This is going to be a dull analytical post, so feel free to skip it. The question before the house is whether longsword and shield is better than longsword and hand axe (In AD&D, only daggers and hand axes are allowed for the "off-hand" weapon in two-weapon fighting and, since the AD&D dagger is almost worthless as a weapon, I'm going to use the axe as the most effective option).
The rules for two-weapon fighting (TWF) are fairly straight-forward: the combatant is treated like any monster with an attack routine and when it is their turn in the initiative scheme, they roll once for one weapon and then once more for the other weapon. Whether these attacks are both against the same target or not is never specified, but there is some reason to believe that they are and that's what I'll assume.
The primary weapon attacks at -2 to hit and the secondary at -4 if the character's dexterity is between 6 and 15. At dex 16 the modifiers are -1 and -3, at 17 they are ±0 and -2, and at 18 the secondary penalty goes to -1.
I'm going to start off with two fighters in platemail, one with a shield and longsword (sword and board as da kidz say) and the other with the longsword and hand axe. I've not calculated these effects ahead of typing this, so the conclusions will be new to me too and may or may not confirm my own feelings. Isn't science exciting?
1 Puny Veterans
We'll start off with no ability score bonuses and 1st level fighters. This is an easy win for sword and board; hardly worth calculating but it serves as a baseline. Calculating the average hit and damage rates gives this:
S&B: 0.9 hp per attack.
TWF: 0.625 per attack.
TWF is doing almost 70% of the damage rate, although in fact the repeating 20's of the combat chart have helped to level the field somewhat here. Let's try bumping the combat ability a bit.
2 Puny Heroes
Same as above but 4th level fighters.
S&B: 1.35 hp per attack.
TWF: 0.85 hp per attack.
The repeating 20's of the combat table are no longer important for these guys and so TWF falls further behind to 63% of the S&B damage.
3 Puny Superheroes
Moving up to 8th level on the "no bonuses" ladder, we find these values:
S&B: 2.25 hp per attack
TWF: 2.45 hp per attack
Now that armour itself is becoming less important, the TWF has overtaken the S&B fighter and is doing almost 9% more damage per attack. Both fighters are now also getting 3/2 attacks per round, but the effect of this cancels out so we don't need to worry about it.
4 Puny Lords
Final run without bonuses: 12th level:
S&B: 3.15 hp per attack.
TWF: 4.05 hp per attack.
TWF is now 28% more effective per attack routine.
This is all nice but unrealistic. Firstly, the fighters are unlikely to not have some bonuses, and secondly, the high level fighters are unlikely to not have magic armour, weapons, and shields.
5 Agile Veterans
We'll give both fighters 18 Dexerity and look at whether this has most effect on TWF or S&B. The TWF now has no penalty for the longsword and only -1 to-hit on the axe. Back to first level:
S&B: 0.225 hp per attack.
TWF: 0.4hp per attack.
Essentially, the off-hand weapon is pure bonus here as everyone needs 20's to hit.
6 Agile Heroes
Moving up to 4th level with 18 Dex:
S&B: .45 hp per attack
TWF: .4 hp per attack
Simple enough: the TWF needs 20's with both weapons to do any damage while the S&B guy is hitting twice as often with his single weapon.
7 Agile Superheroes
8th level, 18 Dex:
S&B: 1.35 hp per attack
TWF: 1.825 hp per attack
TWF is now clearly superior, doing 35% more damage.
8 Agile Lords
Final run - 12th level 18 Dex:
S&B: 2.25 hp per round
TWF: 3.425 hp per round
52% better with two weapons.
Okay, at this point we've learnt something about the big value of 18 Dex as a two-weapon fighter. I'm not going to list every possible combination but I want to look at the possible effect of a magic shield. I'm going to assume that any magic armour and swords cancel out for to-hit purposes and see what a +3 shield does to the superhero and lord levels. I'm going to assume that the hand axe is not magical and that both longswords are +2 for damage.
9 Magic, Agile Superheroes.
S&B now has an effective AC of -5 while TWF has an effective AC of -1.
S&B: 1.95 hp per attack.
TWF: 0.825 hp per attack.
TWF is back to doing only 42% of the damage rate of the sword and shield.
10 Magic, Agile Lords
12th level as previous example.
S&B: 3.25hp per attack.
TWF: 2.825hp per attack.
Improving back up to 87% of the opponent's score, the TWF is still clearly the less effective option here too.
Next, what happens if the S&B fighter decides to put their 18 into STR while the TWF keeps it in Dex? Assuming 18/50 and staying with the magic listed above, the S&B fighter's effective AC drops to -1, while the TWF's is reduced to 0 due to the +1 to-hit for 18/50 strength. How does the increased damage balance against this AC change?
11 Strong Vs Agile Magic Superheroes
Back to 8th level with +2 swords.
S&B: 3.325 hp per attack.
TWF: 2.825 hp per attack.
I have to admit to being surprised; I thought the TWF would win this one. Instead it scores 85%.
12 Strong Vs Agile Magic Lords
12th level as above:
S&B: 5.225hp per attack.
TWF: 4.825hp per attack.
TWF moves up to 92%.
I'm surprised enough by these results to briefly go back and look at the STR Vs Dex values without magic.
13 Strong Vs Agile Heroes
So we're back to no magic and 18 Str Vs 18 Dex. S&B guy has an AC of 2, while S&B is at 0 (again, because of the +1 to hit from Strength).
S&B: 1.425hp per attack.
TWF: 1.825hp per attack.
Advantage to the TWF guy; 28% better.
14 String Vs Agile Superheroes
When we move these two magically-impoverished fighters up to 8th level the pendulum swings again:
S&B: 2.85hp per attack
TWF: 2.625hp per attack.
Now TWF is back to being 92% as effective as S&B
As I suspected, the balance of power between TWF and S&B is quite delicate and it's not an easy thing to call which side it will settle on just by a casual glance over a character's stats and possessions. It's also true that these examples have been quite artificial and I don't expect every fighter to have an 18 in either Dex or Str.
Having said that, even a 16 Dex represents quite a step down for TWF and a quick calculation on some of the above situations suggests that it really isn't very viable.
To really dig into the implications of TWF you'd really need to construct a spreadsheet to allow changing of lots of factors so that more realistic situations could be examined. But "more realistic" equates to a huge number of possible combinations of magic swords, shields, dexterity, strength, weapons (and weapons Vs armour which I've ignored here) and numbers of opponents. Probably there are a lot more situations where TWF is at least marginally better than sword and shield but I'm pretty confident that a shield of the calibre of the +3 one in the examples here will more than negate the usefulness of TWF in any situation likely to arise in play.
Of course, "likely to arise" is of no consequence to the DM who has to handle the situation when it does arise and a fighter with 18 Str and 18 Dex would be mad not to take TWF and there's no value in the DM saying "well, that was really lucky rolling!". Of course it was. But rather than worrying too much about how such an unlikely thing might be "out of balance", I think it's more productive to say "this character could become one of the really memorable ones" and try to embrace it without allowing it to overshadow the other members of the party. All characters have weak points (usually the player behind them) and so long as the game doesn't become a long string of "well, Dexto the Barbarian handles it" then every one should still be able to have fun with such a fluky character in the party.
One other thing I've noticed while going through this process is that the two-handed sword is not as bad an option as I thought, if the character has 18/50 strength. With an average adjusted damage of 8.5 against a human and 13.5 against large creatures, it's probably worth consideration at low levels. At high levels, a magic shield is still a much better option, I think, especially as a magic longsword will probably have made an appearance by then.
As I've mentioned on Dragonsfoot, I charge a proficiency slot for fighting with two weapons over and above any used to learn how to use the weapons singly, although a character can learn, for example, "longsword and dagger" without taking either weapon on its own, and simply receives the non-proficiency penalty when forced to use one or the other alone. This is an additional factor in deciding whether to take two weapon combat or not, particularly for thieves who only start with two slots.