Warning--this post will probably turn long and rambly.

Like David, I remember reading these articles a long time ago and remember thinking that the "Odd Theory" article was wrong. I also remember thinking that Pulsipher's article made sense.

Now that I've come back to it after studying more game theory, I have to say, not only are both articles wrong, but they are wrong

on their own terms.

David's point seems to be that we shouldn't care about a "perfectly played" game because humans will never play perfectly, so it doesn't matter. This is an argument that I am very much used to hearing in the context of a different game (poker), and I'd say something even stronger here than I say there--if you understand enough game theory, it actually SUPPORTS the idea that a "perfectly played" game of Diplomacy does not really exist, even in theory!

Pulsipher is doing a few things in his article that I think are mistakes from a game theory perspective. The first one is that he is trying to use the "maximin" idea that game theory usually applies to 2-player games, and extend it to a 7-player game. My understanding of game theory is that this is actually a mistake. Game theory says that in games with more than 2 players, the natural course of the game should be something like:

1. Players form coalitions and create a "sub-game" of one coalition against another.

2. Each coalition plays "maximin"

as a group against the other coalition. This can be VERY different from each player playing to maximize their own minimum result.

3. After the dust settles, any coalition that has won anything divides the shared utility among them.

Furthermore, with Diplomacy being a repeated game (i.e. as many turns as necessary until the result is reached), there is a chance for coalitions to shift at every juncture in the game. This blows up the ideas underpinning both of the articles mentioned in the OP.

Birsan's article seems to be trying to "intuit" the above coalition-based framework, but it's sort of self-contradictory. He assumes that a "well-played" game will naturally begin with a coalition of 5 against 2, then when the 2 are eliminated, the 5 breaks down into 3 vs. 2. Like David, I see no inherent reason why that should be the case. One reason is because the idea of the "coalition" of 5 isn't really a coalition of 5 if the idea is for 2 of them to be eliminated later in the game!

A real-life example might be a "Central Power System"--G/I/A team up with F and R in the opening to kill off E and T, but the idea is that F and R will be the next ones out. Is that really a 5-on-2 coalition? I'd say it's actually a G/I/A coalition against an E/F/R/T coalition where the latter coalition is playing sub-optimally and loses.

Pulsipher's article makes some claims that I think are mistakes as well. His "interest groups" are somewhat, but not really, the same as what I'd call "coalitions"; but from a game theory perspective, there should always be at most two coalitions (a game-ending draw is the only time there can be one). Also, at one point he says this:

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.

I disagree almost entirely with everything in this paragraph. First of all, I see no reason why the opening coalitions in a game of Diplomacy should be divided up by the stalemate line. From a game theory perspective, one coalition is formed to defeat the other coalition. So for example, if E/F and R/T both ally and start fighting the central powers, Pulsipher might say that the Western and Eastern interest groups are both playing sub-optimally, but I'd argue that an E/F/R/T coalition has formed, and that this is not sub-optimal, because the E/F/R/T coalition can certainly eliminate G/I/A with good tactics.

Second of all, the idea that there are three coalitions is wrong from a game theory perspective. For example, let's say that there's some game where E/F/G buy into Pulsipher's thinking and decide that their optimal play is to form a Western Triple. I would argue that if this happens, the two (not three) natural coalitions to form are E/F/G/T against R/A/I. Italy is not an "odd man out" here, it's a member of a 3-power coalition fighting against 4. The same idea could be argued for anytime you might think there are more than 2 coalitions--if the third coalition is cooperating with one of the first two, then that's just one bigger coalition.

I am of course not trying to argue that stabs can't happen--I'm just saying that when they do, it represents the coalitions changing. And the big question mark in a game like Diplomacy, of course, is that a player can attempt to play optimally within what they believe to be their coalition, but be wrong about what the coalitions are! That is the big reason why "perfect play" doesn't make sense in Diplomacy from a game theory perspective--one hallmark of good play in Diplomacy is to convince other players to misplace their trust, and I think that speaks exactly to what I'm talking about here.

So anyway, one other point is that if the powers on a Diplomacy board sort themselves into two coalitions, even assuming that all powers are not deceived about what they are, it is extremely unlikely that those coalitions will be of equal strength (in fact, if they are, then one power could switch coalitions and automatically be on the stronger one). If one coalition is stronger, and they stick together, they could sweep the board clear of the other coalition. This completely contradicts Pulsipher's point that a game with 7 players "should" have no eliminations--a coalition of 4 could very easily eliminate a coalition of 3 and this does not contradict game theory's definition of "perfect" play.

But in fact, perhaps the single biggest problem with trying to evaluate Diplomacy from a game theory perspective is more fundamental than any of this. At the end of the article, Pulsipher claims that "Diplomacy is a zero-sum game". I don't know whether this was considered uncontroversial when the article was published (I wasn't even born yet), but it certainly is controversial now! Not only is there no rigorous argument that Diplomacy is zero-sum--and I think actually most serious players would say that it isn't--there is no universally accepted way to attach utilities to the outcome of a game, so it is not even possible to prove whether it is or isn't.

That means that anyone trying to determine how "perfect" play looks would have to begin by asking what the utility of all the different outcomes are--even BEFORE taking everything else into account. And if no answer can be arrived at, then there can't be any such thing as a "perfectly played" game.