Chatswood Kendo Judo Karate Club

武侯問曰有師甚眾
既武且勇背大險阻
右山左水深溝高壘
守以彊弩退如山移
進如風雨糧食又多
如何以對千乘萬騎
兼之徒步分為五軍
各軍一衢五軍五衢
敵人必惑莫之所加

敵若堅守以固其兵
急行間諜以觀其慮
彼聽吾說解之而去
不聽吾說斬使焚書
分為五戰子吳子越
戰勝勿追不勝疾歸
如是佯北安行疾鬭
一結其前一絕其後
兩軍之枚或左或右
而襲其處五軍交至
必有其力擊彊道也

The passage describes a strategy that uses the principles of deception, signaling, and multi-front coordination to disrupt the Defender’s strong position. The Attacker leverages psychological and physical maneuvers to force the Defender into suboptimal responses, ultimately seeking to gain an advantage without committing to a costly direct assault. This approach exemplifies how complex strategic decisions can be modeled and analyzed using game theory, particularly in scenarios where asymmetric information and psychological factors play crucial roles.

Further Expansion on Multi-Front Coordination:

In the context of the passage, multi-front coordination is a critical strategy employed by the Attacker to exploit the Defender's vulnerabilities.

This approach involves dividing the Attacker’s forces into five separate armies, each advancing from different directions to create confusion and overwhelm the Defender.

On how this strategy can be understood and optimized using game theory:

Purpose of Multi-Front Coordination:

Overwhelming the Defender:

By attacking from multiple directions, the Attacker forces the Defender to spread their resources thin across various fronts. This dilutes the Defender’s ability to concentrate their forces and respond effectively to any single attack.

The Defender, unsure of where the main assault will occur, may struggle to allocate their troops optimally. This leads to suboptimal defensive positions, making it easier for the Attacker to achieve local superiority at critical points.

Creating Confusion and Uncertainty:

Multi-front attacks introduce significant uncertainty into the Defender’s decision-making process. The Defender must constantly assess where the primary threat lies, which can lead to indecision or mistakes in their strategic responses.
The Attacker can exploit this confusion by feigning attacks on certain fronts while concentrating their true strength elsewhere, thus forcing the Defender to commit resources to false threats.

Deception and Misdirection:

The Attacker can use some of their armies as decoys to mislead the Defender about the true direction of the main attack. For example, one or more of the five armies might engage in skirmishes or visible maneuvers to draw the Defender’s attention, while the main force prepares for a decisive strike from an unexpected direction.
This misdirection can cause the Defender to make poor strategic choices, such as moving their forces away from the actual point of attack, thus weakening their defenses where the Attacker intends to strike hardest.

Game Theory Analysis of Multi-Front Coordination:

Strategic Diversification:

Attacker’s Mixed Strategy:

By dividing their forces, the Attacker essentially adopts a mixed strategy, where they are not fully committing to a single course of action. This keeps the Defender uncertain about the Attacker’s true intentions, increasing the chances of the Defender making errors.

Defender’s Response:

The Defender must also adopt a mixed strategy, deciding how to allocate their forces across multiple fronts. If the Defender allocates too many resources to one front, they risk leaving other fronts vulnerable; if they spread their forces too thin, they may be unable to repel a concentrated attack.

Nash Equilibrium in Multi-Front Warfare:

The Nash Equilibrium in this context occurs when both the Attacker and Defender adopt strategies that minimize their respective losses, given the actions of the other. For the Defender, this might involve keeping a reserve force that can be quickly redeployed to counter the Attacker’s main thrust once it becomes clear where that is. For the Attacker, achieving equilibrium might involve committing to a point of attack only after observing the Defender’s initial reactions.

However, the equilibrium is dynamic. The Attacker’s use of deception and multi-front pressure is designed to push the Defender out of equilibrium, forcing them into a reactive rather than proactive stance.

Coordination and Timing:

Synchronized Attacks:

The success of multi-front coordination heavily depends on timing. The five armies must synchronize their attacks to prevent the Defender from concentrating their forces on any one front. If the Attacker's forces strike too far apart in time, the Defender may be able to defeat them piecemeal.

Adaptive Strategies:

The Attacker must be prepared to adapt their strategy based on the Defender’s movements. If the Defender concentrates their forces on one front, the Attacker can adjust by redirecting their main effort to a weaker point. This adaptability is key to exploiting the confusion caused by multi-front operations.

Psychological Impact:

Pressure and Fatigue:

The psychological toll on the Defender of having to monitor and respond to multiple threats can lead to fatigue and mistakes. The Attacker can use this to their advantage by maintaining pressure on all fronts, increasing the likelihood that the Defender will eventually crack.

Moral and Cohesion:

Constant pressure from multiple directions can also erode the morale and cohesion of the Defender’s forces. Soldiers who are unsure where the next attack will come from are more likely to become disheartened, reducing their effectiveness in battle.

Optimal Strategies for Both Sides:

Attacker:

Diversify Attack Vectors:

The Attacker should ensure that their forces are capable of attacking from all directions, with each army having the flexibility to switch to a more promising front if needed. They should also employ deception to disguise the true main effort.

Exploit Timing:

The Attacker must carefully coordinate the timing of their assaults. Delaying the main attack until the Defender has committed their forces elsewhere can significantly increase the chances of success.

Defender:

Reserve Forces:

The Defender should maintain a mobile reserve force capable of rapidly reinforcing any front that comes under heavy attack. This reserve should be kept out of sight to prevent the Attacker from accurately assessing the Defender’s full strength.

Intelligence Gathering:

The Defender should prioritize intelligence and reconnaissance to better understand the Attacker’s intentions. By identifying the main attack direction early, the Defender can better allocate their forces and disrupt the Attacker’s coordination.

Conclusion:

Multi-front coordination, as described in the passage, leverages game theory principles of strategic diversification, deception, and timing to create a scenario where the Defender is constantly on the back foot.

The Attacker’s ability to confuse, overwhelm, and outmaneuver the Defender is key to overcoming even a well-prepared and strongly positioned enemy.

By understanding and applying these principles, the Attacker can turn the complexity of multi-front warfare to their advantage, achieving victory through superior strategic planning and ex*****on.

以一擊十莫善於阨
以十擊百莫善於險
以千擊萬莫善於阻
敵眾我寡為之柰何
避之於易邀之於阨
用眾務易用少務隘

The passage describes a strategic military scenario in which one is faced with a stronger enemy.

The response advises using terrain advantages to neutralize the enemy's numerical superiority.

It suggests that when outnumbered, it is crucial to:

Avoid Open Ground:
Engage the enemy in narrow or difficult terrain where the numerical advantage of the enemy is minimized.

Leverage Ambush Tactics:
Use the element of surprise and the environment to create panic and disorder in the enemy ranks, even if they are in large numbers.

Theory Analysis:

In game theory, this scenario can be understood as a zero-sum game where the gain of one player results in the loss of the other.

The strategies recommended in the passage can be analyzed as asymmetric strategies in response to the power imbalance.

1. Avoidance and Engagement in Terrain:

Strategy:
The weaker side should avoid direct confrontation in open areas (where the enemy's numerical superiority can be fully utilized) and instead choose to fight in narrow or difficult terrain (阨, 險, 阻).
This can be modeled as a strategy of limiting the enemy's options.

Payoff Matrix:
By engaging in narrow terrain, the weaker side increases its chances of success. The enemy's numerical advantage is mitigated, reducing their effective strength and potential payoff.

2. Exploiting Psychological Factors:

Strategy:
The passage suggests creating confusion and panic within the larger force by utilizing surprise attacks (e.g., sudden noise or ambush in narrow terrain).
This corresponds to psychological warfare in game theory, where the goal is to disrupt the opponent's decision-making process.

Game Dynamics:
This introduces an element of uncertainty into the game.
The larger force, expecting a straightforward engagement, might overreact or make poor decisions when confronted with unexpected situations, thereby shifting the equilibrium in favor of the weaker side.

3. Minimizing Risk with Minimal Forces:

Strategy:
The passage advises using fewer forces in situations where the terrain is narrow (務隘). This strategy is akin to resource allocation in constrained environments. The idea is to commit minimal resources in high-risk areas while maximizing the strategic impact.

Optimal Strategy:
The weaker side aims to fight only where they can maximize their payoff by choosing battles where the terrain and situation minimize the enemy's advantages.

Nash Equilibrium

In this scenario, the Nash equilibrium would involve both sides adjusting their strategies based on the terrain （environment) :

The stronger side might choose not to engage in narrow terrain or might attempt to change the battlefield to open terrain, where their numerical superiority is more effective.

The weaker side would continually seek to force engagements in narrow or difficult terrain, where their chances of success are higher.

However, the equilibrium will be dynamic, with both sides continually adapting based on the terrain and other situational factors.

The weaker side's optimal strategy is to avoid direct conflict in open ground and instead engage in situations where their chances of success are maximized due to the environment, aligning with the idea that "用眾者務易，用少者務隘" (those with many troops should seek easy terrain; those with few troops should seek narrow terrain).

總文武者軍之將也
兼剛柔者兵之事也
凡人論將常觀於勇
勇之於將數分之一
勇必輕合輕不知利
此未可也子吳子曰
將所慎五一曰理法
二曰備戒三曰果斷
四曰戒慎五曰約化

理者治眾如治寡矣
備者出門如見敵矣
果者臨敵不懷生矣
戒者雖克如始戰矣
約者法令省而不煩
受命不辭敵破言返
將之禮也師出之日
有死之榮無生之辱

理 (Principle):
Treating a large group as if it were a small one.

Application:
This principle suggests that a leader must manage both large and small units with the same level of precision and efficiency. In game theory terms, this is akin to maintaining consistent strategies across different scales. The strategy should be robust enough to handle varying levels of complexity without losing effectiveness.

備 (Preparation):
Preparing as if facing the enemy.

Application:
This involves anticipation and foresight. In game theory, this can be related to the concept of strategic anticipation where a player prepares for potential moves of opponents, ensuring that strategies are resilient to various scenarios. This is similar to a "minimax" strategy where the leader plans for the worst-case scenario to ensure success regardless of the opponent's actions.

果 (Decisiveness):
Facing the enemy without hesitation.

Application:
Decisiveness can be linked to commitment strategies in game theory, where a player must make binding decisions that force the opponent into a predictable response. In competitive scenarios, making decisive moves can force the opponent to react in a way that favors the decisive player's position.

戒 (Caution):
Even after victory, maintain vigilance as if in the initial battle.

Application:
This relates to maintaining strategic consistency and not becoming complacent. In iterative games, staying vigilant and continually reassessing the opponent's strategies prevents them from exploiting any perceived weaknesses or changes in strategy.

約 (Simplicity):
Ensuring laws and regulations are straightforward and not overly complex.

Application:
Simplicity in rules can be related to clear signaling in game theory. When the rules or strategies are simple, it reduces the potential for misunderstandings and miscalculations, leading to more predictable outcomes and fewer opportunities for opponents to exploit complex rules or loopholes.

In summary, a commander must:

Develop scalable strategies applicable to both large and small groups.
Anticipate and prepare for the worst-case scenarios.
Make decisive moves to force favorable outcomes.
Remain vigilant and adaptable even after achieving success.
Keep strategies and rules simple to avoid unnecessary complexity.

By integrating these principles, a commander can enhance their strategic decision-making and improve their chances of success in various competitive and conflict situations.

端末未見人莫能知
天地神明與物推移
變動無常因敵轉化
不為事先動而輒隨
圖制無疆扶成天威
康正八極密定九夷
如此謀者為帝王師

"The enemy's end has not yet been seen, and no one can know it. Heaven, earth, and the gods; shifting with all things. Change is unpredictable; respond to the enemy's transformations. Do not act prematurely; move and adapt accordingly. Thus, one can plan and strategize without limits, supporting the power of Heaven. Steady the eight extremes, and firmly establish the nine regions. Only one who can devise such strategies will be fit to advise the emperor."

1. Uncertainty and Incomplete Information:

Theory Context:

The passage highlights the uncertainty of the enemy’s intentions and outcomes, akin to games of incomplete information where players do not know all the factors influencing their opponent's decisions.

Strategic Implication:
To deal with this, you can use mixed strategies, where your moves are not entirely predictable.
This mirrors the idea of adapting to the enemy’s transformations without revealing your true intentions, maintaining strategic ambiguity.

2. Dynamic Adjustment (Responding to Change):

Theory Context:

The passage stresses the importance of responding to the enemy's transformations, similar to dynamic games where the players’ strategies evolve over time.

Strategic Implication:

In game theory, this can be related to strategies in repeated games, where players adjust their strategies based on the opponent’s previous actions. Flexibility and responsiveness are crucial, making use of feedback loops to continuously adjust your strategy based on the latest information.

3. Optimal Timing (Not Acting Prematurely):

Theory Context:

The idea of not acting prematurely aligns with concepts from timing games or "real options" theory in game theory, where the value of waiting for more information before committing to an action is recognized.

Strategic Implication:
This suggests adopting a wait-and-see approach, where you delay irreversible decisions until more information becomes available, allowing you to capitalize on the most opportune moment.

4. Planning Without Limits (Strategic Foresight):

Theory Context:

The passage suggests planning and strategizing without limits, which corresponds to the concept of extensive form games where strategies are devised by considering all possible future scenarios.

Strategic Implication:

You should build a comprehensive decision tree, analyzing all potential moves and counter-moves to ensure that your strategy remains robust against any possible enemy action.

5. Establishing Dominance (Steady the Eight Extremes, Nine Regions):

Theory Context:

This relates to the concept of Nash equilibrium in game theory, where the stability of a system is achieved when no player has anything to gain by changing their strategy unilaterally.

Strategic Implication:

To achieve this, you must create conditions where your position is so strong that the enemy cannot benefit from changing their strategy. This could involve securing key positions or resources, ensuring that your dominance is unchallenged.

6. Becoming the Master Strategist:

Theory Context:

The passage ends by describing the ideal strategist as one who masters these principles, akin to a player who consistently achieves the best possible outcomes in various games.

Strategic Implication:

This suggests aiming to become a player who not only responds to immediate challenges but also shapes the broader strategic environment to your advantage, effectively becoming a "master" of the game.

Conclusion:

Applying these game-theoretic concepts to the passage results in a strategy that is adaptive, responsive, and built on deep foresight.

It advises against rigid plans, instead advocating for a fluid approach that can adjust to the unpredictable nature of conflict.

The ultimate goal is to establish a position of such strength that it becomes unassailable, making you the "master of the strategic game".

子吳子曰進兵之道
先明四輕二重一信
使地輕馬使馬輕車
使車輕人使人輕戰
明知陰陽則地輕馬
芻秣以時則馬輕車
膏鐧有餘則車輕人
鋒銳甲堅則人輕戰
進有重賞退有重刑
行之以信令制遠之
此勝主也子吳子曰

On the importance of understanding and prioritizing key factors to ensure success.

Breakdown of the Principles:

(1) Four Light
(四輕)

地輕馬 (Terrain Lightens the Horse):
Understanding the terrain allows you to optimize the mobility of your cavalry.

馬輕車 (Horse Lightens the Cart):
Proper care of horses (timely feed and rest) ensures they can effectively pull carts.

車輕人 (Cart Lightens the Soldier):
Well-maintained vehicles reduce the burden on soldiers.

人輕戰 (Soldier Lightens Combat):
Equipped soldiers (sharp weapons, strong armor) make the actual battle less burdensome.

(2) Two Heavy
(二重 ):

進有重賞 (Great Rewards for Advancement):
Incentivize soldiers with rewards for advancing.

退有重刑 (Severe Punishment for Retreat):
Deterrence through the threat of severe punishment for retreating.

(3) 一信 (One Trust):

行之以信 (Execute with Trust):
Consistent and reliable enforcement of orders and promises ensures trust in leadership.

Application Theory:

In game theory, these principles can be analyzed through the lens of incentives, resource allocation, and strategy under uncertainty.

Resource Allocation (Four Light):

Game Theory Perspective:
The principle of "四輕" is about efficient resource management.
Optimizing the use of terrain, animals, and equipment can be modeled as a resource allocation game.
The objective is to minimize the cost (in terms of energy, time, and risk) while maximizing the effectiveness of the military forces.

Example:
Consider a zero-sum game where two generals compete for dominance on a battlefield.
Each general must decide how to allocate resources (e.g., troops, vehicles) across different terrains.
The one who better understands the terrain and resource management will have a strategic advantage.

Incentive Structures (Two Heavy):

Game Theory Perspective:
The "二重" aspect focuses on creating a strategic incentive structure. In a dynamic game, where soldiers (players) make decisions based on their payoffs (rewards or punishments), the commander must set up a system that motivates desirable behavior (advancement) and deters undesirable behavior (retreat).

Example:
This can be modeled as a principal-agent problem, where the commander (principal) needs to design a contract (incentive scheme) that aligns the soldiers' (agents') actions with the overall goal of victory. The balance between rewards and punishments affects the soldiers' strategies.

Trust and Credibility (One Trust):

Game Theory Perspective:
Trust is a key component in any cooperative game. "一信" emphasizes the importance of credibility and consistent ex*****on of orders. In repeated games, maintaining trust through consistent actions ensures cooperation and prevents defection.

Example:
In a repeated game scenario, if soldiers observe that rewards and punishments are consistently applied, they are more likely to cooperate with the commander’s strategy. This credibility serves as a deterrent against betrayal or desertion.

Strategic Implications:

Sequential Decision Making:
The principles suggest a sequence where resource optimization precedes combat engagement. This aligns with backward induction in dynamic games, where the commander anticipates the outcomes of different resource allocations before making a final decision.

Mixed Strategies:
In situations where the enemy is uncertain of the terrain or resources, the commander might employ mixed strategies, allocating resources in a way that is not entirely predictable to the enemy, thereby maintaining a strategic advantage.

Equilibrium Analysis:
The ultimate goal is to reach a Nash equilibrium, where both the commander and the soldiers have no incentive to deviate from the prescribed strategy, ensuring coordinated and effective military action.

Conclusion:

By understanding and applying these principles through game theory, one can craft strategies that optimize resource use, motivate soldiers effectively, and maintain trust, leading to a higher probability of success in military campaigns.

用兵必須審敵虛實
而趨其危遠來新至
行列未定敵人可擊
既食尚未設備可擊
奔走可擊勤勞可擊
未得地利可擊勿疑
失時不從可擊勿疑
旌旗亂動可擊勿疑

涉長後行未息可擊
涉水半渡可擊勿疑
險道狹路可擊勿疑
陳數移動可擊勿疑
將離士卒可擊勿疑
心怖可擊子吳子曰
凡若此者選銳衝之
分兵繼之急擊勿疑

On scenarios where an opponent might be vulnerable and the optimal strategies to exploit those vulnerabilities.

Theory Application:

Information Asymmetry and Timing:

Scenario:
The enemy is newly arrived and their formation is not yet established.

Strategy:
Exploit the disorganization by attacking before the enemy can solidify their defenses. This is akin to capitalizing on an opponent's lack of preparation in game theory, where acting early can provide a strategic advantage.

Concept: This relates to the principle of first-mover advantage, where being the first to act can often result in a more favorable outcome.

Resource Allocation and Supply Lines:

Scenario:
The enemy has not yet secured resources or is struggling with logistics (e.g., they are hungry or ill-equipped).

Strategy:
Attack when the enemy is weak due to lack of resources. This exploits the enemy’s inefficiencies and lowers their capacity to retaliate.

Concept:
This is similar to exploiting an opponent’s weaknesses in resource allocation, which can lead to a strategic advantage.

Operational Fatigue:

Scenario:
The enemy is fatigued from long marches or crossing difficult terrain.

Strategy:
Target the enemy when they are exhausted and less capable of effective defense. This focuses on taking advantage of an opponent’s decreased performance due to operational strain.

Concept:
This relates to exploiting an opponent’s diminished capacity, akin to capitalizing on their reduced utility or payoff.

Disorganization and Disarray:

Scenario:
The enemy is disorganized (e.g., confused banners, disrupted formations).

Strategy:
Attack when the enemy’s command and control are compromised. This leverages the opponent’s lack of coordination.

Concept:
This can be viewed through the lens of disrupting an opponent’s strategy, making their responses less effective and thus easier to defeat.

Vulnerability During Critical Transitions:

Scenario:
The enemy is crossing a river or traversing difficult terrain.

Strategy:
Strike during these vulnerable transitions to maximize the impact of the attack. This targets the enemy at a time when they are least capable of effective defense.

Concept: This is similar to targeting an opponent during periods of transition or when their strategy is less stable.

Psychological Warfare:

Scenario:
The enemy’s morale is low or their leaders are separated from their troops.

Strategy:
Exploit psychological weaknesses by attacking when the enemy’s morale is compromised. This leverages the psychological aspects of warfare to undermine the enemy’s resolve.

Concept:
This involves influencing an opponent’s psychological state to induce mistakes or lower their overall strategic effectiveness.

Optimal Strategy Synthesis:

In game theory, the optimal strategy involves understanding the opponent’s weaknesses and exploiting them at the most opportune times. The principles outlined in the passage suggest a dynamic approach to strategy, where the adversary’s vulnerabilities are assessed continuously and attacks are timed to capitalize on those vulnerabilities.

By combining these principles, a strategist can develop a plan that integrates the following elements:

Timing of Attack:
Exploit moments when the enemy is least prepared.

Resource Exploitation:
Target the enemy when their resources are strained.

Psychological Impact:
Use psychological tactics to reduce the enemy’s effectiveness.

Conclusion:

In sum, the game theory approach to these principles emphasizes flexibility and adaptability, ensuring that actions are taken in response to the enemy’s current state and vulnerabilities.

子吳子曰料敵不卜
而與戰者籠統有八
一者曰之疾風大寒
早興寤遷刊木濟水
不憚艱難二者曰之
盛夏炎熱晏興無間
行驅飢渴務於取遠

三者曰之師既淹久
糧食無有百姓怨怒
祅祥數起上不能止
四者曰之軍資既竭
薪芻既寡天多陰雨
欲掠無所五者曰之
徒眾不多水地不利
人馬疾疫四鄰不至

六者曰之道遠日暮
士眾勞懼倦而未食
解甲而息七者曰之
將薄吏輕士卒不固
三軍數驚師徒無助
八者曰之陳而未定
舍而未畢行阪涉險
半隱半出子吳子曰

諸如此者擊之勿疑
有不占而避之者六
一者曰之土地廣大
人民富眾二者曰之
上愛其下惠施流布
三者曰之賞信刑察
發必得時四者曰之
陳功居列任賢使能

五者曰之師徒之眾
兵甲之精六者曰之
四鄰之助大國之援
凡不如敵避之勿疑
見可而進知難而退

- strategic guidelines for engaging or avoiding conflict based on an assessment of both your own and your enemy’s situation.

Theory Framework

Payoff Structure:
We need to assess the potential outcomes of engaging in or avoiding battle.
Each decision (engage or avoid) has a different payoff based on the conditions described.

Information Asymmetry:
Both sides have different levels of information about their own strengths and weaknesses and those of their opponent.

Mixed Strategies:
The passage suggests situations where a commander should engage or avoid battle without hesitation.
This implies that in some cases, a mixed strategy (choosing between multiple options probabilistically) might be optimal based on the specific conditions.

Conditions to Engage in Battle (Exploit the Enemy's Weaknesses)

Harsh Weather Conditions:
If the enemy is exposed to harsh weather and is struggling with mobility and logistics, this creates an opportunity.
The payoff for attacking under these conditions is high because the enemy's fighting capability is diminished.

Overextension in Summer Heat:
If the enemy is pushing their forces too hard under extreme heat, they are likely to suffer from exhaustion and dehydration.
This increases the likelihood of their defeat, making it a favorable time to engage.

Prolonged Engagement with No Supplies:
If the enemy has been engaged in battle for an extended period without sufficient supplies and the morale of their troops is low, the payoff for attacking becomes high because they are weakened.

Resource Depletion:
When the enemy has depleted resources and is in a desperate situation with no prospects for resupply, an attack is highly advantageous as their capacity to resist is minimal.

Small and Sick Forces:
If the enemy's forces are small, disadvantaged by terrain, or suffering from disease, attacking them yields a high payoff because their defenses are compromised.

Fatigue and Low Morale:
If the enemy is tired, hungry, and demoralized, especially after a long march, the likelihood of their defeat increases significantly, making it an optimal time to strike.

Poor Leadership and Disorganized Troops:
If the enemy's leadership is weak, and their troops are disorganized and demoralized, they become easy targets, thus maximizing the payoff for an attack.

Incomplete Formations and Risky Terrain:
If the enemy is in the middle of forming their battle lines or navigating difficult terrain, they are vulnerable. Attacking at this moment yields a high payoff.

Conditions to Avoid Battle (Recognize the Enemy's Strengths)

Enemy's Strong Position:
If the enemy has the advantage of a large, well-resourced, and well-populated territory, the cost of attacking is high, and the likelihood of success is low.

Strong Leadership and Morale:
If the enemy’s leadership is beloved by their troops and has successfully distributed rewards and benefits, the morale and cohesion of their forces are high, making battle risky.

Effective Command and Timing:
If the enemy has a track record of delivering timely commands and maintaining strict discipline, the risk of engaging them is significant due to their efficiency and preparedness.

Merit-Based Leadership and Skilled Troops:
If the enemy's leadership is based on merit and they have placed capable individuals in charge, their forces are likely to be highly effective, increasing the cost and risk of an attack.

Superior Troop Quality and Equipment:
If the enemy's troops are well-equipped and numerous, engaging them would likely result in a costly and potentially unsuccessful battle.

Allied Support:
If the enemy has strong alliances and the backing of powerful neighbors, the strategic environment becomes unfavorable, making avoidance the better option.

Analysis

Dominant Strategies:
The decision to engage or avoid battle should be based on the relative strengths and weaknesses outlined.
The dominant strategy in this context would involve attacking only when the payoff is clearly in your favor (e.g., when the enemy is weak and vulnerable) and avoiding battle when the enemy is in a strong position.

Best Response:
In a dynamic situation, where both sides are adjusting strategies based on ongoing assessments, the best response is to continuously evaluate the conditions and choose whether to engage or avoid based on maximizing your payoff while minimizing risk.

Minimax Principle:
When deciding to avoid battle, you're essentially employing the minimax principle—minimizing the potential maximum loss (i.e., avoiding a battle where you're likely to be defeated).

Mixed Strategy:
If the conditions are not clearly in favor of either side, adopting a mixed strategy might involve probing the enemy with smaller skirmishes to gain more information before committing to a full engagement.

Conclusion:

This application of game theory helps to clarify the decision-making process and supports the strategic insights provided in the passage in determining when to engage in or avoid battle.