Space Invading is Hard

Assuming an interstellar war between 2 Earth-sized civilizations, a question could be asked, “How many troops would I need to invade another world in another star system?” 

To answer this question would need several assumptions.  Just considering raw troop-numbers needed to invade that world one can arrive at a needed effectiveness ratio of said troop against an enemy defender.  Technology disparities between the antagonists would obviously be needed to reach a match between the invaders numbers vs. home world troop numbers as it can be conceived that home world defenders would vastly out-number invaders.  Thus invaders would need to make up the difference in technology superiority as well as better strategy.  An example may illustrate the thought experiment. 

Assuming Earth would invade another world of similar size and population in Alpha Centauri, how many operators would it need to do so?  Assuming number of operators would be 0.01% (according to 1^st Law of Interstellar Warfare) of current world active duty personnel would yield approximately ~2000 “shooters” that would go invade Centaurian home world.  On the flip-side, Centaurians would use all available combat power to defend themselves.  More or less all military able population between ages of 15 to 49 years of age.  Again only 10% would be combat troops while remaining 90% would make up logistical tail including manufacturing of war materiel to carry on the fight.  If A-C has a similar population to Earth of 7B people that yields about 350M combat troops to defeat in order to subjugate the Centaurian home world.  So a combat effectiveness of a Terran space marine would in fact need to be ~200k times a Centaurian warrior for rough tactical combat parity between defender and invader.  But in general, defenders always have an advantage, sometimes up to 3 to 1, so the effectiveness may well need to be much more.  So from a technology point-of-view, Earth technology would need to be on same order of betterness or 200k times more advanced than Centaurian.  Flipping it around with Centaurians invading Earth, they would need to be 200k more advanced technologically than Earth.  And that would just be for tactical parity which does not guarantee victory.  That comes to another consideration of strategy.    

To invade Centaurian home world, Earth would need superior technology (200k) plus superior strategic thought.  To make sure of victory strategies would need to be 10X better at least.  But why not just send robots to do it?  Right now, our computers can win at chess, but still even have a hard time with Go and those are games with known rules and no hidden information.  Artificial Intelligence would need to advance a fair amount to be able to compete in messy strategic conundrums such as warfare, let alone interstellar war.  And we sure don’t want to commit the Ludic Fallacy (hat tip Dr. Taleb) confusing games with Black Swanny real-life.   So in light of this, it could be thought that these 2,000 troops would be superior strategic thinkers (My name is KHAN!) who would be the generals-admirals directing fleets of robotic soldiers who could manage for themselves on a tactical-level. 

Concluding this thought experiment, one reason we haven’t been invaded by aliens is that being the invader would be ‘ahem’ challenging.  One’s civilization has to be advanced technologically which may take millions of years, while at the same time needing to span the gulf of space to get here.  One may conclude the best route strategically is not to fight and just look for uninhabited planets to colonize while avoiding any inhabited ones.  Of course, none of the above considers interstellar politics and rationales of why one would want to invade another star system.