1v1 Puzzles

[bozotheclown]

I think I'd probably revise my earlier note, since France has three rather than two options in the event that F Baltic is busy doing something other than supporting or tapping Berlin...

France move sets
A) Ber-Mun supported by Bur/Ruhr/Kiel
B) Bur-Mun supported by Ruhr/Ber, Kiel supports Ber H
C) Bur-Mun supported by Ruhr/Kiel/Ber

Austrian move sets
1) Pru-Ber supported by Mun, Tyr/Boh/SIl support Mun H
2) Pru-Ber supported by Mun/Sil, Tyr/Boh support Mun H
3) Mun-Ber supported by Sil/Pru, Tyr-Mun supported by Boh
4) Mun-Ber supported by Pru, Tyr-Mun supported by Boh/Sil
5) Mun-Kiel, Tyr-Mun supported by Boh, Pru-Ber supported by Sil

France move set A:
v Aus1: Austria takes Berlin, Munich holds, Prussia empty space, Austria gets 18
v Aus2: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space
v Aus3: France takes Munich and doesn't lose Berlin, hits 18
v Aus4: France takes Munich and doesn't lose Berlin, hits 18
v Aus5: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space

France move set B:
v Aus1: no change in unit positions
v Aus2: no change in unit positions
v Aus3: Austria takes Berlin, Munich empty space, Austria gets 18
v Aus4: no change in unit positions
v Aus5: Austria takes Berlin, Munich empty space, Austria gets 18

France move set C
v Aus1: no change in unit positions
v Aus2: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space (also A Ber destroyed, but A Par presumably backfills Bur in the considered order set)
v Aus3: Berlin/Munich control flipped, A Ber destroyed
v Aus4: Austria takes Berlin, Munich empty space, Austria gets 18
v Aus5: Berlin/Munich control flipped, A Mun destroyed, Prussia is empty space (also A Ber destroyed, but A Par presumably backfills Bur in the considered order set)

From Austrian perspective:
Move set 1: 1/3 chance of winning, 2/3 chance of 0 net change
Move set 2: 3/3 chance of 0 net change (therefore sub-optimal, wouldn’t take it)
Move set 3: 1/3 chance of winning, 1/3 chance of losing, 1/3 chance of 0 net change
Move set 4: 1/3 chance of winning, 1/3 chance of losing, 1/3 chance of 0 net change
Move set 5: 1/3 chance of winning, 2/3 chance of 0 net change

Move set 2 does nothing, is sub-optimal, you can chuck it
Move set 5 is strictly better than move set 3 (against A, it draws instead of loses; against B, it’s same outcome; against C, it’s basically the same outcome)

So you can narrow it down to France move sets A/B/C against Austrian move sets 1/4/5
Each has a 33% chance of Austrian outright victory, where if France chooses move set A, he gets a shot at an outright win himself.

So it's not at the 50-50 level of "Austria has an outright coin flip to win the game if he predicts accurately when France will try for Livonia" but it's in the ballpark of 33% (adjusted for chance of outright loss that turn).

That said, there could certainly be some other tactical option I'm missing here. But I THINK that in order to take Livonia and then Prussia, France is required to accept the risk of an outright loss (also, if you get a status quo turn, but Sev gets into Moscow, you then add in Mos-STP and Mos support Liv-STP as possible moves, which makes it harder on France and probably creates the risk of an Austrian army in STP which also wins the game for Austria)

You are missing the threat of Kiel to Mun. supported by Ruhr and Bur., which is a significant risk to Austria trying moves 3 or 5 if France is willing to wait him out.

[mhsmith0]

That is functionally the same as France move C (especially presuming that Paris backfills burgundy).

[RoganJosh]

Maybe to clarify, this is exactly how "by probability 1" should be interpreted:

P.S. For those confused, a probability 1 win means that for all p<1, France can find a strategy that wins with probability at least p.

***

Would it be correct to interpret the solution for puzzle 3, "probability 1" as you are looking for some strategy by France that should allow him to win eventually? So Austria would have to continually guess right to perpetuity to stop it?

If Austria has a sequence of guesses where the best outcome is to re-guess and worst outcome is a loss, then of course Austria will loose with probability 1. But, no, this is not what the solution is to this puzzle looks like. As correctly pointed out above, conditioned on that France decides to attack, Austria has a positive probability of obtaining a positive outcome.

[RoganJosh]

One thing I now realized is that posting these as screen shots from actual games was maybe not the best idea, because in an actual games there are situations all over the board, and often a lot of possibilities. Which makes solving the puzzle a tedious task of working through cases, rather than puzzling.

So, here is a redux of puzzle 3. Same idea behind the solution - fewer involved pieces and fewer possibilities.

Puzzle 5

East of Switzerland there is a settled stalemate line. What remains is only the battle for Spain and Marseilles.

If icons are unclear: there are armies in Marseilles and Piedmont, there are fleets in Portugal and Spain. Not that it matters.

France to win with probability 1.

https://imgur.com/gallery/DojtaQw

[bozotheclown]

That is functionally the same as France move C (especially presuming that Paris backfills burgundy).

Look at France C vs. Austria 4 for the difference.

[bozotheclown]

That is functionally the same as France move C (especially presuming that Paris backfills burgundy).

Look at France C vs. Austria 4 for the difference.

Never mind, I was thinking of a different move, but I think what you need to consider is the result of all of the Austria moves if France is using the Baltic Sea to protect Ber. If France enters safe move repeatedly, Austria can't afford to randomly cycle through all of the move combinations that have a positive result if France does not use the Baltic to protect Ber., because France could then win without ever risking Ber.

[jay65536]

If you are basing your claim that puzzle 5 has a solution on batogian's definition of probability 1, that's not good, because it's wrong.

That definition says, a probability 1 win means that for all p<1, there exists a French strategy that wins with probability greater than p.

The correct definition is, a probability 1 win means there exists a French strategy such that for all p<1, that strategy wins with probability greater than p.

If such a strategy does not exist, then it is incorrect to say that France can win with probability 1.

[Tugster]

It's the same bullshit, me thinks that someone does not understand what probability 1 is, there is no certain way for France to win, he has to guess. If Austrian guesses right for enough turns, he then writes the mods and forces a draw. France does not get infinite guesses. And if Austrian guesses right a few turns in a row, the mods force draw the game. NOT a CERTAIN WIN, NOT probability 1.

Give it a rest folks.

[Tugster]

I will tell you some actual probability 1 events.

Someday, each of us will die. => probably 1

If princess leia is ejected into space from the bridge of her star-ship, her body will explode. probability 1

Shall I go on?

[RoganJosh]

If you are basing your claim that puzzle 5 has a solution on batogian's definition of probability 1, that's not good, because it's wrong.

That definition says, a probability 1 win means that for all p<1, there exists a French strategy that wins with probability greater than p.

The correct definition is, a probability 1 win means there exists a French strategy such that for all p<1, that strategy wins with probability greater than p.

If such a strategy does not exist, then it is incorrect to say that France can win with probability 1.

This is not really relevant in terms of solving the puzzle, but out of curiosity..

If for all p < 1 there exists a French strategy that wins with probability (at least) p, then the supremum of France's expected payoffs over all strategies is 1.

If there is a French strategy that wins with probability 1, then the maximum of France's expected payoff over all strategies is 1.

As we all know, if the maximum is attained, then it is equal to the supremum. So the only time the distinction is relevant is if the maximum does not exist. Which might very well be the case in this example since the strategy space is non-compact.

So, if the maximum does not exist, would you simply consider "France's probability to win" to not be well-defined?

[Tugster]

In order to take Spain, France must put Marseilles at risk, if Austria makes the correct defensive guess that turn, he takes Marseilles, swapping it for Spain, and then it's a permanent stalemate. If France gives Marseilles the full defense, and Austria defends correctly, France cannot take Spain. You really have NO IDEA what probability 1 means it means => with CERTAINTY. The above guesswork, is not certainty. If you think France gets to guess "forever" you are wrong. Because after the SC do not change hands for 2 years in a row, Austria writes the mods and they force the game to draw.

[jay65536]

Without spoiling the "solution," I'm pretty sure I know what RJ has in mind when he claims a win with probability 1.

However, if it's what I think it is, then I would disagree that that is allowed in a game like Diplomacy. It hinges on a mathematical technicality and, if nothing else, is more of a math puzzle than a Diplomacy puzzle, which I'm sure is not what the OP had in mind.

With that out of the way, Tugster is also wrong, for two separate reasons. First of all, rulebook Diplomacy does not have a turn limit. So any argument based on webDip games having a bounded number of turns is invalid. Puzzles like this are perfectly fine to consist of an unbounded number of turns. My issue with the puzzles is the mathematical difference between "unbounded," which does apply to Diplomacy, and "infinite," which does not. They aren't the same.

The second reason is that a different puzzle construction could result in something that can only be called a probability 1 win. As someone else brought up before, suppose instead that F/I have a guessing game where no matter what France does, Italy guessing incorrectly results in a French win, but Italy guessing correctly results in a repeat of the same position. (These puzzles feature the former, but not the latter.) Now France clearly does win with probability 1, since France can fix a discrete strategy and we can compute that Italy's chances of guessing correctly every time converge to 0. That is a probability 1 win for France even though it is theoretically not certain not to go on forever.

To answer your question, RJ, I suppose I'd agree with "not well-defined".

[RoganJosh]

I'm sure is not what the OP had in mind.

I think I am the OP... Anyway, I was gonna say, maybe we should postpone that conversation until you know what I have in mind.

suppose instead that F/I have a guessing game where no matter what France does, Italy guessing incorrectly results in a French win, but Italy guessing correctly results in a repeat of the same position. (These puzzles feature the former, but not the latter.)

If Puzzle 3 was the 4x4 version, and Puzzle 5 was the 3x3 version, then here is the 2x2 version:

France chooses between 1 and 0
Italy chooses between A and B

{0, A} or {1,B} is a French win.
{0, B} and they play another round.
{1, A} is a draw

In this game, for any p < 1, France has a strategy that gives a probability of a French win which is at least p.

The turn this discussion took, you are also overlooking the point of the puzzle. If you threat this 2x2 game as a simple guess, choosing your option with probability .5, then the probability of France winning the game is .66. However, already at a turn limit of 2, there is a strategy which gives France a .75 probability of winning the game. At a turn limit of 5 there is a strategy which gives France .90 probability of winning the game. That France can do much better than .66 is not some obscure thing that happens in the limit.

The limit value is = 1. Sure, we can discuss whether the limit being = 1 justifies saying that "France wins by probability 1" or not. But I find it rather silly if you are going to dismiss the puzzle based on that. Try to solve it instead.

[RoganJosh]

Ooops, put the wrong numbers. You need 3 rounds for .75 and you need 9 rounds for .90.

Of course, if you are playing Tugster, then you just play 0 every time and that's it. You'll win it within 3 rounds or so.

[Tugster]

THIS =>

France chooses between 1 and 0
Italy chooses between A and B

{0, A} or {1,B} is a French win.
{0, B} and they play another round.
{1, A} is a draw

I agree with, but that's not how this puzzle is set up.

[RoganJosh]

I agree with, but that's not how this puzzle is set up.

Oh, but it is! You should try again.

[jay65536]

France chooses between 1 and 0
Italy chooses between A and B

{0, A} or {1,B} is a French win.
{0, B} and they play another round.
{1, A} is a draw

In this game, for any p < 1, France has a strategy that gives a probability of a French win which is at least p.

The turn this discussion took, you are also overlooking the point of the puzzle. If you threat this 2x2 game as a simple guess, choosing your option with probability .5, then the probability of France winning the game is .66. However, already at a turn limit of 2, there is a strategy which gives France a .75 probability of winning the game. At a turn limit of 5 there is a strategy which gives France .90 probability of winning the game. That France can do much better than .66 is not some obscure thing that happens in the limit.

The limit value is = 1. Sure, we can discuss whether the limit being = 1 justifies saying that "France wins by probability 1" or not. But I find it rather silly if you are going to dismiss the puzzle based on that. Try to solve it instead.

After having done a bunch of math on the 2x2 game to confirm my intuition, which I already put up in an earlier post, I've confirmed that every claim you made in the penultimate paragraph is true. If we set a turn limit of N, there exists a French strategy that results in a win with probability (1 - 1/N).

However, the last paragraph, especially the bolded, shows you missed my point. I am not trying to debate "whether the limit being 1 justifies" anything. I am TELLING you that your solution is invalid because you have done your limits wrong.

What this argument is doing is creating a sequence p_N of probabilities of French wins based on turn-limited versions of the same problem, and then saying "the limit is 1, so France wins with probability 1". But in fact, the big problem with your argument is a subtle error having to do with how you're taking the limit.

You've actually created two sequences, not just one. You are considering the sequence of probabilities, but what about the sequence of strategies? That sequence--the one from which the probabilities are derived--also has a limit, in the form of one single strategy. Without giving it away for anyone still trying to follow this on their own, if France plays the limiting strategy, that is NOT necessarily a win with probability 1. The probability of a French win playing the limiting strategy is...well, I would say undefined. In other words, the limit of the probabilities is 1, but the probability of the limit does not exist.

This is mathematically subtle, but it greatly affects the puzzle. If someone is trying to find a single French strategy that wins with probability 1, which is what the puzzle is clearly asking for, they won't be able to, because it doesn't exist.

So far I've only done thorough work for the "2x2" game, but I feel very confident that the same error exists for the "3x3" game. There should be a proof by contradiction that no strategy exists that wins with probability 1.

[jay65536]

Where I wrote (1 - 1/N), should have been (1 - 1/(N+1)).

[RoganJosh]

We are discussing the meaning of the sentence

France wins by probability 1.

Your interpretation is that this sentence is equivalent to "France has a strategy whose expected payoff is 1." You have not provided any arguments for why your interpretation is reasonable.

My interpretation is that this sentence is equivalent to "The supremum of the expected payoffs for all French strategies is 1". I find this to be the only reasonable interpretation.

This is literally the difference between taking maximums and taking supremums. And that just about settles the issue. No sane person would try to do real analysis using maximums.

I have never claimed that France has a strategy whose expected payoff is = 1. If you solve the puzzle by giving a sequence of mixed strategies, then that sequence converges to a mixed strategy which allows Italy to stall the game forever. That is a fairly simple observation.

[mhsmith0]

fwiw, i think puzzle 3 is interesting in that I'd basically just assumed that Munich was a stalemate line from the south by itself, so it's helpful to know that this isn't actually the case, and that it's Berlin who is the key middle of the board SC to hold to establish/keep up a stalemate lien
(though Munich is enough by itself from France's end at least)