Some games have no stable outcome in fixed choices at all. Chase the reasoning around and it never settles, because the moment your move becomes predictable, your opponent can exploit it.
A game with no resting point
Two players each hide a coin and reveal at once. If the faces match, the first player takes both; if they differ, the second does. Whatever one wins the other loses.
Look for an equilibrium in fixed choices and there is none. If you always play heads, your opponent plays tails and beats you every time. If you switch to tails, they switch too. Every combination has somebody who wants to move, so nothing is stable.
Randomising on purpose
The resolution is to stop choosing and start randomising. A mixed strategy assigns a probability to each option, and the player commits to the odds rather than the move. Here, both players choosing heads with probability is a Nash equilibrium: neither can improve by shifting the odds, because against a genuinely even opponent every mix pays the same.
The reason unpredictability works is that it denies information. Any pattern in your play is something an opponent can read and beat. Randomness is not indecision here; it is the only defensible policy.
Finding the odds
The equilibrium mix has a curious property: you pick the odds that make your opponent indifferent between their options. If any option paid them better, they would take it always, and you would be exploitable.
This is why a penalty taker who is stronger to their left does not therefore shoot left more. The odds have to leave the keeper with no better guess. It is also why the right mix depends on the payoffs rather than on ability alone, and why real competitors, from tennis servers to poker players to wartime patrol planners, have been shown to randomise at rates close to the theoretical ones.
Being genuinely random is hard
People asked to produce a random sequence make it too alternating: they avoid long runs because runs feel non-random, and that avoidance is itself a readable pattern. Serious competitors therefore outsource it, keying their choice to the second hand of a watch or a card cut, rather than trusting themselves to be unpredictable.
Zero-sum games like this one are the natural home of mixed strategies, but they are also the exception. Most real situations are not purely opposed, and the players meet again tomorrow.