You are the much-loved ruler Joe Bob of anywhereland. You won the election with all of the votes except for one: from your annoying brother Bob Joe. But that is not relevant. You rule with supreme power, but you are not a dictator. You have advisors, a court system, etc. You can do anything to the army. Your technology is 21-st century American.
You have a very large army of a million people (soldiers, not planners or generals (under, say rank of captain). However, most of them are, well, mediocre. You wouldn't say they're horrible and they do an okay job but you need an elite legion. You want the elite legion to have a thousand people in it.
Your army is well-armed. It's the soldiers' ability to fight that you're worried about. When building your elite legion, you want the absolute BEST soldiers you can have. You do this by (somehow) rounding up all of the soldiers and putting them in trials (Fortunately, your country is in total peace. No war is going to happen anytime soon). You don't want so many people to die during these trials (though when you put a million people through extensive trials, some of them are going to die) but you also want the elite legion to be the best that it can be.
So, what trials should you do to have the best fighters while reducing deaths? Let's assume the members of your elite legion can do everything but all specialize in one specific detail.
And another question(s)
What should members of your elite legion specialize in and how many of them should specialize in ****? Where **** is replaced with what they'll be specializing in, like "strategy" or "piloting".
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I disagree with your training concept: It is not testing, but training that generates elites.
You will screen a several thousand volunteers. Screening methods can be some combination of things that are simply hard (long Marches) tests of attitude, skill. but these are screening tests. You're trying to find the ones worth bothering with.
Physical training continues, but it's a mind game too. You want the non- gung-ho ones to wash out.
You ramp up the requirements. You split it into groups that compete with each other.
Send them into combat.
Now, use THIS group to make the next group. You don't just make 1 group of elite and stop. It's a continuous process, each round getting better.
Seal/ranger is a young man's game. You have fairly high turnover in terms of your line forces.
Do it right, however, and the ones that wash out, are still far more capable than the general troups. Don't waste them. Put them back and get them to train the next set of candidates. Similarly, use your elite troops as training cadre when they get old enough to start losing their edge.
To keep an elite force of 1000, with an active service time of 5 years, I suspect that you have to start with 3-10,000 troops a year.
Go and read about Ranger, Seal, Top Gun. Don't limit to the U.S. Britain's SAS, whatever Israel calls theirs now.
Study the way the Navy works up flight operations on carriers; the way carrier fleets train.
Read Clancy's books on various chunks of the armed forces.
One of ideas that comes out of this: Train as if you were fighting, Fight as if it just another training exercise.
Supreme ruler but not dictator. A distinction without a difference. What tests and skills should a 1000 man field force have? It depends on what you want to do with them. You've made no effort to define an enemy, so obviously we can't define a solution to an undefined problem.
If, as you imply, you want them to outscore the 999,000 soldiers of the rest of your army, well then, only accept those who score above 99.9% in the testing your regular army does. Why would they need special testing? Testing is (or should be) designed to predict real world performance. If your Army's testing doesn't, fix it. If it does, then you have your answer.
You also aren't clear about what support this 1000 men unit will have. Do they need the typical 3 to 20 support staff for each infantry soldier? Does that mean you actually only have a hundred or two of front line soldiers?
For infantry, you need a command structure, you need communications, intelligence, first aid, maintenance, supply, snipers/infiltrators, assault troops, mortar/rocket/anti-armor/anti-aircraft troops. For some reason you didn't indicate what technological level they have. Flintlocks or drones?
Ask for volunteers that pass rigorous tests to qualify to be given the most comprehensive training. Elite forces are about training and morale.
So physical and mental tests, just to get in they would need to be committed and already doing above average. The rest is hardcore training weeding out any who cannot keep up physically, mentally or are not committed enough. You'd need to look at a substantial number not making it through all the way. You scoop off the cream and you have an elite force.
This answer is necessarily very broad, because you don't go into specifics of what this force will be made to tangle with. So this is just broad strokes. Basically you want highly skilled, highly intelligent, flexible fighters able to respond or learn in a short time. Justifiably proud of their elite status and willing to lay down their lives if need be.
There are several pretty big questions to as here that define the answer.
Does this country have a real-world history? How much time and other resources are you willing to expend getting your 'Legion' up to scratch? Is this a self-contained unit, requiring it's own specialists?
Assuming this Legion is self-contained, it will need many different types of specialists, so any single battery of tests is going to be pretty useless on the individual level. But..
First thing's first you need to determine if an individual is physically capable of basic soldiering. Your priority here is allergies, & internal organ defects. We already have methods of sight & hearing (etc) correction & enhancement, so I'm not going to worry about those.
Meta-fitness: If you're willing to train your Legion, general health isn't really an issue, what you will be most concerned with is an individual's attitude. Pretty much every aspect of a person's health can be improved or mitigated for with the investment of time, energy & technical expertise.
Once you've established that an individual's attitude is acceptable or reasonably corrected, it's pre-existing sills & talents.
You can train a person to discern which type of helicopter they're looking at, but the fact is that some people will take whole seconds longer to notice something if they're not specifically listening out for it..not because their hearing is inferior, but because they get 'too' focused on whatever it is their attention is on and lack holistic awareness. The converse is equally true. Training people out of this sort of ingrained behavior is sketchy at best, but..
Take the old barracks room cliche from movies of the spotless boot camp, wherein drill instructors seek to catch out recruits on any and every infraction, checking them for any failure to meet a perfect standard. If you look at this from the perspective of teaching a recruit to be aware of his surroundings, of how his every action affects his surroundings and how punishment reinforces this awareness..you can see perhaps how something that just looks pretty mean & pointless begins to make sense as a training regimen.
So, you've begun a testing of attitude. You've said a war is not coming any time soon, so you have plenty of time for training..but the fact is that an 18 yr old recruit has been alive for 18yrs..and all upbringings are certainly not equal. Some people are incredibly passive, or indulge themselves as much as possible in fiction that has no bearing on reality whatsoever. Now maybe they've grown up a bit in the last few years and decided to dedicate themselves to the defense of their people and nation, but chances are their knowledge of how the world works will require as much effort unscrewing as it does filling in. This kind of thing is why selection processes are done by people rather than pc's, it's hard for a system to judge just why a person is screwing up, whilst people can generally make educated guesses. Still.
Physical: Suitability (Incapacities, I'm assuming you're willing to overlook imperfections and things that can be medically corrected)
Physical: Capacity (A rifle & pack carried by a 120lb person weighs as much as it does by a 220lb person)
Mental: Dedication (Your tests have to be administered by people and these people have to be the judges of merit, one source of dedication can look as strong as any other until it breaks, a person dedicated to the training regimen in the pursuit of proving themselves capable of it, is not going to be as good a soldier as one who is dedicated to proving and preserving the primacy of their nation.)
Mental: Composure & Decision making (Deciding to do the right thing is more important in almost every scenario than actually being able to do it)
Cutting short, hopefully useful in forming some more comprehensive answer :)
If you are in the Bronze Age, 1000 champions would be great. You could send your Achilles to fight the best Hector the enemy could put forth. The strength of a modern army is its ability to function as a unit, use its tech, and be practical, rational and quick. Hand to hand combat skills are not as useful in the modern era.
Pick your top 6 generals. Have each pick 5 colonels they trust and respect. Each colonel gets to pick the soldiers serving under him or her and so on down. Each member of your army is chosen by his immediate superior: the people who know them best and who want to continue working with them. Separate the wheat from the chaff.
Then split the result in 2: north and south or whatever. Split duties; defense of north and south etc. A big standing army is dangerous to a polity. A little intramural rivalry is good. Plus they can practice against each other; war games simulating the types of conflict you anticipate.
You might want to consider the effect on your economy of sending 900,000 working age adults back to civilian life. If you can do this gradually it might be good; let anyone go home who wants to (if some of your chosen want to go, sweeten the deal for them a little. But an army of volunteers is better than an army of conscripts.) For those without employment prospects at home form a public works corps or something of the sort.