From: Columbia, MD. USA
S&I: Yes, if only 1 country invests for a side, it does provide S&I 'defense' for the whole side. There are advantages to having multiple countries invest though: 1) Higher chance that one will get a break through to provide a bonus vs. the enemy and a higher defense vs. the enemy; and 2) You get more Intelligence enemy unit reveals, as that is cumulative or a separate roll (I think if 2 countries have it, it would be 2 rolls/chances to reveal per enemy unit). I think the reveal chance is 1% per level so 3% at maximum.
Yes, the manual isn't clear which is why there was a long discussion which I wanted to point you to, but can't find. :(
'Catch Up' applies a bonus when your level is behind that of your opponents. I also believe it is cumulative. I think the only restriction is that you don't get a 'catch up' bonus if you are behind a neutral power on the other side. e.g. Germany (level 0 Industry) does not get a catch up bonus from a neutral USA (level 1 or higher Industry); However, the USA does get the bonus from Germany (since Germany is not neutral).
'Sharing' works like 'Catch up', but on friendly sides and only when the countries are at war. It only applies when 1 country has a higher level than the other (e.g. If Italy researches Inf Weapons 1 before going to war while Germany already has Inf Weapons 1, Italy will get no bonus. However, once Italy goes to war she will get a 1% sharing bonus (or 2% if Germany is at level 2).
I think, but could be mistaken, that the USSR no sharing is one way not both. I think they get a sharing bonus from UK & USA, but will not provide one if they are ahead in an area. It's not a big deal because it is pretty rare that the USSR is ahead (perhaps Advanced Tanks or Anti Tank later in the war).
The Catch Up rule used to apply always no matter war status, until it was pointed out that Germany would get a bonus for USA Industry research before the UK would.
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