Good day, class.

Last time, we finished by asking what are the French chances of winning, and what is France's best strategy, when the situation on the board reduces to the following table (where 'w' indicates a French Win, 'd' a Draw, and '-' means that the position is unchanged and the game proceeds to the next move with pieces unmoved).

Apb | Amb | Amun | |
---|---|---|---|

Fsaf | |||

Fber | |||

Fbur |

Actually, I revealed that France has a range of strategies, which give her chances of winning up to, but not including, 1: a virtual win. But what are these strategies?

Intuitively, and informally, France must use her safety move,
**Fsaf**, for a while, to make sure that Austria isn't going
to play **Amb**. If/when she can eliminate the Austrian move
**Amb**, the table reduces to this one:

Apb | Amun | |
---|---|---|

Fsaf | ||

Fber | ||

Fbur |

And then France has the usual 'epsilon' strategies available,
namely, she plays **Fbur** with chance 1-epsilon, and **Fber**
with chance epsilon, for some small-valued epsilon.

But this is too informal. Let's make it rigorous. The strategy
(strategies) for France are what we might call (following Dan
Shoham) 'second degree epsilon strategies'. France wants her probability
of playing **Fsaf** to overwhelm her probability of playing
either of the other options, and her probability of playing **Fbur**
to overwhelm her probability of playing **Fber**.

Then her approach is this:

PlayFsafwith probability (1-e-e^2);

PlayFburwith probability e;

PlayFberwith probability e^2.

Let's recall what this means, in terms of the positions we actually studied. The French Berlin position turns out to be a Virtual Win for France. And this means the original position, in which France holds Munich and not Berlin, is also a Virtual Win, for we could force with chance 1-epsilon either the French Berlin position or an outright Win, depending on Austria's strategy. Barring a few loose ends, the original position is now fully analyzed.

Are there any questions? Ah, I see there are some questions!

**Q1**: This sort of situation seems awfully arcane. I thought
we were going to have some *useful* advice, some lessons
we could apply. What gives?

**Answer**: The position in question is not arcane at all,
but rather quite common. Admittedly, you will not very often find
the *exact* position we've analyzed, but you will find positions
very close to it. And just as important, you will find positions
that are evolving toward our analyzed position -- now you know
whether you want the evolution to continue or whether instead
you have to do something to prevent it.

Furthermore, a Second Degree Epsilon position extends beyond the situation we have analyzed. I know of one other such position, a position which was on the horizon in a game played by Dan Shoham. (Master Class students are encouraged to try this Chopin Etude.) Here it is:

Austria: F Bla, A Gal, F Ion, F Lyo, A Mar, F NAf, A Pie, F Por, A Rum, A Sev, A Tri, A Trl, F Tun, A Ukr, A Vie, F Wes.

England: A Boh, A Bre, A Bur, F Eng, A Gas, F Hel, F Iri, F MAO, A Mos, A Mun, F NAO, F Nth, A Pru, A Sil, A Spa, A StP, A War.

Ownership of Supply Centers:

Austria: Ank, Bud, Bul, Con, Gre, Mar, Nap, Por, Rom, Rum, Ser, Sev, Smy, Tri, Tun, Ven, Vie.

England: Bel, Ber, Bre, Den, Edi, Hol, Kie, Lon, Lvp, Mos, Mun, Nwy, Par, Spa, StP, Swe, War.

Each side owns 17 centers; the position east of Switzerland is locked up. Austria wins if she takes Spain without losing Mar or Por. A little thought about the position will show that Austria has all the chances here, and England is just hoping to hold on to the draw.

Work out the Relevantly Different sets of orders for England and Austria, and plot them on a table. You will find that the table is isomorphic to the one for French Berlin. Austria has a Second Degree Epsilon strategy.

**Q2**: Are there Third Degree Epsilon strategies? Fourth Degree??

**Answer**: In theory, yes, but not on the actual Diplomacy
board. Actually, I am not perfectly certain that there are no
Higher Degree Epsilon positions on the actual Diplomacy board;
maybe someone can discover one! I have constructed fanciful maps
on which Higher Degree Epsilon positions can arise, and a recipe
for making maps to illustrate any degree. But these are novelties,
or of merely theoretic interest.

**Q3**: In the set-up for the French Berlin position, and for
the original position to be analyzed, you weaseled around a bit.
You *assumed* that France could always order A Pie-Trl, but
you admitted that this assumption is unrealistic. Won't you come
clean now?

**Answer**: Uh, ahem, yes. (Conrad, take off that mask, I know
it's you.) Barring some preposterous, hopelessly unrealistic gerrymandering,
the position on the board will admit *some* chance for Austria
to dislodge the French army from Piedmont. We can make the chance
small, and we can begin with Austria needing to jockey her units
around in a very tedious, arduous way in order to have any hope
of dislodging that unit. In most games, and most situations, we
would be justified in writing off the possibility. However, we
are considering Virtual Wins, situations in which one side can
get its chances up as high as it wants. The complication of Piedmont
will mean that there is a theoretic ceiling, lower than 1.0, above
which France cannot get her chances after all. Teacher is too
tired of the position to investigate what this ceiling might be
for plausible variations of the Mediterranean portion of the French
Berlin position. Perhaps a reader might like to have a shot at
it?

**Q4**: Speaking of realism, isn't this epsilon deal a fairytale?
In real life, no one is going to be willing to wait a thousand
game years for a win. How might this fact be added to the analysis?

**Answer**: Conrad Minshall has suggested the following plausible
apparatus. Having a .999 chance of winning in 1000 game years
is, intuitively, worth a whole lot less than having a .99 chance
of winning in 10 game years. There is a kind of discount rate,
according to which the same result is worth less now when that
result is projected into the future. In economics, the value of
money now is related to the value of the same quantity of money
later according to the (projected) interest rate; this phenomenon
is fairly well understood. Now, the discount for futurity in Diplomacy
outcomes is going to depend in large part on the psychology of
the given player. But there are some manageable factors, too.
For instance, you might find it important to get your Hall of
Fame points before September 1st, when Nick Fitzpatrick does the
next calculation to determine who is invited to play in Hall96.
Given a discount function, we will be able to optimize the value
of epsilon to maximize your expected value.

One further note is relevant. When France is playing an epsilon
strategy, what is Austria's *best* counter-strategy? Unless
epsilon is very high (which would be pretty dumb), Austria's best
strategy is to throw caution to the winds and play her riskiest
move on the very first try. This is a happy fact, for it means
that if you steadfastly play your epsilon strategy, and your opponent
plays her best counter-strategy, you are very, very likely to
win right away.

**Jamie Dreier
Brown University
(**

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