Short Swords or Hand axes?

Question

I'm building a dual wielding+archery character as a "recommended" build for a friend. I don't know whether it would be better for him to build for handaxes/throwing axes or short swords.

I know this is probably a long asked question of higher critical modifiers or wider critical ranges, but I want to figure it out for THIS CHARACTER. I think that the character themselves are the most important thing for it, so here's his backstory he's told me.

"I lived a normal life until my father killed my mother when I was young. I ran into the forest to hide from my murderous parent. The only reason I survived (rather than dying from disease) was because I was found by a small clan of Druids. I learned the way of the predator in my years. I also made a connection with the spirit animals of the world, the wolf, eagle, and bear, from my time with the druids. I was found by (the other party members) the same day I learned my father was a member of a chaos cult that they (the other party members) were working to fight against. I now use my skills gained from my time in nature to fight the chaos cultists and hope to kill my father to avenge my mother".

level: 1 Race: Wood elf (+2 dex, +2 int +1 wis -2 con) Class: Ranger/fighter Feats: Able sniper and point blank shot. available wealth: 99 gold equipment already bought: leather armor and a longbow

Answer 1

Unless you have specific features that interact with one other other, it really does not matter. The difference in expected damage is absolutely minuscule; critical hits simply don’t happen often enough. Both 20/×3 and 19-20/×2 are close to meaningless.¹ So unless you want to take some feat or class that says something special happens only when you wield one or the other, just take whatever you feel like.

Or, better, whatever the actual player feels like. You can just say to him, “so this character works equally-well with handaxes or shortswords; which would you prefer?” Don’t bother mentioning the mechanical differences; again, they don’t matter. Purely a matter of imagery and style.

In reality, the character sounds like a druid, and sounds like he’s going to prefer Wild Shape anyway; claws and fangs are likely to be his truly-favored weapons. Or if you were going for all-out power, spells.

1. ^{For that matter, 18-20/×2 and 20/×4 are also basically identical with one another, and even though they are strictly superior to 19-20/×2 or 20/×3, the difference between all four is small enough that it isn’t worth worrying about. For this reason, spending any significant resources (feats, levels, thousands of gold) “upgrading” from 19-20/×2 or 20/×3 to 18-20/×2 or 20/×4 is not worthwhile. Even larger die sizes, which tend to be worth just +1 damage on average and therefore also aren’t worth spending any significant resources, are worth more than the critical stats.}

Answer 2

It is complete preference, there is no correct answer to which is "better" because both weapons are pretty much even. The same thing is true about any weapon with a 19-20/x2 crit when being compared to a 20/x3 crit.

Here is the math: (Assuming that your critical hits are confirmed)

On a roll of a 1 you auto miss, 2-18 = 3.5 average damage, 19-20 = 7 average damage. (1*0)+(17*3.5)+(2*7)/20 = 3.675 average base damage with a shortsword

On a roll of a 1 you miss, 2-19 = 3.5 average damage, 20 = 10.5 average damage. (1*0)+(18*3.5)+(1*10.5)/20 = 3.675 average base damage with a handaxe

The only time that this alters is when you are against an opponent who requires a natural 20 to hit. One is piercing and one is slashing, some feats and some class features provide benefits to one over the other.

Answer 3

Somebody else did the "too low of a chance to matter" part, I'm going to expand on the math since the existing answers don't satisfy me.

On a purely statistic basis, axes are slightly better

A sword threatens a critical 1/20 of the times if it only hits on a 20 and 2/20 otherwise. An axe always threatens 1/20 of the times. Any weapon hits x/20 of the times and confirms a critical x/20 of the times, where x is the number of d20 facets that allow you to hit.
This means that a) since the two weapons deal the same damage (1d6+modifiers), and they hit with the same frequency, we can ignore the base damage from our calculations and only focus on the extra damage from the criticals actually happening, and b) the chance of a critical happening is (x/20)(1/20) or (x/20)(2/20). The mean damage is ×1*2/20*x/20 for a sword versus ×2*1/20*x/20 for an axe.
Those figures are exactly the same, except when you only hit on a 20, as @DanceSC already shows in his answer. In this case (1/400 chance!) the axe is better.

You don't roll that many dice to trust statistics

As we all know, the fact that tossing a coin gives head 50% of the time does not prevent the coin from always falling on tails. Sooner or later, if you make enough rolls, you're deemed to see every number on an unbiased die show up with the same frequency. Not in a single combat. This means that having a 10% chance of critting with a sword is better than having a 5% chance of critting with an axe. The axe is swingy - you might never see that awesome critical in a combat. What's better? Dealing an occasional 3x or having an almost reliable 2x?
Or you might have a biased dice and favor an axe if 20s are more frequent than 19s and viceversa. You cheater ;)

One hit, one kill

On the other hand, that 3x crit might outright kill an enemy. Enemies whose lives are shorter deal less damage. You might be lucky and get that high damage early, shortening combat. Whoho axes. If combat lasts for long, you might crit twice with a sword, making this unimportant.