Diplomacy Paradoxes

Andy Schwarz

Introduction

As far as I can tell, all Diplomacy paradoxes stem from non-simultaneity introduced into what is an otherwise simultaneous game. In particular, these paradoxes arise when the success of one type of move is made contingent on the success of another move during the same turn. Not all contingent movement is rife with paradox, of course. Successful movement of a unit is in some sense contingent on the success or failure of other units. In Spring 1901, A Mar-Pie will fail if A Ven-Pie is also ordered, but will succeed otherwise. However, the resolution of these events is simultaneous, in that we don't look at whether A Ven-Pie succeeds before deciding the fate of A Mar.

However, any rule in Diplomacy -- like the dislodgment rule for disrupting convoys or the various Suez Canal control rules in Colonial Diplomacy -- which make the success of one unit's actions contingent on the success of some other event during the same season will kill simultaneity and introduce room for paradox. This is a subtle distinction, which I may not be making clear enough, but consider the famous convoy paradoxes, such as Pandin's Paradox, which all rely on the issue of whether the convoyed units can cut the very support needed to ensure the success of their convoys, to beleaguer a garrison, and so on. If the fleet is disrupted, the convoy fails, making the army's move contingent on the success of the convoy move. Unlike a bounce resolution, we now have to see if the convoy succeeds before we can move the convoyed army. So, if we decide it does succeed and transport the army across the water, it can have an impact on the adjudication we had to perform to determine if the convoy succeeded. This is what I mean my non-simultaneous contingency.

Since the Diplomacy convoy has been repeatedly analyzed, I would like to cite two new milieu in which the rules introduce non-simultaneous contingency and how that leads to paradox. From Colonial Diplomacy, I'll use the Suez Canal rule as an example. And from Machiavelli, I'll discuss the Sicilian Straits rule (known as the "Piombino/Sicilian Straits" rule in the first edition of Machiavelli).

Part 1: Schwarz's Suez Paradox

Consider the following rules from Hasbro's Colonial Diplomacy:
"9.31 You must have a unit in Egypt the entire turn in order to use the canal"

"9.33 The unit in Egypt may have an attack order so long as it results in a standoff and the unit in Egypt stays in Egypt."

Clearly, this introduces non-simultaneous contingency. The attack may be ordered but it must result in a stand off, meaning we need to resolve the move from Egypt before we can determine the success or failure of a move through the canal. This opens the door for what I modestly have named Schwarz's Suez Paradox. For example, 

England: Army Egypt -> Mecca.
England: Army Syria SUPPORT Army Egypt -> Mecca.

Turkey: Army Arabia-Mecca.
Turkey: Fleet Red Sea SUPPORT Army Arabia -> Mecca.

If the unit in Egypt were not required to stay put by rule 9.33, this would be easily resolved; the Med-Red move would cut the Red Sea's support and Egypt would move to Mecca having two strength to Arabia's one.

However, by rule 9.33 and rule 9.31, we know that this easy resolution would result in the Mediterranean fleet not being able to use the Suez Canal because that would allow the unit in Egypt to leave, violating 9.31, and thus not preventing the attack on the Red Sea fleet.

So, then the Red Sea's support is not cut, which means both Egypt and Arabia move to Mecca with strength of two, resulting in a standoff.

Which, by Rule 9.33, means the move from the Mediterranean Sea to the Red Sea is legal, which means the support is cut, which means the Egyptian move succeeds, which means the support is not cut, ad infinitum.

But it gets worse!

"9.36 If a superior force dislodges the unit controlling Egypt, a fleet ordered to use the canal will be prevented from doing so."
Again, we've introduced non-simultaneous contingency. Just like a convoy's dislodgment determining the success or failure of a convoyed armies move, now the dislodgment of a unit in Egypt determines the success or failure of a "Suezing" units move. This creates the Schwarz-Woo Paradox, found by Andy Schwarz (me), with assistance from Alexander K. ("Smiley") Woo. Consider the following: 

Russia: Fleet Mediterranean Sea -> Red Sea.
Russia: Fleet Egypt Hold.

England: Army Sudan -> Egypt
England: Fleet Red Sea SUPPORT Army Sudan -> Egypt.

Turkey: Army Syria-Egypt.
Turkey: Army Mecca SUPPORT Army Syria -> Egypt.

If we ignore the 9.36 rule, the move by the Mediterranean fleet cuts the support of the English fleet in the Red Sea on Egypt, making the Sudan army attack with a strength of one and thus allowing the Turkish assault, with strength two to succeed. This of course, dislodges Egypt. However, now we need to consider rule 9.36, which says that dislodging Egypt invalidates F Med-Red. So the support of Red Sea is not cut, which means both England and Turkey attack with strength two, which invokes the beleaguered garrison rule, meaning Egypt is not dislodged. So, thus we should allow the Med-Red order to go through after all, which in effect, dislodges Egypt. Once again, repeat ad infinitum.

One solution for these two paradoxes is to add a rule which says that a "Suezing" unit cannot cut support of a unit in the Red Sea (or Med Sea) that is supporting an action in or into Egypt. This is how the convoy paradox was removed in standard Diplomacy (saying that a convoyed unit does not cut the support of any unit supporting an action into a body of water), but the result is that one can interpret this to mean that even if the Red Sea is dislodged its support is not cut. For an example of this, consider the same set up as above, but now add an additional Russian Fleet in Eritrea: 

Russia: Fleet Mediterranean Sea -> Red Sea.
Russia: Fleet Egypt Hold.
Russia: Fleet Eritrea SUPPORT Fleet Mediterranean Sea -> Red Sea.

England: Army Sudan -> Egypt
England: Fleet Red Sea SUPPORT Army Sudan -> Egypt.

Turkey: Army Syria-Egypt.
Turkey: Army Mecca SUPPORT Army Syria -> Egypt.

If you don't prevent Suezing units from cutting support of units in the Red or Med Seas, the paradoxes will remain, but if we do interpret it this way, we get a "Ghost Ship" problem where a dislodged fleet ends up adding valid support, which is unsavory from an intuitive point of view. In the above adjudication, for example, a blanket prohibition on Suezing fleets cutting support would mean that Med-Red succeeds and dislodges the English fleet in the Red Sea but that the dislodged Red Sea fleet still provides support for the move from Sudan into Egypt. Otherwise, we're back in a paradoxical loop.

Part 2: Schwarz's Straits Paradox

Machiavelli has a similar problem. Although Avalon Hill's second edition removes the straits around Piombino (from which this example is derived), the Sicilian Straits at Messina remains (and the Judge's implementation still use the first edition), so this has more than just curiosity value.

The First Edition Rules state:

Piombino: This province includes the offshore island of Elba. A fleet in this province controls the straits between the mainland and the island. An army unit in this province does not control the straits. The straits are not a separate sea, but are part of the Eastern Tyrrhenian Sea. If they are not controlled by a Fleet unit in Piombino, a hostile Fleet could move directly from Pisa to the Eastern Tyrrhenian Sea or vice versa, or transport an army from one to the other.

Sicily: A fleet in Messina similarly controls the straits between Messina and Otranto. As long as there is a fleet in Messina, no hostile fleet could move directly from the Gulf of Naples to the Messina and Otranto. As long as there is a fleet in Messina, no hostile fleet could move directly from the Gulf of Naples to the Ionian Sea or vice versa.

If a fleet successfully advances into Piombino or Messina in the same turn as a hostile fleet attempts to use the straits, the movement through the straits is blocked. As long as a fleet unit is in the controlling province, no other player's units may use the straits without the controlling player's permission. If a player wishes to allow another player to use the straits, he must include an "alliance power" command with his movement orders.

The second ediion rules run as follows:
Also, a Fleet (not an Army) in Messina controls the straits between Messina and Otranto. As long as there is a Fleet in Messina, no enemy fleet may advance from the Ionian Sea to the Gulf of Naples, or vice versa, or convoy an Army through one of these seas to the other.
Notice again how this rule introduces non-simultaneous contingency, namely the status of any unit "using" either strait (for movement or support) is contingent on who ends the turn in the critical space. Focusing on Piombino, consider Schwarz's Straits Paradox: 

France: Fleet Pisa SUPPORT Fleet Piombino -> Eastern Tyrrhenian Sea.  (*blocked*)
France: Fleet Piombino -> Eastern Tyrrhenian Sea.

Naples: Fleet Eastern Tyrrhenian Sea Hold.
Naples: Fleet Sienna -> Piombino.
(The term blocked is used by the judge to indicate that an otherwise legal move is disallowed because the unit in question did not have permission from the owner of the critical space for controlling the Straits).

Note that Naples owns Piombino because it gets control of it during the course of the turn, as proper according the second paragraph above. However, if Naples has control of the Straits, then Pisa is effectively blocked from supporting into East Tyrrhenian Sea, because Pisa and the Eastern Tyrrhenian Sea are only contiguous if the player is allied with the Piombino owner. This is why the Pisan order is marked as (*blocked*), indicating Pisa was not granted permission to use the straits for support. So, in theory, the move by France from Piombino ought to have failed, since its support was blocked/invalidated. But in that case, F Sienna-Piombino would bounce, leaving Piombino in French hands. Of course, that would make the support by Pisa valid, dislodging the ETS fleet, and vacating Piombino, thus allowing Sienna into Piombino. But that would invalidate the support, killing the attack, and so on, ad infinitum. The Classic "Non-Simultaneity Paradox" is seen here in all its glory.

In response to this paradox (which really came up in a judge game I was moderating), Charlie Eldred, posed a different sort of paradox arising from the same set of rules. All of the paradoxes I have discussed until now involve loops of inconsistent logic. Jamie Dreier, Professor of Philosophy at Brown University and an all-star Diplomacy player, makes the analogy with the liars paradox:

L: This sentence, L, is false.
The adjudication of a Diplomacy (or Machiavelli) turn is akin to determining which of a series of statements are true (moves succeed) and which are false (moves fail). Statement L cannot be properly adjudicated. Assume it is true, then it is false (it says it is false, after all). But if it is false, then it is true. Jamie calls this an ungrounded and paradoxical statement. This is the Schwarz's Suez paradox (and the Pandin's Paradox and the rest) in a nutshell.

However, following S. Kripke's landmark paper, "Outline of a Theory of Truth," Jamie offers up a second type of paradox (or quasi-paradox) which is ungrounded but not (technically) paradoxical. For example, consider the following statement:

T: This sentence, T, is true.
On the surface, it seems simple enough. If it is true, then it is true. Viola. Except, what if we say it is false? Then it is also false. So it is ungrounded (we have no other way to determine if it is true or false) but it is not paradoxical, because once we assume it is true (or false) it remains true (or false).

The Eldred Piombino Paradox is this sort of quasi-paradox (an ungrounded, non-paradoxical move with two equally valid adjudications), as seen below: 

France: Fleet Pisa -> Eastern Tyrrhenian Sea. (*blocked*????)
France: Fleet Piombino SUPPORT Fleet Pisa -> Eastern Tyrrhenian Sea.

Naples: Fleet Sienna -> Piombino.
Naples Fleet Eastern Tyrrhenian Sea SUPPORT Fleet Sienna -> Piombino

The move Sienna to Piombino cuts the Piombino fleets support of Pisa-ETS, without question.

If we assume Piombino is successfully dislodged by the supported attack from Sienna, then the support of ETS is not cut, and the Piombino fleet is legally dislodged, fully consistent with our assumption of dislodgment.

But if we say ETS's support is cut by Pisa, then Piombino is not dislodged, cutting the support by ETS, again fully consistent with the assumptions of cut support.

So, rather than creating an infinite and unresolvable loop (a paradox), Charlie has created two equally valid solutions, which are contradictory (an ungrounded statement).

Nonetheless, both the Schwarz's Straits paradox and the Eldred Piombino Quasi-Paradox are the result of a non-simultaneous contingency, in this case who ends the turn in Piombino (or similarly, in Messina for the Sicilian version).

Part 3: What Do We Do?

I think that introducing more conditional, non-simultaneous moves into Diplomacy via Colonial's Suez (in addition to the long-standing convoys) was a step in the wrong direction. Removing such moves, as we've seen in the removal of the Piombino Straits rule with the release of the second edition of Machiavelli (but which sadly remains in the Sicilian Straits), is the correct approach. The rule on using the Suez canal (or the Sicilian Straits) should be something to the effect of: "To use or permit the use of the Suez canal (Sicilian Straits), a power must start the turn with a unit in Egypt (Messina)." That's it -- no rules on bouncing or having to stay in Egypt the whole turn. The can of worms that the current rules open is far worse than any benefits of perceived realism. By making use in Fall conditional only on the status at the end of Spring retreats (Colonial actually uses years rather than seasons, but you get the idea) or Spring moves conditional only on the status at the end of winter, you do not introduce non-simultaneity/contingency, and thus you avoid paradoxes like the Schwarz Suez and the Schwarz-Woo. Ditto on the Schwarz Straits and the Eldred Piombino.

The only other solution is to live with the paradoxes, which is often phrased as "All units involved hold." However, try neatly defining "all units involved" and you need an algorithm that boggles my mind, though it does exist. My feeling is that we are stuck with convoy issues because the game is over 35 years old and convoys have been there since 1959. But, we ought to root out the Suez evil while the game is still young, and fix things in Machiavelli while the second edition is still hot off the press. Heck, if we can add a land bridge to Sakhalin, we can change the Suez rule a tiny bit, replacing 9.31 with "You must have a unit in Egypt at the start of the turn in order to use the canal" and eliminating 9.33, 9.36, and perhaps the now innocuous but unneeded 9.37.

By the way, when I first brought the Suez paradox to the attention of Avalon Hill, they seemed ready to make a quick fix via The General. However, after Smiley and I developed the second paradox, AH told me they had gone back to the designer to rework things, and a pronouncement would eventually follow.

Has anyone had seen such a pronouncement? I have not.

Part 4: The Trans-Siberian Railroad

In case anyone was wondering about the Trans-Siberian Railroad rule in Colonial Diplomacy, here's a short analysis:

Rule 9.23 begins by saying "A unit using the TSR may only be transported to an unoccupied province." If we interpret this to mean that it is not even valid to order a unit on the TSR to attempt to enter a space that **began** the turn as occupied, then I do not think the TSR order will create paradoxes, because since TSR units are never allowed to reach an SC that is occupied, it is impossible to use TSRs to cut support, dislodge other untis, or to support a beleaguered garrison, which are the main causes of paradoxes through non-simultaneous contigency.

Assuming my interpretation of 9.23 is correct and TSR units can't be ordered to spaces that begin the turn as occupied, then the TSR rule basically just makes the TSR unit get to make a series of mini-moves contingent only on the success of its own previous mini-moves. Therefore, it is possible that the TSR has avoided the pitfalls of the Suez Canal.

However, it has been pointed out to me that Avalon Hill (in The General, Volume 30, Number 4) has clarified that TSR units can attempt to enter occupied provinces in the hope that they become unoccupied as the turn moves on. This would normally open up some interesting paradoxes. Consider the simple: 

China: Army Krasnoyarsk -> Mongolia.
China: Army Mongolia -> Irkutsk.

Russia: Army Omsk TSR Irkutsk
This normally would allow for a paradox: If Omsk bounces Mongolia-Irkutsk, then Krasnoyarsk cannot leave and so the TSR cnnot go through, so it does not bounce Irkutsk, so Krasnoyarsk can leave, so it does go through, so it bounces, so Kra stays, so the TSR move fails, etc.

However, and very wisely I might add, Avalon Hill added the following proviso and example:

9.25 - A curious example of this rule applies if an enemy unit attacks from one province to antoher along the line of the TSR at the same time that a Russian unit is railed through these provinces in the same dircetion, such as:

Russia - A VLA-TSR-OMSK

China A KRA-OMSK

This results in a standoff in Omsk. The Chinese unit stays in Krasnoyarsk and the Russian unit stays in Irkutsk. The theory being that Krasnoyarsk is left empty by the Chinese at the same time the Russian unit moves through. Thus both units attack Omsk ands cause a standoff.

In other words, the very attempt (and not the success) of a unit to leave a space is enough for the TSR to be allowed to try to get through. Notice that by linking the success of the TSR's attempt to move to attempted movement, rather than successful movement, rule 9.25 turns non-simultaneous contigent movement into standard Diplomacy movement. By 9.25, in my example above, the TSR is allowed to try its luck in Irkutsk and it fails, bouncing Mongolia, which bounces Krasnoyarsk and no one moves. Paradox avoided.

However, while I think that this successfully avoid paradoxes, it creates a situation where I unit can jump over a foreigner on the TSR, or so it seems. Consider the Avalon Hill example in rule 9.25, but with a slight twist: 

Russia: Army Irkutsk TSR Omsk.
Russia: Army Perm SUPPORT Army Irkutsk TSR Omsk.

China: Army Krasnoyarsk -> Omsk.
As far as I can tell, this is a fully legal move and Russia's Irkutsk army is allowed to try for Omsk because China tried to vacate Krasnoyask. And now, Russia has legal support (it is legal to support a TSR'ing unit into a vacant square, just not to support an TSR'ed attack on a unit), so Russia gets Omsk. So Krasnoyark stays in place and the Russian has leap-frogged over the Chinese unit on the TSR route. And if you scoff at this and say that's ridiculous, that the general rule of no leapfrogging has precedence, remember if you try to resolve this without that 9.25 interpretation, you trade this silly result for a paradox without any resolution. Which is better: ridiculous consistency or non-ridiculous unresolvability?

Finally, let me add an interesting side effect of this rule. In the sitution above, if China had ordered its army in Krasnoyarsk to hold, the TSR move would have failed and the Irkutsk army would not have been allowed to move. So in this case, while this is not an adjudication paradox, we have a situation in which if China tries to bounce Russia, Russia does not bounce, but if it does not try to bounce it, Russia does bounce. This is a very deep thing, beyond the scope of this article. I think this is like the Zen belief of only getting something you want if you really don't want it, and I will leave the TSR here for others to ponder.

Andy Schwarz
(schwarz@haas.berkeley.edu)
University of California at Berkeley

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