Global Victory Conditions MATH analysis

Gargantua had it the correct way around.

So to get to 50/50, it has to be about a 29% chance to win on each board for the axis, assuming an even chance on each side.

@Gargantua:

This is the response I was looking for! Haha epic!

You’re welcome. Thanks for starting this statistical fire.

I’m certainly no math expert – not even a talented amateur – but it seems to me that two different concepts are being confuzzled here.

On the one hand, there’s the following question: what is the correct percentage number which accurately expresses a particular power’s chances of winning on either the Europe or the Pacific side of the board?  I think it’s a fair question, and that the forum’s rule experts and math experts will have an interesting discussion about how these figures ought to be calculated.

On the other hand, there’s the question raised by Garg’s original post; in generic terms, it could be expressed as: If Power A has an X% change of winning on the Europe side of the board, and if Power B has a Y% chance of winning on the Pacific side of the board, how does this translate into a Z% chance of victory by the team (Axis or Allied) to which Power A and Power B both belong?  I find the question problematic in several respects.

First, Garg’s post (if I read it correctly) uses Germany as Power A, 50 as X, Japan as Power B, 50 as Y, and 75 as Z; assuming for the sake of argument that 50 is an accurate figure for both X and Y, how does that figure get translated into Z = 75?  Is it by adding 100 and 50, and then dividing by two?  If so, what’s the rationale behind that equation?

Second, the calculations (whatever their basis is) seem to assume that X and Y are static and unrelated figures.  But is that necessarily the case?  Let’s say, for the sake of argument, that we’re dealing with a standard game in which all the players follow the standard models for playing their respective powers as effectively as possible, and let’s say that this situation does indeed result in X being 50 for Germany and Y being 50 for Japan.  Now let’s assume that the US player throws all of his resources into the European theatre.  Wouldn’t that result in the value of X decreasing and the value of Y increasing?  Would the two changes precisely offset each other, leaving the calculated value of Z unchanged, or would the situation be more complicated?

Third, and most fundamentally, I’m wondering about the concept of adding together two “individual power winning on one side of the board” situations and translating them into the result of “one particular team winning on the entire board”.  Global 1940 is played between two coalitions, the Axis and the Allies, on a map representing the entire world, not just between five major powers and several smaller ones on two halves of the global map.  To give an exaggerated example: if, let’s say, the Axis has a 100% chance of winning on the Europe side of the board and the Allies have a 100% change of winning on the Pacific side of the board, what does that work out to mathematically for the global game as a whole?  A draw?  A half-win and a half-loss by both sides?  This sounds a bit like the Schrodinger’s cat paradox in quantum physics, in which a cat is mathematically described as simultaneously being both dead and alive.

@CWO:

First, Garg’s post (if I read it correctly) uses Germany as Power A, 50 as X, Japan as Power B, 50 as Y, and 75 as Z; assuming for the sake of argument that 50 is an accurate figure for both X and Y, how does that figure get translated into Z = 75?  Is it by adding 100 and 50, and then dividing by two?  If so, what’s the rationale behind that equation?

Given the assumption that Germany wins 50% of the time and Japan wins 50% of the time in their specific theaters, the outcome of a global game is as follows:

| Germany | Japan | Chance |
| loses | loses | 25% |
| wins | loses | 25% |
| loses | wins | 25% |
| wins | wins | 25% |

If you then add the three lines containing an Axis win that results in 75% win.

Unfortunatly you may only do this breakdown in 25% chances when both events are independent. Meaning that the chance of Germany winnig doesn’t influence Japan’s changes of winning which doesn’t hold for global. Which statistical magic holds in this scenario, I do not know.  :-D

Um, now I’m even more confused.  The rationale seem to be that 75% comes from adding three selected lines from a table representing the Axis coalition.  What if I were to select different lines, however?  Or what if I were to pick the equivalent three lines from a table representing the Allied coalition (minus the USSR, for simplicit’s sake) – wouldn’t that give the Allies a 75% chance of winning?

US UK Chance
loses loses 25%
wins loses 25%
loses wins 25%
wins wins 25%

@CWO:

Um, now I’m even more confused.  The rationale seem to be that 75% comes from adding three selected lines from a table representing the Axis coalition.  What if I were to select different lines, however?  Or what if I were to pick the equivalent three lines from a table representing the Allied coalition (minus the USSR, for simplicit’s sake) – wouldn’t that give the Allies a 75% chance of winning?

US UK Chance
loses loses 25%
wins loses 25%
loses wins 25%
wins wins 25%

That was exactly my point a few post earlier:

If you’re logic would hold you could as well argue that:

If Germany has a 50% chance of losing the Europe board.
and if Japan has a 50% chance of losing the Pacific board.

Doesnt that mean the Axis has a 75% chance of losing any given game of Global?

Or

If Allies has a 50% chance of winning the Europe board.
and if Allies has a 50% chance of winning the Pacific board.

Doesnt that mean the Allies has a 75% chance of winning any given game of Global?

@Gargantua:

For all you math genius’s out there.

If Germany has a 50% chance of winning the Europe board.
and if Japan has a 50% chance of winning the Pacific board.

Doesnt that mean the Axis has a 75% chance of winning any given game of Global?

Is that why bids are 45+ these days?

I think Gargantua is messing with us. He says germany has a 50% chance of winning Europe and Japan a 50% chance of winning the Pacific and hence it is a 75% chance that axis win the game. If you play OOB with no bid. Seems like a good ballpark number

BUT allies must win both europe and pacific to win wheras axis only has to win one side. So it means we must also add a draw factor in the numbers, say axis are 50%, allies are 25% and undecided is 25% (or whatever number you feel) That means one map can be undecided whereas the other map is decided by the axis and hence the speculation of 75% allies win is not valid given a 50% chance for both Germany and Japan to win (I think)

@Gargantua:

US UK Chance
loses loses 25%  = game lost
wins loses 25%  = game lost
loses wins 25%  = game lost
wins wins 25% = game won

You have to win on BOTH sides of the board to win,  so 25% is right!

Tjoek’s Axis table has the following structure and numbers:

Germany Japan Chance
loses loses 25%
wins loses 25%
loses wins 25%
wins wins 25%

My Allied table has the following structure and numbers:

US  UK  Chance
loses  loses  25%
wins  loses  25%
loses  wins  25%
wins  wins  25%

In other words, both tables are identical except for the names of the two powers in each table, which means that both the Axis and the Allies have exactly the same numbers being applied to them.  How, therefore, can a table for the Axis be interpreted to mean that the Axis has a 75% chance of winning and an identical table for the Allies be interpreted to mean that the Allies have a 25% chance of winning?

Axis can win on either side by collecting victory cities. Three  of four scenarios have a win for the axis - therefore 75%

Allies on the other hand have to take all three capitals. That means they have to win on both sides - that’s only in one of those four scenarios, therefore 25%.

Ok folks. Here’s some math

Chance of event A and event B both happening.( allied win)

= p(A) x p(B)

That is .5 x .5 = 0.25 or 25% chance of allied win.

Chance of Axis win. Since condition is OR.

= 1 -  (   p(A) X p (B)  )

= 1 - (.5  x .5 )

= 1 - 0.25

= 0.75 or 75% chance axis win.

Underlying assumption is that axis win = 50% on either board.

The revalation from this perhaps is for Japan to give up on Australia or Hawaii and attack egypt instead. Then build a few factories in middle east and join the assault on Russia. Should be pretty easy to get the three russian ones with joint assault from all three axis powers.

@Elsass-Lorraine:

Axis can win on either side by collecting victory cities. Three  of four scenarios have a win for the axis - therefore 75%

Allies on the other hand have to take all three capitals. That means they have to win on both sides - that’s only in one of those four scenarios, therefore 25%.

Thanks for your clear and concise analysis, which points to the terminology problem that’s been at work here.  The thread has been discussing Axis versus Allied victories in terms of “chances”, and has been quantifying them as percentages.  What victory really hinges on in the game is hitting a certain number of benchmarks (i.e. victory conditions), which is an additive situation rather than a percentage-based situation.  And what makes the Axis different from the Allies is that, in case of each side, they don’t have to hit the same types of benchmarks nor the same number of benchmarks to achive victory.  And I guess that the point to which Garg was drawing attention was that the benchmarks are not distributed equally on the two halves of the game map – which is a valid point, but a point which I think got a bit lost when it was expressed through the concept of percentage victory chances rather than though the concept of additive victory conditions.

I once came across something similar in a book I was reading about the American Civil War, in which the author stated that the Confederacy had a two-thirds chance of winning.  I remember blinking when I read it, and taking a moment to think before going on to the next sentence.  I wondered what the basis of the author’s reasoning could be.  If he was going by the relative military competence of both sides in the first half of the war, and nothing else, then the argument made some sense, but if he was going by the economic, industrial and demographic imbalance between the two sides, then the argument made no sense to me.  I looked at the next sentence, and I saw that the author wasn’t basing himself on either of those factors.  His argument was basically that, in order to win, the North had to physically occupy the territory of the Confederacy; the Confederacy, by contrast, could win – defined by the author as “continue to exist” – either by physically occupying the North or by achieving a stalemate.  In other words, the author was giving the North one way of winning and giving the Confederacy two ways of winning (which is arguably correct), but he was then translating this state of affairs (“2 is bigger than 1”) into a two-to-one advantage for the Confederacy (which is faulty reasoning because it disregards the question how achievable those winning conditions were for each side).

@CWO:

[…](which is faulty reasoning because it disregards the question how achievable those winning conditions were for each side).

And here we have this taken into account because of the assuption that axis win in 50% of the cases on the european resp. pacific side.

So all four outcomes have a 25% chance of happening. Therefore axis win 75% of the games.

So if we say on the Europe map the axis will get 5-6 quite easily

Berlin. Rome. Paris. Warsaw.
Then 2 in Russia.  Leningrad/ Stalingrad

Victory depends on getting 2 of Cairo/Moscow/London

On the pacific map it comes down to India/Australia or Hawaii.

The real question is what are the odds of capturing each of these individual cities and how can gameplay or purchases or cooperation affect the odds of individual cities.

Eg is Japan better off attacking the 6 territories around Soviet Far East and ignoring China. This will have more impact on the Germans collecting the 3 russian victory cities.

How can the European Axis distract UK so Japan can push through India.

This is something i havent thought of before but lets take this one step further. The axis could go completely bonkers on one board in order to gain victory on the other and win the game.

Eg germany invades canada. Lets say Germany ignored russia beyond maybe a token effort. But over turns 4-8 could go all out after America. The goal to is  get US forces distracted and committed to going Atlantic. Russia takes the opportunity to invade germany but as a consequnce Japan gets hawaii.

Another scenario might be germany building strategic bombers to do Japans dirty work and bomb Australia. Germany again loses on the Europe map but with german help Japan walks into Sydney picking up the last Victory City.

@thespaceman:

This is something i havent thought of before but lets take this one step further. The axis could go completely bonkers on one board in order to gain victory on the other and win the game.

Eg germany invades canada. Lets say Germany ignored russia beyond maybe a token effort. But over turns 4-8 could go all out after America. The goal to is  get US forces distracted and committed to going Atlantic. Russia takes the opportunity to invade germany but as a consequnce Japan gets hawaii.

Another scenario might be germany building strategic bombers to do Japans dirty work and bomb Australia. Germany again loses on the Europe map but with german help Japan walks into Sydney picking up the last Victory City.

All good points, and they raise an interesting conceptual issue about the Global 1940 victory conditions, whose OOB form reads:

The Axis wins the game by controlling either any 8 victory cities on the Europe map or any 6 victory cities on the Pacific map for a complete round of play, as long as they control an Axis capital (Berlin, Rome, or Tokyo) at the end of
that round.

The Allies win by controlling Berlin, Rome, and Tokyo for a complete round of play, as long as they control an Allied capital (Washington, London, Paris, or Moscow) at the end of that round.

These victory conditions are actually rather abstracted when you look at them from the perspective of how the actual powers fighting the actual Second World War might have seen them.  One part of the problem is that victory is defined in terms of the control of certain cities (capitals and/or non-capitals, depending on which team is involved).  This concept has both merits and questionable elements, but in my opinion a more serious conceptual problem arises from the fact that the game is trying to deal with two aspects of WWII which can both be viewed in two different ways.  The first aspect is: was WWII fundamentally a single global war or was it fundamentally two separate (though linked) theatre-scale wars occurring pretty much on opposite sides of the planet?  The second aspect is: was WWII fundamentally a war between two coalitions – the Axis and the Allies – or was it fundamentally a war involving multiple individual powers who, at various times, were either out of the war or were in on one side or were in on the other side?

I raise this point because the game’s concept of an Axis win (or an Allied win) would probably have looked strange to the actual participants under some outcome scenarios.  Let’s take, for example, a hypothetical scenario that fits the requirements for an Allied win: a scenario in which the Allies control one of their capitals (Moscow), plus all three Axis capitals, while the Axis controls Washington, London and Paris.  Would the Americans, the British and the French have considered this to be an Allied victory?  Perhaps.  A Soviet victory?  Probably.  A victory for themselves?  I wonder.

Let’s take another example.  In May 1945, the Allies “won on the Europe side of the board” when the parts of Germany and Italy that were still under Axis control surrendered.  Let’s say that this situation get replicated on the Global map.  The OOB rules says that, for the Allies, “winning on the Europe side of the board” (as happened historically in May 1945) isn’t enough for them to achieve victory; they also have to control Tokyo.  (That’s probably the reason why that rule is there: because historically, having the Allies “win on the Europe side of the board” wasn’t enough to complete the Allied victory over the Axis.)  Now let’s say that Japan, improbably, somehow manages at this point to defeat the Allies in the Pacific theatre and fulfil the game’s requirements for an Axis victory.  Would the defeated Germans and Italians (in real life ) have regarded this outcome as an Axis victory?  I rather doubt it.

My point here is that WWII was, in many ways, a complex set of interrelated localized conflicts with different starting and end dates and different participants.  To give just a few examples: the war between Japan and China lasted, non-stop, from 1937 to 1945; the war between Germany and France lasted, arguably, from 1939 to 1940 (and was lost by France); from the American and Soviet perspectives, WWII began in 1941 rather than 1939.

And WWII also had a complicated geography – not just because of its overall European theatre / Pacific theatre structure, but also because of the geographic differences between the participants themselves.  Some have very small home territories (the surface area of Great Britain is smaller than that of Kansas) while others (like the USSR, the largest country in the world) have vast ones; some had no “remote” territorial possessions beyond their home territories, some had a few, and some had vast holdings of this kind.  Just out of curiosity, I assembled this table (I hope the formatting won’t be too wonky when I post it into this thread) to describe each of the game’s powers situation in this regard:

Player          Home territory on                Has any remote colonial / territorial
powers        which side of board?            holdings? If so, on which board side?

US              Both                                    Yes / Both

USSR          Both                                    No

UK              Europe                                  Yes / Both
France        Europe                                  Yes / Both

ANZAC      Pacific                                  Yes / Pacific
Japan          Pacific                                  Yes / Pacific

China          Pacific                                  No

Germany      Europe                                No

Italy            Europe                                Yes / Europe

Non-player  Home territory on                Has any remote colonial / territorial
entities        which side of board?            holdings? If so, on which board side?
with map
roundels

Canada        Both                                      No
[Same situation as USSR]

Holland        Europe                                  Yes / Both
[Same situation as UK and France]

I guess what I’m wondering at the end of all this is: have any of the folks on the forum who’ve developed house rules come up with a more nuanced set of victory conditions which take these various factors into consideration?

Marc, you make some great points in your previous post. I think in order for the game to mesh most effectively, the Axis victory conditions need to be more global (e.g. 11-14 VCs on the whole board, perhaps 1 MUST be a major capital, numbers need to be play tested with), while the Allies conditions should remain roughly the same. Perhaps the Allies and/or Axis have to hold 2 major capitals (E.g. the big 5/6) in order to ensure that winning is more realistic. As for the territorial possessions, I think that, while important, they should in no way play into the victory conditions. I mean, UK lost most of their possessions soon after the war (albeit not ALL because of the war necessarily, but it still happened), and of course they still saw the outcome of the war as a victory. I think the most pressing matter is, as you mentioned, the fact that the idea of victory for Japan was much different than Germanys or Italys. But for gameplay purposes, I think this must be overlooked, and both sides must be viewed as completely united bands of belligerents.

Hypothetical scenario: Tokyo is taken by Allies. Axis are one VC short of the new global victory condition of 11-14 cities total. Think about it. If the Axis take the last one (say, Moscow or some more important, thus more difficult since it will be the last one they need), then I think it would be fair to say that the Axis win. Theoretically, in “real life” this would put the Allies in a position where they would surrender and, if the axis were truly the good partners in the war that they are made out to be in Axis and Allies, conditions of surrender would be Japan is brought back under Japanese (thus Axis) control.

Sorry, this is messy. Just my random thoughts.

Setting a number of global victory cities would be a good idea for both sides instead of requiring the Allies to hold Axis capitals to win. If the Axis dedicates itself to it, it can never lose by making sure it’s two big capitals (no one counts Italy really :-)) are completely impregnable without three or four dozen turns of Allied play.

I would suggest that 13 is a viable number for both sides. If the Allies have held 13 VCs for a full round, the Axis is pretty much toast.

Marsh

But dont the allies start with 13? UK: London, Calcutta, Cairo, Hong Kong, Ottawa (5) US: San Fransisco, Manila, Honolulu, Washington (4), Russia: Stalingrad, Leningrad, Moscow (3), ANZAC: Sydney, and even Paris for 14, that is if I’m not missing any

Edit: I guess maybe you could say US and Russia aren’t officially Allies at the beginning, but still. I also wouldn’t say the Axis are toast after round 1 if they dont take any besides Paris because the only really other possible ones are Hong Kong and Manila and that is only if Japan does a DOW on J1, which the 13 VC condition would force them to. Perhaps instead of strictly taking capitals the allies have to ‘neutralize’ the Axis Powers. For example, Japan only has Tokyo left (only territory), allied convoy surrounding it, so Japan can literally do nothing since they gain no IPCs, thus neutralized. Of course this would have to be played with, but just a thought.