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Subject: Is it possible to have a pure abstract strategy game where no side is favored and it doesn't draw? rss

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Richard Hutnik
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I am under the impression in regards to my subject question that the answer is no. But is it?
 
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Russ Williams
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docreason wrote:
I am under the impression in regards to my subject question that the answer is no. But is it?

Assuming by "pure abstract strategy game" you mean "combinatorial game" in the sense of 2-player, no randomness, no hidden information, alternating turns (i.e. not simultaneous, not real-time), guaranteed termination (e.g. no infinite cycles), and no ties possible (i.e. exactly one player wins when the game terminates), then indeed the answer is no; it's a basic theorem of combinatorial game theory findable in any book on the subject (e.g. Winning Ways, Lessons in Play) that one of the following 4 statements is true of any position in any such game between players 'black' and 'white':

1. the player whose turn it is has a guaranteed win regardless of whether they're black or white

2. the player whose turn it is not has a guaranteed win regardless of whether they're black or white

3. the black player has a guaranteed win regardless of whether or not it's their turn

4. the white player has a guaranteed win regardless of whether or not it's their turn

(assuming competent play by the player, of course - in real life for "interesting" games, real human players will often not be sufficiently competent to invariably find optimal moves...)

PS: this is noted in the BGG wiki page combinatorial games.
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Thomas Chu

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In Go, the white player, who moves second, has a 5.5 points handicap (referred to as komi). Therefore, in general, there is no draw. However, the game can still end up as a draw because of repetition such as a triple ko. In practice, this happens like 1 in 1000. This may fit your criterion.

By the way, if one does not like the 5.5 komi, one can also use an auction system with players bidding secreting for the amount of komi that he or she would give to play black (the first player). This has been used in some international tournaments.
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christian freeling
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russ wrote:
docreason wrote:
I am under the impression in regards to my subject question that the answer is no. But is it?

Assuming by "pure abstract strategy game" you mean "combinatorial game" in the sense of 2-player, no randomness, no hidden information, alternating turns (i.e. not simultaneous, not real-time), guaranteed termination (e.g. no infinite cycles), and no ties possible (i.e. exactly one player wins when the game terminates), then indeed the answer is no.

It is known as Zermelo's Theorem. The wiki link is:
http://en.wikipedia.org/wiki/Zermelo's_theorem_(game_theory).

It proves the existence of a winning strategy, but usually holds no clues with regard to it. Some games have been solved strongly, like Pentago, just recently. It means that the truth of any position in the game tree is known and the implied strategy is executable.

As I've pointed out previously, Symple is no exception, but given its size, its move protocol and its embedded balancing mechanism, the answer to the question "Is it possible to have a pure abstract strategy game where no side is favored and it doesn't draw?", though it is No in a game-theoretical context, might be considered Yes where actual play is concerned.
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Russ Williams
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Right, I assume Richard was asking about theory, not practice.

Yeah, there are plenty of games where there's no visible statistically significant advantage for the first or second player, in practice.

E.g. I recall reading an analysis of Arimaa based on very many (tens of thousands? more?) archived games showing no statistically significant advantage for the first or second player.
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christian freeling
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russ wrote:
Right, I assume Richard was asking about theory, not practice.

I suspect Richard was asking a question to which he knew the answer.

russ wrote:
Yeah, there are plenty of games where there's no visible statistically significant advantage for the first or second player, in practice.

Countless indeed, but Symple is simple and homogeneous and its balancing rule would appear to be very refined. I would say 'more so than a pie-rule', if that wouldn't bring me on a slippery slope because the truth of any position is either a win or a loss. So if I say that, the context should be considered to be games between humans or less than perfect bots.

But I agree, even in the category of simple homogeneous games, there are plenty of games where there's no visible statistically significant advantage for the first or second player.
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Russ Williams
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christianF wrote:
russ wrote:
Right, I assume Richard was asking about theory, not practice.

I suspect Richard was asking a question to which he knew the answer.

I confess I am surprised at the question, whether it was about theory or practice, since the answer seems well-known in either case... Perhaps I am missing something.
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Michael Boggs
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Hex?
 
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Mo Bell
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Yes!
Numhex (a new hex variant based on numbers and infinite moves) I invented is drawless and does not advantage any side.

 
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Moshe Callen
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The theorem argues that a purely combinatorial game must either have an advantage for one of the two players or allow a draw. Fine.

If we make the game favor neither player, how improbable can we make a draw scenario?

If we make draws impossible, how negligible can we make the advantage of either player?
 
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Moshe Callen
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Massakra52 wrote:
Yes!
Numhex (a new hex variant based on numbers and infinite moves) I invented is drawless and does not advantage any side.


So you can prove the cited theorem wrong? Fantastic! Can you show us?
 
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Markus Hagenauer jr.
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beanieboy007 wrote:
Hex?


No, the pie rule is an advantage for the second player. As there is no draw, one colour must have a winning strateyg for sure. The second player can choose this colour and will win (if he plays perfect).
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Russ Williams
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Ah, the communication problem of ambiguity in a thread where it's unclear if people are talking about theory or practice and both are being discussed in parallel...

I assume Hex and Numhex were mentioned as examples of drawless games which are balanced in practice (with sufficiently large boards) since surely we all understand and agree that drawless pure abstract strategy games favor one position or the other in theory... right? ...
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christian freeling
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whac3 wrote:
The theorem argues that a purely combinatorial game must either have an advantage for one of the two players or allow a draw. Fine.

If we make the game favor neither player, how improbable can we make a draw scenario?

If we make draws impossible, how negligible can we make the advantage of either player?

I'm sure I'm not breaking any new ground for you here, but for clarity's sake it may be good to realise that "advantage", though commonly and rightfully used in the theory of many combinatorial games, is a term that has no meaning in game theory as a mathematical discipline, because the truth of any game position can only be a win for one player or a draw.
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Moshe Callen
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christianF wrote:
whac3 wrote:
The theorem argues that a purely combinatorial game must either have an advantage for one of the two players or allow a draw. Fine.

If we make the game favor neither player, how improbable can we make a draw scenario?

If we make draws impossible, how negligible can we make the advantage of either player?

I'm sure I'm not breaking any new ground for you here, but for clarity's sake it may be good to realise that "advantage", though commonly and rightfully used in the theory of many combinatorial games, is a term that has no meaning in game theory as a mathematical discipline, because the truth of any game position can only be a win for one player or a draw.

True but there have at least been attempts to try to mathematically describe advantage, although none successfully I'm aware of.
 
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Mo Bell
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whac3 wrote:
Massakra52 wrote:
Yes!
Numhex (a new hex variant based on numbers and infinite moves) I invented is drawless and does not advantage any side.


So you can prove the cited theorem wrong? Fantastic! Can you show us?


You surely know Mertens summatory function.
It is not random function.
Sometimes it is negative sometimes it is equal to zero and sometimes it is negative. Until now we can not predict how it behaves.
Now imagine an abstract game starting from state 0 to state k.
Each state corresponds to some round i (i varying from 0 to k).
At the end of each round there is always only one winner. The game continues indefinitely starting each time from the previous state. As k grows to infinite the game goes to draw state (like summatory Mertens function).

I let you work with this new concept.

Good luck!
 
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Moshe Callen
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Massakra52 wrote:
whac3 wrote:
Massakra52 wrote:
Yes!
Numhex (a new hex variant based on numbers and infinite moves) I invented is drawless and does not advantage any side.


So you can prove the cited theorem wrong? Fantastic! Can you show us?


You surely know Mertens summatory function.
It is not random function.
Sometimes it is negative sometimes it is equal to zero and sometimes it is negative. Until now we can not predict how it behaves.
Now imagine an abstract game starting from state 0 to state k.
Each state corresponds to some round i (i varying from 0 to k).
At the end of each round there is always only one winner. The game continues indefinitely starting each time from the previous state. As k grows to infinite the game goes to draw state (like summatory Mertens function).

I let you work with this new concept.

Good luck!

So its limit is a draw state. That does't disprove anything then.
 
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Mo Bell
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whac3 wrote:
Massakra52 wrote:
whac3 wrote:
Massakra52 wrote:
Yes!
Numhex (a new hex variant based on numbers and infinite moves) I invented is drawless and does not advantage any side.


So you can prove the cited theorem wrong? Fantastic! Can you show us?


You surely know Mertens summatory function.
It is not random function.
Sometimes it is negative sometimes it is equal to zero and sometimes it is negative. Until now we can not predict how it behaves.
Now imagine an abstract game starting from state 0 to state k.
Each state corresponds to some round i (i varying from 0 to k).
At the end of each round there is always only one winner. The game continues indefinitely starting each time from the previous state. As k grows to infinite the game goes to draw state (like summatory Mertens function).

I let you work with this new concept.

Good luck!

So its limit is a draw state. That does't disprove anything then.


Limit is draw state does not mean it reach draw state.
(1/n) goes to zero when n goes to infinite BUT it will never be equal to zero.
 
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Moshe Callen
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but again you're still overstatng your own claim.
 
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Mo Bell
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Numhex is played on (at least) 11x11 hex board.
So the only innovation is on numbered pawns with a new protocol of moves.
Anyone who has understood my new concept can build dozens of similar games.
Have good day.
 
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Steve
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Rock Paper Scissors - with an extra round if the round is a draw.

So, you don't take turns, but does that mean it's not a pure abstract strategy game?
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Nick Reymann
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slashing wrote:
Rock Paper Scissors - with an extra round if the round is a draw.

So, you don't take turns, but does that mean it's not a pure abstract strategy game?


Even simpler, and with no draws period, is Matching Pennies:

One player is "Same", the other "Different". Each selects either Heads or Tails simultaneously. If both faces are the same, "Same" wins. If they are different, "Different" wins.
 
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Nick Reymann
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Massakra52 wrote:
Yes!
Numhex (a new hex variant based on numbers and infinite moves) I invented is drawless and does not advantage any side.


What are the rules? I'm interested.
 
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Russ Williams
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slashing wrote:
Rock Paper Scissors - with an extra round if the round is a draw.

So, you don't take turns, but does that mean it's not a pure abstract strategy game?

A pure abstract strategy game consists of alternating turns. (Of course it's trivially easy to define fair games with simultaneous turns.)

Also RPS is not guaranteed to terminate, though of course it probably will terminate within a few rounds if the players are not intentionally making it last forever.
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Steve
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russ wrote:
slashing wrote:
Rock Paper Scissors - with an extra round if the round is a draw.

So, you don't take turns, but does that mean it's not a pure abstract strategy game?

A pure abstract strategy game consists of alternating turns.

Ahh, my mistake; I didn't realize that was part of the definition.
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