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I hope your not gaming on a crappy Cox gateway. If so, that is probably half your problem. What port does PS4 use for Host share? Have you tried a port scan? If not a Cox gateway, what does your network consist of? I have a PS3 hooked up right now on Cox and everything is working fine.
Thanks so much for replying. I am visiting family that has Frontier as an ISP and I plugged in my PS4, set up the internet connection on all automatic settings, and my friend was able to connect to me via Shareplay for the very first time.
I will go home in a couple of days and do the exact same thing I did here and see what's what.
Ok, friends, I'm back hope. I plugged in my PS4 and set it up exactly the way it was at the other place and my friend got the same message he always gets.
THE HOST'S INTERNET CONNECTION IS UNSTABLE
or something very close to that. No connection, even though I show HIGH up and down speed.
Next I will factory reset my router and see what's what.
AFAIK, Host play uses P2P, so the first thing I would suggest is check the latency between your home IP and the IP of the person you are trying to share from. If you see high latency, post a tracert to the IP address to see where the latenct is. In general, both PS4 should be on WIRED or ethernet connections and both of you should be get internet speeds above 15Mbps. If one of you is restricted to slow internet, try decreasing the resolution from 1080p. The lower the resolution, the less data you need to share between PS4 to play. See here for more info.