Your way of looking at it is perfectly sensible. But a sensible person might also say that a uniform prior over strategies is appropriate in the absence of any information about your opponent. Choice of priors is always going to be partly subjective, and reasonable people can make different choices.
I would point out that in a real world scenario, a uniform prior is barely any less realistic than the a prior derived from the assumption of mutual knowledge of perfect rationality. Is your poker opponent equally likely to be doing their utmost to lose as to win? No. But on the other hand, they're not likely to be tracking 100% of available information and making the mathematically correct choice all the time.
In general, the question of what you should assume about an unknown opponent's strategy is a tricky philosophical question, and reasonable people can gravitate to different points of view on this issue. I think that might be why so few people give the answer you want. I was just suggesting that explicitly mentioning mutual knowledge of perfect rationality might help.
Do you think it's equally likely that a random opponent will redraw below 0.01 as it is that they will redraw below 0.5 or below phi?
Most people do not give the right answer not because of philosophical issues around the Nash equilibrium but because they have a hard time understanding that looking to maximize the expected value is not optimal.
>Do you think it's equally likely that a random opponent will redraw below 0.01 as it is that they will redraw below 0.5 or below phi?
If random really means random, then that's presumably true by definition. If we're talking about an opponent randomly selected from a representative group of people, then no, but equally, I wouldn't assume that they're going to pick the golden ratio as their threshold either.
The thing is that maximizing the expected value is optimal in the absence of any information about the opponent's strategy. But sure, I can easily believe that many people would get the question wrong anyway.
> If random really means random, then that's presumably true by definition.
No it's not. Randomness is always with respect to a given distribution. There is nothing special about the uniform distribution over a threshold. The space of strategies also include strategies where your opponent redraw only when they hit a number between 0.1 and 0.2, or where they redraw if the second decimal digit of the first draw is even.
> The thing is that maximizing the expected value is optimal in the absence of any information about the opponent's strategy.
Your statement is meaningless without a prior on the strategy. Under which prior is maximizing the expected value optimal? If you assume the other party randomly picks a threshold between 0 and 1, then the optimal answer is (sqrt(39)-3)/6 not 0.5. Not that this prior makes any sense. It's far more sensible to assume a strategic opponent.
Rethrowing below 0.5 is typically the optimal strategy when the opponent has no strategy at all, i.e. always or never rethrows.
If a strategy is any function from the preceding game state to keep/redraw, then I don't have the mathematical chops to specify a uniform prior over all such strategies and compute the optimal strategy that way. (In fact it may not necessarily be possible to do this in a sensible way, as Bertrand's paradox suggests, as another poster pointed out.)
However, there is a much simpler argument suggesting that maximizing EV would be the right approach in the absence of any knowledge about the opponent's strategy. Say the opponent chooses to redraw at random. Then in effect they are drawing once. If we redraw when the value is < a, then we just want to maximize the probability of winning, (1-a)(a + 0.5) + 0.5a, giving a = 0.5.
Modeling a process that we know nothing about as a random process seems reasonable. And the reasoning above does not actually involve computing EVs, though it happens that the right answer is the answer that maximizes the EV.
To justify this a bit further, we can reason as follows. If we know nothing about the opponent's strategy, then their initial draw tells us nothing about whether or not they will choose to redraw. So our estimate of their probability of redrawing, p, must be independent of their initial draw. Then the probability of winning is 0.5ap + (1-a)(1-p)(a + 0.5) + (1-a)p(a + 0.5) + 0.5a(1-p), which is invariant with respect to p, and which is at its maximum when a = 0.5.
>It's far more sensible to assume a strategic opponent.
Well, this is going round in circles, but I would say that this is sensible only if you know something about the opponent! I don't see how it's sensible to assume anything about your opponent if you know nothing about them. And if we're talking about the "real world", then of course you do know something, but you certainly don't know that they're perfectly rational.
I would point out that in a real world scenario, a uniform prior is barely any less realistic than the a prior derived from the assumption of mutual knowledge of perfect rationality. Is your poker opponent equally likely to be doing their utmost to lose as to win? No. But on the other hand, they're not likely to be tracking 100% of available information and making the mathematically correct choice all the time.
In general, the question of what you should assume about an unknown opponent's strategy is a tricky philosophical question, and reasonable people can gravitate to different points of view on this issue. I think that might be why so few people give the answer you want. I was just suggesting that explicitly mentioning mutual knowledge of perfect rationality might help.