The axiom of determinacy, which implies some of (or all?) of the statements in op, and is more or less stated at the end. AD implies ~AC but they're not equivalent.

If I open up a pack of biscuits, and we each take turns eating a biscuit, AD says that there's a dominant strategy that can ensure that I eat the last biscuit. (e.g. there's only 1 biscuit; I win, or there's an odd number of biscuits; I win)

i.e. AD says you can rig games like this from the start