Optimal strategy for iterated real life coin tosses?

Let’s say someone asks you to make iterated predictions for the result of a series of real life coin toss. The “real life” part has two important implications: First, that the coin might not be perfectly fair (that is, it might be biased towards one side) and two, that it is not magical, and thus does not have magical properties, such as being able to remember what your previous predictions were.

I’d actually like to place one more restriction, which is that the person flipping the coin isn’t antagonistic against you. That is to say, even if the person flipping the coin figures out what your strategy is, he will not then subsequently intentionally try to influence the coin to increase your odds of losing. If this is too informally specified, we can switch from coin tosses to a scratch-card with two squares to scratch, one winning and one losing, all of which are printed in advance.

One good strategy that I’ve come up with is to pick randomly for the first prediction, and then for each subsequent prediction, to pick whatever won last time. This strategy may sound like Tit for Tat, but I believe the resemblance is purely cosmetic. In particular, many of the preconditions that make Tit for Tat effective (e.g. an unbalanced payoff between “cooperate” and “defect”) are simply not present in the coin-toss game.

One reason this strategy is more effective than the purely random strategy is that your guesses will follow the distribution of the coin. If the coin is perfectly fair, then my pseudo-Tit-for-Tat strategy (pTfT) reduces to the purely random strategy. However, if the coin is biased for, say, 80% head and 20% tail, then pTfT will also guess head 80% of the time and tail 20% of the time.

pTfT is not optimal, however, because for a coin which lands 80% head and 20% tail, the optimal betting strategy is to bet on head 100% of the time. See the following table:

  Pure Random Pseudo Tit-for-Tat Always Head
Coin Lands Head: 80% You bet head 40% 64% 80%
You bet tails 40% 16% 0%
Coin Lands Tail: 20% You bet head 10% 16% 20%
You bet tails 10% 4% 0%
Success Rate 50% 68% 80%

So a “better” strategy — better in the sense that it gives a better success rate — is to check the entire history of the coin thus far, and always bet on whichever side had the higher probability thus far. One reason you might prefer pTfT in real-life is that it might be too much effort to actually remember the entire history of the coin thus far, and compute the probabilities. So even if pTfT yields a lower success rate, it’s much easier to implement. And given that humans are notoriously bad at being random, pTfT is probably even easier to implement than pure random, and yet yields better success rates.

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