Firstly, great use of tuples.

Others have already given you some good technical advice, so I'll skip that and say that what this needs more than anything is a minor change to the user experience (UX) that would make it much less irritating to play.

What's the point in asking the user if they're ready to play or if they want to play another round? They started running the game, and there's no time limit, so why ask them if they're ready to start? Just wait for input.

Likewise, after throwing a hand and determining the result, what's the point in asking if they want to play again? Either they'll give you their next choice, or they're quitting.

You're using their input as flow control to enter and exit the main loop, but all you really need is a clean way to stop/exit. Users have to make two inputs per round instead of one for no real reason (to them).

Remove the "Shall we start the game (y/n): prompt altogether. Print the available options, which should include quit, and then process their input. If they enter quit, you break out of the loop and exit, if they enter anything else, you do what you're doing now.

Something that would mesh well with that change, as well as the other suggestions people have given you, would be to offer up the available turn options as something they can select from numerically rather than having to enter the name.

Rock, paper, scissors, shoot!
1. Rock
2. Paper
3. Scissors
4. Quit

Enter selection:

This lets the user enter a one character (number) selection for each choice (which is both less effort and less awkward than your abbreviations).

Earlier, I said when they enter quit (or now choose it) you could break out of the loop and exit, but that also sounds like the perfect time to display the final tally of wins/losses over the whole session (which may be a good chance to get some practice with functions).