The Endgame in Diplomacy: An Electronic Conversation
by Mark Nelson
Introduction
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!
Are Endgames Boring?
In response to the idea of an endgamevariant 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' .
What happens in the endgame when three players remain?
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 threeway 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 rulebook 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 rulebook 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 
Table One. The number of centers owned by the winning player in
Electronic Protocol games which finished in a rulebook
victory.
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 danwin. This showed that
"Only 2 of them had 5 or more players within a year of the victory.
If we ignore 12 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 nondan 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) 
Table Two. The number of eliminatees in games which finished in
a win. The bracketed figures refer to the corresponding
figures for a game which ended in a rulebook win.
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
threeway 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 3way 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 twoway draw, in
addition to win and threeway draw; there were 39 twoway 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 twoway
draw is 27.27% and that it finishes in a 3way 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.
