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題記：本文是唐世平教授于2022-01-25日公開(kāi)預(yù)測(cè)“俄羅斯將大概率出兵烏克蘭”的英文原文的中文翻譯版本（建議閱讀英文原文）。為了方便讀者對(duì)照閱讀，英文版附在中文版之后。鑒于這個(gè)包含了模型的短文并不是一個(gè)學(xué)術(shù)文章，因此行文風(fēng)格并沒(méi)有那么“學(xué)術(shù)”。俄羅斯出兵烏克蘭之后，唐世平教授并未對(duì)文章的內(nèi)容做任何實(shí)質(zhì)意義上的改動(dòng)。

對(duì)2022年烏克蘭危機(jī)的博弈論模型

2022-1-24

唐世平（復(fù)旦大學(xué)復(fù)雜決策分析中心）

通過(guò)在俄烏邊境集結(jié)10萬(wàn)戰(zhàn)備部隊(duì)，普京本質(zhì)上是向拜登/北約/烏克蘭發(fā)出（準(zhǔn)）最后通牒。

1.做出可能（中規(guī)模到大規(guī)模）進(jìn)攻烏克蘭的態(tài)勢(shì)，而目標(biāo)未定（也很難界定）。

2.向拜登/北約/烏克蘭提出一系列要求，包括：

a)?停止北約的擴(kuò)張（沒(méi)有烏克蘭、格魯吉亞等）

b)?烏克蘭承認(rèn)其被分裂：東部地區(qū)歸俄羅斯所有

c)?烏克蘭成為（準(zhǔn)）中立國(guó)家：與俄羅斯保持友好關(guān)系

3.利用冬季（包括天然氣管道和價(jià)格）作為時(shí)間窗口：因?yàn)閮鐾劣欣谲娛滦袆?dòng)，特別是機(jī)械化作戰(zhàn)。

通過(guò)拒絕其他選項(xiàng)【“坐以待斃”，讓烏克蘭變得更頑強(qiáng)且成為北約的（準(zhǔn)）成員國(guó)】，普京表示他不打算妥協(xié)，否則對(duì)俄羅斯來(lái)說(shuō)這將意味著慢性死亡和又一次羞辱。屆時(shí)普京有可能已經(jīng)離任。甚至情況可能更糟，比如，新的清潔能源更加普及，天然氣/石油價(jià)格再次下降。

這是預(yù)防性戰(zhàn)爭(zhēng)的邏輯。

我用一些數(shù)字來(lái)表示這個(gè)博弈的收益（為了更容易求解和詮釋?zhuān)?/p>

收益：(A，B，C) = (俄羅斯，北約/拜登，烏克蘭)。

核心假設(shè)：

1）戰(zhàn)爭(zhēng)對(duì)所有參與者而言都是有代價(jià)的。

2）會(huì)有負(fù)收益，因?yàn)闀?huì)有損失，包括聲望和其他物質(zhì)財(cái)富的損失（例如：領(lǐng)土）。當(dāng)某一方退縮時(shí)，它的威信會(huì)受損。

3）先升級(jí)后退縮，威信（或聲望）還是會(huì)受損。

4）對(duì)俄羅斯的制裁會(huì)給俄羅斯帶來(lái)代價(jià)。

兩個(gè)額外的核心假設(shè)：

1.這個(gè)博弈具有內(nèi)在的預(yù)防性邏輯：隨著普京的老去和俄羅斯在技術(shù)上逐漸落后，同時(shí)，北約不斷武裝烏克蘭和其他近鄰國(guó)家，俄羅斯即便保有核威懾力，也很難再與烏克蘭和睦相處。因此，在普京80歲之前（普京生于1952年，現(xiàn)在70歲），每一年都有“折舊”系數(shù)（比如說(shuō)0.1），共計(jì)十年。

2.拜登（和北約）可能希望普京進(jìn)攻并陷入泥潭之中。而普京對(duì)此心知肚明，但他仍會(huì)采取行動(dòng)，因?yàn)樗坏貌挥?jì)算賠率（odds）。

烏克蘭的決策：極不可能屈服（即α=0）。

當(dāng)烏克蘭堅(jiān)定立場(chǎng)時(shí)，拜登/北約極不可能向?yàn)蹩颂m施壓使其屈服（這將是另一個(gè)“慕尼黑”！）。換言之，1-γ≈0

然而，如果烏克蘭確實(shí)屈服了，拜登/北約可能同意避免戰(zhàn)爭(zhēng)（但增加制裁）。換句話(huà)說(shuō)，1-β≥β，或1-β≥1/2

當(dāng)烏克蘭（拜登/北約）拒絕普京的要求時(shí)，進(jìn)攻的概率幾乎是肯定的：如果普京退讓?zhuān)敲粗辽倏隙ㄊ堑诙畹慕Y(jié)果：可能不會(huì)有第二次機(jī)會(huì)。

解（有兩個(gè)版本，一個(gè)比較簡(jiǎn)單，另一個(gè)略復(fù)雜）

簡(jiǎn)單版本

假設(shè)α=0且γ=1（這非常合理）。換句話(huà)說(shuō)，我們假設(shè)烏克蘭不會(huì)屈服，且當(dāng)烏克蘭拒絕普京的要求時(shí)，北約/拜登將（只能）支持烏克蘭。情況如此時(shí)，普京的計(jì)算結(jié)果將是：

-5≤10δ+(-15)(1-δ)

或δ≥0.4。換句話(huà)說(shuō)，只要普京相信俄羅斯能占上風(fēng)并保住烏克蘭東部的機(jī)會(huì)有40%，進(jìn)攻就會(huì)獲得至少與其“坐以待斃”一樣多的收益（后者是很確定的，特別是從長(zhǎng)遠(yuǎn)來(lái)看）。

更復(fù)雜的解

從左邊的子博弈（即當(dāng)烏克蘭決定屈服時(shí)），我們得到普京的潛在收益為：

10α(1-β)+10αβδ+(-15)αβ(1-δ)=10α-25αβ+25αβδ

從右邊的子博弈（即當(dāng)烏克蘭決定堅(jiān)定立場(chǎng)時(shí)），我們假設(shè)γ=1（即拜登只能支持烏克蘭）來(lái)簡(jiǎn)化博弈。那么，我們得到普京的潛在收益為：

10(1-α)δ+(-15)(1-α)(1-δ)=25δ-15-25αδ+15α

結(jié)合左邊的子博弈和右邊的子博弈，我們得到普京的總計(jì)算結(jié)果為：

-5≤10α-25αβ+25αβδ+25δ-15-25αδ+15α

我們得到

10≤25α-25αβ+25αβδ+25δ-25αδ

或

2≤5α-5αβ+5αβδ+5δ-5αδ

解δ，我們得到：

δ≥(2-5α+5αβ)/(5-5α+5αβ)

我們能容易就估計(jì)出，當(dāng)α非常小（即接近0）或β非常大（即接近1）時(shí)，我們?nèi)缘玫溅摹?.4。這就回到了先前的估計(jì)。

當(dāng)α和β都等于0.5時(shí)，我們得到δ=0.75/3.75=0.2

簡(jiǎn)而言之，即便俄羅斯（在戰(zhàn)場(chǎng)）取勝的幾率只有約40%，俄羅斯也會(huì)進(jìn)攻。普京已把可能的制裁考慮在內(nèi)。

這個(gè)博弈的關(guān)鍵驅(qū)動(dòng)力其實(shí)是當(dāng)前局勢(shì)（在展望未來(lái)時(shí)），對(duì)普京而言，現(xiàn)在進(jìn)攻而非簡(jiǎn)單的等待是合理的。換句話(huà)說(shuō)，如果烏克蘭和北約拒絕普京的要求，他將無(wú)路可退：因?yàn)檫@樣一來(lái)他會(huì)顯得很軟弱（且損失慘重）。

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附錄I（英文版）：

A game theoretical model of the 2022 Ukraine crisis

2022-01-24

Shiping Tang (Center for Complex Decision Analysis, Fudan University)

By amassing 100,000 combat ready troops alone the Russian-Ukraine border, Putin is essentially delivering a (quasi-)ultimatum to Biden/NATO/Ukraine.?

1. Building a posture of possible (medium to large scale) invasion of Ukraine, with objectives undefined (and it is hard to define).

2. Presenting Biden/NATO/Ukraine with a list of demands, including:?

a) Freezing NATO expansion (no Ukraine, Georgia etc.)

b) Ukraine admits its divided up: East part to Russia

c) Ukraine becomes a (quasi-)neutral state: maintaining friendly relations with Russia?

3. Using the winter season (including gas pipeline and price) as a time-window: frozen terrain facilitates military operation, especially mechanized movements.?

BY rejecting the other options (sitting still and letting Ukraine become more fortified and (quasi-) NATO member states, Putin has signaled that he does not intend to do so: it will be a slow death and another humiliation to Russia, and by then, Putin may be gone (and the situations may be even worse, say with new clean energy goes up and gas/oil price goes down again.)

This is the logic of preventive war.

The game (for easier solution and interpretation), I use some numerical denotations for payoffs

Payoffs: (A, B, C)=(Russia, NATO/Biden, Ukraine). Key assumptions:?

1) War is costly to all participants.

2) There will be negative payoffs because there will be losses, including losses of reputation and other material possesses (e.g., territory). When one backs down, one loses prestige.

3) Escalate and then back off incur loss of prestige (or reputation).

4) Sanctions against Russia carry cost for Russia.

Two key additional assumptions

1. The game has an inherent preventive logic to it: with Putin aging and Russia slowly falls behind technological advances, and with NATO gradually arming Ukraine and other near abroad states, Russia, even with nuclear weapon, may not have an easy time with Ukraine again. So there is: a discount factor (say 0.1) for each year passing, before Putin reaches 80 yrs old (Putin is now 70 yrs old, born 1952), with ten more years.

2. Biden (and NATO) may want Putin to invade and face a quagmire and Putin knows it. But this alone does not stop Putin from acting: Putin has to calculate this odds.

Ukraine’s decision: very unlikely to submit (i.e., α=0).?

When Ukraine stands firm, it is highly unlikely that Biden/NATO will pressure Ukraine to submit (this will be another “Munich”!). In other words, 1-γ≈0

However, if Ukraine does submit, there is a good chance that Biden/NATO may agree to avoid war (but adding sanctions). In other words, 1-β≥β , or 1-β≥1/2)

When Ukraine (Biden/NATO) rejects Putin’s demands, the probability of invasion is almost certain: backing down is at least a second worst outcome for certain if Putin backs down: there may not be a second chance.

Solutions (with two versions, one is more simple, the other slightly more complex)

Simple version

Assume?α=0 and γ=1 (very reasonable). In other words, we assume that Ukraine will not submit and when Ukraine rejects Putin’s demand, NATO/Biden will (can only) support Ukraine. When this is the case, Putin’s calculation will be:?

-5≤10δ+(-15)(1-δ)

Or: δ≥0.4. In other words, as long as Putin believes that is a 40% chance that Russian will prevail and hold Eastern Ukraine, he can believe that he can attack and gain as much as if he sits still (which is quite certain, especially from the long run).

A more complex solution

From the left subgame (i.e., when Ukraine decides to submit), we have Putin’s potential gains as:?

α(1-β)10+αβδ*10+(-15)αβ(1-δ)=10α-25αβ+25αβδ

From the right subgame (i.e., when Ukraine decides to stand firm), we assume γ=1 (that is, Biden can only support Ukraine) to simply the game. We then have Putin’s potential gains as:

(1-α)δ*10+(1-α)(1-δ)(-15)=25δ-15-25αδ+15α

Combine the left subgame and the right subgame, we have Putin’s overall calculation as:

-5≤10α-25αβ+25αβδ+25δ-15-25αδ+15α

We have:

10≤25α-25αβ+25αβδ+25δ-25αδ

Or:?

2≤5α-5αβ+5αβδ+5δ-5αδ

Solve for δ, we have:

δ≥(2-5α+5αβ)/(5-5α+5αβ)

We can easily estimate that when α is very small (i.e., close to 0) or β is very large (i.e., close to 1), we still have δ≥0.4. This returns to the earlier estimation.

When both α and β are equal to 0.5, we have δ=0.75/3.75=0.2

In short, Putin may invade even though there is only about 40% chance that Russia can win (on the ground). He has already taken the possible sanctions into account.

The key driver of this game is really that the present situation (when looking into the future), it is rational for Putin to invade now rather than simply wait. In other words, for Putin, he cannot afford to back off if Ukraine and NATO rejects his demands: he looks weak (and loses much).