how many 9.1 speakers are you planning on connecting? two? four? eventually a whole 5.1 set?
how many other sources are you going to connect? i dont think you listed any of the others besides the ps4.
one reason why i suggested a receiver is that those speakers are rated at 100w @6ohm maximum power handling. most small cheap amplifiers are not capable of that and the ones that do are encroaching on the cost of a cheap receiver.
also bear in mind that to use an amplifier you need an analog input (typically rca). you would need to either convert the optical to rca or split off the sound and convert it to rca from the hdmi cable with a dac box. this can add $40 easily to the cost which brings the cost of an amp solution in line with cheaper receivers. now if your tv does have analog output (3.5headphone jack) an amplifier could work without this box but not all tvs have them.
also keep in mind that some of the cheap amplifiers you see are rather limited in function (some do not even have a volume control) and have limited output capability which means you wouldnt get the most from those speakers. there are models with a DAC (which you need if you wanted to route optical or coax spdif to them however they are typically getting close to the cost of a receiver. in general while a cheaper solution it can also be more of a headache to connect.
another thing to think about is that a stereo amplifier will limit your future upgradeability. you will be limited to stereo only. you will have older inputs and need conversion boxes for everything to work.
going with a receiver makes everything simple and may actually be cheaper in the long run. my parents needed an amplifier for the tower speakers they had (100w @8ohm) and i searched for an amplifier since that is what they wanted however it just was not going to be the best solution. we ended up with a sony strdh540 5.2 receiver which was on sale for $200. considering a good amp was going to cost $100-150 and a dac was going to cost $50 it was a no brainer.
with a receiver you have the option to route video through it as well. this means less cables out of the back of your components if you intend on using the same tv.
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TLDR:
i would suggest a receiver since it is easier to connect, gives you more future upgradeability and doesnt add much to the budget.
you could use an amp however you would need to use a dac as well and your options might be more limited in the future
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in any case... you can pick whichever route you want... we are just here to provide options.