What not to do...

One of the most common mental exercises the cheated upon may experience after the discovery of an affair is the expectation that they will try harder to win back the cheater. This expectation either comes from the cheater directly -- "I cheated because you're a lousy [partner, housekeeper, lover]" -- and how are you going to up your game to keep me? Or it comes from the codependency of the betrayed spouse -- "What did I do to make them cheat? And how can I be a better partner to make them stay?" Often both dynamics are at play and feed into each other. The cheater, of course, is quite happy to pin this crap on you.
When terrible things happen, it's very natural to want to feel a sense of control. To think, oh if I'd only done X, Y would not have resulted. If you are at fault, the reasoning goes, then you could fix this. (Codependents love to fix things.) So you will take this crappy situation and think you can control the outcome by just trying harder.
This is a bad idea for several reasons. First, you aren't at fault for another's cheating. That's on them. As they say in therapy about people behaving self destructively, there's a Three C Rule -- "you didn't cause it, you cannot cure it, and you cannot control it."
Second, if you see the affair as a competition that you must try harder to "win," the marriage becomes a bidding war between the betrayed spouse and the affair partner. The best response is to fold, because the game is rigged. They want to sit impassively while you do the humiliating dance of "pick me!" This makes them feel powerful, central, special.
Cheating comes from a sense of entitlement. All you do when you compete for your marriage is solidify that entitlement -- that it is your job to ensure the happiness of the cheater, and hey, you missed a spot. Healthy relationships are based upon reciprocity and mutual respect. Infidelity is a toxically lopsided situation. Cheaters want the scales tipped in their favor (more attention, more ego stroking, more sex, more materialism) at your expense. They just don't want to try that hard, and they're gonna sulk if you make them.
What does the humiliating dance look like?

• Mounting a defense of the marriage and trying too hard sell your cheater on What You Have Together.

• Eating the sh*t sandwich. Not bringing up the affair. Stuffing your emotions so as not to upset the cheater with your distress.

• Believing that the cheater's need for "happiness" is paramount to the commitment they made to you. If they want to break that commitment, fine, there are honest ways to do that, beginning with a divorce lawyer. If they want to work on happiness, there is therapy, God, and working at pet shelters. But they cannot have all the benefits of marriage and a side emotional or physical relationship because they aren't "happy."

• Let's make a deal! This the bargaining stage of grief, that as long as you try harder to make the cheater happy, they won't betray you. Their happiness, however, is an ever moving target. If you do not want to be in an open marriage, don't be. It's one thing to be presented with that from the start. It's quite another thing for a cheater to renegotiate the terms after an affair is discovered. Be true to yourself and what you need.

• Super spouse! Are you having hysterical bonding sex, going to the gym, and dressing spiffier? If you're trying to be a better you to "win," you're just rewarding them. Be a better you for you. Your next partner will appreciate it a lot more than they will.

Finally, don't beg. Don't grab their ankles as they walk out the door. Don't drape yourself over furniture weeping. Let them go. Don't ever do the humiliating dance of "pick me." You're better than that.