If we go by math, not picking the coin flip is illogical but I would still take the cash. Funny how our brain works.
it is not necessarily illogical. the investment with the highest intrinsic value is not always the ideal investment. risk and uncertainly can be quantifiable, and in this hypothetical, are perfectly quantified. no intelligent investor is putting 100% of their assets into a single equity that they believe is most promising. and in fact, most investors even put money into low return assets (relative to equities) likes bonds and commodities to reduce risk, even with full knowledge that they should expect lower long term returns in doing so. and this is perfectly logical for someone who values a certain level of risk.
another way to look at it; if for example the change in your quality of life if given 1 million dollars goes up 1000%, but the change in your quality of life with a billion dollars only goes up 1800%, and the quality of your life is the primary measure you use in your decision making, then it is inehrently logical to select the 1 million dollars, as you would have a guarunteed 1000% increase vs. a average 900% increase with the higher risk option.
These arbitrary hypothetical values arent ridiculous either. there are many people out there who 1 million dollars will absolutely change their quality of life drastically. and simultaneously, a billion dollars would not necessarily make their quality of life drastically better than a million, as what is most important is stability and access to necessities.
Hell even speaking for myself, i dont think anything above ~9 million would make any appreciable difference in the quality of my life, probably much less than that even.
and in the previous hypothetical, these values are such that you dont even need to account for the value of risk, as the equity in terms of quality of life is already enough to select a choice. but in theory, lets say 1 million gives 1000% increase while 1 billion gives 4000% increase in QoL, making the coin flip twice as equitable. now is where a calculation of risk might make a difference, because if youre able to quantify your risk tolerance, and the cost of this massive risk is seen as being more than the value of 1000% increase in QoL, then you would still logically take the guaranteed mil.