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堅持你鋼鐵般的信念:再論凱利賭徒!



摘要:本文再度探討凱利公式(Kelly Formula),我們從二元期權出發,進而探討多重事件賭局。在多重事件賭局裡,凱利賭徒的精神在於“堅持你鋼鐵般的信念”!信念對了,利潤就來;信念越對,利潤就越大!

二元期權(Binary Option)

近年來流行一種新的期權商品,叫做二元期權(Binary Option),既然是二元,想必概念非常簡單。沒錯,所謂二元指的就是發生or不發生。簡單的說:下注一場單一事件的賭局,結果只有兩種:事件發生,賺到固定的獲利金額;事件不發生,損失下注的籌碼。

上一篇介紹凱利賭徒時,我們研究的賭局皆類似二元期權的概念。在本文裡我們先稱此為單一事件賭局:也就是賭的是單一事件會不會發生?例如前篇所提的擲硬幣賭局(圖一左圖),數字出現為贏,賭的事件只有"數字出現",如果數字不出現 (也就是人頭出現) ,那就算輸。然而,萬一今天賭徒想賭的是"人頭出現"為贏,那就不適合玩前面所述的賭局。因此,我們轉而考慮多重事件賭局。

多重事件賭局

生活中更常遇到的是多重事件賭局,例如賽馬比賽(n匹馬比賽,賭最後誰是冠軍)、運動彩卷(n個隊伍比賽,賭最後誰是冠軍)、公益彩卷(n組號碼開獎,賭哪個號碼開出)…甚至是總統大選(n個候選人,賭誰當選總統)。在多重事件賭局裡,至少超過一種以上的事件結果會讓你贏錢,只有你將籌碼壓在對的事件上。(不同於前面討論的單一事件賭局,只有數字出現會讓你贏錢。)

圖一左圖為前面所述擲錢幣的單一事件賭局,當然我們也論述了在此賭局裡,凱利賭徒如何計算出最佳的資金下注比例。圖一右圖為擲硬幣的多重事件賭局,賭徒可以選擇把籌碼壓在人頭上,也可以選擇壓在數字上。如果事件結果是數字出現,則壓在人頭上的賭金全部輸光,壓在數字上的賭金翻為2.25倍(2.25=1+1.25)。

這裡必須提醒的一點是在多重事件賭局底下,所有事件的賠率乘積一定要等於1(例如圖一右圖的0.8*1.25=1)。若乘積大於1,此賭局會有套利機會,賭徒可以將全部資金壓下去穩賺不賠!相對的,若乘積小於1,那此賭局對莊家有利,賭客玩越多次,莊家賺越多。

圖一:單一事件賭局與多重事件賭局
介紹完多重事件賭局,厲害的讀者可能已經觀察出,其實多重事件賭局就是多個單一事件賭局的合併而以。我們甚至可用凱利公式分別去計算拆開後的單一事件賭局下注方式,但這裡我們要說明一個出乎意料的結果:當你用凱利公式去下注多重事件賭局時,你的最佳下注方式竟然跟事件的賠率無關。這點很出乎意料之外。畢竟,我們複習一下單一事件賭局的凱利公式為 [p(1+b)-1]/b,而賠率b是下注資金比例的變數之一。在多重事件賭局裡,凱利公式告訴我們最佳的下注方式就是


拿事件發生的機率當作下注資金的比例!

這裡我們不講複雜的數學推導證明,我們直接從圖二與圖三看凱利公式在多重事件賭局上的結果:


圖二:賭局:人頭數字機率皆為50%;賠率分別為0.8與1.25
圖二的賭局為一個多重事件賭局,賭徒可將籌碼壓人頭贏或是數字贏。因為人頭出現的機率為50%,數字出現的機率亦為50%,根據凱利公式,最佳的下注方式就是把資金分成兩等份,各為50%分別壓下去!

您可能會感到很驚訝,明明就是數字的賠率比較好,直觀上應該壓在數字的資金比例要比人頭多?怎麼會各壓50%會是最好的資金比例下注方式呢?

事實不然,壓數字的賠率雖高,但若壓在數字的資金比例多,一旦失敗也將損失更多資金。很神奇吧!我們雖不在這論述這定理的證明,但有興趣的讀者,還是可以私下寄信給牧清華(Email: mn@bituzi.com)。

現在問題來了,不是毎場賭局都確實知道事件發生的機率,至少很多比賽型的賭局就不知道。例如球賽、賽馬...等。沒有確定的機率,這時該怎麼賭?

賭你的信念 (Bidding Your Belief!)

謀權奪利真英雄,牧清華就是喜歡用英雄人物做比喻,我們來看下面應用情境:

情境:自從鋼鐵人3票房開出紅盤後,蜘蛛人實在越看越不爽,準備找鋼鐵人大幹一架以決勝負!當然其他英雄人物:蝙蝠俠、超人哥也來看樂鬧,開了賭盤。雖然鋼鐵人最近真的很夯,但賭盤似乎是蜘蛛人的賠率略勝一籌,為0.8 (註:市場越看好的賠率越低),也就是若蜘蛛人贏,壓1塊錢可拿回1.8元),而鋼鐵人雖然氣勢很旺,賠率卻高達1.25。(如圖三所示)
鋼鐵人 大戰 蜘蛛人
圖三:鋼鐵人(賠率1.25) v.s. 蜘蛛人(賠率0.8)

回到看熱鬧的兩位英雄人物 -- 蝙蝠俠與超人哥。蝙蝠俠認為鋼鐵人贏的機率為70%、蜘蛛人贏的機率為30%;另一方面,因為超人哥跟蜘蛛人是好朋友,超人哥認為蜘蛛人實力應該比鋼鐵人好才對,所以鋼鐵人贏的機率只有40%,而蜘蛛人贏的機率有60%。假設蝙蝠俠與超人哥都各有10元,這時他們該如何下注?

根據凱利公式,蝙蝠俠會拿出7元在鋼鐵人身上,且壓3元在蜘蛛人身上;而超人會壓4元在鋼鐵人身上,壓6元在蜘蛛人身上。(如圖四所示)

他們的認知,決定他們的最佳下注方式,與賠率無關。凱利公式告訴您最有利的下注方式,就是堅持你鋼鐵般的信念!


蝙蝠俠、鋼鐵人、蜘蛛人、超人
圖四:根據凱例公式,最佳的下注方式就是按照自己認為的機率分佈下注。
注意到上述論點有個前提,你對你自己的認知要有足夠的信心!事實上,沒有人有把握自己認為的機率分佈是絕對正確 (除非是內線)。凱利公式只是在說明,當你對自己的認知有足夠信心時,你該怎麼下注?就是賭你的認知(Bidding Your Belief)。


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看到這裡,相信讀者最後都會有個疑問,誰知道自己的信念到底對不對阿?尤其是摘要的最後一句話 “信念對了,利潤就來;信念越對,利潤就越多!”。這不是廢話嗎?

問題是,當你信念對了,你知道怎麼賭嗎?當你信念不對,你還期望你能獲利嗎?

未來有機會我們介紹,凱利賭徒信念的準與不準決定你獲利的多寡!

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