A Myth Defended

by Lew Pulsipher

I thought of titling this “The Even Theory” as a counter to Edi Birsan’s “Odd Theory”. Edi’s theory is one way of playing, but it is not necessarily the way perfect players would play, and he has offered no definitions, proof, or examples, only assertions, to show why he believes that a perfectly played Diplomacy game will be a 3-player rather than a 7- or other-player draw. I believe that I can satisfactorily show that a 3-player draw is not the necessary outcome of a perfectly played game, though a 2 or 3-way draw is. I think this will also show that the “theoretical” outcome of the standard Diplomacy game is a 7 or 6-player draw.

Why is a common 3-player draw not likely to break down among “balance of power” players? Simply, no single player can chance attacking another, for fear that the third, un-engaged player will take advantage of both of the others and win. What we really have here are three separate interest groups as well as three players; each player is one interest group, The game is drawn when the three players-interest groups resolve into one interest group – each player must act in such a manner that both of the others remain viable powers, or else he may fall and another may win,

Unwillingly, the three must work together for the same end as they work for themselves; this is forced on them by the nature of the game. A resolution down to a single interest group embracing all surviving players is the outcome of any Diplomacy gene which does not involve a stalemate line defence, (I do not include concessions, by which a player outside the winning interest group may survive because the game is not played out; the same is true for a rulebook win – the game is not played out, so the losers survive even though they are not part of the winning interest group.) Two interest groups remain when one is defending a stalemate line against the other in a drawn game.

Edi’s mistake is in assuming that each player automatically represents a separate interest group. As we have seen above and will see later, more than one player may be part of a single interest group. This may be forced on the players. A good example of such a situation is the recently completed demonstration game. Turkey, Italy and Austria were part of a single interest group which had to remain together in order to withstand the other two players, who were each an interest group but were working together to take advantage of the disorganisation of the third interest group on the board. Turkey chose to perpetuate this disorganisation by stabbing his allies. He hoped to be able to eliminate them and set himself up as a one-player interest group in place of the three-player interest group. This was a bad play, as we saw.

In a perfectly-played game, the south-eastern interest group would have held the stalemate line and a five-player draw would have resulted -three ways, five players. Given especially good play by Italy, Austria and Turkey, they might even have held on to draw without a stalemate line, for Germany and France could not trust each other fully. Play of this calibre is not to be expected even in a demonstration game, however. In effect, Italy, Austria and Turkey had to act as a single player in order to avoid losing, and much co-operation is difficult to establish.

If you look at drawn games which do not involve stalemate lines, you will find that they are virtually always draws of three interest groups, resolving into one when the game ends, no matter how many powers are involved. Stalemate line draws tend to be of two interest groups. For example, a 17-17 draw (which is a stalemate line draw in most cases) can be between two one-player interest groups, or between a one-player and a two-player interest group, or various other combinations, This combination into interest groups explains why a perfectly played Diplomacy game ought to be a seven-player draw: no player will be willing to break up his interest group, because the chance will be too great that another interest group, which is playing perfectly, will move in and destroy his interest group and his own country.

This interest group concept applies throughout the game. The eastern and western spheres (Austria-Russia-Turkey, England-France-Germany, Italy in both or perhaps neither) are each an interest group at the beginning of the game. Their interest is in resolving their conflict before the other interest group can. If they do so, one or more of their number will win the game, (Of course, a “win” by a multi-member interest group is actually a draw unless it is further resolved within the group – unlikely in a perfectly played game.) In a perfectly played game it is unlikely that one interest group will triumph. Either, players will perceive that they must work together or else the other sphere interest group will gain the upper hand.

One can see that no player can begin a conflict in his sphere unless he can be sure that the conflict will be resolved in time to set up a defence against a threat from the other sphere. And ignoring for the moment the existence of stalemate lines of less than 17 units, this means that no player can begin a conflict in his sphere-interest group because he cannot be sure that the other sphere-interest group will not immediately begin an advance that will destroy his own interest group before its conflict can be resolved. In this manner, a seven-player draw would be the most likely outcome. Given the availability of stalemate lines of 14 or so units, a 6-player draw may be more likely than a 7-player draw. I am inclined to think that a 7-player, 3-way draw (with Italy being the third interest group) is the theoretical outcome.

Perhaps part of my disagreement with Edi stems from our definitions of a perfect player. A perfect player is one who maximises his minimum gain – this is the definition used in game theory. This means that a perfect player will always strive to avoid losing, If he has a choice between a win and six losses on the one hand, and seven 7-player draws on the other, he will take the latter because he has a minimum gain of a 7-player draw (1/7 of a win), while in the former he has a minimum gain of a loss. (Naturally I refer to “balance of power” players–any draw is preferable to any place, even second with 16 units,) If this perfect player is playing against imperfect players, he will do better than his minimum gain, some one or more of his opponents will err, and fewer will participate in the draw. The perfect player (again referring to game theory) assumes perfect play on the part of his opponents; then by maximising his minimum gain, he is assured of that minimum gain but can expect better if his opponents play less than perfectly – this is derived from the game theory definition of the best game strategy,

I think Edi would say that a perfect player, offered the above choice, would take the win and six losses. This means that he would take more chances 1/7 of the time he would succeed, 6/7 of the time he would fail (given perfect opposition), and as a perfect player he would not take the chances that gave him a greater than 6/7 risk of losing. I think that against perfect players this man would never find an opportunity to take such a risk, and he would have to accept a draw just as every other perfect player would.

But against imperfect players, this strategy would result in more outright wins and some losses than would the “draw” strategy. This difference is reflected partially in our play. My ratio of draws to outright wins is much higher than Edi’s, while my ratio of losses to non-losses is much lower. I have been completely eliminated only once, while Edi is eliminated relatively more often – an inevitable consequence of taking many chances. Given Edi’s type of player, someone in an interest group may take a chance, by breaking up his interest group, he may actually allow himself to come out on top if another interest group is broken at the same time. But he may also be committing suicide.

The advice given in Edi’s article is not good – far more often than not, you will find that you fail because you’re not a perfect player yourself. It is probably wiser to maintain the integrity of your own interest group and wait for someone from the other (or another) interest group to make an error first. This “error” of breaking up a sphere-interest group is so common in postal Diplomacy, where one can be certain that play will be less than perfect even in demonstration games, that the average player can often work to form an interest group of two players within his sphere by eliminating one of the sphere members, without much fear that the other interest group will stick together. In those cases where one interest group does remain together at game start, we see that it quickly crushes the other, broken interest group. The only thing that can save a broken interest group is imperfect play in the other interest group.

In defending the “myth” of the 7-player draw I have unintentionally presented an “even theory.” Most games will not follow this pattern of eastern vs. western interest groups because most games are not well played (and that is an understatement). The only 7-player draw I ever played (FtF) followed this pattern, even though it was far from perfectly played (which may be why Italy was intact in the eastern sphere while Turkey was down to one unit in Tunis holding the line for the west).

(I might note there that I’m using definitions from zero-sum game theory. Diplomacy is a zero-sum game, as one player’s loss is directly another’s gain. These ideas don’t necessarily apply to non-zero-sum multi-player games.)

Reprinted from Diplomacy World No.5 (Sept 1974)