I’ve been fiddling with a couple of Diplomacy variants based on the Baltic area during various times in European history. My interest has mostly been in figuring out how to make Diplomacy work well for different numbers of players. This is a non-trivial task, as most Diplomacy variants tend to leave different players in grossly different situations. While this isn’t alone much of a problem, it tends to leave a bad taste for players who didn’t expect the situation. A recent concrete experience had me play an 8-player variant from the variant banks called Medieval Diplomacy with the teenagers around here, and I can’t say that I’d been entirely happy with it. The variant has half the players fighting over the Italian centers in the middle, while the rest go about calmly picking and choosing their targets. Especially Turkey is in an idiotic position; an attack from the north takes a full two years to reach a Turkish center by land, which basically means that the Turkish player never needs to worry about such an attack.
Anyway, my topic: as part of my variant exploration efforts I’ve produced a number of graphs concerning the triangle theory of Diplomacy map construction. I’ll present those and explain the triangle theory, too, in case the reader is not familiar with it already.
So, triangle. The significance of the triangle in board analysis of the traditional Diplomacy board is a cliche, or near enough: in this analysis the Diplomacy board is reduced into a graph of “significant interactions” between the various Powers of the board. I’ve attached a picture of how the situation is usually viewed. As can easily be seen, I’ve emphasized a couple of triangles here. The reason is not, or so the triangle analysis claims, that I decided to draw triangles on the board, but rather that the more important relationships between the countries of the board actually naturally form triangles on the Calhamer board.
And why is that? Why triangles? There are two complementary explanations: the first is that Allan Calhamer designed it this way because the triangles structure interesting Diplomacy play. The second is that triangles form naturally because players need them to conduct the matter of Diplomacy at all. This latter explanation, which I confess to be of my own making, has it that the most fundamental act of Diplomacy play is actually conducted between three players who all have the capability and motivation to prey upon one another: no other small-scale structure suffices to produce meaningful play of Diplomacy, and therefore the triangles are an absolute necessity of a functioning, or at least traditionally functioning, Diplomacy board. What’s more, the triangles actually leach significance from other potential interactions.
Taking a closer look… the two triangles I’ve emphasized are often considered the most pressing issues of the Diplomacy board: all the thick lines represent relationships where Powers will most likely consider it to their benefit to either negotiate or attack the other party. For reasons of simplicity I will now assume that these relationships are always equal (both have equal opportunity) , which of course is not the case in reality. These particular six relationships are all equal enough to function as a triangle, anyway, which is what I want to explain right now: what happens when three players are in an equal position to strike at one another?
The most fundamental act of success in Diplomacy, the one all Diplomacy players must learn after learning the basics of tactics, is to choose their alliances and betray their victims. It is due to the triangle that I call this one act: betraying victims is as efficient as it is exactly because the player could seek gains elsewhere, against his other neighbor. Likewise, allying is only effective if the players have a common neighbor. This is why the triangle is imperative for the tension of play to appear: a player needs to be forced to choose one of his neighbors as an ally, while using the other’s trust as a weapon, while simultaneously fearing that the same could be done to him. This is only possible if all three are in a triangle arrangement relationship. If the relationship were a chain A-B-C, then player B would be attacked without fail by players A and C at no risk to themselves, which produces little interest.
Anyway, that’s the basics of why triangle relationships are the most important ones in Diplomacy. Lots has been written about how the resolution or delay of resolving the triangle relationships on the board affects strategy in the game, so I’ll leave that for now and present my original subject of interest: what the triangle theory says of good variant design. I’m curious about the topic because I’m right now designing a 5-player variant, and it’s just not getting balanced. The reason might be found in these graphs:
Triangle-optimized Power relationship graphs for different numbers of players
3 players: The simplest case has only one even nearly sensible arrangement of players, assuming that we want to have diplomacy in the game instead of just war. It is, of course, the triangle. I’ve never played a Diplomacy variant with only three players, but I’d assume that it would be very pure, annoying and perhaps, if the variant was too large, repetitive. The situation would only worsen if all three players did not have equal or near-equal opportunity to attack each other, though.
4 players: A very difficult practical variant design case, as the players will feel it very natural to ally into teams of two players, with little in the way of opportunity for shifting alliances. Leaving that aside, however, there is only one arrangement of players that allows for triangles, the cut diamond. Here we find the first problem of the exercise: two of the players (marked “Austria” and “Turkey” here) have three neighbours, while the rest only have two. This migth require extra attention from the variant designer to relatively de-emphasize the urgency of the “Austrian” and “Turkish” situations.
It is also notable that if the map is not planar, all four players might easily be connected (as per 4a), in which case all players participate in three triangles equally. This might or might not be preferable, depending on the exact nature and urgency of the Power relationships. The non-symmetric situation might be preferable simply for having more inherent non-symmetry.
5 players: The symmetric balance gets even worse with the addition of “Germany”, giving “Austria” four neighbours and, presumably, four relationships in need of immediate resolution. An obvious solution for the situation in variant design is to split the board into two triangles with a common point or two (5b), so as to reduce the intensity. The Calhamer board uses this solution, as it de-emphasizes some of the country relationships with awkward troop compositions, convenient tactical borders or simple distance. If all the countries that have avenues of attack against each other were able to campaign against each other from year one, the game would lose much of its predictability and charm.
An example of how the triangle problem cannot be resolved is when a five-player board is created from the 4-player one by adding a Power with borders against the two border-Powers of the cut diamond, as shown in 5a. The idea here would be to add strong relationships to only the countries that only have two. Not only is this set-up a bit awkward to create as a planar map (possible, though), but it’s also inviting trouble: because the two players bordering “???” are not themselves neighbours, they suffer negligible diplomatic risk in agreeing to attack “???” in unison. Meanwhile “???” has the unenviable position of negotiating for peace: that’s the only thing he can negotiate after all, for he can proffer no aid against a common enemy for either of his primary neighbours.
5b, on the other hand, is an example of a sub-planar solution, where an expected neighbour-relationship has been removed between Germany and Russia, leaving Austria alone in the middle of the map. Everybody can attack Austria and Austria can attack everybody, which implies a need to bolster Austria a bit to help him be a credible factor in both triangles he participates in. This is not necessarily a bad solution, and it might well do as an alternative to the more connected one, above.
6 players: graph 6a shows the suboptimal triangle arrangement (with three Powers having four neighbours), while graph 6 shows the optimal. As can be seen, six players can be arranged into triangles with the maximal vertex degree (sorry, got a bit carried away with the lingo there) no higher than four, again, but this time two Powers suffer from four neighbors instead of just one. This rising number of neighbours among the Powers when the number of players rises is very troubling for design, as having three strong neighbour relationships is just about the maximum a normal 3-center starting Power can handle; even that is really too much, as can be seen from the Calhamer board Austria. The triangles simply need to be broken up somehow, while ideally still leaving the farther off Powers some ability to influence the internal life of each triangle; if the triangles are separated completely (six players could just be put into two triangles on different sides of the board), the game could as well be two separate three-player exercises.
6b shows one such example of a (relatively) split relationship map of three triangles. This also just happens to be the closest approximation of what the Calhamer map might look like, dynamics-wise, with the removal of Italy. As can be seen, not much changes; the most important bit is probably the fact that Austria now doesn’t really have to worry about his rear, so he can go and fight with Russia and Turkey on equal terms. And when the time comes to cross the triangles, Russia, Austria, Germany and England are the places where it happens. It’s no wonder that this is a well-played variant among the hobbyists.
7 players: Well, OK, no need to continue this, really. What is obvious for the exacting reader by now is that I can’t have both a solidly connected map and two neighbors per player if I want to have the strong neighborhoods in triangles. I have the use the Calhamer tactic of having separate triangles of different importance and detachment. The other option would be to follow a triangle lattice of some sort, which would end up with Powers having as many as six immediate neighbours. Considering that Austria is on the brink of disaster with three, it’s a pretty safe bet that a set-up like that wouldn’t work in practice, except as some kind of “ultra-powerful dark lord in the middle” kind of scenario. Or, alternatively, set the map on a torus surface and have all players neighbour each other, leading to another kind of idiocy.
It seems to me that it is not reasonable to create all-strong triangles on a board with more than 3-4 players; the number of connections per player simply grows too large to handle in a reasonable manner. Just imagine if, instead of having to choose between attacking Russia or Austria, you had to choose between all the other countries on the board as well. And not only that, but they all could attack you! Not conductive for paced play, that.
The way Calhamer Diplomacy works around this problem is to simply de-emphasize some of the relationships by adding distance or geography, as I explained previously. Even in this case it is important to preserve the triangles; one might even argue that the Calhamer board supports a proposition here: a country in one strong triangle is pleasurable to play, while three strong relationships are simply too much. Italy is weak because it has only one strong relationship, which makes attacking Austria a too obvious measure. Meanwhile, Austria is weak because it has three strong neighboring relationships, which makes it too easy for two or even all three neighbors to attack him. All the other countries apart from these two only have two strong relationships (in a triangle, note!).
If the above is correct, then we affirm a fairly self-evident proposition: a good Diplomacy board variant needs to include distance and other mitigating factors that force players to priorize their treatment of other players, negotiating and scheming upon one part of the board and within a limited set of options to begin with. A variant that allows any player to attack any other is just as bad as one where players do not have the freedom of choice in their allies and enemies. The best balance is struck when tactical issues like time and trust-convenience play into the diplomatic decisions; this is where Diplomacy is a pretty unique game compared to many others that include “scheming”, as most games of that ilk fail to force the players to consider any practical issues apart from whom they want to vote out of the game next.
Meanwhile, it is important for the variant designer to not forget matters of historic accuracy and simple originality while designing the variant. Theory on this level cannot answer as to what is a pleasurable variant map to play. As an example, I’ll end here with the relationship map for my “early Baltic” variant-in-development as it stands now; as can be seen, it’s an absolute catastrophe as far as the triangle theory goes. I’ll probably have to change it, but at least I can’t be taken to task for too much dogma in this matter!