I'll do it for three million dollars. Might be less, depending on complexity and platforms.

The best way to get where you want to go, is to build the game using an engine that already supports multiplayer, and then use the platform for matchmaking.
So, for example, if you build your game on the Unreal Engine 4, and then host on SteamWorks or Xbox 1 or PlayStation Network or Google Play or Apple Game Center, then you can use the built-in multiplayer support of Unreal, and the built-in match-making of the platform.
There's still work to be done (your game loop/activities must know that things happen on the network) but it's much, much easier than building it from scratch, or trying to add it to an existing game in some engine that doesn't support multiplayer.

To get a flavor for what multiplayer development is like on Unreal Engine, check out these video tutorials:
https://www.youtube.com/playlist?list=PLZlv_N0_O1gYwhBTjNLSFPRiBpwe5sTwc
Even if you're not interested in Unreal Engine in particular, it may give you a bit of the flavor of the tasks.
Note that this tutorial only goes through the "what you should do" once -- you then have to do that for each and every thing in your game (anything you throw, blow up, plant, ride, steer, control, in any way.)
But that's still a lot simpler than also having to build all the code that Unreal does for you under the covers.