In AN 4.171, what is “reincarnation where the intention of others is effective”?

Per commentary, this passage refers to "devas corrupted by the mind" (Manopadosika-deva), living in the world of the Four Great Kings. When they get angry with one another their mutual anger leads to their mutual deaths.
Here are descriptions of Manopadosika-deva from different sources:
Buddhist Dictionary. Manual of Buddhist Terms & Doctrines (4th Edition) by Ven. Nyanatiloka:

Manopadosika-deva: ‘the celestial beings corruptible by temper’, are a class of devas (q.v.) of the sensuous sphere. “They spend their time in becoming annoyed with one another, and getting into a temper, and thus by being bodily and mentally exhausted, they pass from that world” (D. 1; 24).

The "Dictionary of Pali Proper Names" by G. P. Malalasekera:

A class of devas living in the Cātummahārājika world (DA.i.114; AA.ii.544; MNid.108). They burn continually with envy one against another (Buddhaghosa relates a story illustrating this in DA.i.114) and their hearts become ill disposed and debauched. Their bodies thus become feeble and their minds imbecile, and, as a consequence, they fall from their state (D.i.20; iii.32). The Manopadosikā were among those present at the preaching of the Mahāsamaya Sutta (D.ii.260). They are so called because their minds are debauched by envy (dosassa anudahanatāya) (VibhA.498).

Brahmajāla Sutta (DN 1) translated from the Pali by Bhikkhu Bodhi:

There are, bhikkhus, certain gods called 'corrupted by mind.' These gods contemplate one another with excessive envy. As a consequence their minds becomes corrupted by anger towards one another. When their minds are corrupted by anger, their bodies and minds become exhausted and consequently, they pass away from that plane.

Brahmajāla Sutta (DN 1) translated by Bhikkhu Sujato:

There are gods named ‘malevolent’. They spend too much time gazing at each other, so they grow angry with each other, and their bodies and minds get tired. They pass away from that group of gods.

So, one deva's hatred towards another deva provokes hatred in that deva towards the first, causing a vicious circle, and this very hatred exhausts their spirits until they fall from their godly state.
One deva's hatred is not his own deliberate intention but is merely a reaction to another deva's hatred. Therefore the deva falls from his state and comes to a different state through an "intention of other".
It's interesting how the vicious circle of judgement starts not through one of the devas performing some unwelcomed act to another, but rather through both spending too much time wondering about the other's attitude to oneself.
I suppose this basically means that not all karma is strictly personal. While in the simplest cases sentient beings inherit their own karma, there are cases when they create (bad) karma for one another, and there are cases when a karmic pattern emerges spontaneously from a background of ignorance - not through anyone's fault.
On a side note, per my (Russian) dictionary, attabhāvapaṭilābhe is not "reincarnation" but "a state of personal existence".
by Andrei Volkov @https://buddhism.stackexchange.com/a/30153/10100.

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I agree with Andrei Volkov explanation, which follows to it's atthakathā. However, for the perfect understood by sutta, not only atthakathā, I write this answer.

1. There is a reincarnation where one’s own intention is effective, not that of others. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe attasañcetanā kamati, no parasañcetanā.

By atthakathā, above text refers to DN Brahmajālasuttaṃ:

7. ‘And what is the second?

‘There are, brethren, certain gods called the “Debauched by Pleasure” [155]. ‘For ages they pass their time in the pursuit of the laughter and sport of sensual lusts. In consequence thereof their self-possession is corrupted, and through the loss of their self-control they fall from that state [156].

8. ‘Now it might well be, brethren, that some being, on his falling from that state, should come hither. And having come hither he should, as in the last case, become a recluse, and acquire the power of recollecting his last birth, but only his last one.
9. ‘And he would say to himself: “Those gods who are not debauched by pleasure are stedfast, immutable, eternal, of a nature that knows no change, and they will remain so forever and ever. [20] But we-who fell from that state, having lost our self-control through being debauched by pleasure-we have come hither as being impermanent, mutable, limited in duration of life.”

By atthakathā, above text refers to MN kandarakasuttaṃ:

2. There is a reincarnation where the intention of others is effective, not one’s own. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe parasañcetanā kamati, no attasañcetanā.

Bhikkhus, which person torments others yoked to it? Bhikkhus, here a certain person is a killer of pigs, goats, birds, a hunter of wild animals, a fisherman, a robber, a highway robber, one who keeps a prison, or engaged in any other vicious activity. This person torments others yoked to tormenting others.

3. There is a reincarnation where both one’s own and others’ intentions are effective. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe attasañcetanā ca kamati parasañcetanā ca.

By atthakathā, above text refers to DN Brahmajālasuttaṃ:

10. ‘And what is the third?

‘There are, brethren, certain gods called "the Debauched in Mind [157].” They burn continually with envy [158] one against another, and being thus irritated, their hearts become ill-disposed towards each other, and being thus debauched, their bodies become feeble, and their minds imbecile. And those gods fall from that state.

11. ‘Now it might well be, brethren, that some [\q 034/] being, on his falling from that state, should come hither; and having become a recluse should ‘as in the other cases, acquire the power of recollecting his last birth, but only his last one.
12. ‘And lie would say to himself: “Those gods who are not debauched in mind do not continually burn with envy against each other, so their hearts do not become evil disposed one towards another, nor their bodies feeble and their minds imbecile. Therefore they fall not from that state; they are stedfast, immutable, eternal, of a nature that knows no change, and they will remain so forever and ever. [21] But we were corrupted in mind, being constantly excited by envy against one another. And being thus envious and corrupt our bodies became feeble, and our minds imbecile, and we fell from that state, and have come hither as Being impermanent, mutable, limited in duration of life.”

‘This, brethren, is the third case.

4. There is a reincarnation where neither one’s own nor others’ intentions are effective. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe nevattasañcetanā kamati, no parasañcetanā.

By atthakathā, above text refers to DN Dānamahapphalasutta:

"Having given this gift seeking his own profit — with a mind attached [to the reward], seeking to store up for himself, [with the thought], 'I'll enjoy this after death' — on the break-up of the body, after death, he reappears in the company of the Devas of the Four Great Kings. Then, having exhausted that action, that power, that status, that sovereignty, he is a returner, coming back to this world.
"Having given this gift with the thought, 'Giving is good,' on the break-up of the body, after death, he reappears in the company of the Devas of the Thirty-three. Then, having exhausted that action, that power, that status, that sovereignty, he is a returner, coming back to this world.
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Why I have to link? Theravāda tradition never allow to explain tipitaka without reference, linking. It's weird and not professional. If buddhaghosa wrote something without the reference, he wouldn't be allowed to translate atthakathā back to pāli. So, the link, which I have given, show tipitaka memorizer's thought.