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First Move Advantage in Pente
Posted:
Jun 29, 2011, 4:30 PM
Hi, guys,
I have reviewed a bit of the Universal First Move Advantage topic. I apologize in advance if this rehashes old territory. I'm not trying to argue that P1 will not always have an advantage.
Here is the question: On P1's second move, at what distance will the first move advantage be made negligible?
Here is a brief synopsis of the thought process:
1. If P1 forgoes the second move, then P2 gains the advantage.
2. If the board were infinite, there would be a distance at which the advantage would shift to P2, at least in practical terms (say 180 moves).
3. At what distance would the P1 advantage be negligible in practical terms?
The puzzle for me in this is that the tournament rule requiring P1's second move to be at least four points away is for the purpose of reducing the P1 advantage. Should consideration be given to trying other distances for P1's second move in order to reduce the advantage?
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Re: First Move Advantage in Pente
Posted:
Jun 29, 2011, 10:20 PM
in terms of P1's stones, the second must be such that a third stone can connect all 3 into a build-able pattern (or at the very least something of a 3, or post 3 ext, but this only works vs a P2 stone it can attack with its 3, long range P2 placements not reachable by the 3 attack will trump a 3 by basic principle, meaning P2 will simply make a structure before P1 can, and P1's 3 will fail to divert P2 by way of key stone capture). however even when this is done, some of these 3 stone shapes will fail depending on where P2's first stone was placed. some second moves can handle any thing, and some second moves only work vs certain P2 first moves. if P2 squanders its second move then the placement of P1's second doesn't matter as long as it's 3rd is well placed.
F9 for P1's second can work some times, because H11 for P1's 3rd will tie them all together into a build-able structure. however this building shape can be trumped dependent upon where p2's early stones are in comparison. thus, when choosing a second and third move we generally design it around P2's move. so, P2 asks a question, then P1 answers, over and over again. there are several things to consider, but one is the basic threat of K10 being attacked via capture and having P1's building diverged into blocking P2's 4 in a row, resulting in P2 snatching away either the initiative, or perhaps just a winning tempo in position. i could continue to try and answer this further, however from here with out me putting visual examples it may become confusing to the reader.
Scire hostis animum - Intelligere ludum - Nosce te ipsum - Prima moventur conciliat - Nolite errare
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Re: First Move Advantage in Pente
Posted:
Jun 29, 2011, 10:47 PM
1. If P1 forgoes the second move, then P2 gains the advantage.
yes, correct.
2. If the board were infinite, there would be a distance at which the advantage would shift to P2, at least in practical terms (say 180 moves).
this is over kill. a larger board is not needed to prove the point. a box can be made to show a basic death zone for P1. the following 4 lines make this basic death zone box where you can't move on it nor beyond it.
P line 15 line E line 5 line
now, inside of this box, any of these places can win vs certain P2 1st moves. but some of these places will lose if P2's first move is correctly placed. beyond those P1 moves found inside of the box that can only win sometimes, there are some moves that will win always. so i suppose these two types of moves with in the box could be divided into 2 categories. i suppose.
the number of P1 seconds that some times win out weigh the number of P1 seconds that always win.
one of our best examples of a P1 move that always wins is N10. N10 can handle anything thrown at it.
3. At what distance would the P1 advantage be negligible in practical terms?
again, as far as distance refer to the box stated above. however distance is not the only factor. the other factor is the placement of P2's first stone. with in the box and away from the outer death zone void surrounding it, there are still principles and rules of logic to follow in determining if P1's advantage is becoming negligible or no.
example, if P2's first move is K9 and P1 goes K14, then P1 loses. but, if P2's first move is N8 or K8 for examples, and P1 goes K14 then P1 now has the win.
so sometimes K14 wins, and sometimes loses. it just depends on P2's placement.
however there are some P1 second moves that are always a win no matter where P2's first stone was.
thus we seem to have 2 different types with in the box as far as P1's second move choices.
confused yet? lol
Scire hostis animum - Intelligere ludum - Nosce te ipsum - Prima moventur conciliat - Nolite errare
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Re: First Move Advantage in Pente
Posted:
Jun 30, 2011, 12:11 AM
On P1's second move, at what distance will the first move advantage be made negligible?
in short; a 5 jump, I.e; P10
your question was of distance so there you go.
when distance is removed from the equation, meaning white has moved in close enough, the question changes. the question then would be, which P2 firsts trump which P1 seconds?
the answer to this becomes lengthy and requires much study. i along with several others have done much of the ground work to answer most of these with in reason, and some are still up in the air due to lack of research.
Scire hostis animum - Intelligere ludum - Nosce te ipsum - Prima moventur conciliat - Nolite errare
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Re: First Move Advantage in Pente
Posted:
Jun 30, 2011, 6:07 PM
Thanks for the information. It will take a bit of time for me to digest. After the post I was thinking a bit more on the issue and was approaching the conclusion that you state here.
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Re: First Move Advantage in Pente
Posted:
Jul 1, 2011, 4:22 PM
This is a good question. Indeed, the Tournament Rule was created in an attempt to eliminate the P1 advantage. However, shortly afterwards (early 80s), the expert players agreed that P1 still had the advantage, but at least the rule does do a fine job of significantly reducing that advantage to the point where it requires expert play to cash in on this edge -- since it was so effective and is a very simple rule change, it basically became standard for competitive Pente. This style was technically termed Pro-Pente, but because it is now used so commonly people just refer to it as Pente (with the Tournament Rule). In the early 80s, once it was agreed that the Tournament Rule was still somewhat advantageous for P1, several more variations (other rules changes) were invented and discussed as possible alternatives for future competitions. Of these, the main ones were Keryo-Pente, which I believe was created by Rollie Tesh and perhaps a few others, and D-Pente, which was created by Don Banks. While Keryo-Pente can be more complex, it is generally considered too different of a game to still be called competitive Pente and so it has become a new game in its own right that people can learn to play. D-Pente involves a swap rule, which, quite frankly I'm personally very surprised that this has not caught on a lot more among the expert players. This simple variation does the best job of reducing either player's advantage, and yet it is still not used in major Pente competitions and is rarely played even in exhibition matches. Of course, other variations could be created that use multiple swaps, and/or swaps at different points in the game to further even things out, similar to how Renju or other stone games are played at the highest levels, but part of the appeal of Pente is the simplicity of the rules and the more you get away from that, the more you get away from the original intent of the game as invented in the late 70s by Gary Gabrel.
(Here's a great article that I had not seen before, at least I don't remember reading it. Don't miss the links to all 5 pages of the article! http://www.inc.com/magazine/19830701/1772.html )
Anyways, more to your question. As discussed in some recent threads at this site, it is strongly theorized, based on certain math and logic concepts, that there is no point within any possible game of Pente at which there is no advantage for either player. In other words, at all times, one player or the other always has an advantage and will win the game if it continues to the end without error. Whether or not certain positions contain "more" or "less" of an advantage is purely a matter of perception -- in terms of the inherant complexity of the position and the ability of the human mind to correctly analyze it. However, mathematically speaking, it's black and white ... P1 either does or it does not have an advantage -- or more correctly, either P1 has the advantage or P2 has the advantage. There is no such thing as the degree of advantage except psychologically.
In terms of the distance of the P1 second stone, in general terms it could be described like this: Without the Tournament Rule P1 has the advantage after a reasonable second move just about every time. With the Tournament Rule, moving three spaces away (2 gapper), P1 has the advantage most of the time. Moving 4 spaces away (3 gapper), P1 sometimes has the advantage. Moving more than 4 spaces away, P1 almost never has the advantage (in which case, P2 has the advantage). No distance exists which would "eliminate" the advantage for both sides.
However, in terms of your line of thinking, there is another variation available at this site which was created quite a bit later than the others. It is called G-Pente, invented by Gary Barnes perhaps 10 - 15 years ago. Again, it's a bit surprising that this has not caught on more among expert players and in competitions. This variation includes the Tournament Rule, and in addition, it prohibits P1 from moving exactly 3 or 4 spaces (2- or 3-gappers) away from center in a horizontal or vertical direction, which encourages generally more complex "off-axis" second moves. This increased complexity gives the perception of further reducing P1's advantage while still keeping the rules of the game relatively simple. This might be the variation you're looking for.
Re: First Move Advantage in Pente
Posted:
Jul 2, 2011, 11:08 AM
up2ng: Anyways, more to your question. As discussed in some recent threads at this site, it is strongly theorized, based on certain math and logic concepts, that there is no point within any possible game of Pente at which there is no advantage for either player. In other words, at all times, one player or the other always has an advantage and will win the game if it continues to the end without error. Whether or not certain positions contain "more" or "less" of an advantage is purely a matter of perception -- in terms of the inherant complexity of the position and the ability of the human mind to correctly analyze it. However, mathematically speaking, it's black and white ... P1 either does or it does not have an advantage -- or more correctly, either P1 has the advantage or P2 has the advantage. There is no such thing as the degree of advantage except psychologically.
It is also strongly theorised in the opposite case. However strongly some may feel about this point, mathematically this is not black and white at all. No mathematical proof has been produced for arguments either for or against.
As for the psychological point, in practice, it may be just as unlikely that someone can perceive a situation that is perfectly balanced as such, as it is likely for them to perceive it as perfectly balanced when it isn't. Also, it may be that such balanced situations are so unlikely as to make the point moot, and so they need not be considered in practice, however this does not in itself preclude the possibility of it. As far as I can tell, no conclusive logical argument has been made despite extensive effort.
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Re: First Move Advantage in Pente
Posted:
Jul 2, 2011, 7:54 PM
"...mathematically this is not black and white at all. No mathematical proof has been produced for arguments either for or against. "
Incognita, that's true, there does still seem to be some debate about this concept. That's why I carefully chose the term "theorized" as opposed to something like "fact" or "law" (think in terms of the difference between Theories and Laws in science). I agree that no formal mathematical proof has been created to support the concept. To do so would likely require a PhD level thesis paper that is written by someone specializing in game theory and logic. For example, a mathematical proof at this level has been done that "solves" Gomoku (I've lost the link to it, but it can probably be found with search engines).
I will say that I have a strong understanding of the game and that I personally know this concept to be fact, but I don't have the time or the desire to create such a proof, and I'm not even sure how to go about writing one anyways.
As for my use of the description that it's "black and white" -- I could have probably used a better phrase. I was actually trying to describe the logic state of something being either "off" or "on" and there is nothing inbetween, like a switch. Like bits in computer memory that are either 0 or 1, there is no other option and they are not designed to be able to hold any other option. There is no 0.913 which might be "strongly on-like". It either is or it isn't. So, when speaking of advantage in this game purely mathematically -- it either is or it isn't. There is no "slightly advantageous" as opposed to "strongly advantageous". In both cases, the player with the advantage wins if the game is played out without error. The result is the same, and if a computer were able to see all possibilities until the end of the game and saw that it will win the game, it would not care about complexity or any other descriptor which would make humans "feel" that the position is "more balanced" than another position -- to a computer that already knows the outcome, either it has the advantage or it does not. That's what I meant by the phrase "black and white", even though saying it that way put things into a bit of a gray area (hehe).
Re: First Move Advantage in Pente
Posted:
Jul 3, 2011, 2:30 AM
up2ng,
Thanks for that Gary Gabrel article. I had not read it either. I think it is worth pointing out that at basically the exact same time that the article was being written, not published-but written, Gabrel sold the game to Parker Brothers for $1 million if memory serves. Unfortunately they didn't promote the game the way that Gabrel did, and the "world" championships ended, and soon the Pente boom ended too. At least until the internet changed things a bit....
Re: First Move Advantage in Pente
Posted:
Jul 3, 2011, 5:48 AM
Hi up2ng, I thought I'd just pop in briefly to respond to this. If for no other reason then to offer an alternative view and perhaps stir up a hornet's nest again!
I think the term you are looking for is 'discrete', meaning that by your definition of advantage there are only two possible states and no continuity from one to the other.
Consider though, how many moves might be required in order to convert an advantageous position into a win. Setting aside the specifics of Pente for a moment, in a given zero sum game we can easily see that without a concrete mechanism that forces the game to finish after N moves, and assuming an infinite playing surface and infinite life-spans, there exists the potential for a situation to exist that requires an infinite number of moves to convert such an advantageous position to a win. Using these assumptions we could state that the advantage could never be realized and hence is no advantage at all - or at least a very (infinitesimally) small advantage... .
Mathematically the advantage A could be restated as being Win-able Advantage WA, with the Advantage A being held as proportional to its conversion factor CF. The conversion factor may be comprised of several components but for the moment we will define it as the number of moves N required to convert A into a win (A => W).
A => W requires A to be a 'win-able' A or WA, proportional to the conversion factor CF
WA = A/CF.
As CF tends toward infinity, WA becomes increasingly small. We can say that the limit of the function F(x), where F(x) = A/CF is 0. Thus in a game requiring infinite moves to complete, there could exist the possibility of a zero advantage. In other words the win-ability of A, WA would be zero. WA = 0.
Obviously Pente comes with certain constraints, not least of which is that it does not have an infinite playing surface and there are features of the game such as a capture limit that ultimately ensures a game will finish in a finite amount of time. So, it may be that there always is a way - with perfect play - to convert a position within a finite time.
Yes, ultimately, if a Pente has a finite length, then you can say that the end is still predetermined, and counting the number of moves required to convert is splitting hairs, and hence CF cannot be infinite. Therefore by the above equation F(x) > 0, (i.e. Advantage is always greater than zero).
But a practical measure of advantage ought to be meaningful for human play and thus take into account the capacity of a player to not only convert to a win in a reasonable time, but to be able to perceive that such an advantage exists in the first place, plus the ability to identify specifically what aspects of the position constitute that advantage, and also the ability to decide on the perfect series of moves that will exploit it. Under this definition, a strong advantage would be one where the win is obvious and quickly executable. A weak advantage would be where a long and difficult to discern series of moves is required.
To go back to our equation then, we could define CF as being not simply the number of moves required to covert to a win, but the capacity of a player to do the converting. This could be equated say, to the obscurity of the advantage OA and the capacity of the player CP to be able to find the exploit and execute it. Thus instead of the conversion factor being the number of moves played in searching for a way to convert the advantage, we now define it as being proportional to CP with respect to OA thus we have CF = CP/OA.
F(x) CF = CP/OA
Clearly if the obscurity of the position's advantage is very large, and the capacity of the player is small then CF will be very large. Without a formal definition of obscurity we can only surmise that it might be possible for the obscurity of a position to be beyond human capacity to identify and thus perfect play would be impossible. So it follows that if OA is beyond human skill than in practice OA is infinite. By definition then CF would be infinite.
So in practice our original function would be limited not by the number of moves required to seal the deal, but by human performance levels.
Theoretically then the limit of F(x) WA = A/CF for human performance is 0. So there could exist situations for which no win-able advantage exists.
As for the question of whether a balanced position could be achieved, such a question would be dependant on your definition of balance. But, by the above definition of win-able advantage, if no win-able advantage exists then the position must be balanced.
I would produce a full description of the game-theoretical calculus of this problem space if I had an infinite amount of time to do it. But alas....
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Re: First Move Advantage in Pente
Posted:
Jul 3, 2011, 10:49 AM
for the readers whom are confused and unable to absorb all this technical stuff.
i think we can maybe sum it up like this;
we can't currently show a formula proving that P1 has the winning advantage in pente, but it is something through experience as humans once at a certain level of understanding, we just know it to be true, with out tangible proof being required.
Scire hostis animum - Intelligere ludum - Nosce te ipsum - Prima moventur conciliat - Nolite errare
Re: First Move Advantage in Pente
Posted:
Jul 3, 2011, 3:42 PM
Mine is a game-theoretical argument which may not actually apply to real Pente as played. I am not suggesting that what I wrote is any kind of mathematical proof, but rather a framework for understanding the question.
The Reds are the experts in actual play. But just remember that history is full of things that people just 'knew' to be true which later were shown not to be.
