiment is applicable to scenarios where a collaborative swarm of multiple unmanned surface vessels (USVs) performs real-time intent inference and collaborative decision-making against another USV swarm with an information advantage. For example... Figure 1 As shown, the core lies in addressing the information disadvantage of observation vessels through an iterative cycle of "observation-inference-decision-execution".

[0016] S1: Construct a Bayesian Stackelberg game model, model the observed boat as the leader of the game with private type information, model the observer boat as the follower in the game, and establish the private type space of the observed boat, the strategy space of the observed boat, the strategy space of the observer boat, the payoff function of the observed boat, the payoff function of the observer boat, and initialize the prior beliefs of the observed boat. Specifically, S1 includes: S11: Define the private type space of the observed vessel as follows: ; In the formula: A private type space set for the observed vessel; The private type of the observed vessel; The total number of private types of the observed submarines; An index for the private type of the observed vessel; S12: Define the strategy space of the observed vessel: ; In the formula: Indicates the strategy of the observed vessel; The strategy space of the observed vessel is the set of all possible actions. S13: Define the strategy space of the observation vessel: ; In the formula: This indicates a joint strategy involving all observation vessels; The total number of observation boats; Indicates the first The strategy of a single observation vessel; The strategy space of the observation vessel is the set of all possible joint strategies of the observation vessels. For the purpose of indexing observation vessels; This is a transpose.

[0017] S14: Define the payoff function for the observed vessel: ; In the formula: Let be the payoff function of the observed vessel; Indicates the strategy of the observed vessel; This indicates a joint strategy involving all observation vessels; The private type of the observed vessel; An index for the private type of the observed vessel; To get closer to the profit; The cost of interception; For the cost of maneuver; Among them, proximity gain measures the benefits that the observed vessel can obtain by approaching the target area, specifically: ; In the formula: for The weights; The shortest distance that the observed vessel is expected to reach the mission area (in this embodiment, the mission area of ​​the observed vessel is a prohibited area); This is the reference distance used for normalization; The cost of interception measures the penalty incurred by the observed vessel for being intercepted, specifically: ; ; In the formula: for Risk aversion weight; For the purpose of indexing observation vessels; The total number of observation boats; Risk sensitivity coefficient; For an exponentially decaying function, when The smaller the value, the higher the risk and the easier it is to be intercepted; For the first The time difference between the arrival of the observation vessel and the observed vessel at the predicted rendezvous point; For the first The expected meeting point and distance between the observation vessel and the observed vessel; For the first The speed of the observation boat; The current speed of the observed vessel; The maneuver cost measures the penalty incurred by the observed vessel for changing course, specifically: ; In the formula: This is the mobility cost coefficient; For the observed vessel at all times Selected heading angle; For the observed vessel at all times Selected heading angle; It is an absolute value.

[0018] S15: Define the observation boat's revenue function: ; In the formula: This indicates a measurement of the strategy adopted by the observed vessel. The observation vessels adopted a joint strategy. And the private type of the observed vessel is At that time, the total benefits paid by the observation vessel; Let be the payoff function of the observed vessel; To counteract the weighting coefficient; For the cost of collaboration; This comes at the cost of fuel / energy consumption. Among them, the coordination cost measures the degree of cooperation between observation vessels, specifically: ; In the formula: The weighting coefficient for the cost of collaboration; The standard deviation is used to calculate the “dispersion” of the predicted interception times for all observation vessels. For the first The predicted rendezvous distance between the observation vessel and the observed vessel; For the first The speed of the observation boat; A set of predicted interception times for all observation vessels.

[0019] Fuel cost measures the energy required for the observation vessel to execute its current strategy, specifically: ; In the formula: This is the fuel cost weighting coefficient; For the first The square of the speed of the observation vessel; S16: Initialize prior beliefs: In this embodiment, prior information refers to the observer's initial judgment or guess about the type of the observed vessel before any observational evidence is available. This occurs at the initial moment of the game. When there is no prior information about the observed vessel, it is assumed that the prior belief about the type of the observed vessel is uniformly distributed, that is: ; In the formula: The total number of private types of the observed submarines; The initial a priori belief, that is, the initial private type of the observed vessel, is: The probability of.

[0020] S2: Based on the Bayesian Stackelberg game model, obtain the posterior concept of the private type of the observed vessel; This step is based on the game theory model constructed using S1, combined with the observed historical strategies of the observed submarine at the current moment. Historical strategies of observation boats Dynamically update beliefs. At every decision-making moment. The observation vessel updates its posterior belief on the type of the observed vessel based on Bayes' theorem: the private type of the observed vessel is dynamically inferred through Bayesian updates based on real-time interactive observations.

[0021] ; In the formula: The posterior concept and posterior belief representing the private type of the observed vessel, i.e., the moment... The private type of the observed vessel is The probability of; Let be the likelihood function, representing the likelihood when the actual private type of the observed vessel is . At that time, its actions were observed. The possibility; In order to be in The data observed in real time includes the speed and heading angle of the observed vessel; For the private type of the observed vessel, the posterior concept and a prior belief, that is, at time... The private type of the observed vessel is The probability of; Specifically, the likelihood function is based on the payoff function of the observed vessel. Construction based on the principle of rational decision-making: A rational observed vessel, if its type is The strategy at the moment of facing the observation boat When choosing a function, one should tend to select the one that maximizes its return. Optimal strategy Therefore, the likelihood function is defined as: ; ; In the formula: This is a normalization constant; For sensitivity parameters, ; This is a difference metric function used to measure the gap between two actions; For at any time The strategy adopted by the observed vessel; The theoretically optimal strategy for the observed vessel; for The payoff function of the constantly observed vessel; The strategic space for the observed vessel; The parameter is the argument when the function reaches its maximum value.

[0022] S3: Based on the posterior concept of the private type of the observed vessel and the Bayesian Stackelberg game model, obtain the joint strategy that minimizes the expected cost of the observed vessel, i.e., the cooperative defense strategy of the observed vessel. Specifically, posterior beliefs based on S2 output The Bayesian Stackelberg game model is solved to generate a cooperative defense strategy for the observation vessel. The formula used is as follows: ; ; In the formula: For at any time The joint strategy that minimizes the expected cost of the observation vessel; This provides strategic space for observation vessels; To and Calculate the expected cost of the observation vessel; To measure the strategy adopted by the observed vessel The observation vessels adopted a joint strategy. And the private type of the observed vessel is At that time, the total benefits paid by the observation vessel; For private type The observed vessel adopts a joint strategy when facing the observation vessel. Theoretically optimal strategy at that time; S4: The observation vessel executes the joint strategy that minimizes the expected cost of the observation vessel. If the interception of the observed vessel is successful or the observed vessel leaves the protected area's warning range, the simulation of multi-vessel cooperative defense is completed; otherwise, S2-S3 are re-executed until the interception of the observed vessel is successful or the observed vessel leaves the protected area's warning range.

[0023] Specifically, the observation vessel follows the optimal cooperative strategy obtained from solution S3. The observer will perform corresponding actions, including but not limited to issuing warnings to the observed vessel, adjusting its course for surveillance, or conducting coordinated interception. After completion, the observer will update the current maritime situation information, including the shortest distance the observed vessel is expected to reach the mission area and the expected meeting point between the two vessels. The observer will also record the observed vessel's response behavior and mission progress during the interaction to provide a basis for the next round of decision-making.

[0024] Specifically, the observation vessel determines whether the current collaborative defense mission meets the termination conditions: if the observed vessel has left the protected area's alert range, or has been successfully intercepted and lost its threat capability, the collaborative defense mission is considered to have ended, and the final defense result is output; otherwise, it returns to S2 and enters the next round of the "observation-inference-decision-execution" iterative loop based on the updated situation information, until the termination conditions are met. The determination of whether the observed vessel has been successfully intercepted is existing technology in this field, and this implementation does not elaborate on the determination method.

[0025] To verify the effectiveness of this invention, the following description is provided in conjunction with specific embodiments and accompanying drawings. The simulation experiment was conducted on the MATLAB platform, simulating a dynamic area patrol mission.

[0026] Example: Multi-ship collaborative defense simulation with three types of private vessels (including malicious vessels, probe vessels, and benevolent vessels) under observation.

[0027] Initial settings are as follows Figure 2 At the initial moment, eight observation vessels are evenly distributed around the perimeter of the protected area, while four vessels to be observed (consisting of one malicious vessel, two probing vessels, and one benevolent vessel) approach the mission area from different directions. The prior beliefs of the observation vessels about the private types of the vessels to be observed are uniformly distributed.

[0028] The process and results are as follows: Figure 3 and Figure 4 As shown: Different types of observed vessels initially adopt an approach strategy. The observer vessel, by observing the behavior of the observed vessels and updating its belief in them based on Bayesian rules, issues a warning. Among them, the benevolent observed vessel A1 (purple) will stop approaching the protected area after receiving the warning; the probing observed vessel A4 (yellow) will hesitate after receiving the warning, choosing its next course of action depending on the situation; the malicious observed vessels A2 and A3 (red) will ignore the warning with a high probability and quickly head towards the mission area. Based on this belief, the solved Bayesian Stackelberg equilibrium strategy guides the observer vessels to collaboratively intercept the observed vessels that ignore the warning and head towards the protected area. Specifically, the malicious observed vessel A3 is collaboratively intercepted by observer vessels D7 and D8, and the malicious observed vessel A2 is collaboratively intercepted by observer vessels D1 and D2. Ultimately, the encirclement and interception are completed before the observed vessels approach the protected area. Simulation results show that this embodiment enables the observation fleet to accurately infer the true intentions of the observed vessels under conditions of incomplete information and generate an effective cooperative interception strategy, demonstrating good engineering applicability.

[0029] The asymmetric information collaborative defense simulation method based on Bayesian Stackelberg game provided in this embodiment has the following beneficial effects: 1. The model closely reflects the essence of sequential interaction: A Bayesian Stackelberg game framework with the observed boat as the leader is constructed, which accurately describes the interaction logic of "first move - then response" under information asymmetry.

[0030] 2. Possesses information disadvantage compensation capability: Dynamically infers the private type information of the observed vessel through an online Bayesian update mechanism, effectively compensating for the information disadvantage of the observation vessel. The update mechanism is clear, efficient, and suitable for real-time applications.

[0031] 3. High scalability and robustness: The distributed collaborative decision-making mechanism avoids single points of failure, supports flexible expansion of the number of observation vessels, and exhibits good robustness in scenarios with multiple interactive objects and complex environments.

[0032] In summary, this implementation effectively addresses the information asymmetry problem in dynamic interactions, possessing excellent real-time performance, adaptability, and scalability. It is suitable for dynamic scenarios requiring multiple observation vessels to infer and collaboratively manage the real-time behavioral intentions of an observed vessel possessing private information, and for practical mission scenarios requiring multiple unmanned surface vessels to collaboratively address uncertain interactions. This invention is applicable to collaborative scenarios such as dynamic area patrols and cooperative mission execution.

[0033] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A simulation method for multi-ship cooperative defense based on Bayesian Stackelberg game with asymmetric information, characterized in that, Includes the following steps: S1: Construct a Bayesian Stackelberg game model, model the observed boat as the leader of the game with private type information, and model the observer boat as the follower in the game. At the same time, establish the private type space of the observed boat, the strategy space of the observed boat, the strategy space of the observer boat, the payoff function of the observed boat, the payoff function of the observer boat, and initialize the prior beliefs of the observed boat based on the Bayesian Stackelberg game model. S2: Based on the Bayesian Stackelberg game model, obtain the posterior concept of the private type of the observed vessel; S3: Based on the posterior concept of the private type of the observed vessel and the Bayesian Stackelberg game model, obtain the joint strategy that minimizes the expected cost of the observed vessel; S4: The observation vessel executes the joint strategy that minimizes the expected cost of the observation vessel. When the interception of the observed vessel is successful or the observed vessel leaves the protected area's warning range, the simulation of multi-vessel cooperative defense is completed. Otherwise, repeat S2-S3 until the observed vessel is successfully intercepted or leaves the protected area's warning range.

2. The asymmetric information multi-ship cooperative defense simulation method based on Bayesian Stackelberg game as described in claim 1, characterized in that, The payoff function of the observed vessel is expressed as follows: In the formula: Let be the payoff function of the observed vessel; Indicates the strategy of the observed vessel; This indicates a joint strategy involving all observation vessels; The private type of the observed vessel; An index for the private type of the observed vessel; To get closer to the profit; The cost of interception; For the cost of maneuver; in, In the formula: for The weights; The shortest distance that the observed vessel is expected to reach the mission area; This is the reference distance used for normalization; In the formula: for Risk aversion weight; For the purpose of indexing observation vessels; The total number of observation boats; Risk sensitivity coefficient; For an exponentially decaying function, when The smaller the value, the higher the risk and the easier it is to be intercepted; For the first The time difference between the arrival of the observation vessel and the observed vessel at the predicted rendezvous point; For the first The expected meeting point and distance between the observation vessel and the observed vessel; For the first The speed of the observation boat; The current speed of the observed vessel; In the formula: This is the mobility cost coefficient; For the observed vessel at all times Selected heading angle; For the observed vessel at all times Selected heading angle; It is an absolute value.

3. The asymmetric information multi-ship cooperative defense simulation method based on Bayesian Stackelberg game as described in claim 1, characterized in that, The observation vessel's profit function: In the formula: This indicates a measurement of the strategy adopted by the observed vessel. The observation vessels adopted a joint strategy. And the private type of the observed vessel is At that time, the total benefits paid by the observation vessel; Let be the payoff function of the observed vessel; To counteract the weighting coefficient; For the cost of collaboration; This comes at the cost of fuel / energy consumption. in, In the formula: The weighting coefficient for the cost of collaboration; Indicates standard deviation; For the first The predicted rendezvous distance between the observation vessel and the observed vessel; For the first The speed of the observation boat; The set of predicted interception times for all observation vessels; In the formula: This is the fuel cost weighting coefficient; For the first The square of the speed of the observation vessel.

4. The asymmetric information multi-ship cooperative defense simulation method based on Bayesian Stackelberg game as described in claim 1, characterized in that, The formula used to obtain the posterior concept of the private type of the observed vessel is as follows: In the formula: This represents the posterior belief in the private type of the observed vessel, i.e., the moment... The private type of the observed vessel is The probability of; It is the likelihood function; In order to be in Data observed at all times; The a priori belief of the private type of the observed vessel, that is, at time The private type of the observed vessel is The probability of; in, In the formula: This is a normalization constant; For sensitivity parameters, ; This is a difference measurement function; For at any time The strategy adopted by the observed vessel; The theoretically optimal strategy for the observed vessel; for The payoff function of the constantly observed vessel; The strategic space for the observed vessel; The parameter is the argument when the function reaches its maximum value.

5. The asymmetric information multi-ship cooperative defense simulation method based on Bayesian Stackelberg game as described in claim 1, characterized in that, The formula used to obtain the joint strategy that minimizes the expected cost of the observation vessel is as follows: In the formula: For at any time The joint strategy that minimizes the expected cost of the observation vessel; This provides strategic space for observation vessels; To and Calculate the expected cost of the observation vessel; This indicates a measurement of the strategy adopted by the observed vessel. The observation vessels adopted a joint strategy. And the private type of the observed vessel is At that time, the total benefits paid by the observation vessel; For private type The observed vessel adopts a joint strategy when facing the observation vessel. Theoretically optimal strategy at that time; This represents the expression that maximizes the value of the variable. .

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