The Price of Wisdom - 高山雪 Snow Hill

I have already written many articles regarding those forecasters, who claim that they had ' accurate ' forecasts in the past, you should pay special attention to. In particular to those who use their blog posts, YouTube channels or Facebook pages for money making purpose.  Look! Those who claimed that the stock market would be rising with the inflation rate, where are they now? Where have they gone? Listen to what they said then and what they are saying now. You can check it out yourself by tracking their words... < 經濟學家論 | 利息上升, 股市會上升? >  Or, some will tell you that they have some great tips for how to purchase UK houses at a below market price at some up-and-coming postcode areas. Don't forget, they tell you so is for a reason, just as they set up their FB pages, YouTube channels is for a reason, which is - money - all about money! Such overseas property investment businesses have been blooming on FB. Some of them will convince you to take their courses (some are on...

Benjamin Graham  將股票投資比喻為投票機器，而凱因斯  John Maynard Keynes  則將股票投資比喻為選美投票。 凱因斯選美 某遊戲：選美投票人要從 100 張照片中挑選最美的 6 位，而選出的最接近平均的便可得獎。 凱因斯說：「這時候，投票人選的不是自己認為最美的 6 位，而是選擇其他投票人最有可能認為是最美的 6 位。因為每個人想法不一，問題不在於你的最佳判斷（不是你自己認為哪 6 位是最美），也不是普羅大眾一般認為哪 6 位是最美，而是在於 預測平均的其他 投票 人的平均預測 （即： 預測 ， 其他 投票 人預測的平均預測結果的平均 ）。」這是 第三層思考 。

在不確定性情況下，機率應該怎樣判斷？ 我在＜ Subjective Probability Estimation ＞一文已寫過就主觀機率，如何 optimise 優化、calibrate 校正、mindful of hypothesis 警惕假設、error  budget 預留錯誤空間、quantify verbal expression of probability 量化人們的語言上的真正機率、clarify 澄清機率。

 心理帳戶 (Mental Accounting) 是 Richard Thaler 主張的一個決策框架。Thaler 說：「當人類在進行與金錢有關的決策時，並不是評估不同因素和選項後，才理性地決定，而是自己在心裏形造一個狹窄框架後，才進行決定。」Thaler 將這個建造框架的行為，叫做「心理帳戶」。 心理帳戶是一個人類的行為偏誤，把自己的主觀價值加諸在自己的錢之上，從而影響自己的決定，再影響消費行為。

框架效應 (Framing Effect) 框架效應 就是指，當人們要決策或回答、分析問題時，人會 選擇把問題以某種方式呈現， 這 種呈現的方式或方法叫 做 「框架」(Frame)。 因為用了不同的框架，而導致我們對同一問題有不同的判斷，稱為「框架效應」 (Framing Effect)。 框架效應是以框架來限制人的注意力在框架以內 。

在 < 貝氏定理 (7): 事後機率分布最大概似估計法 (Maximum a Posteriori Estimation, MAP) > 中，我已經介紹過如果是 continuous distribution 的話是怎樣找到 Posterior Probability。

賽局理論 (Game Theory) 賽局理論（又稱博奕論）是一個決策思考模型，適用於有兩個或多個對手在同一個 競爭 # 下的策略制定。而囚徒困境 (Prisoner's Dilemma) 是賽局理論中，一個最基本的 靜態競爭賽局 *  的思考模型。 *   靜態賽局 (Static Game Theory)  的意思是，競爭參與者皆不知道對手各自的選擇，所以是不受對方選擇影響，可以是一次性的同時行動 (one-off simultaneous actions)。 *   動 態賽局 (Dynamic Game Theory)  的意思是，是相繼或連續性或互動性的重覆行動 (consequential, sequential, or interactive repeated actions)。 #  所謂「競爭」便是賽局，又或所謂「賽局」便是競爭： 競爭 = 賽局 = 博奕  賽局的意思是，一個競爭參與者的選擇，要考慮另一個競爭參與者的選擇，而作出對自己最有利的決策。假設每個競爭參與者也希望把自己利益最大化。賽局理論就是幫助競爭者計算，如何把其他競爭者的選擇納入考慮，而選擇對自己最有利的行動。

在  < 機率思維 | 貝氏定理 (1): 理論 (Bayesian Theorem) > ，我們知道不同的人面對一樣的消息，有著不一樣的信念；而基於不同的信念，我們應該怎樣去投資而不會有太大的損失呢？   我們 在決策時 ，務必要 用最少的資金，把期望值最大化。 我在 < 風險決策的兩個理論: 期望值 & 期望效用 > 已列出過以下公式及計算概念：           期望值 = 機率 × 價值                     Eq. (1)        總期望值 = ∑ 機率 × 價值              Eq. (2)          where 價值   (V) = 收益 (E) - 本金 (C)  亦即是：                            總期望值 = 贏的期望值 - 虧的期望值                  Eq. (3)    值博 的 總期望值 ≥  $0  即是要： 總期望值 ≥   $0         ⇒    贏的期望值 > 虧的期望值   風 險較低 的 總期望值  ≥   本金 ...

聯合機率 Joint Probability  如果 X 的發生是需要有 A 和 B 的出現，那就是：            P(X ) = P(A, B) 如果：  A  = {a1, a2, a3}； B  = {b1, b2, b3} 那麼： P(A, B) =  {P(a1, b1), P(a1, b2), P(a1, b3), P(a2, b1), P(a2, b2), P(a2, b3), P(a3, b1), P(a3, b2), P(a3, b3)}

Continuing from the previous article < Subjective Probability Estimation >, a nother thing we should deal with caution is other peoples' subjective probability. Clarifying Expert's Estimation As mentioned, we should understand what people mean when they speak "chance" - a descriptive probability.  If an expert or a consultant says his estimation for a possibility of an event happening is "more likely", you have to be very cautious about this word, and should seek further clarification by what does he mean by that. Because you will never know if there is any difference between your understanding about "more likely" and his. If you don't clarify, you'll never know.   You can guide him through the following example conversation. You should start from the average and guide him through by questions: You: Would you say it is 50%? Expert: No, more than that. I said it is more likely. You: 70%? Expert: I would say it is a bit more than 70%. Y...

在不確定情況下，我們可以利用 Principle of Insufficient Reason  來先進行平均機率假設，即：把每個「假說 Hypothesis」 的 Prior Probability 假設為一樣的平均機率，再計算 Posterior Probability (事後機率分佈)。 如有 N 個假說 Hypothesis ，它們各自的 Prior Probability  便假定為 1/N 。 然後把新資訊納入計算，把第一次計算出來，每個 Hypothesis 的 Posterior Probability 變成下次更新的 Prior Probability，再計算每個 Hypothesis 的新的 Posterior Probability。如此類推，一直更新。這就是「 貝氏更新 (Bayesian Updating) 」。

In this article, we will continuously go through Mr K's decision making process in Example 4 in < Decision Tree > by examining his risk tolerance. Risk Tolerance Risk tolerance is the willingness of a person to take risk for better consequences. It means how the person weigh the downside of the consequences as compared to the upside, i.e. how significance of the downside of the consequences he would be impacted?  Most   risk averse people will weigh the downside (loss) heavier than the upside (gains), even the probability for both sides are equal.  Risk tolerance is a crucial factor to determine the diversification of an asset portfolio, which also is an attribute to tradeoff of risk/return. So, understanding the role of risk tolerance  is important not only for making daily decisions but also beneficial for making investment decisions.  

In the previous article < Subjective Probability Estimation >, I have mentioned the difficulty in assigning chance for decision making. One of the crucial tools I mentioned is to the Principle of Insufficient Reason (also known as Principle of Indifference ) to assign probability of events in the absence of knowledge about the situation.   

Decision Tree Another way to present the consequences together with probabilities and outcomes is decision tree . A decision tree can be a more efficient representation of the present uncertainties, alternatives and consequences than a consequence table.  When drawing a decision tree, using quantitative measures, the hardest part is how to assign chance (probability) because it involves a high degree of subjective probability estimation   where it is easily affected by heuristics and biases .     

This article is an elaboration on assigning chance subjectively - the hardest bit for many people - in the article < Uncertainty and Chance >. This article is by no means serves as a deep-dive about risk or whatever, it is about pointing out which parts of chance assignment is hard and how to resolve them.  

A good quote from the book of Smart Choices, "The quality of our decision making is not judged by the quality of consequences." The authors put it, you may end up: A smart choice, a bad consequence : the decision is fine, but it is only things happen. The rare bad case scenario happens which ruins the result. How do you know the rare bad thing could happen before make the decision? We never know! This is uncertainty. A poor choice, a good consequence: even the decision is poor but outcome is good, it is just because of luck.  Once you have made the decision right, the outcome may be subject to uncertainties.     What uncertainty means to us? The authors put it: " you may know what might happen, but you won't know what will happen ." This is because of uncertainty .  Uncertainties can be arisen from:  (a) information/ data uncertainties,  (b) model uncertainties,  (c) calculation/ mathematical uncertainties,  (d) result/ outcome uncertaintie...

We have difficulty in decision making is because very often there are more than one alternative to choose from, where one alternative would better satisfy one objective but worse in the other. In a real world situation, we can't always have all goods in one alternative. In such a situation, we have to make trade-off, i.e. seek balance amongst all alternatives and make a relatively balanced decision. The more objectives and the more alternatives you have in your decision making process, the more trade-offs you will have to make, and more difficult a task the decision maker has to do.  An excellent method called "Even Swap Method" is useful to get one or more objectives removed from your consequence table so as to simply a complex decision, which adopts the concept of trade-off  . I term it "Cancel-Out" method .  The key to it is "compensation" . 

  In order to construct a consequence table, we must list out objectives (on rows) and alternatives (on columns) to be able to build a consequence table.   Objectives Setting Objectives are the criteria and your guide for your decision, so they are important in your decision process. Think about:  What are the things you most want, The concerns you want to address,  The outcomes you most want to avoid ,  Always ask "why" can help you to find out the fundamental objectives.  Using the fundamental objectives to evaluate and compare alternatives, From the fundamental objectives, you can think about some mean objectives to achieve those fundamental ones. Or you can grow some sub-objectives from one fundamental objectives (see Figure 1). Ask yourself, " are you happy with the alternative that satisfies the objectives you've listed? " If no, there may be some objectives missing, Ask yourself, "are the objectives you wrote just the means or the end?"....

  The following is the summary of  the PrOACT approach taken out from the book, Smart Choices, by Hammond et al (1999).  The first part of the book review: Smart Choices (1) | Focus and Defining Problem   As said in the first part, this PrOACT approach is to resolve a complex decision problem by breaking it into the following eight elements, think systematically, focus on those that are critical  to the problem.

不確定時 如果在完全沒有資料和數據下，我們唯有用自己的 本能 (instinct) 或 直覺 (gut feeling) ，作最合理的決策。因為有時有些事情，人的腦子不能逐一拆解，但能記下那複雜系統引伸的後果或記下用來處理的手法，這就形成了人的本能 (instinct) 或 直覺 (gut feeling)。這套思考習慣、捷徑又叫「捷思 Heuristic」。 Gerd Gigerenzer 提倡用直覺來解決複雜系統內的問題。例如：攬球員是用凝視捷思 Gaze Heuristic 來接下飛來攬球，而非運算那複雜數式來計算飛球的軌跡。所以在決策上，要善用自己的本能或直覺。 筆者要強調 用 直覺 進行的決策， 只限於 : 1) 高不確定性 high uncertainty 而 時間緊迫 時使用， 而非 在有明確數據資料顯示是高風險 high risk 又時間充裕時使用。因為如數據資料清晰，而後果又影響深遠時，我們必須經過適合思維模型進行理性分析、估 算 可能 和 最有可能 出 現的後果，判斷風險及自己的承受範圍後, 才可以下決定。相反，如要做出 即時決定 ，我們只能靠自己的直覺 (Heuristics) 作最快的計算來得出結果。 2) 樣品數量太少，不夠代表性 時 (Number of samples is too small to be representative)。因為數量太少我們不能依賴數據分析，得出 statistical significant 的機會率來協助做決定。在 沒有曆史數據或太少數據 的情況下，我們只能夠用 Uniform Distribution 來形容機率分布。亦即是說，假定是相同機率。Refer to < Principle of Insufficient Reason > 要知道， 可思考的時間愈短，犯錯的機會愈高 。因為時間充裕，你可以反覆思考關鍵條件，例如，需要甚麼的條件會導致災難，甚麼條件會是成功？那你可以轉注那些關鍵條件的出現。但若然要當下做 即時決定 ，可用 直覺 並以 保守 為原則來作決策。(保守的意思是盡量減低風險。) 若要運用到捷思來做即時決策時，最好要假設這個決策有佷高可能性會出錯，想像其後果，自己能否承受？若非，最好還是 Status quo。 即時決策可能只有數分鐘內便要下決定，請不要在這類性況下，作重...