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 Post subject: Re: Sente, gote and endgame plays
Post #201 Posted: Sat Apr 28, 2018 8:20 am 
Tengen

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Code:
......A...........
...../.\..........
..../...\.........
.0.X.....B.-15|7..
......../.\.......
......./...\......
.-7|8.C.....Z.-22.
...../.\..........
..../...\.........
.1.Y.....D.-15....


C is a simple gote. B is Black's simple sente.

We have t(B) = 7 and t(C) = 8.

We test the tentative gote traversal move value aka local temperature t'(A) = (x - d) / 2 = 7.5. Bill, this contradicts the conditions you suggest: t'(A) <= t(B), t(C) <=> 7.5 <= 7, 8 are partially violated. Therefore, according to your conditions, A is not White's long gote but can only be a simple gote or simple sente.

Next, we test the tentative gote move value t'_gote(A) = (x - b) / 2 = 7.5 and tentative sente move value t'_sente(A) = x - c = 7. The condition t'_gote(A) > t'_sente(A) <=> 7.5 > 7 identifies White's simple sente and excludes 'ambiguous'. This contradicts t'_sente(A) = t(B) <=> 7 = 7 identifying 'ambiguous'. Due to the contradiction, we do not have a simple sente, either.

The condition t'_gote(A) > t(B) <=> 7.5 > 7 identifies a simple gote. This contradicts t'_gote(A) > t'_sente(A) <=> 7.5 > 7 identifying White's simple sente. Due to the contradiction, we do not have a simple gote, either.

Using your suggested conditions, t'(A) <= t(B), t(C), the local endgame does not have any type. Since this contradicts that each local endgame has a type, your conditions are wrong! The example is a counter-example for them.

***

Code:
......A.-7.5|7.5..
...../.\..........
..../...\.........
.0.X.....B.-15|7..
......../.\.......
......./...\......
.-7|8.C.....Z.-22.
...../.\..........
..../...\.........
.1.Y.....D.-15....


Make the hypothesis of White's long gote with m'(A) = (x + d) / 2 = -7.5 and t'(A) = (x - d) / 2 = 7.5.

Let us study the profits of the moves of White's alternating sequence: P1 = 7.5, P2 = 8, P3 = 8.

Let me again suggest the conditions t'(A) <= P1, P2, P3 as the requirement for calculating traversal values of a long gote.

Applying them, we find the conditions t'(A) <= P1, P2, P3 <=> 7.5 <= 7.5, 8, 8 fulfilled.

I suppose, we have analogue conditions for longer long gotes. How about long sentes? Can we keep move values aka local temperatures for them or do we also need profits?

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 Post subject: Re: Sente, gote and endgame plays
Post #202 Posted: Sat Apr 28, 2018 9:40 am 
Judan

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Code:
......A...........
...../.\..........
..../...\.........
.0.X.....B.-15|7..
......../.\.......
......./...\......
.-7|8.C.....Z.-22.
...../.\..........
..../...\.........
.1.Y.....D.-15....


There is no question that A = {0 | -15}, i.e., that White's move to B reverses through C to D. m(A) = -7½, and t(A) = 7½.

However, as you point out, the temperature of B is 7, which is less than 7½. Therefore, there will be times that Black will want to save B as a ko threat and not immediately continue to C. But since 7 is close to 7½, there will also be times that Black should continue to C to prevent White from getting the reverse sente from B to D. Because of the half point difference, this is a close call.

Edit: In either case, A is gote. :)

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 Post subject: Re: Sente, gote and endgame plays
Post #203 Posted: Mon May 07, 2018 9:54 am 
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Bill, I know too little about history of research in using counts and move values for evaluating gote and sente after Sakauchi Jun Ei and until 2016. CGT (and Mathematical Go Endgames) studies a lot but, AFAIK, not in terms of count and move value, as go players use them: unchilled, without infinitesimals. Has everything in between been your invention? I wonder because everything I read had been written by you: comparing counts or move values, gains, distinguishing types, conditions for move order in environments etc. What of that has been your invention and what has been invented by others (whom)?

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Post #204 Posted: Mon May 07, 2018 11:27 pm 
Judan

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RobertJasiek wrote:
Bill, I know too little about history of research in using counts and move values for evaluating gote and sente after Sakauchi Jun Ei and until 2016. CGT (and Mathematical Go Endgames) studies a lot but, AFAIK, not in terms of count and move value, as go players use them: unchilled, without infinitesimals. Has everything in between been your invention? I wonder because everything I read had been written by you: comparing counts or move values, gains, distinguishing types, conditions for move order in environments etc. What of that has been your invention and what has been invented by others (whom)?


Among the first go books I bought were Sakata's Killer of Go series and Takagawa's Go Reader series. One of the Sakata books deals with tsumego and yose, one of the Takagawa books is about the yose. Both mention miai counting, but Sakata regards it as useful only in special cases. Takagawa is clearer, and simply mentions both deiri and miai counting. Both authors, however, start with finding the count. Neither mention the problems with double sente.

My own efforts were mainly based upon my understanding of Takagawa. Most go books start out with assuming that a play is a double gote (sic!), a one-way sente (sic!), or a double sente (sick! ;)), and make the calculations accordingly. On my own I discovered that if you start out assuming that all plays were simple gote, you could derive a contradiction when the value of the opponent's reply was larger than the assumed value of the supposed gote. Then you got a sequence of plays that was sente or gote depending upon when the size of the plays dropped below that of the original play. With this method I was able to get all the temperatures and mean values of non-ko thermography, a few years before thermography was invented. I don't know whether I improved on Takagawa or not, since I had donated the Go Reader set to the Yale library before I concentrated on yose calculation. It took me a few years before I abandoned the idea of local double sente. I had never actually calculated one, only assuming that plays that had humungous follow-ups for both players were double sente. But I managed to prove, to my satisfaction, that they did not exist. (Before 1976.) I even sent an article to the Go World saying that they did not exist, but Bozulich did not bite. ;)

I developed my own theory of ko evaluation, but it is not very practical. You have to know too much to apply it, as a rule. I touch on it at the start of This 'n' That. It is at the root of the CGT idea of komonster, and my classification of types of ko threats, and the idea of the ko ensemble. :) After studying CGT I came up with the idea of ambiguous plays. I also discovered how to evaluate multiple kos and superkos, in 1998. And a few years later I discovered the relation between simple approach kos and Fibonacci numbers (Edit: in a neutral threat environment). (Edit: Earlier I had regarded approach kos as a kind of sente. In an environment with sparse ko threats, that might be more accurate. :) For instance, the proverb says that a three move approach ko is no ko at all. In an NTE, it is worth 1/13 of the swing, which is often worth fighting. As a sente, it is worth 1/24 of the swing, which is closer to 0.)

Talking about how much a play gains, on average, is just another way of talking about miai values. Less scary and unfamiliar, I think. Colored thermographs add a bit of clarity. They make it easy to describe privilege, for instance.

I owe a lot to Takagawa's clarity. I doubt if I would have gotten very far on my own without that. Like most players, I probably would have remained mired in deiri values, deciding between sente and gote by the seat of my pants. ;)

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 Post subject: Re: Sente, gote and endgame plays
Post #205 Posted: Tue May 08, 2018 2:01 am 
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Very interesting history!

RobertJasiek wrote:
Application I: From move 2 on, increasing or constant move values every second move identify a long gote sequence. Treat it like one move. Then, for the first move, verify the simple sente condition.

Application II: From move 1 on, for each 2-move sente sequence part, verify its simple sente condition. And verify increasing or constant move values every second move to link the parts.

Which application is correct?


Neither is correct. Currently, I am successfully testing my idea of increasing or constant gains.

Luckily and probably, this also means that we do not need my conjecture "Proposition 2" in forum/viewtopic.php?p=229609#p229609

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 Post subject: Re: Sente, gote and endgame plays
Post #206 Posted: Tue May 08, 2018 5:40 am 
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Bill, I would like to continue to encourage you to produce that endgame book you occasionally threaten to write! I get bits and pieces of the theory here and on Sensei's Library (and Robert has been very helpful in this thread by forcing you to clarify things :)) but I would really love to be able to work through it in a logical fashion, from fundamentals on up. I'm sure it would sell dozens of copies :)


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 Post subject: Re: Sente, gote and endgame plays
Post #207 Posted: Tue May 08, 2018 6:28 am 
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dfan, please see viewtopic.php?p=230911#p230911

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 Post subject: Re: Sente, gote and endgame plays
Post #208 Posted: Wed May 16, 2018 5:05 am 
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Assume a local endgame without complex kos, not doubly ambiguous, with Black's alternating sequence creating followers with the counts B1, B2, B3,... and White's alternating sequence creating followers with the counts W1, W2, W3,... Calculate the gains of the moves. Testing longer before shorter sequences worth playing successively, determine the count C and move value M of the local endgame so that M is at most each gain.

The method has a theoretical problem: we must show that there is only one solution. We must prove that two solutions (one for a longest black sequence, one for a longest white sequence) cannot exist. Maybe prove by contradiction. (A proof can rely on the already proven non-existence of local double sente.)

Have CGT or thermography already proven this unequivocality?

Bill, you often say that any assumption can be made for the type and values of the local endgame because contradictions occur until we find the correct values. Is there a proof why necessarily at least one contradiction occurs?

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 Post subject: Re: Sente, gote and endgame plays
Post #209 Posted: Wed May 16, 2018 6:38 am 
Judan

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RobertJasiek wrote:
Assume a local endgame without complex kos, not doubly ambiguous, with Black's alternating sequence creating followers with the counts B1, B2, B3,... and White's alternating sequence creating followers with the counts W1, W2, W3,... Calculate the gains of the moves. Testing longer before shorter sequences worth playing successively, determine the count C and move value M of the local endgame so that M is at most each gain.

The method has a theoretical problem: we must show that there is only one solution. We must prove that two solutions (one for a longest black sequence, one for a longest white sequence) cannot exist.


It is quite possible that the solution involves the longest sequence for each player. With thermography, without double ambiguity you can show that the solution for move values is unique. The solution for territory values is unique, anyway. That is easy to show, because the right wall cannot decrease as the temperature increases and the left wall cannot increase as the temperature increases. So when the scaffolds meet, we have the territorial count and the minimum temperature, and when they cross we have the maximum temperature. (That is not the case with ko thermographs, OC. ;))

Testing longest sequences first can be efficient, and you won't miss any reverses. :)

Quote:
Bill, you often say that any assumption can be made for the type and values of the local endgame because contradictions occur until we find the correct values. Is there a proof why necessarily at least one contradiction occurs?


If the assumptions are correct, no contradiction occurs. If you will notice, my pre-thermography methods always start with counts. Move values are derived, not assumed.

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 Post subject: Re: Sente, gote and endgame plays
Post #210 Posted: Wed May 16, 2018 8:51 am 
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This is very good news, thank you! For now, I have to believe you because I have not studied thermography enough to do the proof or imply it from an algorithm of drawing a thermograph. However, I find the underlying constructive reasoning ("the right wall cannot decrease as the temperature increases and the left wall cannot increase as the temperature increases") convincing.

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 Post subject: Re: Sente, gote and endgame plays
Post #211 Posted: Tue Jun 05, 2018 3:24 am 
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I thought to have understood tally. Now that I think about it more I am not sure any more.

We have Black's and White's sequences. For Black's sequence, we use Black's perspective of counting plays: b := the number of its black plays minus the number of its white plays.

For White's sequence, we can use either Black's or White's perspective. If we use Black's perspective, we define w := the number of its black plays minus the number of its white plays and might define the tally X := b - w. If we use White's perspective, we define w := the number of its white plays minus the number of its black plays and might define the tally X := b + w.

However, b and/or w can be negative. This raises the following questions:

1) If we use Black's perspective for White's sequence, is X := b - w well-defined or should it be X := |b - w|? Why is which well-defined?

2) If we use White's perspective for White's sequence, is X := b + w well-defined or should it be X := |b + w|? Why is which well-defined?

3) If tally is well-defined without taking the absolute value, we can have a negative tally. What does this mean for counts and move values? Are they negated indeed and, in particular, can we get a negative move value in this manner? How to interpret both?

Example 1:

Black's sequence has 1 black play so b = 1. White's sequence has 1 white play followed by 4 black plays. Using Black's perspective for White's sequence, we have w = 4 - 1 = 3 and X = b - w = 1 - 3 = -2. Using White's perspective for White's sequence, we have w = 1 - 4 = -3 and X = b + w = 1 + (-3) = -2. Taking the absolute value negates this to become X = 2.

Example 2:

Black's sequence has 1 black play so b = 1. White's sequence has 3 white plays. Using Black's perspective for White's sequence, we have w = 0 - 3 = -3 and X = b - w = 1 - (-3) = 4. Using White's perspective for White's sequence, we have w = 3 - 0 = 3 and X = b + w = 1 + 3 = 4.

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 Post subject: Re: Sente, gote and endgame plays
Post #212 Posted: Tue Jun 05, 2018 7:08 am 
Judan

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RobertJasiek wrote:
I thought to have understood tally. Now that I think about it more I am not sure any more.

We have Black's and White's sequences. For Black's sequence, we use Black's perspective of counting plays: b := the number of its black plays minus the number of its white plays.

For White's sequence, we can use either Black's or White's perspective. If we use Black's perspective, we define w := the number of its black plays minus the number of its white plays and might define the tally X := b - w. If we use White's perspective, we define w := the number of its white plays minus the number of its black plays and might define the tally X := b + w.

However, b and/or w can be negative. This raises the following questions:

1) If we use Black's perspective for White's sequence, is X := b - w well-defined or should it be X := |b - w|? Why is which well-defined?

2) If we use White's perspective for White's sequence, is X := b + w well-defined or should it be X := |b + w|? Why is which well-defined?

3) If tally is well-defined without taking the absolute value, we can have a negative tally. What does this mean for counts and move values? Are they negated indeed and, in particular, can we get a negative move value in this manner? How to interpret both?

Example 1:

Black's sequence has 1 black play so b = 1. White's sequence has 1 white play followed by 4 black plays. Using Black's perspective for White's sequence, we have w = 4 - 1 = 3 and X = b - w = 1 - 3 = -2. Using White's perspective for White's sequence, we have w = 1 - 4 = -3 and X = b + w = 1 + (-3) = -2. Taking the absolute value negates this to become X = 2.

Example 2:

Black's sequence has 1 black play so b = 1. White's sequence has 3 white plays. Using Black's perspective for White's sequence, we have w = 0 - 3 = -3 and X = b - w = 1 - (-3) = 4. Using White's perspective for White's sequence, we have w = 3 - 0 = 3 and X = b + w = 1 + 3 = 4.


A negative tally indicates a ko or superko. Which means that the ko threat situation matters.

The position can be a "number". (I put number in quotes because kos are not combinatorial games. But the ko threat situation is such that the position acts like a number.) For instance, suppose that b = 0 and w = 1, so that the tally is -1. Black to play is sente, and White to play lets Black take and win a ko. Let's also suppose that the result when Black plays first is 1 and the result when White plays first is 2. Then White should not play first, but let Black do so, for a score of 1. We evaluate the position as 1.

A negative tally can also indicate a mistake. Suppose that Black's play is actually gote, as is White's. The final scores are the same. Then we may find that at a high ambient temperature White should play to the Black ko, but at a low temperature White should win the ko instead, so that the final result is 1 instead of 2.

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Post #213 Posted: Tue Jun 05, 2018 8:27 am 
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So you are saying that we do not take the absolute so that we can appreciate negative tallies?

In your ko example with the black sente follower's count S = 1, white follower's count W = 2 and tally X = -1 , the initial count is given due to White's decision to avoid the mistake of local play to the larger, less favourable white follower's count W = 2 and choose the more favourable, smaller black sente follower's count S = 1. Hence the correct calculation of the initial count C is to inherit it from the sente follower's count S, that is, C = S = 1. If we calculate the move value as (S - W) / X = (1 - 2) / (-1) = 1 by dividing by the tally, it does, however, not express White's gain, as it would for a reverse sente. The move value is meaningless here (or expresses the contrary of the usual because of White's mistake)! Instead, we better calculate White's gain (a loss, to be honest) as G(W) := C - W = 1 - 2 = -1 (a negative gain so it is a loss, as promised). Have I understood this right?

Depending on study purposes, a negative tally need not indicate a ko or mistake but can also indicate a certain non-ko local playing which might be correct in a global context of ordinary tenukis.

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 Post subject: Re: Sente, gote and endgame plays
Post #214 Posted: Tue Jun 05, 2018 10:48 am 
Judan

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RobertJasiek wrote:
Depending on study purposes, a negative tally need not indicate a ko or mistake but can also indicate a certain non-ko local playing which might be correct in a global context of ordinary tenukis.


In that case you need to include the other positions as part of the analysis.

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Post #215 Posted: Thu Sep 20, 2018 10:05 pm 
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Bill, you have shown the following two multiples:





You rely on the CGT definitions of local temperatures and mean. Can you explain the relation between the definitions and their application, please?

You show application for a local endgame with one player's simple follow-up and without kos. For such, application is straightforward. How about local endgames with more complicated follow-ups and without kos? Is the method exactly the same, except for needing more multiples?

Can you write down your general method as a procedure applicable to all examples of a class?

Do we know a priori how many multiples we need at least? How do we find out the minimal necessary number of multiples? (Why) is it always 4 for a local endgame with one player's simple follow-up and without ko?

How would you define as a procedure the gote strategy and the sente strategy a) for a local endgame with one player's simple follow-up and without ko and b) for an arbitrary local endgame possibly with follow-ups and without ko?

For a local endgame with both players' follow-ups, how to determine for whom to test a sente strategy? Must a procedure be more complicated by possibly having to test either player's gote strategy and either player's sente strategy?

Suppose we test one player's gote versus sente strategies with the resulting counts G versus S. Does the more favourable count for the starting player determine the correct strategy? Is the count of one local endgame the resulting count of the more favourable strategy divided by the number of multiples? Why is this so explained by the CGT definitions? If we determine a local sente, is the reverse sente gain S - G (with counts calculated in favour of the starting player)? Why is this so explained by the CGT definitions?

Suppose we have a local endgame with follow-ups and without ko and have to test both players' gote versus sente strategies with the resulting counts Gb, Sb, Gw and Sw. How to determine the correct strategy, calculate the correct count, explain by the CGT definitions, possibly calculate a reverse sente gain and explain by CGT definitions?

For all cases, how to calculate the move value and explain it by CGT definitions?

In summary, your examples look convincing but how and why can they be generalised as methods?

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Post #216 Posted: Thu Sep 20, 2018 11:20 pm 
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Many questions, which I will gloss over, if you don't mind too much. ;)

In college, when I was trying to come up with a convincing argument for the equivalent of the Monty Hall problem, I realized that, instead of asking what the correct probabilities were, I could ask what the winning strategy was. People could argue about probabilities, but they could not argue about the results of the different strategies. OC, the winning strategy was consistent with a certain probability. Much later I used this strategy with various puzzles, and found, for instance, that the two envelope puzzle cannot be turned into a finite game. :)

With the method of multiples I had simply accepted the fact the mean value for sente games is a limit as the number of games goes to infinity. That's fine in terms of mathematics, but unsatisfactory for those whose eyes glaze over. Also, correct play in multiples is not always obvious, so proving that the mean value is correct is also not obvious.

However, this past winter I realized that you can compare strategies without worrying whether the result is correct. And since there are only two strategies with these simple positions, you can find out which (if either) is better and classify the position without calculating mean values or temperatures or doing any math except counting the score. :) Once you know the correct classification you can figure out the mean values and temperatures. :)

I am pretty sure that the correct number of multiples (sans ko, OC) is 2^d, where d is the depth of the tree. However, increasing depth increases the number of possible strategies, and increasing the number of options at the same depth also increases the number of possible strategies. Things can quickly become unwieldy.

Right now, I think that the main value of this method is to get the basic concepts across. Currently a lot of players resist the idea of local sente. The fact that the sente strategy wins in certain positions shows that local sente exist. A lot of players also resist the idea of sente and gote positions. I think that this illustrates that we can classify positions in that manner. :)

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Post #217 Posted: Fri Sep 21, 2018 12:03 am 
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Why do you think that many would be sceptical about local gote versus sente positions? There is no fundamental difference between a local position (not) having a sente move / sequence and the local position (not) being one having a sente move / sequence.

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Post #218 Posted: Fri Sep 21, 2018 12:24 am 
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Hi Bill,
Quote:
In college, when I was trying to come up with a convincing argument for the equivalent of the Monty Hall problem, I realized that, instead of asking what the correct probabilities were, I could ask what the winning strategy was. People could argue about probabilities, but they could not argue about the results of the different strategies. OC, the winning strategy was consistent with a certain probability. Much later I used this strategy with various puzzles, and found, for instance, that the two envelope puzzle cannot be turned into a finite game. :)
The MH puzzle is interesting to me (and to many others, apparently, who wrote passionately to what was it, Scientific American?) -- could you explain a bit more about "why people couldn't argue over the results of different strategies", using MH as an example. (For me, a nice way to explain MH, even to people with very little background, is the variation to change it to from 3 to a million doors, and then to open (1M-2) of them; I found this variation much easier to digest intuitively.)

I also don't know what's the 2-envelope puzzle. :)

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Post #219 Posted: Fri Sep 21, 2018 2:03 am 
Judan

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Hi, Ed. :)
The Monty Hall problem is usually presented in an imprecise way. In the late 70s or early 80s Monty, who had retired, was interviewed and asked about the Monty Hall problem. He had not heard of it. But his reply was that he could always manipulate the person to make the wrong choice. ;) Marilyn vos Savant presented the problem in one of her columns without restricting Monty's actions. ;) You have to assume that Monty always gives you the choice to switch doors. Anyway, even making that assumption she got letters from math professors telling her that her answer was wrong, that switching doors was a 50-50 proposition.

For those who have not heard of the problem, Monty Hall ended his show, Let's Make a Deal by offering a contestant the choice of three doors. Behind one door was a fabulous prize, behind each of the other two doors was a "goat", a prize you might not want to take. Typically after the contestant chose a door, Monty would open one of the other doors to reveal a goat, and then would ask the person if they wanted to switch doors and take what is behind the other unopened door. Whatever the person said, Monty would usually try to get them to change their mind, offering cash for them to do so. These extra offers are not part of the Monty Hall Problem. The question is, should you switch doors or not?

The usual answer given is that you should switch doors, because the probability is ⅔ that the fabulous prize is behind the other door. This is where the arguments arise. Many people will insist that it is a 50-50 choice, that the probability is ½.

What I realized in college was that arguing probabilities was usually futile, but you could easily show that the switching strategy paid off. You do not have to appeal to Bayesian probability or the Principle of Restricted Choice or anything except the idea that you have chosen the prize ⅓ of the time. Monty can always show you a goat behind one of the other doors, by assumption he must always do so. In two of the three cases switching gets the prize, in one it loses it, so switching is the superior strategy. Once you have found the superior strategy you can work out the probabilities. ;)

In the Two Envelope Problem you are informed, correctly, that each envelope contains a valid check that you can cash for a certain amount of money. One envelope contains a check for twice as much as the other one. You get to choose an envelope, open it, and then decide whether to take the other envelope. You choose an envelope and inside you find a check for $100. Should you take the other envelope? Logically, it seems that it does not matter. Half the time you will get more money, half the time you will get less. How much money it is should be irrelevant.

But there is an argument for switching. Half the time the other envelope will contain $200, half the time it will contain $50. So your expected payoff for switching is $125. Should you switch?

As against that there is the argument that if you compare the strategy of always keeping the envelope, no matter what is in it, with the strategy of always switching, their expected payoffs are the same. Should you switch?

Making this a finite game can spoil it. For instance, if the maximum payoff is $100 you should obviously not switch, if the minimum payoff is $100 you should switch. For the infinite game there are any number of good strategies that are better than always switching and never switching, but none known to be best. The first one I came up with was to switch if the amount was less than $1,000, otherwise not to switch. ;)

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Last edited by Bill Spight on Fri Sep 21, 2018 2:26 am, edited 5 times in total.
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 Post subject: Re: Sente, gote and endgame plays
Post #220 Posted: Fri Sep 21, 2018 2:06 am 
Judan

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RobertJasiek wrote:
Why do you think that many would be sceptical about local gote versus sente positions?


Experience. :(

Did you notice in our recent discussion how when I talked about the value of a position some people interpreted me as talking about the value of a play? Years ago on SL even good players with mathematical backgrounds insisted that sente and gote only applied to plays, not positions.

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