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數學家的賭局:凱利公式推導證明!




本文摘要:上篇文章推出後,許多人來信希望能提供凱利公式詳細的推導過程。牧清華一向不喜歡講解複雜的數學公式,原本想一一回信就好。但想了想,若讀者能"大略瞭解"凱利公式的推導,相信在資金管控上的修練肯定更高一層。因此我還是試著用儘量簡單的方式陳述,希望不會澆滅投資者對交易的熱忱。當然,目的只有一個 --
祝大家都賺大錢!


凱利公式推導 (單一事件賭局)

單一事件賭局勝率為p,賠率為(1元贏了可拿回1+b元,輸了1元全賠光)

假設這賭局可不斷的重複玩下去,且每次都壓手上全部資金的比例 f  (例如 f 60%)。我們的工作是去決定這個  f  該選多少,使得在玩過多次賭局後,資金成長最快。

假設 At 表示玩到第 次賭局後的資金,我們分成下面兩個CASEs討論:

CASE 1:若第 - 1次賭局的結果為,則 At = At-1(1 + bf) 
(說明) 因為每次都壓原來資金的 比例。換句話說,在時間點 - 1時一共壓了At-1那麼多資金。因為賭局結果為贏,且賠率為b,所以會淨賺At-1fb,再加上原來的資金At-1,故在時間點 的資金變為 
At = At-1 At-1fb = At-1(1 + bf)

CASE 2: 若第t - 1次賭局的結果為,則 At = At-1(1 - f)
(說明) 因為每次都壓原來資金的 比例。換句話說,在時間點 - 1時一共壓了At-1f  那麼多資金。因為賭局結果為輸,且所壓的資金是全部輸光,所以一共賠了At-1f故在時間點 的資金變為 
At = At-1 At-1f = At-1(1 - f)

有了上面兩個CASEs後,我們可以開始計算每一次賭局後的資金變化:

只要下一個時間點贏,就將原來的資金乘上(1 + bf);只要下一個時間點輸,就將原來的資金乘上(1 - f)

我們假設總共玩了T次。T次的賭局裡,贏了W次,輸了L次 (也就是L)。因此,從一開始 (時間點為 t = 0)手上的資金為 A0,到時間點 的總資金可以表示如下:
A= A0(1 + bf)W(1 - f)L

再來要做的工作便是決定 多少,使得A可以最大化。這就完全是微積分求最大值的計算問題。我們用下面圖一展示這個計算推導。

單一事件賭局
圖一:單一事件賭局的凱利公式推導
結論:由上面推導可知,每一次賭局所要投入的資金比例為期望淨利除上賠率。注意到當期望淨利為正的時候(分子為正),變是第一篇所提的有利賭局。只有在有利賭局的時後,才值得下注,而上面的推導告訴你該怎麼下注?以上為單一事件凱利賭徒的解釋與證明。

凱利公式推導(多重事件賭局)

多重事件賭局:一枚硬幣賭局,人頭出現機率為 p1,賠率為 b1;數字出現機率為p2,賠率為 b2

假設每次下注的方式為壓資金的f比例在人頭,壓資金的f比例在數字。(註:pp= 1;bb= 1),則 f與 f要如何決定,可以使得玩過多次賭局後,資金成長最快。(註: f1 f2界在0與1之間 (包含))

類似單一事件賭局的推導過程,我們假設At 表示玩到第t次的總資金,我們分成下面兩個CASEs討論:

CASE 1:若在第- 1回合人頭出現,則 At = At-1(1 + b1f1 f2)
(說明) 因為每次都壓原來資金的 f1 比例在人頭上,f2 比例在數字上。如果時間點 - 1時人頭出現,且賠率為b1,則可淨賺At-1b1f1,但是壓在數字上面的金額At-1f會全部輸光。最後,再加上原來資金At-1,故在時間點 的資金變為 
At = At-1 At-1f1b1  At-1f2 = At-1(1 + b1f1 – f2)

CASE 2:若在第- 1回合數字出現,則 At = At-1(1 + b2f2 – f1)
(說明) 此部份的推導過程完全與CASE 1對稱, f1f2b1b2對調而已。

有了上面兩個CASEs後,我們可以開始計算每一次賭局後的資金變化。

只要下一個時間點人頭出現,就原來的資金乘上(1 + b1f1 – f2);只要下一個時間點數字出現,就原來的資金乘上(1 + b2f2 – f1)

我們假設賭局進行回合,人頭出現W1次,數字出現W2次 (也就是W1 W2)。因此,從一開始(時間點t = 0) 手上的現金為A0,到時間點的總資產可以表示如下:
A = A0(1 + b1f1 – f2)W1(1 + b2f2 – f1)W2

再來要做的工作便是決定多少,使得A可以最大化。一樣是微積分求最大值的計算問題,只不過在這我們要用偏微分跟一些計算的小技巧(在此省略)。我們用下面圖二展示這個計算推導。
多重事簡賭局的凱利公式推導證明
圖二:多重事件賭局的凱利公式推導
結論:由上面推導可知,每一次多重事件賭局所要投入的最佳資金比例就是每個事件發生的的機率。對於不確定發生機率的多重事件賭局,上面的推導結果告訴你該怎麼下注?Bidding Your Belief!

星期五;一天一錠,效果一定,歡迎訂閱「幣圖誌Bituzi電子報」

本篇獻給那些對凱利公式想要更深入瞭解的朋友們。

看不懂,絕對不會影響您原本操作績效的好壞;
看的懂,希望對您資金控管有更實質的幫助。

若您看到數學公式會頭暈,但又支持本專欄的,本篇可以跳過,但請讚一下!

您的鼓勵,是牧清華持續成長的動力!

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