The Results of the War | 战争的结果

📚 The Results of the War | 战争的结果

In armed conflict, commanders attempt to maximise their gains while minimising losses; this strategic struggle can be captured by a branch of mathematics called game theory. A zero-sum game models a situation where one side’s victory is exactly the other’s defeat, so the ‘results of the war’ can be distilled into a single numerical value—the value of the game. By analysing payoff matrices, maximin strategies, and mixed strategies, we can predict the optimal outcome when both adversaries play rationally.

在军事冲突中，指挥官既想扩大战果又想减少损失；这种战略博弈可以用数学中的博弈论来描述。零和博弈模拟的正是“一方所得即为另一方所失”的局面，因此所谓“战争的结果”最终可以浓缩为一个数值——博弈的值。通过分析收益矩阵、最大最小策略以及混合策略，我们就能在双方都采取理性行动时预测出最优结果。

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1. Zero-Sum Games and Payoff Matrices | 零和博弈与收益矩阵

A two-person zero-sum game involves two players, usually called Row (Player A) and Column (Player B), whose interests are strictly opposed. The outcome is summarised by a payoff matrix, where each entry aᵢⱼ represents the gain to Row when Row chooses strategy i and Column chooses strategy j. Since the game is zero-sum, Column’s payoff is simply −aᵢⱼ.

双人零和博弈包含两名玩家，通常称为行玩家（甲）与列玩家（乙），他们的利益完全相反。博弈的结果由收益矩阵概括，矩阵元素 aᵢⱼ 表示当行玩家选择策略 i、列玩家选择策略 j 时行玩家获得的收益。由于是零和博弈，列玩家的收益就是 −aᵢⱼ。

For example, consider a simplified war scenario where two generals choose between ‘Attack’ and ‘Defend’:

例如，设想一个简化的战局，两位将领可选“进攻”或“防守”：

	B: Attack	B: Defend
A: Attack	3	−1
A: Defend	−2	4

Here, a positive number is a victory for Row (gain of territory, for instance), while a negative number is a loss. The results of the war depend entirely on which strategy pair is chosen.

正数代表行玩家获胜（如夺得领土），负数代表其失利。战争的结果完全取决于双方所选策略的组合。

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2. Maximin and Minimax Strategies | 最大最小与最小最大策略

When facing an intelligent opponent, a cautious player will assume the worst and choose a strategy that guarantees the best possible floor. For Row, this means finding the maximin: for each row, note the minimum payoff, then select the row with the largest of these minima. Formally, maximin = maxᵢ (minⱼ aᵢⱼ).

面对聪明的对手，谨慎的玩家会做最坏打算，并选择一个能保证最高“保底”收益的策略。对行玩家而言，就是寻找最大最小值：先标记每一行的最小值，再从这些最小值中选出最大者。公式表达为 maximin = maxᵢ (minⱼ aᵢⱼ)。

Column, on the other hand, wishes to hold Row’s gain as low as possible. The column player will apply the minimax principle: for each column, find the maximum payoff (worst case for Column), then choose the column with the smallest of these maxima. Minimax = minⱼ (maxᵢ aᵢⱼ).

相反，列玩家希望把行玩家的收益压得越低越好。列玩家会采用最小最大原则：先看每一列的最大值（这是列最不利的情况），再从中选出最小的那个最大值，即 minimax = minⱼ (maxᵢ aᵢⱼ)。

In our war example, Row’s row minima are min{3, −1} = −1 and min{−2, 4} = −2, so maximin = −1. Column’s column maxima are max{3, −2} = 3 and max{−1, 4} = 4, so minimax = 3. Since maximin ≠ minimax, a stable pure-strategy solution does not yet exist.

在上述战局中，行的最小值分别为 min{3, −1} = −1 和 min{−2, 4} = −2，故 maximin = −1。列的最大值分别是 max{3, −2} = 3 和 max{−1, 4} = 4，故 minimax = 3。因为 maximin 不等于 minimax，不存在稳定的纯策略解。

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3. Saddle Points and Stable Solutions | 鞍点与稳定解

If maximin equals minimax, that common value is called a saddle point, and the corresponding strategies form a pure-strategy equilibrium. Neither player can unilaterally change strategy to improve their outcome; the results of the war become fixed as that value.

如果 maximin 等于 minimax，这个公共值就称为鞍点，对应的策略构成纯策略均衡。任何一方单方面改变策略都无法改善自己的结果；此时战争的结果就被锁定为该数值。

Consider a modified payoff matrix where both sides have a natural defensive advantage:

考虑一个修改后的收益矩阵，双方都有明显的防守优势：

	B1	B2
A1	2	1
A2	3	0

Row minima: {1, 0}, maximin = 1. Column maxima: {3, 1}, minimax = 1. The saddle point occurs at (A1, B2) with value 1. Here, the war results in a guaranteed small gain for Row, provided both play optimally.

行最小值为 {1, 0}，maximin = 1；列最大值为 {3, 1}，minimax = 1。鞍点出现在 (A1, B2)，值为1。在这种情况下，只要双方都采取最优策略，战争的结果就是行玩家一个确定的微小收益。

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4. Mixed Strategies When No Saddle Point Exists | 无鞍点时的混合策略

When maximin < minimax, as in our first example, no pure strategy guarantees a stable result. Instead, players randomise their choices with certain probabilities, called a mixed strategy. The results of the war then become an expected value rather than a fixed gain.

当 maximin < minimax 时（如第一个例子），没有纯策略能保证稳定结果。这时玩家需要以一定概率随机化自己的选择，这叫做混合策略。战争的结果也从此变为期望值，而非固定收益。

Suppose Row chooses Attack with probability p and Defend with probability 1−p. Column chooses Attack with probability q and Defend with probability 1−q. The expected payoff E(p,q) is then:

假设行玩家以概率 p 选择进攻、以概率1−p 选择防守；列玩家以概率 q 选择进攻、以概率1−q 选择防守。期望收益 E(p,q) 为：

E(p,q) = 3pq + (−1)p(1−q) + (−2)(1−p)q + 4(1−p)(1−q)

Row wishes to maximise this expected value, while Column aims to minimise it. The optimal mixed strategies are found by ensuring that the opponent is indifferent between their options.

行玩家想将这个期望值最大化，列玩家则希望将其最小化。最优混合策略的获取，关键在于让对手在自己的选项之间无差异。

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5. Solving 2×2 Zero-Sum Games Analytically | 解析求解2×2零和博弈

For a general 2×2 payoff matrix:

对于一般的2×2收益矩阵：

[ a b ]
[ c d ]

If there is no saddle point, Row’s optimal probability p* for the first row satisfies p*a + (1−p*)c = p*b + (1−p*)d. This equation removes Column’s incentive to deviate, because the expected payoff is the same whichever column is chosen.

如果没有鞍点，行玩家对第一行的最优概率 p* 满足 p*a + (1−p*)c = p*b + (1−p*)d。该等式消除了列玩家改变策略的动机，因为无论选择哪一列，期望收益都相同。

Solving gives:

求解可得：

p* = (d − c) ÷ (a − b − c + d)

Similarly, for Column, the optimal probability q* for the first column satisfies q*a + (1−q*)b = q*c + (1−q*)d, yielding:

同样地，列玩家对第一列的最优概率 q* 满足 q*a + (1−q*)b = q*c + (1−q*)d，解得：

q* = (d − b) ÷ (a − b − c + d)

The value of the game v (the results of the war) is then the expected payoff when both use these optimal mixed strategies:

博弈的值 v（战争的结果）即为双方都使用上述最优混合策略时的期望收益：

v = (a·d − b·c) ÷ (a − b − c + d)

Applying this to our initial war matrix where a=3, b=−1, c=−2, d=4, we get p* = (4 − (−2)) ÷ (3 − (−1) − (−2) + 4) = 6 ÷ 10 = 0.6, q* = (4 − (−1)) ÷ 10 = 5 ÷ 10 = 0.5, and v = (3·4 − (−1)·(−2)) ÷ 10 = (12 − 2) ÷ 10 = 1.0. So the results of the war yield an expected gain of 1 for Row, not a predictable single-outcome but a long-run average.

将此法用于最初的战局矩阵（a=3, b=−1, c=−2, d=4），得 p* = (4 − (−2)) ÷ (3 − (−1) − (−2) + 4) = 6 ÷ 10 = 0.6，q* = (4 − (−1)) ÷ 10 = 5 ÷ 10 = 0.5，v = (3·4 − (−1)·(−2)) ÷ 10 = (12 − 2) ÷ 10 = 1.0。也就是说，战争的结果是行玩家获得1的期望收益——不是某次具体的胜败，而是长期重复下的平均值。

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6. Graphical Interpretation of the Results | 战争结果的图形解释

When only one player uses a mixed strategy, we can draw a graph to visualise the payoff. For Row, plot the expected payoff against p for each of Column’s pure strategies. The line for Column’s first strategy is y = a·p + c·(1−p), and the second is y = b·p + d·(1−p). Row’s guaranteed payoff is the lower envelope of these lines, and the maximin point is its peak.

当只有一方采取混合策略时，可以用图形直观呈现收益。对行玩家，以 p 为横轴、分别画出列玩家各纯策略下的期望收益直线。列第一策略对应 y = a·p + c·(1−p)，第二策略对应 y = b·p + d·(1−p)。行玩家可以保证得到的收益是这两条直线的下包络线，最大最小值点就是该包络线的最高处。

In our example, the lower envelope meets at p = 0.6 and gives v = 1, confirming the algebraic result. Such diagrams help strategists visualise the point of indifference and the guaranteed result.

在我们的例子中，下包络线在 p = 0.6 处交汇，收益值为 v = 1，与代数解一致。这样的图形能帮助决策者直观看到无差异点以及可以保证的战争结果。

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7. Dominance: Simplifying the Battlefield | 优势策略：简化战场

Sometimes a strategy is clearly inferior regardless of the opponent’s choice. If every payoff in row i is less than or equal to the corresponding payoff in row k (and strictly less in at least one), we say row i is dominated by row k and can be removed. Similarly, a column j can be dominated by another column if it never gives a smaller cost.

有时某项策略无论对手如何应对都明显更差。如果行 i 的每个收益都不大于行 k 的对应收益（且至少有一个严格小于），则称行 i 被行 k 优势支配，可以将其从矩阵中删除。类似地，某列若永远不会给出更小的代价，也可被其他列优势支配而消去。

Dominance arguments reduce the size of a pay-off matrix without altering the results of the war. A large conflict can thus be analysed more easily once dominated strategies are discarded.

优势论证可以在不改变战争结果的前提下缩小收益矩阵的规模。一旦剔除被支配的策略，大型冲突也能更容易地进行分析。

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8. Application to Business and Economics | 在商业与经济中的应用

While the language here is military, zero-sum games appear wherever two parties compete for a fixed resource: market share battles, sealed-bid auctions, or penalty shootouts in sport. The results of the war become the expected market share for the winning strategy, guiding executives on how often to advertise or to discount.

尽管上文使用军事语言，只要是两方争夺固定资源的场合——市场份额争夺、密封投标拍卖或体育中的点球大战——都会出现零和博弈。此时“战争的结果”就是获胜策略下的期望市场份额，指引企业高管如何规划广告频率或折扣力度。

Decision maths module in Edexcel A-Level expects students to compute optimal strategies, interpret game values, and understand the stability of solutions. The concepts of maximin, saddle points, and mixed strategies thus provide a toolkit for rational decision-making in conflict situations.

Edexcel A-Level 决策数学模块要求学生能够计算最优策略、解释博弈的值并理解解的稳定性。最大最小准则、鞍点及混合策略等概念，为冲突情景下的理性决策提供了一套完整的工具。

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9. Limitations and Real-World Extensions | 局限性与现实扩展

A pure zero-sum model assumes perfect rationality and complete information, which rarely hold in actual warfare. Generals may miscalculate, intelligence is imperfect, and outcomes are not always a simple number. Moreover, real conflicts can be non-zero-sum, where cooperation or third parties alter the structure.

纯粹的零和模型假设完全理性和完全信息，这在实际战争中很少成立。将领可能误判，情报并不完全，结果也不总是一个简单的数字。此外，现实冲突可能是非零和的，合作或第三方的介入会改变博弈结构。

Nonetheless, the framework offers a baseline for understanding the logic of strategy. By extending to games with multiple players or stochastic elements, one can move closer to the complexity of actual battlefields, yet the fundamental question—’What are the results of the war?’—still drives the mathematics.

尽管如此，该框架为理解战略逻辑提供了一个基线。通过扩展到多人博弈或引入随机因素，可以更接近真实战场的复杂性，但“战争的结果是什么？”这一根本问题始终推动着数学分析。

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10. Summary: The Mathematical Peace Treaty | 总结：数学和平条约

The results of a zero-sum war, from a game-theory perspective, are encapsulated by the value v. When a saddle point exists, the value is deterministic; when only mixed strategies yield equilibrium, the value is an expected payoff. Both cases rely on the players acting to secure their maximin or minimax positions.

从博弈论视角看，零和战争的结果浓缩为数值 v。存在鞍点时，该值是确定性的；只有当混合策略带来均衡时，该值才是期望收益。两种情况都建立在玩家采取行动以确保其最大最小或最小最大立场的基础上。

Recognising that mathematics can formalise the outcome of competition deepens our appreciation of strategic thinking. Whether in war, business, or everyday decisions, the concept of a zero-sum game reminds us that the results often emerge not from hope, but from carefully balancing risks and rewards.

认识到数学能够形式化竞争的结果，加深了我们对战略思维的理解。无论是在战争、商业还是日常决策中，零和博弈的概念都提醒我们：结果往往不是源于希望，而是来自于对风险与回报的审慎平衡。

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