Can you decide not to use a shield's AC bonus? Dex bonus?

Question

Let's say that there is a combat. There's is one player character and two enemies- one melee and one ranged. The player character has Armor of Agathys (or some other spell) and wants the melee combatant to hit them- but they also have a shield and light armor. If the initiative was Melee, Ranged, Player-

1. Could the player use their shield to defend the ranged attack, but not the melee attack?

2. Could the player apply their Dex bonus from their light armor to defend the ranged attack, but not the melee attack?

Answer 1

By the rules - probably not, but a DM might allow it as a houserule

The rules say that while you are wielding the shield you automatically get the bonus to your AC. So while you are holding and using the shield you are assumed to be getting that bonus at all times. There is no mechanism in the rules for turning this bonus on and off. The same thing applies to your Dexterity bonus to AC.

If you want to say that your character is foregoing that shield bonus, say by flinging your arms wide open so the shield is not protecting you, then you must ask your DM if they will allow that to reduce your AC for that attack.

I suppose the same could be said for foregoing your Dexterity bonus to AC as well seeing as you can say that narratively you simply stop moving and stand very still.

If the DM allows it as a houserule, then the could probably allow you to block one attack and then leave yourself open for the next one, Though they might also rule that once you leave yourself open that you are vulnerable to any other attacks as well. Since this is not specified in the rules it will be up to them entirely as to how to manage it.

Answer 2

By RAW: no. A shield you're wielding, as well as your Dex bonus (except special cases like barkskin and Tortles), always apply to your Armor Class.

By reasonable houserule: Yes for shield, no for Dex. It absolutely makes sense that you can focus on shielding against ranged attacks while ignoring the melee attacker in front of you - but even selectively dodging the ranged attacks will make it harder for the melee to hit you, so dropping your Dex bonus for the melee attack should drop your Dex bonus for the ranged attack as well.

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By RAW, revisited: To get the effect you're looking for without any rules-lawyering...
Drop prone! It's free to do, it grants melee attackers advantage and ranged attackers disadvantage, and it accomplishes your end goal (get hit by the melee, don't get hit by ranged) in a far superior way than even selectively applying AC bonuses would.

Answer 3

RAW silliness: technically, you might be able to go one better.

This is dealing with AC, and in 5e, AC is determined by formula. By the rules, if you have more than one formula for AC, you get to pick which one you use, and there's no indication that this takes any particular time or action. If you somehow manage to have multiple AC formulas, there's nothing technically stopping you from using the AC calc of your choice on each attack.

So... technically, RAW forgoing your dex bonus is really quite difficult, and forgoing your shield bonus requires an action, but under certain circumstances you might be able to forgo your armor bonus, and a monk/barbarian can forgo their wisdom/con bonus pretty easily.