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In his seminal essays, Don Rae suggests that a retreating player always leaves one infantry in the territory she is retreating from:

Retreat and regroup when faced with overwhelming odds against your forces, always leaving at least one infantry defending in a territory - give them NOTHING for free.

I agree this advise makes sense if the opponent has tanks adjacent to the territory you are retreating from and can capture two territories in one turn, or reach an otherwise un- or under-defended territory through the vacated territory, but that aside, why does it make sense to leave one infantry in a territory you're retreating from?

In other words, if you know your enemy will attack (and conquer) a territory whether you have one infantry in it or not, why would you leave one in there? That's one less infantry you have to help defend the territory you just retreated to.

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In answer to your subquestion of "why does it make sense to leave one infantry in a territory you're retreating from?"

The core of the analysis around this hinges on the assumption that you will be trading back and forth over the contested territory with small numbers of units- that you have a large, constantly growing stack in A, your opponent has a large, constantly growing stack in C, and you keep trading off B between the two territories with as small stacks as you can manage. Each player has a steady supply chain providing new troops from the factory. The reason why you do this is because if you move your large stack into the territory, it will be missing the current turn's set of reinforcements, and be unable to use your planes for defense, and your opponent can attack with their whole stack and use all their planes for offense. This is the whole idea of Dead Zones. You want to control them because they're worth IPCs, but with as few units as possible because those units will be destroyed by a counterattack next turn.

So why leave a lone infantry there? Because if it's empty, your opponent can take it with the bare minimum of units, and their planes are free to attack elsewhere. If it isn't, they are forced to commit some of their planes to it, and it's economically in their interest to send 2 infantry. Even if that is a good trade for him (you will lose your 1 infantry for only a 33% chance of killing one of his) the next turn, you will be able to kill both his infantry, an even better trade for you!

So then what about the title question: Should you always leave one infantry?

No. Almost always, but not always. If your opponent will move in with his whole stack next turn (because it's big enough that yours is no longer a threat) the lone infantry is no longer economically in your favor. You will be trading an infantry for a 33% chance to kill an infantry, which is not effective. You might still do it to prevent a blitz but it's no longer automatically good.

Also, if the territory is only worth 1IPC. Generally, the territories you have these big-stack standoffs over are worth more than 1, so this doesn't come up. For a 1IPC territory, it no longer makes your opponent want to send in 2 infantry. The expected value for him attacking your lone infantry with 1 infantry + 1 fighter will be approximately 7/6, while the value he'll get from just waltzing into the territory uncontested will be 6/6. You might still do it to block a blitz or just to force the opportunity cost of him using his planes. But if he has more planes than he knows what to do with and no blitz threat, you wouldn't want leave the guy there.

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Yes, never leave a territory undefended

I never leave a territory undefended if my enemy is in attacking range. A single infantry can capture an undefended territory, think of what risk and reward this exposes them to:

  • -3 IPCs: they might lose the unit if you counter-attack (this is pretty much the worst that can happen to them, as the value of a single infantry elsewhere on the board is likely not as large as the benefit they're about to get by taking this territory from you)
  • +1 IPCs: they might get a hit when you try and take it (this is their worst case: only one round of combat killing on of your infantry. If you're dumb enough to attack without cannon fodder or if you go in too light and it takes multiple turns this expected value goes up.)
  • +1 (or more) IPCs: the income from the territory that turn and any future turns until you take it back
  • +? IPCs: how many IPCs in units it will cost you to take it back (see below)
  • priceless: the opportunity cost of what you're not going to be able to attack with the next turn because they are not going to leave it undefended.

Now think of what it takes to capture a territory defended by only 1 infantry:

  • If you attack with just 1 infantry you're likely to completely fail, mutual destruction is also quite likely--in which case they still own the territory.
  • If you attack with 2 infantry, you still only have about a 60 percent change of taking the territory, and even then with some risk of losing one of your units.
  • If you use armor, then you're likely to win but leave it in a dead zone. Armor should be treated like fine China: bring it out to impress, but don't put it in harm's way.
  • Your best option is to use airpower, but now we're back to opportunity cost. A fighter that spends it's attack taking back border countries you never should have given away could have been used in a land attack, a naval attack, or just moved into a better position on the board.

So the lesson is when you realize one of your territories is a dead zone, don't abandon it. Leave one infantry there and force your opponent to decide how much he's willing to devote to take it back.

If your enemy presents you with an open territory take it. If you don't have an infantry on your front lines territory (for shame), then blitz in and out with a tank just to get the free IPCs. But unless it's worth more than 5 IPCs don't leave a tank there by itself.

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what if you are 99% certain your enemy will attack the territory with everything she has, not just 1 or 2 infantry or a tank or whatnot? –  Scott Mitchell Jun 10 '11 at 2:50
    
@Scott: then it would probably depend on the rest of the board and I won't say I'd never do it, but I'd still be inclined to leave an infantry behind. The reason is that I'm always wary of an open territory giving my opponents options. Perhaps they can now blitz through the territory I left open and throw everything they've got against an adjacent territory instead--something they perhaps can't afford in they have to siphon off a few units and actually attack both. Perhaps this enables them to just blitz into it to get the IPCs and delay their attack one more turn to gather reinforcements. –  Adam Wuerl Jun 10 '11 at 12:16
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My question is should you always leave an infantry in a territory you are retreating from? It seems to me that if you are positive your opponent will attack with full force and there is not the threat of a blitz or other attack, then it makes no sense to sacrifice that poor infantry. –  Scott Mitchell Jun 10 '11 at 16:03
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@Scott: Always is a hard thing to say about boardgame strategy because it rejects the notion of corner cases. It's much more instructive to use heuristics and realize that if gaming was all about inflexible rules then computers would be better at it than they are. I'm highly unlikely to ever leave something unoccupied for the reasons discussed above. I'm hard-pressed to imagine the particular set of circumstances you've described manifesting very often, and if they did I'd be wary that my assessment of the board was correct. I say better to risk the one unit on the off-chance you're wrong. –  Adam Wuerl Jun 10 '11 at 18:11
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Ordinarily, I'd say, leave one infantry in the country. But if it is about to be attacked by overwhelming force, them it might make sense to pull it out.

One reason is to establish a "line of retreat." That is, you, as the UK, pull forces out of India in the early going to have an infantry in Persia, one in Syria, all the way to Egypt.

And the other thing to think about is that the opponent who concentrated the overwhelming force may be doing so as a bluff, planning to move heavily in another direction, and glad to get a "free" territory for one infantry.

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