[請益] 群集行為的賽局分析

看板Economics (經濟學)作者warep (我不知道)時間18年前 (2007/11/29 15:22)推噓0(0推 0噓 0→)

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這是一篇分析為何會有一窩蜂行為的賽局文章其中有一些內容我不太懂它的邏輯內文有些長我盡量把段落切割方便閱讀煩請見諒 (我把問題寫在有疑惑的段落下) There is a population of agents of size N each of whom maximizes the identical risk-neutral utility function VNM defined on the space of asset returns. For convenience we shall just assume that this utility is the same as the monetary amount received by the person. There is a set of assets indexed by numbers in [0,1]. Call the ith asset a(i).The physical return to the ith asset to the nth person investing in that asset is z(i). Let us assume that there is a unique i* such that z(i) = 0 for all i =\= i* and z(i*) = z, where z > 0. This is essentially the assumption that the excess return on one asset to the people investing in it is strictly greater than that on all other assets. Of course, everybody, given these payoffs, would want to invest in i*. The trouble is no one knows which one it is. We assume uniform priors so there is not even a likely candidate for i*. However, some people have an idea of which one it might be. Formally, there is a probability A that each person receives a signal telling her that the true i* is it. The signal need not, of course, be true, and the probability that it is false is 1- B. If it is false, then we assume that it is uniformly distributed on [0,1] and therefore gives no information about what i* really is. 最後一句(If...is.) 我的解讀是: 如果這個signal是錯的作者假設這個錯的signal的分配是0到1之間的均勻分配也就是每一個錯的signal出現的機率都相同但這和"therefore gives no information about what i* really is" 有什麼關係呢? The decision making in this model is sequential; one person chosen at random takes her decision first (she cannot decide to delay her decision). The next person, once again chosen at random, takes her decision next but she is allowed to observe the choice made by the previous person and can benefit from the information contained in it. However, she is not allowed to find out whether or not the person before her actually got a signal. The rest of the game proceeds in the same way, with each new decision maker making her decision on the basis of the history of the past decisions and their own signal if they have one. After everybody has made her choice, all the alternatives that have been chosen are tested, and if any of these turn out to work, those who have chosen it receive their rewards. If no one has chosen an option that works, the truth remains undiscovered, and no one gets rewards. It will be assumed that the structure of the game and Bayesian rationality are common knowledge. Each person's strategy is a decision rule that tells us for each possible history what that person will choose. We are looking for a Bayesian Nash equilibrium in these strategies. The nature of the equilibrium play, however, turns out to depend on certain critical tie-breaking assumptions. Some of these assumptions may be dispensed with by strengthening the equilibrium concept, but it seems more natural to introduce these as explicit assumptions. These assumptions are listed below; the relevance of these assumptions will become clear in the appropriate context. It should also be possible to see that each of these assumptions is made to minimize the possibility of herding. ASSUMPTION A. Whenever a decision maker has no signal and everyone else has chosen i = 0, she always chooses i = 0. ASSUMPTION B. When decision makers are indifferent between following their own signal and following someone else's choice, they always follow their own signal. ASSUMPTIONC. When a decision maker is indifferent between following more than one of the previous decision makers, she chooses to follow the one who has the highest value of i. The first decision maker's decision will clearly depend on whether or not she has a signal. If she has a signal, she will certainly follow her signal. While if she has no signal, by our Assumption A she will choose i = 0. This choice minimizes misinformation: the only case where this will cause confusion is when i* = 0, but since this happens with probability 0, we can ignore this possibility. 請問為什麼這裡會說i* = 0的機率為0呢? 如果i* = 0的機率為0 那沒有signal的參與者為何還要選i=0 選其它的i不是還有機會猜中真正的i*嗎? 反正猜錯了的報酬都一樣是0 何必要選一個已知報酬是0的資產呢? If the second decision maker has no signal, then she will of course imitate the first decision maker and invest in the same asset. However, if she has a signal and the first person has not chosen i = 0, she has a problem. She knows that the first decision maker had a signal and this signal is as likely to be right as her own signal. She is therefore indifferent between following the first decision maker's signal and following her own signal. In this situation our Assumption B becomes relevant. By invoking this assumption, we determine that the second person will, in this case, follow her own signal. The third decision maker can observe four possible histories: one or both of her predecessors may have chosen i = 0, and if neither of them had chosen i = 0, they could have still either agreed or disagreed. If they both chose i = 0, the third person should follow them if she has no signal and follow her signal otherwise. In all the other cases, if she does not have a signal, she should follow the person who has not chosen i = 0. If both the others have chosen i =\= 0 but have not agreed with each other, of course this does not determine a course of action. Since she is indifferent, however, we can invoke our third tie-breaking rule, Assumption C, which tells us that she should follow the person with the highest i. On the other hand, if the third decision maker does have a signal i', she will follow her own signal, unless both people before her have chosen the same option and this option is neither i = 0 nor i = i'. When both of her predecessors have chosen i = 0, this is obvious. When only one of them has chosen something different from i = 0 and i = i' and the other has chosen i = 0, this is a consequence of our Assumption B. And, of course, when the third person's signal matches the choices made by one or both of her predecessors, she must choose to follow her own signal since this could not happen unless her signal was correct. The last point is much more general than this specific situation and deserves to be emphasized. Whenever some person's signal matches the choice made by one of her predecessors, she should always follow her signal. This follows from the fact that the probability that two people should get the same signal and yet both be wrong is zero. 最後一句(This...zero.) 我的理解是: 兩個人有相同的signal但同時都是錯的這種情況的機率是0 但我不懂這句話的邏輯為什麼不可能兩個人都得到同樣的錯signal? 此外我覺得這很難和實際情況做結合文章中的敘述: 第一個人如果沒有signal會選擇i=0 之後的人如果沒有signal會選擇i=\=0且人最多的選項 (如果之前沒人選i=\=0的選項,才會繼續選i=0) 如果最多人的選項不只一個,會選擇i最大的那個選項我在想所謂"i最大的那個選項"到底是什麼意思如果我要選擇去臨近的三家餐廳中哪一家用餐餐廳1假設就是"i=0"的選項,有6個人在裡面吃飯餐廳2和餐廳3各有5個人在裡面吃飯而我沒有任何signal得知哪一家可能是最好吃的那我到底要選餐廳2還是3呢? 非常感謝~~~ -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.114.208.88 ※ 編輯: warep 來自: 140.114.208.88 (11/29 15:55) ※ 編輯: warep 來自: 140.114.208.88 (11/29 15:56)