In AN 4.171, what is "reincarnation where the intention of others is effective"? 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"?
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 &amp; 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".