I have found a way to get the multi-player working again, but this may not work for all people. So I will describe the symptoms that my game had and if you are having the same then this should work;
1 Single player is working ok
2 Multi-player does not work;
Application starts, shows the splash screen then the screen goes black and then nothing........... application hangs and you have to press ctrl, alt and del to start task manager and end iw4mp.exe that is no longer responding.
3 you start the application again and get the option for safe mode and go back up to 2.
4 you remove game content through steam and re-download or restore from backup and multi-player still does not work.
if this is sounding familiar then this is what I did to get the game working, which is possible the long way but here goes;
In steam, select "Backup game files" for Call of Duty: Modern Warfare 2 and multi-player, (select both titles in the backup list). Then select Delete local content for both single player and multi player.
Next, locate your steam common folder, eg; D:\Steam\steamapps\common\ there should be a folder called"call of duty modern warfare 2"
I deleted this folder, you may wish to move this folder to another location like your desktop.
Then restore modern warfare from the backup you made earlier and allow it to update and resync
lastly, start multi-player as normal ....... you may get a message "completing installation" .......... just retry and it should start up allowing you to set key bindings etc...
I am guessing that simply renaming the dir "players" to something else could also work and may be quicker but as my game is working ok i am unable to verify, however the single player profiles are also in the players dir so its worth keeping it in a safe place.