This is a summary of an electronic discussion of the endgame in Diplomacy. The contributors were Mark Nelson (amt5man@amsta.leeds.ac.uk), Douglas Van Belle (dvpo@uno.edu) and Dan Shoham (shoham@hnc.com).
The discussion started when rec.games.diplomacy readers stated that they were interested in improving their endgame play and wondered if it would be possible to design a variant where the game started in an interesting endgame. The simple answer is, if you want to improve your endgame play then look at the list of games which need replacement players and join any which are in the endgame!
In response to the idea of an endgame-variant Douglas commented that the endgame in Diplomacy can often be boring. Dan's response was immediate
"End games are far from boring. Some of the best tactical plays come out at end games. Once the alliances and battle lines are drawn out clearly, the good tactician finally has enough information to work with and bring home the victory."
Are endgames boring? Far from it, they can be intellectually stimulating. Once the alliances have set, the leader usually has a secured set of centers and can readily identify the centers needed to win. In such circumstances it is often possible to analyse the position several years ahead, because it is easy to identify correct tactical play, and to determine what the optimal strategy is to win the game.
However if the good tactician "has enough information to work with and bring home the victory" the position is, by de facto definition, boring! An 'interesting endgame' is a position where the leader's units are scattered across the board, amongst those of the defending side. In these positions the leader not only has to concentrate on securing centers, he has to concentrate on avoiding the loss of centers. Such positions are considerable rarer than the 'boring positions' .
Douglas mentioned that he has some
research which
"seems to indicate that once the game gets
down to three players it almost always ends in a draw. Four players much
less often".
He went on to comment that
"Wins seem to come the most when there are five players
or more on the board just a few moves before the end" and that
this phenomenon "fits very well with a formal theoretic analysis of
anarchic systems."
I do not have any stats to hand about the state of the board just prior to the end of the game. However I do have stats on the number of eliminated players in games which finished in a win. In my database of 260 internet games, 105 finished in a win and 68 in a three-way draw.
In analysising this data a few approximations were made. First of all I assumed that the games were DIAS. This is not as unreasonable as it seems because my dataset is for games played according to the Electronic Protocol houserules which specified that games should be DIAS. Of the games which finished in a win 14 were concessions rather than rule-book wins. In including these games in my dataset I am assuming that if these games had continued to completion the number of eliminated players would have remained the same. This is not too unreasonable because in six of the games the winning player had 17 centers and in four of them the winning player had 16 centers. As shown in Table One, in most rule-book victories the winner has 18 or 19 centers. In most cases moving from 16/17 centers to 18/19 centers would not result in the elimination of an additional player. Therefore for simplicity I include the data from games which ended in a concession.
| # Centers | # Frequency | % Percentage Frequency |
|---|---|---|
| 18 | 57 | 62.6 |
| 19 | 25 | 27.5 |
| 20 | 5 | 5.5 |
| 21 | 3 | 3.3 |
| 22 | 1 | 1.1 |
The number of eliminations in games which finished in a win are shown in Table Two. Note that there were three or four players remaining at the end of the game in a significant majority of the games finishing in a win. Although this does not tell us how many players there were remaining "a few moves before the end" (is it possible to define this in a more rigorous manner?) it does suggest that Douglas is right with his comment that "Wins seem to come the most often when there are five players or more on the board just a few moves before the end".
Dan provided some opposing stats by examing 10 Judge games which finished in a dan-win. This showed that "Only 2 of them had 5 or more players within a year of the victory. If we ignore 1-2 SC powers, than none of the 10 had 5 or more powers within 2 years (up to 10 turns) of victory." This could be a reflection of the fact that these were games which Dan had won, and not a reflection of non-dan games.
| # Eliminations | Sample Size | Percentage | ||
|---|---|---|---|---|
| 1 | 6 | (5) | 5.7 | (5.5) |
| 2 | 17 | (13) | 16.2 | (14.3) |
| 3 | 40 | (37) | 38.1 | (40.7) |
| 4 | 37 | (31) | 35.2 | (34.1) |
| 5 | 5 | (5) | 4.8 | (5.5) |
In response to Douglas' post Dan commented that "Given that most wins pass through a point where there are only 3 players left, it follows that it is untrue that `once the game gets down to three players it almost always ends in a draw'."
Examining Table Two we observe that only 40% of rulebook wins passed through a point where there are 3, or fewer, players left. These games comprise 13.85% of the database (260 games). Recall that three-way draws comprise 26.15% of the database.
Assuming that all the games were DIAS we conclude that games which reach a stage where there are only three players remaining (assuming that all games which finished with two players remaining passed through a stage when three players remained) are twice as likely to finish in a 3-way Draw than a win.
In addition note that games which pass through a stage in which there are three, or fewer, players remaining can finish in a two-way draw, in addition to win and three-way draw; there were 39 two-way draws in my database.
Based on these numbers once a game reaches a stage where there are three players remaining (assuming for simplicity that all games which finished with only two players remaining passed through a stage where there were three players remaining) the probability that a game finishes in a win is 25.17%, that it finishes in a two-way draw is 27.27% and that it finishes in a 3-way draw are 47.55%. So the probability that once the game gets down to three players it finishes in a draw is 74.82%.
Of course these figures are derived making a DIAS assumption and by examining the number of eliminatees at the end of the game. Douglas comments that there is a significant difference in examining the number of players remaining at the end of the game and the number remaining the year before the game was won. "It is significant for, in theory, the concern is the decision making context prior to the end which would be the year before the game was won."
Clearly there is scope here for a more indepth analysis of Diplomacy game results.
Mark Nelson
(amt5man@amsta.leeds.ac.uk)
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