Alice and Bob each are affected once by the Thought Shield feature.
The Thought Shield feature is a game effect with a permanent duration (until the character/creature is dead) that grants the following effects as part of its permanent duration:
- Your thoughts can’t be read by telepathy or other means unless you allow it.
- Resistance to psychic damage.
- Whenever a creature deals psychic damage to you, that creature takes the same amount of damage that you do.
After the first bounce, the Thought Shield feature's same effect creates an overlap. The most potent instance of the effect will be applied. Apply the first instance of damage as it is the highest by default. Only one effect can be applied at any time, regardless of the number of ways that try to apply the effect.
Alice and Bob are affected once by the Thought Shield feature. Bob takes half of the damage of synaptic static, which is reflected towards Alice who takes half of the reflected damage, which then is reflected back to Bob who takes half of that damage. There are no futher bounces because the Combining Game Effects rules take place.
Consider the rules as described in Round Down (PHB 7) when you calculate your fractions.
DMG errata version 2.0 page 1:
Combining Game Effects (p. 252). This is a new subsection at the end
of the “Combat” section:
Different game features can affect a target at the same time. But when
two or more game features have the same name, only the effects of one
of them—the most potent one—apply while the durations of the effects
overlap. For example, if a target is ignited by a fire elemental’s
Fire Form trait, the ongoing fire damage doesn’t increase if the
burning target is subjected to that trait again. Game features include
spells, class features, feats, racial traits, monster abilities, and
magic items. See the related rule in the “Combining Magical Effects”
section of chapter 10 in the Player’s Handbook.
The basics for Damage Resistance and Vulnerability are found in the PHB (197).
Alternatively, it loops infinitely, even when the damage is reduced to 0 by the rules of resistance and rounding down.
The duration overlap here is the trigger. A different Thought Shield feature could damage either one and trigger a recursion. The effect overlap is the triggering of the recursion that is based on the duration of the feature's effects.
This game feature is not an instantaneous spell like a fireball. A fireball hits instantaneously, while the Thought Shield feature has a permanent duration. If you think that it is an instantaneous effect separate from its feature, then the infinite loop happens.
If the DM rules that the damage caused by the Thought Shield feature is not subject to feature but a new instance of an instantaneous effect of damage separate from its feature instead of an effect with a duration that grants you telepathy protection and resistance and also happens to causes damage, then the recursion loops even after the resistances have reduced the damage to 0.
Sage Advice Compendium V.2.3 (11):
Can damage be reduced to 0 by resistance or another form of damage
reduction? There is no damage minimum in the rules, so it is possible
to deal 0 damage with an attack, a spell, or another effect.
So if you follow this alternative ruling-way, Alice and Bob will forever be affected by Bob's foolish cast of synaptic static. Albeit the initial loops quickly reduce the looped damage to 0. Only death will break the loop because the corpse of the former creature has no Thought Shield feature.