It's possible to win with all 34 centres, (theoretically at least) since there is no centre bordering only other centres (slightly more complicated but there's no real problem).

As far as number of possible games, do you rule out fluctuation (i.e. move back and forth), or simple "rotation" i.e. we just keep trading s.c.'s in a circular way as being permissible? Unless you place limits on repeating a situation then I think you'll find there's a (countably) infinite number of possible games. As for the limitations you apply, I'm not sure what you'd want, nor how you'd formulate them. If you say "reasonably logical", without defining that better, then I still think you'll find there's an infinite number of possible games, though many of them will be boring. e.g. you get in a situation where everything bounces, and the cycle is broken only after n turns, you've already got an injection from the natural numbers into set of all possible games.

emmmm - i thought we were talking about final positions sorry my mistake

Hi Riff, i imposed the clause of reasonably intelligent different players specefiacallu to rule out games with fluctuations and rotations, let us define reasonably inteeligent players as players who want to
first, win a game and
second, finish the game as soon as possible.
Is their an upper bound on such games, i have searched the net on this but did not find any such analysis, hence curious.

should we start with the maximum number of mathimatical endings and work backwards.

Well, if you ban any loops and such like there must be an upper bound, however it will still be extremely large

If you want to count how many possible ways in which the SCs can be divided so that someone has won. If we assume the winner has 18 out of the 34 SCs and the rest are divided among the remaining players, (all are occupied) by my reckoning the 18 SCs can be selected in 34C18 = 2.2x10^10 ways the remaining SCs can be distributed among the remaining 6 players in 6^14 = 7.84x10^10 ways. So there are 1.73x10^20 winning positions where the winner has 18 territories.

Assuming we are only talking numbers IE Italy with 35 and france with 1 is the same result regardless of where france is, then....
If we take 1 country say Italy and look at his endings first.....
He can have only 1 way of ending with 34 SCs, 6 ways of ending with 35 SCs, 21 ways of ending with 34.....errrr that's as far as I've got.

Assuming we are only talking numbers IE Italy with 35 and france with 1 is the same result regardless of where france is, then....
If we take 1 country say Italy and look at his endings first.....
He can have only 1 way of ending with 34 SCs, 6 ways of ending with 33 SCs, 21 ways of ending with 32.....errrr that's as far as I've got.

amathur, you're definition still isn't sufficient to reject my example, because it could be that the only rational thing to do is continue in a stalemate (i.e. anything else reduces your chances of winning). Your second condition of ending the game as soon as possible then determines when the stalemate is broken (i.e. depending on how weak/strong we make your second condition compared to the first).

Moreover, drawing is entirely optional in diplomacy, and if we're simpyl stating that rationality is wanting to win, then there's no reason to draw. (I know I'm being picky, but to do justice to the question, we have to be).

I think that on any decent definition of reasonably logical (=rational?) we're going to have an infinite number of possible games, although I think a proof of that would be more than I could give here (or perhaps am able to give).

My feeling that this is true is because until the game is inevitably won (i.e. someone gets into a position from which he cannot lose so long as he plays optimally), then I doubt we have a finite number of ways to reach this stage. Adding a rationality / reasonably logical clause doesn't help too much with regards to diplomacy because we have to take into account what goes on behind the scenes as to what actions are rational. If we limit what actions are rational (e.g. always take a centre when you're guaranteed to do so, always defend your centres where possible unless this conflicts with rule 1, always prefer to support a move/support a hold than simply hold, or some other things) then we're ignoring the psychological component of diplomacy.

Now, I'm a little tired, so they could be a flaw in my logic, but surely the maths is actually quite simple:
There are 34 supply centre's, and 7 nations. Each supply centre could be owned by any one of the 7 nations. Therefore there are 7^34 possible endings. However, this includes options where every supply centre is owned by the same nation, which is not possible with rational players. Calculating a reasonable value is far harder, because it would have to take into a number of points, including:
1) Position and type of units, including non-SC's
2) The winner normally has 18 or 19 units, but the spread of these changes a lot, and different nations have different numbers of realistic winning positions (for England they are all roughly the same, but for Austria it really depends which way you go first!).
3) What happened to the enemies: the remaining SC's could be owned all by one person or as a mix, but again they are likely to be in a roughly continuous shape.
4) If you did assume all nations are continuous, then you have to consider the possibility of rogue units, such as those forced behind enemy lines
5) etc etc
--------------
Maniac - do you mean simply the combinations, ie the 34th row of pascals triangle? That would work, although you'd then have to take into account the combinations of how the other supply's would be distributed between other players, and again that assumes every SC is equally likely to be in a winning set, which they are not.

I think there are an infinate number of solutions, and indeed in a 'perfect' game would never finish. With 7 fully rational and emotionless players, as soon as any one player took a lead or was set to take a lead they would be targeted. Thus the other players would reduce their size until someone else was the most powerful nation, when the 'target' would change.

On a math level there are calculations out there for say the number of opening moves.
On a greater level there are philosophical approaches to what would be the result of 7 elite and perfect players playing with full diplomatic/strategic/tactical skill operating under a achievement background that is oriented towards Draws Include All Survivors and all draws are equal. There was an article on this called the 'Odd Theory'
done about 35 years ago and often reprinted that makes the case that the perfect game would logically end with a three way draw, see it here:

http://www.diplom.org/~diparch/resources/strategy/articles/odd.htm

Now one of the biases is what do you 'scale' the results as? If you use supply center counts or draws are equal?
The main philosophy on this web site is not only supply center bias, but in the case of a solo win the non winner secures a substantial bonus relative to the solo winner which is a significant difference from most commonly accepted view points of achievement in the hobby. Though this sort of structure makes it appearance now and again it rarely survives the test of time before people in the local support culture shift towards more extreme views of bringing the value of those in a game which is won by someone else down to a minimal value.
But that is another topic of discussion on WHAT IS ACHIEVEMENT.

very interesting, thanks all for your views, the problem with the purely mathematical solutions is that they clearly discount the diplomatic spirit of the game. Additionally, not all 34 sc's are equally likely to be taken by any power. Hence i do belive there is a fairly small upper bound on 99.9% of the games, however those weird 0.1% games ( eg with turkish fleets occupying stp, or russian armies in brest) can be infinite.

The mathematical and logical answer to the question differ.

Enough said on the maths by others in this thread, I would like to explore the logical answer a bit, from the perspective of having "rational players" involved in a WTA game.

Given that, the algorithm of the rational player is "do the move with the best ratio between benefits and risks" one could be tempted to say that the answer to the question could be deterministically given by knowing if - at any given moment and for any given players - there will be a situation in which two possible moves will have the same ratio benefits/risks. The "tempted person" could hence argue that each such case constitute a "fork" in the tree of possible games and thus that the final number of possible configuration will increase exponentially with the number of such possibilities.

In reality further scrutiny will dissuade this "tempted person" to think that way, as the rational player will consider the ratio benefits/risks not only for the turn "X" but also for the options that each option of turn "X" will generate for turn "X+1". A rational player - indeed - should perform this evaluation routine recursively till the end of the game, and then will pick the option in turn X that will generate the option in turn X+1 that will generate the options in turn X+2 ... that will generate the best options in the final turn of the game.

Since each of the 7 rational players will play with the same logic, regardless of how they will estimate benefits and risks, the orders that will be issued in spring 1901 can only be one set that will generate the best options for the future, and all the rest of the game will be determined until the very last turn, in which - should there be more than one options with the same ratio - any of the orders could be given.

So, according to logic, the number of possible "rational outcomes" will be very limited: In the case of many players still be involved in the game, this number could potentially be 1 (as all units will have to defend some SC). In the case of only 2 players left in the last turn, several units will have no active role in getting further SC's or protecting acquired SC's and could therefore move anywhere, generating a few more possibilities (precisely: unit*number_of_options_open_to_that_unit*number_of_units_having_more_than_one_option).

Another interesting point is trying to imagine how this final scenario will look like: will it be a solo victory? a two-ways draw? a seven ways draw? Edi's article suggest it will be a three-way draw, but the premise of that article is that players will not be "rational" (this is for sure a case closer to reality, but it is not the case we are considering).

The answer to that question can be researched by examining the diplomacy Board. In fact, if the board was symmetrical, the logical answer would certainly be: 7 ways-draw. But - luckily - Dip board is everything but symmetrical.

The answer to that question will therefore have to be investigated by studying if the "imbalance" between the geographical strengths of different players are minor enough not to affect significantly the "benefit/risks" ratio of each turn moves.

Should this be the case, a diplomacy "rationally played game" will lead to a 7-ways-draw, and therefore the "rational player's" spring 1901 orders will be:

/draw

The reasoning would vary a bit with a PPSC game, but the outcome would be very similar, indeed.

NOTES:
1. For those interested in the balance of the Dip-board I suggest this serious stuff: http://www.diplom.org/Zine/F1999R/Windsor/dipmap.html
2. For those interested in playing Diplomacy "rationally" I suggest this other less serious article: http://www.diplomacy-archive.com/resources/humour/diplomacy_and_star_trek.htm

Your first paragraph (first big one) is slightly misleading, especially in light of your second one in which you correctly state that it you're going to look at the entire game and the set of moves in which you'll do the best. Unfortunately, there are two problems with your argument in practice:

1. Diplomacy is not a game of complete information. As in chess, there is a "rationally best" strategy, but we shall never know it.
2. Diplomacy players are not perfectly rational, and I can see that slight errors in optimal play will screw over everyone else's optimal strategy, or in game theoretic terms, the"optimal" strategy is unlikely to be "trembling hand perfect".
3. I think you should consider a set of strategies as being optimal, rather than a single strategy. i.e. it is likely that the optimal opening move for England will result in moving to the channel x% of the time, and moving to the North Sea (100-x)% of the time, so that there is not one single perfect strategy... this is because if everyone is perfectly rational, and has complete information about the game, and knows that you are going to play the single "optimal" strategy, they can deviate from the "optimal" strategy to get better returns.

In conclusion, there is no one optimal move at any stage (well, maybe at a very limited few), but there will be an optimal probability assignment to the range of possible moves. There is therefore a lot more possible games than you consider.

caveat to my point #1, when I say there is an optimal rationally best strategy in chess I mean range of strategies as discussed in point 3, not as you had discussed in your post.

You can also take the view that for every possible opening there is, a rational player can make something out of it based on a certain alignment of personalities and other objectives. Certainly it has been proven here again and again that even with certain countries making NO moves in all of 1901 those countries can come back to win the game.
Now if you want to look only at the 1901 openings then check out these calculations:
http://www.stabbeurfou.org/docs/articles/en/positions_en.html

It comes out to 7.49 x 10 to the 13nth

@RiffArt. I think we are talking about two different scenarios here: the totally theoretical one (me) and the realistic one (you). I agree with you that IRL my reasoning does not apply, but this is the same than as for mathematical answers: the vast majority of "possible endings" can't happen IRL, as they have no strategical significance.

@Edi: I read your article and my comment is on the same line: your article consider a "realistic scenario" and therefore its content is applicable IRL. In a "perfectly logical scenario" - though - things would not go the way you describe, as there is no "logical way" at the beginning of a game to be sure if the promise of a given alliance is trustworthy or not. Again: this is for the sake of academical dissertation as - in real life, and this is what is interesting - your analysis bare a lot of useful information.

In a perfect balanced game system of any type where the objective is to eliminate as many people as possible to achieve either a solo win or to achieve a draw wherein all parties participate in the draw equally then the logical conclusion for a 7 player game would be to get to 3 because 5 would overwhelm 2 then 3 overtake 2. With 3 left the ability of one power to throw the game to one of the others would force the game to be a three way draw in theory.
As an additional support for the argument, in nearly all Diplomacy Face to Face events over the years where the emphasis is on Draws, the most common result of the first two rounds with veteran players is generally a three way draw.

Edi, don't get me wrong here. Your article is the one that really gives useful information and people should consider in thinking to their strategies. My dissertation is more of a game (in the spirit with this thread) playing with logical principles and "abstract perfectly logical players"...

You yourself called the system "perfectly balanced", and this is the key to my point: in a perfectly balanced system the problem is not defining what outcome would be best, but how to break that balance. Please consider that in the scenario discussed in my example all the players are "perfectly logical" (so they also think alike) and - as it is normal in Diplomacy - orders are submitted simultaneously by all players (so there is no "advantage/disadvantage" in giving away information at a different point in time).

If Diplomacy board was symmetrical, then all the players would act the same, and the situation would be a stalemate, or a loop of moves (like: move here, move there, then move back, then repeat).

Diplomacy board is however not symmetrical, and therefore only one of the two could happen:

If the board was balanced enough not to give any player a decisive advantage, then the situation would re-balance itself maybe in a different configuration, but still in a 7 ways draw.

If the board was not balanced enough, then the same players would form every game the same coalition and would go through a process similar to that described by you until the next "balanced situation" would present. Then it would be a stalemate.

Again: the key point is not about agreeing that 5 to 2 and then 3 to 2 is a logical way of conducting your strategy, but it is about (in a "perfect balanced scenario") the inability to decide how to form the coalitions initially.

Hopes this clarify, and hope that I managed to be clear on the fact other players should use your "realistic" article to develop their strategies, rather than my "theoretical" dissertation. :)

I understand what you are saying in that the formation of the first alliance of the 5 could not be logically obtained.
I wonder if there is a solution though. When there is no logical solution would not 7 logical people agree to a random solution that provided them with a potential gain? So wouldn;'t the 7 players then agree that they could write out on a card the 7 names and then pick at random 5 cards and agree that that would be the winning group. Then again on the 3 vs 2 group. The question then is in the final round would the three agree to have a one in three chance at a win or a would they choose a 2 in three chance at a 2 way draw.
That comes down to risk taking. Considering that in each case you would want the potential of a gain to be greater than 50% I suspect that with robots the program would choose an ending that is a 2 way draw.
IF the programing is such that a player would take a 51 per cent chance of a gain over a risk of elimination. Then there is the argument of whether the program should be made so as to avoid loss rather than seek gain.
And we are back to the philosophy of play for which there is no solution.
In short there may not be a theoretical dissertation solution because of that very nature of conflict: Risk taking vs Loss Aversion which cannot be determined in perfect world.

@mac: My first two points were definitely more concerned with realistic rather than theoretical constraints, but even theoretically have an effect:
point 1 (not complete information) : entails that we cannot determine what the "logical" play will be, so we'd need to use other methods (e.g. Edi's Odd Theory) to say what the outcome of the game would be.
point 2 (trembling hand perfection) : by deviating from what rationality "theoretically" could dictate, would itself be the rational choice, hence
point 3 (probability distribution of a strategy) : even theoretically there are (albeit finitely many) possible games which rational players will play. You can see this in simpler games, (this might help: http://en.wikipedia.org/wiki/Game_of_chicken#Best_response_mapping_and_Nash_equilibria)

In determining the total number of diplomacy games that could be played by logical players perhaps we should first accurately determine the accepted logical first moves for each SC. Then we could determine the amount of different permutations of all the logica; first moves. Perhaps Kestas could inform us of any valid move that is rarely (someone needs to define rarely) made. I.e. Lon-Wales. If Kestas can't inform us I don't think it would take toolong for us to agree on a list of illogical first moves :)

@Edi and RiffArt. I think we understood each other on what we mean with our respective demonstrations and I happily acknowledge your points. For the sake of pure academy, I would just mention that logic and rationality are not the same thing. To quote the Stanford Encyclopedia of Philosophy: logic "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality". But again... this is not mean to prove any of us is "more right" than others, I just wanted to point out that solutions to the same problem might be different according to the angle from which the problem is observed.

Spring 01 order stats can be found on several of the large tournaments run in the hobby. The problem with general stats is that it does not reflect the different dynamics of the scoring system or the game type,
for example in a C-Diplo scoring (center based and ends in1907) you get a different pattern than in a Gunboat game that goes forever.
Likewise Draw based systems yield slightly different results.

Generally the most common Spring 01 orders in Face to Face events of all types for this decade seems to be

F EDI-Nrg F Bre-Mid F Kie-Den F Tri-Alb F Nap-Ion
F Lon-Nth A Par-Bur A Ber-Kie A Vie-Gal A Ven-h*
A Lvp- Yor A Mar-Spa A Mun-Ruh A Bur-Ser A Rom-Apu

F Ank-Bla F StP-Bot
A Con-Bul A War-Gal
A Smy-Con A Mos-Ukr
F Sev-Bla

*Some decades Ven-Try is more popular but this decade is not over yet

Seems the columns did not quite work out well... sorry folks but you can figure it out

I like Ven-Tyr but then maybe I'm just out of date or maybe ahead of my time.

@mac, agreed the logic and rationality are different, technically "logical move" doesn't really make sense.

What do you mean by "playing with logical principles" and "abstract perfectly logical players"? I'm assuming the first simply indicates using logic to find the best way to achieve your goals, in a Humean way, and the second simply that the players can apply logic perfectly. You also talk about "rational players" which seems to imply they use logic perfectly to achieve goals which we accept are reasonable, viz. to maximize the chance of winning, minimize the chance of losing, maximize expected returns, etc.

So basically yes, could you clarify your definitions / aim a little? You write that if the board were symmetrical then all the players would act the same, but I still think that game theory shows that's not true (in a lot of simple games even, and diplomacy is so much more complicated, even if we symmetrify the board, my argument holds that the optimal RATIONAL strategy is one with a distribution of possibilities for each move. As per my point above, "LOGICAL strategy" just doesn't make sense, except in the sense that the strategy was arrived at by using logic, and I'll argue that RATIONAL implies LOGICAL (in this instance))

Edi - are those averages, or from the average game. Basically, what I'm asking is whether that is the most likely situation or just a combination of each nations most common moves.
To explain what I mean, on average a human has around 9.8 fingers, but thats because most have 10 some have less, whilst 9.8 is an incredibly rare occurance. (please don't pick up on that statistic - its there to make a point!)

Hir Riff, it is very possible that I have been inconsistent in the use of "logic" and "rational" in my previous posts, as I stop thinking to this difference only in my last reply.

In fact, my aim in presenting the case was purely to entertain myself (end eventually others) in seeing how far one could go given certain premises. Pretty much the same that people who approached the problem from a mathematical point of view did, really.

I would say that a "logical" player in my presentation can be equalled to be a computer: a kind of deterministic machine operating on a strict boolean logic (yes/no). In my example I did not stop thinking for much time to the "goal" of the player: whether the player primarily tries to prevent victory of others, or to eliminate others, or to win himself, or not to loose himself. Of course this would change the final outcome configuration of the game, but still the "logic of the logic" (to choose the best move in a set of possibilities) would not change.

Being these abstract players "perfectly logical" (so no creativity) and "deterministic" (so no randomness), the fact that they would play the same moves is a corollary (things can't be different than that, or the initial assumptions of logicality and determinism would be automatically proved false.

Now, I found this nice little sentence on wikipedia that basically says it all: "A logical argument is often described as "rational" if it is logically valid. However, rationality is a much broader term than logic, as it includes "uncertain but sensible" arguments based on probability, expectation, personal experience and the like, whereas logic deals principally with provable facts and demonstrably valid relations between them."

So, in this case I would definitively say that game theory is rational, but not - strictly speaking - logic or deterministic. I would also agree that Edi's odd theory is rational and that the situations such throwing a game to somebody else as a response to an attack could well be seen as attempt to play along the lines of possible Nash equilibria.

I also agree with your analysis of the term "logical strategy".

So in essence: we described and analysed two different scenarios, both of them with their specific (and most probably correct) outcomes. The one that you and Edi described is surely closer to the reality of playing diplomacy in the real world.

Incidentally, game theory has started not by deductive logic (given certain premises you can deduce this and that) but by analytical observation of real situations in which the success of a "player" was not solely dependant from his own choices, but by the interactions with other "players" ones... so diplomacy seems the perfect ground to verify and experiment with the ideas of it. :)

there are obviously infinite combinations theoretically, as everyone could just hold an infinite number of times before actually doing something.

but you asked logical, so we need to know what you consider logical (ex. A (Par) - Bur, A (Mun) - Ber, stands off every time for the first eighteen years?).

The openings are the most common for each country independent of other interactions. So if you were a betting person and bet on each country playing an opening these would be the one to play for each country to win money.
The chance of ALL of them happening at the same time, in other words that this is the entire opening set up now that I have not given much thought to. I would suppose that it would happen more than any other set but where as the individual country's opening could reach as high at 50 per cent the total combined outcome would probably be lower than 5 per cent though it might be more than any other combination.
Hope that answers you question.

@mac, thanks, I think I understand what you mean now.