Screenshots++

[ElvisJJonesRambo]

Man, I'm dumb. I didn't even know how to screen shot. Well, now I might be dangerous.

August 1944.

Germany has defeated Russia. Meanwhile, in the Pacific, the Japanese have tightened up their position against crazy strong Chinese mass Army with US Air Support. The Philippines are still held, well, the important half Manilla and North.

[ElvisJJonesRambo]

Same game, Germany has defeated Russia. Has now pushed the Allies and Patton into a coffin. There's no escape.

[ElvisJJonesRambo]

Is this even mathematically possible?

On turn #1, Italy sells tech, in order to immediately invest in Infantry.
It's now October 1942, and Italy still doesn't have Infantry level #2.

This kind of crap happens all the time with me.
Even worse, when it's Russia.

Obviously, I immediately invest in Infantry once level #1 is obtained.
This happens all the time with me.
I don't see opponents having level #1 infantry with Russia.
It's October 1942, and no infantry #2.

[Chernobyl]

I believe there are 13 turns a year.

Assuming Italy never gets any breakthroughs:
Italy starts unmobilized, so doesn't benefit from tech sharing until mobilization happens. I believe catch up bonus applies as long as the enemy is fully mobilized, even if you are unmobilized yourself. So this means UK should give +1% / turn for Italy's research. Italy should be averaging 6% research per turn on Infantry Weapons starting in 1939. Once they join in 1940, they should average 7%, until to get to level 1, after which they will only average 5% a turn.

October 1942 is roughly 40 turns into the game. Let's say Italy gets average luck but no breakthroughs. Say they join in the second half of 1940 and UK/Germany take a long time to get to infantry weapons level 2 (~June 1941 for Germany, even later for UK).

So that means Italy will plod along for about 8 turns averaging 6% progress, then at 7% progress average after joining the war. Let's now assume Italy gets WORSE than average luck on the tech rolls, and actually averages about full percentage less than expected. It could take as long as 18 turns to get to Infantry Weapons level 1, especially if it ends at 99% and you need a whole extra turn to finish. That's extremely bad luck, but possible.

So now it's 18 turns into the game, and I don't have the table of turn dates with me but I think that would be like April 1941. So let's say Germany gets Infantry Weapons level 2 right now, and Italy pays MPP to begin level 2. Well, Italy should average 5 + 1 = 6% per turn, but and let's assume average luck here, that's 17 turns to get to 100, which would be about August 1942. That's assuming average luck for level 2, but if Italy also got worse-than-average luck here, it could very well drag on past October as in your screenshot.

But this would take very bad rolls, over the course of around 40 tech rolls, you'd have to be about 30 percentage points or about 5+ turns behind average luck. Even worse luck assuming UK/USSR/USA got to level 2 at some point before Italy did. I don't know how unlikely that is exactly, it doesn't seem totally impossible, but definitely rare. Maybe 1-2% of the time?

The chance of a breakthrough is 5% per turn per chit invested, when you are at 45% or more research progression. Assuming about 20 qualified turns, the chance of never getting a breakthrough before getting to Infantry Weapons 2 is about 36% (0.95^20 = 0.3585) so you should definitely plan on the possibility of never getting a breakthrough.

Assuming average luck but no breakthroughs, Italy should probably get to Infantry Weapons level 2 sometime around early 1942, earlier assuming USSR or UK get level 2 before this.