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Subject: On the pie rule and its use to balance 2-player abstract games rss

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Laurentiu Cristofor
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So, I noticed the use of the pie rule as a balancing method in various board game situations; most recently, I've seen it mentioned as a way to counter the first player advantage in games like Hex or Twixt.

But is this a proper use of the pie rule of is it just a way to unload the balancing task to the players?

I think of the original use of a pie rule as being a clever contract that ensures a fair outcome for both parties that are trying to divide a pie among themselves, by separating the action of dividing a pie slice from that of selecting a pie slice. Many contracts require a third party that is trusted by both parties, to play the role of arbitrator. But the pie rule's cleverness is that is only requires the two parties to agree to the terms of its contract.

But this contract only works as intended when the pie can indeed be cut in half and when the players can easily recognize a larger slice.

But applying this to an abstract game raises several issues:

d10-1 There may be no fair first cut. The first move advantage may be so strong that you either use it or you give it away.

d10-2 If players are of different skill, we have two situations.

d10-2d10-1 If the weaker player starts, then he suffers the burden of making a neutral move with less knowledge.

d10-2d10-2 And if the stronger player starts, he could make moves that are winning but require exact and difficult playing that is harder to pull off by a weaker player - thus trapping the weaker player into having to choose either a weaker side or a side that they don't know how to play well.

In the end, this approach seems to offload the task of balancing the game to the players (a kind of game design externality). But if there were clear balancing moves, why do they not form standard opening positions that players should start the game from? Sort of like Othello's opening setup.
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Russ Williams
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Laurentiu wrote:
But if there were clear balancing moves, why do they not form standard opening positions that players should start the game from?
I'd guess a couple reasons:

1. For opening variety, and to let players try "trick move" openings (i.e. that could be considered a feature, not a bug).

2. Especially for older games, the game's creator may not have known which openings are fairest! (Nowadays computer analysis could help here before a game is published...)

But indeed your question seems good and thought-provoking. Will ponder further...
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Stephen Tavener
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Agreed; the pie rule is a nice - and sometimes necessary - balancing mechanism with two equally skilled players, otherwise it's more likely to be a skill multiplier. If one player is significantly weaker than the other, the weaker player should take the first play and the pie rule should not be used.

As for the balanced opening positions, I have a few objections:
- reduces variety in play
- increases rules complexity; Renju is a good example
- deprives the players of an element of discovery; ideally openings should emerge from the game naturally, not be forced on the players from the designer
- future-proofing; whatever we think when we design a game, will be proven wrong as player strength improves.
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Nathan James
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Inasmuch as Hex is always a win or a loss, there certainly is not a fair first cut. And if we consider the pie rule as part of the game, (why would we not?) then we can say that the player who does the choosing has a winning position every time. He just may not know whether his winning move is to play the light or dark pieces.

When a stronger and weaker player play together the stronger player can allow the weaker to play strong first moves and simply choose not to swap. While not designed for it, it has a possibility of functioning like a handicap mechanism.

In my opinion, the real beauty of the pie rule is that because it puts the balance in the player's hands, they can continually refine it. That makes the pie rule pretty nearly ideal for serious play. It will adjust as strategy develops, and is adaptable to any surprises.

Furthermore, as understanding of the game grows, serious players add more and more opening moves to the soft-banned list, which continually refreshes the game as new openings are explored.

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Richard Moxham
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NJames wrote:
Inasmuch as Hex is always a win or a loss, there certainly is not a fair first cut. And if we consider the pie rule as part of the game, (why would we not?) then we can say that the player who does the choosing has a winning position every time. He just may not know whether his winning move is to play the light or dark pieces.

When a stronger and weaker player play together the stronger player can allow the weaker to play strong first moves and simply choose not to swap. While not designed for it, it has a possibility of functioning like a handicap mechanism.

In my opinion, the real beauty of the pie rule is that because it puts the balance in the player's hands, they can continually refine it. That makes the pie rule pretty nearly ideal for serious play. It will adjust as strategy develops, and is adaptable to any surprises.

Furthermore, as understanding of the game grows, serious players add more and more opening moves to the soft-banned list, which continually refreshes the game as new openings are explored.

Very much this! I think everybody agrees that the rule puts the issue in the players' hands, but that's not a cop-out, as Laurentiu argues. It's the whole point.

And I would further say that for mortal purposes, and for any game remotely in the same complexity department as Hex, the existence or otherwise of a guaranteed theoretical win is immaterial.

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Nathan James
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mocko wrote:


And I would further say that for mortal purposes, and for any game remotely in the same complexity department as Hex, the existence or otherwise of a guaranteed theoretical win is immaterial.

Immaterial to what, exactly? It's impossible to think clearly and accurately about such games without recognizing that quality of them.
 
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christian freeling
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Laurentiu wrote:
In the end, this approach seems to offload the task of balancing the game to the players (a kind of game design externality). But if there were clear balancing moves, why do they not form standard opening positions that players should start the game from? Sort of like Othello's opening setup.
Othello's opening setup was born out of necessity rather than choice. And I have no information about turn order advantage but to compare a rather capricious flipflop game with the permanency of moves in Hex seems hardly fruitful. Certainly not where the pie rule is concerned because in Othello all first moves are equivalent.

The games we discuss here are all completely determined in a mathermatical sense. But 'balance' is a human criterion. There's nothing balanced in the tree - it's just a tree and it contains information about every possible position in terms of win/draw/loss. 'Balance' is in the same category as advantage: there is no 'advantage' in the tree.

So if a game seems slightly unbalanced by turn order, why is it so wrong to have a balancing mechanism based on the players' insight? I can't see how that is 'offloading the task of balancing the game'. And the implication seems to be that every game could be balanced 'objectively'. I'd like to see suggestions of how to go about it.

Meanwhile I wrote a little piece a while ago called Balancing protocols in symmetric two-player games. It features several mechanisms that 'offload' the task in very interesting ways.

 
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Nathan James
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The Chess with pie rule discussion might be of interest.
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Richard Moxham
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NJames wrote:
mocko wrote:


And I would further say that for mortal purposes, and for any game remotely in the same complexity department as Hex, the existence or otherwise of a guaranteed theoretical win is immaterial.

Immaterial to what, exactly? It's impossible to think clearly and accurately about such games without recognizing that quality of them.
Sorry to have been a bit cryptic. What I meant was pretty much exactly what Christian has gone on to say.

 
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christian freeling
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NJames wrote:
The Chess with pie rule discussion might be of interest.
We're about to come up with something better.
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David Bush
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Laurentiu wrote:
So, I noticed the use of the pie rule as a balancing method in various board game situations; most recently, I've seen it mentioned as a way to counter the first player advantage in games like Hex or Twixt.

But is this a proper use of the pie rule of is it just a way to unload the balancing task to the players?
Right there is where you lost me. How is it improper use of the pie rule, to unload the balancing task to the players?
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Nathan James
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christianF wrote:
NJames wrote:
The Chess with pie rule discussion might be of interest.
We're about to come up with something better.
Don't tell anyone, but in that old thread, I used Chess as a gimmick to create conversation about the pie rule. Worked beautifully, by the way.

Threads that open with thoughtful observations and nuanced questions get no action. On the other hand, if you invite people to start throwing out opinions about Chess...
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Laurentiu Cristofor
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twixter wrote:

Right there is where you lost me. How is it improper use of the pie rule, to unload the balancing task to the players?

Good question.

Also, let me clarify that I'm not exactly sure what I am asking about - I am trying to figure that out myself. I'm just feeling a bit uncomfortable with how the pie rule is being proposed and quickly adopted as a solution. I thought that sharing some of the thoughts here might help me figure out more about what the problem is. I was also wondering if maybe others already considered this aspect and came to some conclusions.

Maybe what I'm asking for is: what are the situations when a pie rule can redress a balance issue? What is an appropriate use of the pie rule?

Coming back to your question: if I invent an obviously imbalanced game and then I would say: try balancing it with the pie rule - would that be appropriate? I can imagine some situations in which the answer would clearly be no.

For example, let's say the game is "first player picks a number between 1 and 10, the second player picks a different number; the player that picked the largest number wins". I know, it's stupid, but at least, in this case the rule can change something if the first player goes for "10".

But then we could also have a game in which the first player is doomed from the start and the pie rule wouldn't change anything because the second player would always choose to stay second. Say: Go on a 2x2 board - second player will always win. Just another example of an imbalance that cannot be fixed with the pie rule.

So it seems that at least the game imbalance should not be very extreme.

And Nathan's Chess question reminded us that nobody bothered to use the pie rule in Chess, despite the fact that White is known to have an advantage and even given that only a subset of first moves are preferred by experienced players. So it looks like people don't mind tiny imbalances.

Is there perhaps a certain range of game imbalance for which the pie rule might help? For example, with my stupid games at one end of the spectrum and with games like Chess towards the other end?

Maybe Go could be a good experimentation subject, because we could vary the board sizes from 2x2 to 19x19 and beyond. Is there perhaps a range of small boards where the pie rule might be more effective than using komi, for example? Come to think of it, why is the pie rule not used *instead* of komi for 19x19 boards?

Hope this clarifies, if not what I am asking, at least the kind of thoughts that I was having about this topic.
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christian freeling
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Laurentiu wrote:
I was also wondering if maybe others already considered this aspect and came to some conclusions.

Maybe what I'm asking for is: what are the situations when a pie rule can redress a balance issue? What is an appropriate use of the pie rule?
I already mentioned Balancing protocols in symmetric 2-player games.

Laurentiu wrote:
Coming back to your question: if I invent an obviously imbalanced game and then I would say: try balancing it with the pie rule - would that be appropriate? I can imagine some situations in which the answer would clearly be no.
How do you invent a (non-trivial) 'obviously imbalanced game' if all but a few games discussed here are symmetric?

Laurentiu wrote:
So it seems that at least the game imbalance should not be very extreme.
Barring bad design turn-order imbalance is usually limited. Whether it is in want of remedy and whether the pie-rule would be successful depends on the game. In placement games the chance that it is effective does seem to be greater than in movement games.
But there are clear exceptions like Othello. But it's hard to tell if there's any turn-order imbalance in Othello in the first place, and if there is (of course there is: the tree, being 'completely determined' and all that, but what's it to humans?) then who has the 'advantage'? And there's also the practical problem of all opening moves being equivalent.

So the pie rule works for a limited number of specific games and it certainly lacks the generality that sometimes is suggested.
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Nathan James
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Quote:
Is there perhaps a certain range of game imbalance for which the pie rule might help?
Not really. The key is that there need to be a set of possible opening moves which are close enough to balanced that the players can't tell the difference, or at least don't mind playing from that position. Pie rule is useful when the game functions, but there exists a strong enough first move that the game will quickly become stale.

Suppose we could list the various opening moves in Hex and label them according to how strong both players need to be before they see clearly how to win from that opening. These is not a real list. It's just to illustrate the concept.

move result ELO
F6 win 1200
E6 win 1400
D6 win 2200
C6 win 3400
B6 lose 2800
A6 lose 1800

If this was the complete list of opening moves, without the pie rule, the game would support players up to 1200 ELO. So, some people would still find it an interesting game, at least for awhile. But many people would quickly discover how to consistently win as the first player.

If the designer foresaw this, he could require that the first player start from D6, C6, or B6. This would allow players to reach 2200 ELO, before the game would be players would consistently win with first move.

With the pie rule the game can reach ELO of 3400, which is much higher. It does this without the designer solving it before turning it over to the players. Pie rule lets the players explore the implications of each opening move.

For example, B6 is losing, but the players may not understand that, and until they do, it stays on the list of playable openings.
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David Bush
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The pie rule can be useful in a 2-player game if, without it, there are initial moves that win, and there are initial moves that lose. It helps if there are lots of possible initial opening moves, for example if the grid is large in a placement game. So somewhere in that wide range of choices there ought to be initial moves which afford equal chances for both players. It also helps if draws do not happen often.

Lots of games have some kind of opening protocol which performs the same function as the pie rule. 2038 is a rail game with a space theme where players bid on the various companies at the start. Experience is needed to know how much of a bid is reasonable for each company. This is another way to unload the balancing onto the players.

Tournament protocols for many wargames invlove a bidding system at the start where you decide how many units you are willing to give up in order to play on one side or the other. Go has "komi pie" which is effectively implemented on some servers. One side chooses the komi, and the other chooses which side to play. The plain pie rule is generally regarded as too crude for Go, but choosing the komi is another way to slice the pie.

NJames wrote:

Suppose we could list the various opening moves in Hex and label them according to how strong both players need to be before they see clearly how to win from that opening. These is not a real list. It's just to illustrate the concept.

move result ELO
F6 win 1200
E6 win 1400
D6 win 2200
C6 win 3400
B6 lose 2800
A6 lose 1800

Nathan, what size grid are you using here? You might be interested in some swap maps for sizes 7 through 9. That was four years ago; maybe 10x10 has been solved as well by now, maybe even 11x11?
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Bidding for small variations of either starting state, in-game resources or end/victory conditions is very common in competitive play of wargames (which mostly are not abstract and symmetrical).

Komi bidding in Go is exactly this.

It works if the variable to bid on is fine-grained enough, like in Go or wargames.
 
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Nathan James
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twixter wrote:

Nathan, what size grid are you using here? You might be interested in some swap maps for sizes 7 through 9. That was four years ago; maybe 10x10 has been solved as well by now, maybe even 11x11?
I glanced at the map for 11x11 in Hex Strategy, by Cameron Browne. He has A6 and B6 marked as "bad swap." But my list was just made up to illustrate the concept.
 
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Laurentiu Cristofor
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christianF wrote:

How do you invent a (non-trivial) 'obviously imbalanced game' if all but a few games discussed here are symmetric?

Do you consider Go symmetric on odd-sized boards? Note my last paragraph and the questions in it that I will now also emphasize.

Laurentiu wrote:

Maybe Go could be a good experimentation subject, because we could vary the board sizes from 2x2 to 19x19 and beyond. Is there perhaps a range of small boards where the pie rule might be more effective than using komi, for example? Come to think of it, why is the pie rule not used *instead* of komi for 19x19 boards?

We can restrict the thought experiment to odd-sized Go boards.

The use of komi tells me that there is a recognized imbalance. Obviously, an imbalance would be more obvious in more trivial games and would become harder to quantify in more complex games.
 
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christian freeling
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Laurentiu wrote:
christianF wrote:

How do you invent a (non-trivial) 'obviously imbalanced game' if all but a few games discussed here are symmetric?

Do you consider Go symmetric on odd-sized boards? Note my last paragraph and the questions in it that I will now also emphasize.

Laurentiu wrote:

Maybe Go could be a good experimentation subject, because we could vary the board sizes from 2x2 to 19x19 and beyond. Is there perhaps a range of small boards where the pie rule might be more effective than using komi, for example? Come to think of it, why is the pie rule not used *instead* of komi for 19x19 boards?

We can restrict the thought experiment to odd-sized Go boards.

The use of komi tells me that there is a recognized imbalance. Obviously, an imbalance would be more obvious in more trivial games and would become harder to quantify in more complex games.
Why would you consider Go a-symmetric on any boardsize?

The pie rule has been discussed here off and on as long as I can remember and probably longer. Go has been considered more than once. One of the questions was, so far as I can recall, whether Go has sufficiently bad opening placements. Komi is being established by statistics and concensus. It's not pretty but it works. And there's also the method of establishing komi with the pie rule.

As for experimenting with different board sizes, there's nothing wrong with that. But figuring out what the best balancing method might be for each of these boards seems quite an undertaking as many thousands of games would be needed. And the reward might be that no-one cares because komi works.
 
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Laurentiu Cristofor
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christianF wrote:

Why would you consider Go a-symmetric on any boardsize?

On an odd-sized board there is a central square where a placement cannot be replicated by the other player elsewhere on the board. On even-sized boards there isn't. I wasn't sure if you cared about this difference.

christianF wrote:

The pie rule has been discussed here off and on as long as I can remember and probably longer. Go has been considered more than once. One of the questions was, so far as I can recall, whether Go has sufficiently bad opening placements. Komi is being established by statistics and concensus. It's not pretty but it works. And there's also the method of establishing komi with the pie rule.

But my question is different than all these topics that you mentioned. It translates to asking if Go has good balancing opening placements on some set of board sizes.

christianF wrote:

As for experimenting with different board sizes, there's nothing wrong with that. But figuring out what the best balancing method might be for each of these boards seems quite an undertaking as many thousands of games would be needed. And the reward might be that no-one cares because komi works.

Could be done with computers these days. I'm obviously not asking anyone to do this. It's just an idea that I think may be worth discussing and might lead to other ideas.
 
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christian freeling
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Laurentiu wrote:


On an odd-sized board there is a central square where a placement cannot be replicated by the other player elsewhere on the board. On even-sized boards there isn't. I wasn't sure if you cared about this difference.
No, not particularly. Of course there's symmetric play to consider but in Go it cannot be maintained.

Laurentiu wrote:
But my question is different than all these topics that you mentioned. It translates to asking if Go has good balancing opening placements on some set of board sizes.

Could be done with computers these days. I'm obviously not asking anyone to do this. It's just an idea that I think may be worth discussing and might lead to other ideas.
On small boards any opening placement may be strong enough to swap, I don't know. On larger boards there may be placements weak enough to refrain from swapping. But it somehow never came out as a real alternative for the established way of balancing the game.
 
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christianF wrote:
But there are clear exceptions like Othello. But it's hard to tell if there's any turn-order imbalance in Othello in the first place, and if there is (of course there is: the tree, being 'completely determined' and all that, but what's it to humans?) then who has the 'advantage'?
(Minor tangential nitpick: it is possible (as far as we know) that optimal play leads to a tie in Othello, so then there'd be no turn order imbalance in Othello... right?)
 
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christian freeling
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russ wrote:
christianF wrote:
But there are clear exceptions like Othello. But it's hard to tell if there's any turn-order imbalance in Othello in the first place, and if there is (of course there is: the tree, being 'completely determined' and all that, but what's it to humans?) then who has the 'advantage'?
(Minor tangential nitpick: it is possible (as far as we know) that optimal play leads to a tie in Othello, so then there'd be no turn order imbalance in Othello... right?)
If that is the case then its a good feature. Draughts variants usually have a sea of draws behind the horizon. That's a bad feature but at least there's no turn order imbalance.
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NJames wrote:
Inasmuch as Hex is always a win or a loss, there certainly is not a fair first cut.
While this is true, the same reasoning could be used to show that Hex is an unfair game when played without a pie rule.

Fortunately, among human players, no one knows what the winning line is. This same fortunate limitation means that the pie rule works just fine with human players.

A player who wants to avoid well-trodden paths can try an opening move that is "off the book", at least when they are the one to make the first move.
 
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