群集行為的賽局分析 - 經濟

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~~~
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