Nice, compete with the OM, totally the wrong thing to do.
Listen to the message that conveys: competing with another man for your wife.
Who has all the power in this situation? If you have to compete with another man for your wife, you definitely don't have any power in this situation.
Your wife has all the power and this will never work if you have to compete with another man for her. You communicate that you're ok with competing for another man for your wife, you shouldn't have to compete, you should be choice #1, the only choice, your wife has options, your wife has control.
You have no options in this situation, you will not win in this situation, competing will allow your wife to compare him to you, in ALL ways (read the hidden message there).
Better to let her go.
"Wife I've been thinking about things, I finally get what you're feeling and going through, the message finally clicked in my head, you're not attracted to me anymore and truthfully I'm not attracted to you anymore. So I've decided, you can have him (the OM), I don't want to be married to you anymore. I've decided I won't be with someone who doesn't value me or the relationship they have with me. It's just too much work going through this with you and it's just not worth it to me. You will be much happier with him. Of course we need to tell the children immediately, the sooner the better. I've also decided you will also need to find another place to live because I can't live with you anymore. While you're gone on your trip, I will start packing your things and placing them in the garage for you, when you return, you can make arrangements to live with a friend or some family until you have your own place because I won't be moving out. FYI, I've spoken to a lawyer and I will file for divorce and file for joint custody as well."
And that's it.
Let her go, completely.
It's the fastest way to get them back.
No begging, pursuing, pleading, competing with the OM (crappiest idea ever), etc. You let her go happily and make this your decision to let her go and drop her like a bad habit. No emailing, txting, letter writing, jumping through hoops to change for her (if you want to change to improve you for you, by all means go for it, but don't change to improve for her, jumping through hoops/changing for your wife just communicates that you're not valuable enough as is and you need to supplement the current "you" with changes to make "YOU" better for your wife - don't be that guy, the forums are filled with guys that made a million changes for their wives only to be rejected at every turn, it doesn't work, that's reality).
The only real "change" that will work, something that is unexpected, something that will show your wife that you got yourself together is to show her the door.
Letting them go is the quickest way to get them back.