といてやる is phonetically changed ておいてやる, i.e. ておく＋やる.
Because やる is rough/rude form for あげる, といてやる is synonym for ておいてあげる which means subject gives action of ～ておく to object person for his favor/sake.
～ておく form has two meanings. 1. do something in advance for preparation 2. keeps the situation as it is, status quo, leaves as it is
In the questioned sentence of 負けておく, meaning 2. is used. So speaker is telling that I will give you this situation of my lose for your sake. Quite possibly speaker seems to calm his pride down by saying it as a sour grape remedy.
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