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Bhikkhu Sujato's translation of AN 4.171 says:

Mendicants, there are four kinds of reincarnation. Cattārome,
bhikkhave, attabhāvapaṭilābhā.

What four? Katame cattāro?

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

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

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

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

These are the four kinds of reincarnation.” Ime kho, bhikkhave,
cattāro attabhāvapaṭilābhā”ti.

What is "reincarnation where the intention of others is effective"?

Perhaps this is what refers to in Abhidhamma as prompted and unprompted.
For instance, a person can kill someone with his own intention or intention of another person (prompted by someone else). One can influence another to kill.
I may be wrong.

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".

It seems the compound "attabhāvapaṭilābhe" means "acquisition of self becoming" or, as often translated, "acquisition of personality".

For example, if parents, for example, allow their children to make their own decisions in life, the children will develop a personality based on their own decisions.

But if parents, for example, are very domineering & even abusive, their children will develop personalities reflective of the violence of the parents.

Or when people are brainwashed by political rulers, such as Mao, Stalin, Hitler or Obama; or by the corporate or pornographic mass-media, the acquired personalty will be reflective of brainwashing.

In other words, it appears the translation of "reincarnation" is very imaginative.

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.

...

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.