Which of these two would be more reliable at least in terms of consistently dealing damage?
I'm not going to take the Agonizing Blast Invocation because I'm planning to take other Invocations, so I won't get the +CHA mods on the EB's. I have 14 DEX so the Crossbow would deal 1d10+2 damage. Now, obviously 5th level EB would have out damage the Crossbow on a high roll, but on the lowest roll, EB would deal 2 damage and the Crossbow would deal 3.
There's also the fact that EB has more mods at least in terms of being able to actually hit (I have 18 CHA so +4 to hit mods, NOT including proficiency), but I could also get a Pact Weapon heavy crossbow with the Improved Pact Weapon Invocation. With that, it becomes more accurate and more damaging, but it still doesn't measure up to the maximum roll of 5th level EB, but I think it would deal much more consistent damage.
Taking everything I mentioned above into account, what is statistically the more consistent damage dealer between a Heavy Crossbow (which I have proficiency with due to Hex Warrior and an additional +2 from 14 DEX), or a 5th level Eldritch blast (which is helped by 18 CHA which gives it +4 to hit, not yet counting the proficiency)?
And what is statistically the more consistent damage dealer between a Pact weapon Heavy Crossbow (+1 to attack and damage rolls and being a Pact Weapon gives it +5 to both attack and damage rolls in total without proficiency) vs the 5th level EB I mentioned above?
Best Answer
A Misconception that needs to be cleared up
Hex Warrior lets you use Charisma instead of the normal modifier for either your Hex weapon, or your Pact weapon. If you were to take the Improved Pact Weapon invocation, you'd be able to use +CHA with your crossbow instead of +DEX. So that alters the calculations somewhat.
Levels 5-10: Eldritch Blast is better (but Crossbows can be made to be better)
The DPR on the double blast is always going to exceed the DPR of a single crossbow bolt, even after being boosted with Improved Pact Weapon. There's not really any getting around that. The other posts have already gone into the math explaining why.
However.
Taking the Crossbow Expert feat will (until level 11) make Crossbows reign supreme. This is because the Crossbow Expert feat removes the attack limitation on crossbows, meaning you'll be able to attack twice a turn (with Thirsting Blade), causing the DPR on the crossbow to rocket upwards, even exceeding that of Agonizing Blast.
Here's a table of DPR values, against various ACs of enemies you might fight:
\begin{array}{|l|r|r|r|r|r|r|} \hline & \text{AC 0} & \text{AC 14} & \text{AC 16} & \text{AC 18} & \text{AC 20} & \text{AC 26} \\ \hline \text{Eldritch Blast x2} & 11.000 & 8.250 & 7.150 & 6.050 & 4.950 & 1.650 \\ \hline \text{Agonizing Blast x2} & 18.600 & 13.850 & 11.950 & 10.050 & 8.150 & 2.450 \\ \hline \text{Crossbow x1 (uses DEX)} & 7.400 & 4.775 & 4.025 & 3.275 & 2.525 & 0.650 \\ \hline \text{Crossbow x1 (IPW)} & 10.250 & 8.150 & 7.100 & 6.050 & 5.000 & 1.850 \\ \hline \text{Crossbow x2 (Expert / TB)} & 14.800 & 9.550 & 8.050 & 6.550 & 5.050 & 1.300 \\ \hline \text{Crossbow x2 (IPW / Expert / TB)} & 20.500 & 16.300 & 14.200 & 12.100 & 10.000 & 3.700 \\ \hline \end{array}
detailed statistical breakdown here
In the last row, you can see that against all AC levels, the DPR of taking the Improved Pact Weapon and Thirsting Blade invocations, and the Crossbow Expert feat, will exceed the DPR of even Agonizing Blast on an Eldritch Blast.
So between levels 5 and 10, (which is where most play occurs!) your DPR with a crossbow can be better than your DPR with Eldritch Blast, assuming you take the right invocations and feat.
Level 11: Agonizing Blast is best
At level 11, the third Eldritch Blast improves the DPR by a lot, bringing the DPR ahead of where the Crossbow is at for this level. The regular Eldritch Blast isn't competitive at this level.
Levels 12-16: Crossbows (barely) come out ahead again
The introduction of the Lifedrinker invocation (+CHA to pact weapon damage... again) will set things just barely back in the favor of the Crossbow:
\begin{array}{|l|r|r|r|r|r|r|} \hline & \text{AC 0} & \text{AC 14} & \text{AC 16} & \text{AC 18} & \text{AC 20} & \text{AC 26} \\ \hline \text{Eldritch Blast x3} & 16.500 & 14.025 & 12.375 & 10.725 & 9.075 & 4.125 \\ \hline \text{Agonizing Blast x3} & 30.750 & 26.025 & 22.875 & 19.725 & 16.575 & 7.125 \\ \hline \text{IPW / Expert / TB / LD Crossbow x2} & 31.900 & 28.600 & 25.300 & 22.000 & 18.700 & 8.800 \\ \hline \end{array}
detailed statistical breakdown here
Level 17: Agonizing Blast is best
Finally, with 4 attacks, Agonizing Blast will get up to a DPR of [41|38.9|34.7|30.5|26.3|13.7], which will be better than the (improved) DPR values that the Crossbow gets, stuck at 2 attacks, with [31.9|31.9|28.6|25.3|22.0|12.1].
So realistically, which combo you go with depends on what level of play your campaign will be at. If you actually expect to make it all the way to level 17+, then focus on Eldritch Blast and ignore the Crossbow feat + invocations. If you plan to never play past 16, then get the Crossbow feat and take invocations improving the Crossbow.
Note that getting the Crossbow this high does involve taking a feat and several invocations; if you'd rather dedicate those slots to other features, the Agonizing Blast setup only requires the one Invocation, and very little else. So in terms of economical builds, Agonizing Blast is easier to build around.