Here is the solution for 100% victory for Austria.
Skip to the "SUMMARY" section if you would rather skip the lengthy development of the solution.
| PW Strategies | ||
|---|---|---|
| 1 | 2 | |
| DC Guesses PW Uses Strategy 1 | No Change | Win |
| DC Guesses PW Uses Strategy 1 | Win | No Change |
The PW will just flip a coin to decide which strategy to take. If he fails to win, he will flip it again next turn. Eventually, the DC will mis-guess, and the PW will win. Saying all of this mathematically, the probability of victory within N turns is P(N) = 1-(1/2)^N. The probability of ever winning is the limit of P(N) as N approaches infinity. That probability is 100%, and the expected number of turns to complete the game is 2.
Notice the difference between this position and a Strongly Forced victory. The PW is no longer able to broadcast his orders in advance. If he does, the DC will be able to perpetually prevent him from winning. I refer to this position as "Weakly Forced." Since there is no risk involved, I also like to call it "Zeroth Degree Force."
| PW Strategies | ||
|---|---|---|
| Basic | Daring | |
| DC Guesses PW Uses the Basic Strategy | No Change | Win |
| DC Guesses PW Uses the Daring Strategy | Win | Stalemate |
On the face of it, it might appear that the PW has a 50% or so chance of winning, but this isn't so. The probability of victory is actually as high as the PW would like to set it. For instance, the PW might choose to play the "Basic" strategy 99% of the times, and the "Daring" strategy 1% of the time. The PW can even tell the DC that this is his strategy (but not tell them on which turn he is "Basic" and when he is "Daring"). Now, if the DC decides to defend by guessing the "Basic" counter-strategy, DC has no hope. Either they'll lose that turn (1% of the time), or nothing will happen (99%). The fact that the chance of losing -- on the turn -- is small is no consolation. Eventually, DC will lose. By guessing "Daring," the DC at least has a chance. There is a 1% chance that the DC will gain a stalemate (because there is a 1% chance that the PW will choose to play "Daring" that turn). On the other hand, there is also a 99% chance of losing on the turn. The overall probability of victory for the PW is 100% if DC guesses "Basic" and 99% if they guess "Daring."
Now the expected length of the game is 100 turns if the DC guess "Basic" (since that's how long before the PW takes his 1% "Daring" turn). The game will last exactly one turn if the DC guess "Daring."
Now, before anyone says "1% chance of being stalemated is pretty good, but it ain't 0%," I'll return to the limiting argument. The DW didn't have to choose 1% as the probability of being "Daring". He could just as easily have chosen 0.1% or 0.01%, etc. The probability of being stalemated would be equal to the probability of playing the "Daring" strategy, and is completely settable by the PW to be as low as he would like it to be. In the limit, it's 0%. Note that the expected game length -- if the DC chooses to guess "Basic" (which is a hopeless play, as seen above) is inversely proportional to the probability of playing the "Daring" strategy. If the DW decides to play the "Daring" strategy only once in a thousand turns, then -- as long as the DC insists on always guessing "Basic" -- it is expected to take a thousand turns to complete the game. Of course, the "superior" (in so much as one may talk about "superior" strategies when they all have 0% chances) DC play is to guess "Daring." At least it gives some chance -- however small -- that there be a stalemate.
To put it in (more) mathematical terms, the optimal strategy for the PW player is to play "Basic" with probability "1 minus epsilon" and "Daring" with probability epsilon, where epsilon is an arbitrarily small positive number. The probability of winning is thus "1 minus epsilon", which is arbitrarily close to 100%. This position is also "Weakly Forced." Since it is necessary for the PW to employ a strategy with diminishingly small (epsilon) probability, I like to call this position a "First Degree Force."
In Chopin, there is a "Basic" strategy. I'll call it "A":
A: NAF-MAO, WES-SPA, POR S WES-SPA, MAR S WES-SPA, LYO S MAR, PIE S MAR.The advantage of that strategy is that there is absolutely no risk to Austria (that a stalemate line would be formed). POR can't be taken because the only two units that adjoin it (MAO and SPA) are both attacked. Neither could provide the other with an uncut support. MAR can't be taken since it is supported twice, and the maximum attack on it can only have two supports. WES is equally undislodgeable. Only one hostile fleet adjoins it (and it's impossible to get a fleet into Spain without losing it in the process). The only way WES leaves its present post is if it goes to SPA (which will conclude the exercise). NAF is likewise immovable. The worst that could happen (from Austria's point of view) is for there to be no change.
Austria will lose nothing by playing strategy "A" with an arbitrarily high probability (say, 99%). This will assure a 99% probability of winning against any counter-strategy which does not properly defend SPA without any risk to Austria.
The disadvantage of the "Basic" strategy is that it is easy to defend SPA. There are two possible counter-strategies (other counter-strategies are also possible, but they are equivalent or strictly inferior to at least one of these two). I'll call these counter-strategies "1" and "2":
1: MAO-POR, BUR-MAR, GAS S BUR-MAR, SPA S BUR-MAR, RUH-BUR. 2: MAO-POR, GAS S SPA, BUR-MAR, SPA S BUR-MAR, RUH-BUR.Strictly speaking, care should be taken to assure the non-dislodgement of the MAO fleet. However, since none of the Austrian strategies discussed here even attempt to dislodge MAO, I won't bother. In any event, within one turn (if France cooperates) or two turns (if he doesn't), there will be three fleets behind MAO capable of assuring the non-dislodgement of MAO.
Against "1", Austria could easily counter with "B":
B: NAF-MAO, WES-SPA, POR S WES-SPA, MAR S WES-SPA, LYO S WES-SPA, PIE S MAR.There is a problem with "B", but it's solvable. "B" is vulnerable to the formation of a stalemate line by yet a third potential counter-strategy (the one France mentioned in an earlier broadcast):
3: MAO-POR, SPA-MAR, BUR S SPA-MAR, GAS S SPA-MAR.However, this isn't such a great problem. Since "3" losses against the basic "A" strategy -- which is played 99% of the time -- any attempt by the defenders to employ it is extremely likely to end in an instant Austrian victory. All Austria needs to do is to play "B" with a probability of 1% and that will take care of "1". In that respect, "B" and "3" acts like the "Daring/Guess Daring" strategies.
The problem for Austria then becomes "2". Strategy "2" doesn't form a stalemate line against either "A" or "B", but it does avoid losing to either.
To defeat "2", Austria could do "C":
C: NAF-MAO, MAR-SPA, WES S MAR-SPA, LYO S MAR-SPA, POR S MAR-SPA, PIE-MAR.The problem with "C" is that it is vulnerable to the formation of a stalemate line against defense "1". But, as we have seen, "1" is already defeated by an "AB" 99%-1% combination. All we need to do now is to add a very small probability of playing "C" to also take care of "2". This probability should be smaller than the probability of playing "B" by as much as the probability of playing "B" is smaller than the probability of playing "A" is (99 to 1).
This way, if the defense plays "1", there is a 99 times higher chance that it will be countered with a winning "B" than with a stalemated "C". (It doesn't matter how often it is countered with "A" since nothing will then change.) If the defense plays "2", it could never form a stalemate line, and eventually it will be countered with a winning "C". (It doesn't matter how often it is countered with "A" or "B" since nothing will then change.) If the defense plays "3", there is a 99% chance it will immediately lose to "A".
The existence of a strategy which should only be played with a small probability relative to another strategy which itself is played with a small probability, is why I like to call this position a "Second Degree Force."
To evaluate the probability of Austrian victory, one should ignore
those cases where nothing changes and only compare the likelihood
that the game will end in an Austrian win to the likelihood that
it will end in a stalemate.
If EL(F?) play strategy "1", the probability of an Austrian win
on any given turn (0.99%) is 99 times higher than the probability
of a stalemate line being formed (0.01%). Therefore, the overall
probability of an Austrian win against strategy "1" is 99%.
(It will take, at worst, 100 seasons for the game to terminate
if EL(F?) play strategy "1" every turn.)
If EL(F?) play "2", the probability of an Austrian win on any
given turn is 0.01%. However, the probability of a stalemate
line being formed is 0%. Therefore the overall probability
of an Austrian win against strategy "2" is 100%.
(It will take, at worst, 10000 seasons for the game to terminate
if EL(F?) play strategy "2" every turn.)
If EL(F?) play "3", the probability of an Austrian win
on any given turn (99.00%) is 100 times higher than the probability
of a stalemate being formed (0.99%). Therefore, the overall
probability of an Austrian win against strategy "3" is approximately
99%. (It will take, at worst, only somewhere around one season for
the game to terminate if EL(F?) play "3" every turn.)
These results are outlined in the "OUTCOME" column of the table.
(Do not forget that the 99% is only an example. The actual probabilities
can be set as high as one wishes to. You can be assured that I won't
set it any lower than 99.9% :) So what if it might take a thousand or
a million seasons to finish? I'm in no hurry :) ).
If players do insist on going on for a long time, we might make an
arrangement with the GM to resolve the continuation of the game quickly.
I could provide the GM with a list of the years in which I will be
playing strategies "B" (on average, once every 1000 seasons) and "C"
(on average, once every million seasons). Lowland, could likewise
provide the GM with the strategies he'll play. The GM can then instantly
conclude when and how the game will end. (Saying something like
"Austrian win in the year 534,108.")
Summary
Austria will play one of 3 strategies:
A: NAF-MAO, WES-SPA, POR S WES-SPA, MAR S WES-SPA, LYO S MAR, PIE S MAR.
B: NAF-MAO, WES-SPA, POR S WES-SPA, MAR S WES-SPA, LYO S WES-SPA, PIE S MAR.
C: NAF-MAO, MAR-SPA, WES S MAR-SPA, LYO S MAR-SPA, POR S MAR-SPA, PIE-MAR.
1: MAO-POR, BUR-MAR, GAS S BUR-MAR, SPA S BUR-MAR, RUH-BUR.
2: MAO-POR, GAS S SPA, BUR-MAR, SPA S BUR-MAR, RUH-BUR.
3: MAO-POR, SPA-MAR, BUR s SPA-MAR, GAS S SPA-MAR.
The outcome of each of Austria's 3 strategies against each of EL(F?) as
well as the probability with which Austria will play each is
given in Table 3.
Austria's Strategies Coalition Defense "A"
(99.00%)
"B"
(0.99%)"C"
(0.01%)Outcome "1" No Change Win
Stalemate 99% "2" No Change No Change
Win 100% "3" Win Stalemate
No Change 99%
Practical Execution of Solution
I would request that players concede the game once they are satisfied
that the above presentation is valid. This could save us many years
of non-movement. Of course, Lowland can always (just about) force a
decision by playing "3".
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