I think the HN relevant use case would be looking at this from the opposite side. You've designed 2 games (or algorithms) and both result in a winning state. But when they are used alternately, they lead to worse outcomes. A made up, possibly bad example that's using similar 'rules':

Start with: 1 large fixed size data structure and 1 cache Algorithm A: Checks as it is iterating whether the cache is full. If not, it generates the cache data (SLOW) but can then iterate over the entire data structure quickly. First time it does this is a loss, but 2nd pass through the data structure leads to an overall win. Algorithm B: Prefers an empty cache. If cache is full it will delete it. It can iterate over the entire data structure fairly quickly. Each time is considered a win.

Now you have a program with many different features and everyone knows that it doesn't really matter if you use Algorithm A or B because they are both programmed to work safely together and if you test a feature using one of the algorithms it will be fast either way, so it's left up to personal preference. The fun begins when the full program starts alternating from algorithm A to B.

It's a mathematician discovering why we have integration tests.

Or, from a much more charitable angle:

> It's a physicist¹ proving that we need integration tests.

    ¹ Juan Parrondo is a physicist by training, not a mathematician.

Yeah, this seems pointless. Here's an example of the paradox:

1) Hitting both nails and screws with a hammer is a losing game.

2) Screwing both nails and screws with a screwdriver is a losing game.

Paradox alert! If you hammer the nails and screw the screws you've transformed two losing games into a winning game!

Or:

1) Wearing shorts and sandals all year round is a losing game (in this part of the world); you will freeze to death in the winter.

2) Wearing your warmest clothing all year round is a losing game; you will overheat in the summer.

This is a paradox, because the population of Canada should be zero. But wait! What if you just dress appropriately for the season?

And know imagine you didn't know you can switch the tools.

I think a closer-to-real example is the problem of playing a collection of blackjack tables. All the games are the same, but sometimes the state of the decks (ie, which cards have been discarded) will lead to better odds of winning. If you know the state of the decks at each table, you can always choose to play the table with the best odds of winning. This type of strategy has been used to win piles of money in Vegas, FWIW, though it leads to ejection if you're caught.

This is similar to the 'ratchet' examples in the wikipedia page - you play the game with the best odds, and use one game to 'cool off' until you're in the right state to win the second game again. The games in the wikipedia article are kinda unsatisfying, though - there's too much dependence on player state.

Market timing in the stock market (if you have a little insider info), and card counting in lotteries, are historical examples.

Sure but I want to imagine that some people are only playing Game A and others are only playing Game B, unaware of the relation between them that creates positive outcomes by sometimes losing in one game or the other.

Indeed, there's no way this is the product of actual game theory because it is using a pathologically incompatible definition of "game" where the player is allowed to change the rules.

Probability - and how you need to reason about independent vs dependent events - is poorly understood generally.

Let's try to make it HN-relevant by asking:

can two things that both AB tested positively in independent one-at-a-time testing possibly backfire if you launch both of them?

Only if you realize that the game could be looked at as Game + Game B

Agree completely