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

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Michael Van Biesbrouck
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russ wrote:
I don't know for sure, but I hypothesize that this would bug many players, or at least feel weird. It is arguably irrational, but I think that many players typically get emotionally attached to the side we are playing, and it would feel jarring to have to switch sides after N turns.
(N or more turns.)

If you are attached to your side, you can just let your opponent call for you to choose and you can keep the side. Your opponent may be happy to introduce a mild weakness into 'your' position before calling for the choice. Or you could give 'their' side a small advantage before calling.

You could have players take multiple turns in a row so they don't get attached to either side. Sort of like playing all colours before deciding on your own in LYNGK.

Personally, I think that it is more 'puttering around with the pieces until the position looks interesting'. You could play randomly for a while until a player thinks that the game is balanced, but that is mechanically annoying and you are more likely to veer away from balance. (Bridge results in lots of interesting challenges due to random setup, but is rarely balanced except in comparison to other players when playing Duplicate.)

christianF wrote:
Generalisation of the pie rule isn't new. Actually you can use it to divide a pie between any number of kids. And I wouldn't call it 'Marquisian' either because that method is based on a preexisting initial position made by one player. A position of which he may know every nook & cranny, the twist being that his opponent gets the choice between choosing colour or moving first.
I'm pretty sure that I first saw a generalization in a somewhat famous mid-20th century book, but apparently not the one I thought to consult. My suggestion is the 'Improved Marquisian Method as it attempts to remove the distinctive feature of the original, the studied position (or goes to a position both players studied). Call in the Deep Dish Pie Rule or something else if you prefer.

christianF wrote:
But you're right that players may dislike it because ideally a game should not need it. And that is sometimes unintentionally the case the case, for instance in some Draughts variants.
It's hard to avoid a first/second player advantage in a simple abstract. Games such as Santorini can counterbalance the innate advantages with asymmetric powers. For Jotunheim I used a computer to balance the powers (but then felt a need to use a computer to suggest the better first power). (The balance is rarely perfect. It is more about the game setting you a wide variety of unique challenges that aren't horribly unfair.) For any abstract game I could use similar techniques to find a start position from which a game appears to be balanced. This has a number of disadvantages:

* you have to tell players to set up what looks like a midgame position
* the starting position might be harder to remember than the rules
* there is now less to explore in the game
* if the AI isn't AlphaZero-quality then someone will probably discover an advantage for one side or the other

I like the IMM from the theoretical standpoint that you can see more of the potential of a game than by starting normally from the beginning as you can wander into territory that is infeasible with competent players (missing opportunities to win, bizarre piece exchanges, etc.).
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Ray R.
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mlvanbie wrote:
It's hard to avoid a first/second player advantage in a simple abstract. Games such as Santorini can counterbalance the innate advantages with asymmetric powers. For Jotunheim I used a computer to balance the powers (but then felt a need to use a computer to suggest the better first power). (The balance is rarely perfect. It is more about the game setting you a wide variety of unique challenges that aren't horribly unfair.) For any abstract game I could use similar techniques to find a start position from which a game appears to be balanced. This has a number of disadvantages:

* you have to tell players to set up what looks like a midgame position
* the starting position might be harder to remember than the rules
* there is now less to explore in the game
* if the AI isn't AlphaZero-quality then someone will probably discover an advantage for one side or the other

I like the IMM from the theoretical standpoint that you can see more of the potential of a game than by starting normally from the beginning as you can wander into territory that is infeasible with competent players (missing opportunities to win, bizarre piece exchanges, etc.).
I remember being pleasantly surprised when I read the Santorini rule for selecting powers. I even used it as an example where I attempted to engage a certain "Playing To Win" author in a discussion about using variations of the pie rule to allow the players to create their desired level of (im)balance in asymmetric games. I recently played Santorini with a player who had less interest in studying the game (I don't want to say "weaker player"). To even the playing field a bit, I gave that player the Pan card ("Also win if you jump down 2 levels"). I'd never played against that power and it was fun to add a challenge while also recreating that feeling of initial exploration.

As an aside, what I particularly liked about Pan's power is it doesn't change any of the movement or placement rules. All it does is create a new threat and if your opponent manages to control that threat then the game will appear to progress according to the default win conditions.

And now I need to go learn about Jotunheim!
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Michael Van Biesbrouck
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Yes, asymmetry is also a great way to handicap. The code that powers that Santorini app balanced power suggestions can also be used to select powers with a certain amount of imbalance and tell you which power the weaker player should play, but the app doesn't expose that capability. If I create a power-selector app for Jotunheim then that would probably be its only feature (I'm not a UI guy).

I do like Pan's elegance. Through computer experiment Pan seems very balanced, but that isn't what many new players will tell you.
 
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christian freeling
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mlvanbie wrote:
I do like Pan's elegance. Through computer experiment Pan seems very balanced, but that isn't what many new players will tell you.
I don't know Pan, but this is an interesting observation. Is this a phenomenon that others have encountered too, maybe in other games?
 
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Ray R.
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christianF wrote:
mlvanbie wrote:
I do like Pan's elegance. Through computer experiment Pan seems very balanced, but that isn't what many new players will tell you.
I don't know Pan, but this is an interesting observation. Is this a phenomenon that others have encountered too, maybe in other games?

I think it was in the Imperial Settlers forum where someone said something along the lines of "You know a game is properly balanced when there's an 'Is X overpowered?' thread for every X"

Early on, people were complaining that the Barbarian faction was overpowered. Then, as people learned to counter that, they started complaining that some other faction was too powerful. Then ...

Barbarian advantage
Egypt too powerful?
Aztecs faction: Overpowered?
Japanese faction - play balance vs the other decks?
Atlanteans too strong?

EDIT: I just noticed (thanks to Michael's response) that I misinterpreted the question.
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Michael Van Biesbrouck
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christianF wrote:
mlvanbie wrote:
I do like Pan's elegance. Through computer experiment Pan seems very balanced, but that isn't what many new players will tell you.
I don't know Pan, but this is an interesting observation. Is this a phenomenon that others have encountered too, maybe in other games?

I guess some context was lost as I replied to rayr without quoting. Pan is a power in Santorini. So Santorini(Pan, X) was balanced for most X, but players felt otherwise for most X. I can come up with a lot of cases where Santorini players (including myself) had the wrong intuition about powers or seemingly failed to notice how strong/weak some powers are.

In asymmetric games there are powers/sides that are easy to play and those that require skill. It is common for new players to gravitate to the easy-to-play options. For a first-game experience it isn't optimal, either. Even in expansions the preference for Santorini was to stick to powers that are easy for new players to play. (I would have thought that people buying expansions, assuming they play them at all, would be experienced. They might also need powers that would challenge them against new/weaker players.)
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Russ Williams
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rayr wrote:
I think it was in the Imperial Settlers forum where someone said something along the lines of "You know a game is properly balanced when there's an 'Is X overpowered?' thread for every X"
FWIW this remark has been made through the years, from long before Imperial Settlers existed.
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Ji Dan
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I think it's an excellent rule for games where it applies.

Elwyn Berlekamp's "An Economist View of Combinatorial Games" explicates how bidding can be utilized to balance games in general, but the pie rule is much more simple and therefore widely applicable for proper playgames.

In the case of the pie, it's not a formal bid, but the starting player choosing the least strong advantageous position.

-------------------

My sense is that locking players into a set of legal openings is restrictive, and, in most cases, inhibits strategic creativity, unless doing so directly supports the mechanics.

Othello is not a capture-and-remove game, but a capture-recapture game, where tokens flip back and forth between two polarities. Probably doesn't produce great game play if not bound to a center.
 
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Russ Williams
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DukeZhou wrote:
In the case of the pie, it's not a formal bid, but the starting player choosing the least strong advantageous position.
Couldn't the starting player also rationally choose the strongest disadvantageous position?

E.g. the starting player would seem to prefer to be left with the disadvantageous position worth -1 rather than let the second player take the next stronger position worth +2, if those were the two fairest possible (least extreme) first moves.

(edited to fix typo "for"/"than")
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russ wrote:
DukeZhou wrote:
In the case of the pie, it's not a formal bid, but the starting player choosing the least strong advantageous position.
Couldn't the starting player also rationally choose the strongest disadvantageous position?

E.g. the starting player would seem to prefer to be left with the disadvantageous position worth -1 for than let the second player take the next stronger position worth +2, if those were the two fairest possible (least extreme) first moves.

Absolutely! Great point.

The underlying guideline still applies—choosing the least sub-optimal disadvantageous position. (Minmax vs. Maximin.)

My thought there is this might, might, might be less optimal because if second player doesn't switch, starting player is saddled with the less optimal position. (For instance, where minimax and maximin are close enough for it not to be easily discernible, particularly if P2 is less skilled than P1.)
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dale walton
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Reading all this a little after the fact, a couple possibly missed points.

The pie rule is very clever in its simplicity. Attempts to improve it can yield unfair "pie" rules (good for con men) which essentially leave the possibility of offering a choice of two losing options. For example, if the rule said first player places a piece, second player chooses a color and passes; then the game trees available to the two players are different, and it is not a true pie rule. (First player plays to a useless position, and then starts the game with the required color and a "first player advantage" if the game has a sufficiently useless position. Another example, first player puts a piece of each color on the board, second player chooses which color to play - this is not a true pie rule (e.g. place symmetrically to losing locations).

The difficulty of augmenting a game with creating truly "fairer" starting rules is why one sees either the true pie rule, or else a bidding rule used so often - they work.

Another reason for games having pie rules built in is because whining about imbalance by new players to a new game can damage marketing efforts even though a game may be quite well enough balanced. Few reviewers will make this comment if the rules include a pie rule.

The simplest pie rule: one player chooses the game, the other chooses whether to go first... demonstrates that pie rules inherently are a "2nd player advantage" system.

hope this is helpful.

 
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dale walton wrote:
The simplest pie rule: one player chooses the game, the other chooses whether to go first... demonstrates that pie rules inherently are a "2nd player advantage" system.

A Chess master and a Go master sit down to play a game. I'd think that the first player (choosing the game) has the advantage.
 
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mlvanbie wrote:
dale walton wrote:
The simplest pie rule: one player chooses the game, the other chooses whether to go first... demonstrates that pie rules inherently are a "2nd player advantage" system.

A Chess master and a Go master sit down to play a game. I'd think that the first player (choosing the game) has the advantage.

Yes, the preparation advantage mentioned before applies to this simplest of pies. If the selected game contains its own pie rule that advantage still holds true...

We should try not to conflate balancing player skill with balancing the game structure. Handicaps are addressed to the former, pie rules to the later (and bidding incorporates handicap into the pie rule to seek to address both)

With Go as an example, komi balances the game, kyu attempts to balance the skill levels, when they are within a reasonable range. Knowing how to balance skill requires a track record, and a philosophy of when the skill levels should be balanced (like striving to reward recent improvement, but not reward cons)

 
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Michael Van Biesbrouck
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If a pie rule truly balances a game, then there is an alternative start position that is naturally balanced. Trivial pie rules in games with fixed setup favour players that have studied a single opening. If players alternate contributions to the setup/opening (see Unlur or my earlier suggestion) then it is possible to achieve an unknown position for normal play that players have attempted to balance (or the stronger player could have intentionally misplayed to give a handicap).
 
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Nathan James
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mlvanbie wrote:
If players alternate contributions to the setup/opening (see Unlur or my earlier suggestion) then it is possible to achieve an unknown position for normal play that players have attempted to balance (or the stronger player could have intentionally misplayed to give a handicap).
It is possible to achieve an unknown position provided the players are sufficiently ignorant. Every game has its limits.

It is possible to achieve, but is it likely? Just as opening theory in Chess tends to guide players to known lines even while playing for advantage, theory will eventually guide players to known lines while playing for balance. This is baked into the nature of games. If you don't like it, the only alternative is to switch to a game where you and your opponent understand less.
 
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Michael Van Biesbrouck
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If you can't get to an unknown position, then the game is solved (at least for you and your opponent). One of you would presumably want a different outcome.
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Nathan James
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That's the first paragraph of my previous comment. But notice also the second paragraph.

Edit: don't lose sight of the fact that you're talking about openings, either. And specifically openings while somehow incentivized to find balanced positions. It's a bit ...unreasonable... to expect people to seek out unknown openings while incentivized to maintain balance.
 
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Michael Van Biesbrouck
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I'm claiming that unknown positions should be found by competent players. If every move you take is known, then one player knows they are going to lose. Either that player deviates at a point that seems strategically viable or the player with the known advantage deviates earlier to prevent the presumed-weaker player from deviating at a previously studied point. This is a fight over asymmetric knowledge.

You don't want the situation where a player can trivially get into a position that they have studied more than the other player. You also don't want all of the options before a player chooses sides to have known advantages. That's the nature of trivial pie rules. Both players need to contribute to the opening moves before sides are chosen if you hope to get to an unstudied position where both sides are intriguing to the players.

The outcome of a combinatorial games is only interesting when neither player has the knowledge and/or computing power to be sure of the outcome.
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christian freeling
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mlvanbie wrote:
The outcome of a combinatorial games is only interesting when neither player has the knowledge and/or computing power to be sure of the outcome.
Generally speaking this makes sense. To make it happen at all a necessary but far from sufficient condition is that the game should allow it to happen in the first place. Chess and Go still seem to comply sufficiently and Shogi certainly does. In Draughts most positions are 'known' but that knowledge is concentrated at the top and that's where the problems are. It's fun if you're not too good at it.
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Nathan James
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mlvanbie wrote:
If every move you take is known, then one player knows they are going to lose.
This argument only applies to playing the game in its entirety. It can not be applied to a subsection of the game, for example the opening.
 
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Ji Dan
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Dave Dyer over at Boardspace.net implemented the pie rule for the Mbrane game, to balance the starting player advantage.

It a great innovation, allowing more balance for single games on odd-sized gameboards. (Another method is matched sets of two games, with players alternating starting, because the game duration is so short compared to Chess and Go, but this doesn't help much with one-offs.)

It's exactly what Berlekamp was talking about re: players bidding to go first. Difference here is that in Go, which Berlekamp was analyzing, additional mechanics are required to make the bid numeric, where in Mbrane, the opening position is both spacial and numeric, eliminating the need for additional mechanics.

"At every position of a game such as Go or Domineering, there are two very important questions: Who is ahead, and by how much? How big is the next move?

Following Conway [1976], classical abstract combinatorial game theorists answer these questions with a value and an incentive, as discussed in WinningWays [Berlekamp et al. 1982]. These answers are precisely correct when the objective of the game is to get the last legal move. Values and incentives are themselves games, and can quickly become complicated.

Our ideal economist takes a different view. Following Hanner [1959] and Milnor [1953], he views the game as a contest to accumulate points, which can eventually be converted into cash. Our modern economist monetarizes the answers to our opening two questions into prices—real numbers that can be determined by competitive, free-market auctions. Specifically, after two gurus have completed their studies of a position, the economist might ask each to submit a sealed bid, representing the amount the guru is willing to pay in order to play Black."


The Economist’s View of Combinatorial Games (Elwyn Berlekamp, 1996)

http://library.msri.org/books/Book29/files/ber.pdf
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