Global Victory Conditions MATH analysis

• 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?

• Well, I guess the answer is no, since the Allies can intervene to reduce the odds to less than 50% on either side of the board?    The Axis can’t necessarily help each other, other than to press their position on their respective boards.  However, the allies can switch around from side to side, and not in symmetric ways. Pulling the air units from India to the Middle East round 1 doesn’t increase Japans chances if it’s not at war while it does reduce Italy/Germany’s chances.

My two cents.  :mrgreen:

• the allies can only win by concession because they lack a rational victory condition.  that has to be modded into the game or the axis have to choose to end the game.  defeating both axis capitals is really difficult but taking six vcs on the right hand board really isnt.  this is also why kjf is crucial because you have to stop japan from achieving that victory condition but by doing so you hand it to germany, especially when japan helps by passing onto the left board

• I don’t really believe they each have a 50/50 chance.  Isn’t it predicated on the U.S. making a decision as where it wants to press its considerable power?

• 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?

You better enlist Dauvio Vann. Here, this might help!

https://www.axisandallies.org/forums/index.php?topic=40347.0

• it would be 75% if the chances where statistical independent events. That would mean that the allies could not transfer resources between the maps and what the axis does on one map does not affect the other map.
List of reasons why they are not independent:

1. Allies can transfer resources
2. US does not the the 40 increase in their home terr
3. Russia gets extra troops and income.
4. UK can shuffle troops between the theatres
5. Japanese declaration of war brings US earlier into the war on the europe map.
6. US can suffle resources between theatres.
7. USSR can hit japanese troops in china
8. Japanese can attack USSR
9. Japanese can soften up cairo, to help it fall.

• 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?

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?

With the assumption that gaems never end in a draw all the above if statements describe the exact same situation. So something is wrong here, right? And Kreuzfeld already mentioned the fact these events are NOT statistical independent.

A short explanation:
If Germany is about to win, the changes that Japan is winning as well are really minimal as the US most likely spend most of it’s income in the Pacific. And if Japan is about to win the US most likely went Atlantic heavy.

Regards,
Joost

• Math genius here. The Germans should have a 37.5% chance to win in Europe and Japan had about 37.5% chance in Asia, leading to a combined 75% chance for the Axis with standard rules and optimal game play. There is about 15% chance of something awful happening in the first two rounds that will lead to either an Allied edge or a fair game. There is an additional 30% chance of a game changing event in the latter rounds (send 4 bombers to Moscow and 3 get shot down, etc). Otherwise the Axis can guarantee a victory without need for luck.

• 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?

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?

With the assumption that gaems never end in a draw all the above if statements describe the exact same situation. So something is wrong here, right? And Kreuzfeld already mentioned the fact these events are NOT statistical independent.

A short explanation:
If Germany is about to win, the changes that Japan is winning as well are really minimal as the US most likely spend most of it’s income in the Pacific. And if Japan is about to win the US most likely went Atlantic heavy.

Regards,
Joost

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

• 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.

• 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.

• 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?

• 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)

• @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%

CWO I think you hit the nail on the head.  Look at it this way.

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!

• 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.

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.

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