The discrepancy between intuition and mathematics stems from several unrealistic features in the mathematical description of the game. Given the formal statement of the game, the player wishing only to maximise money should indeed pay any price at all to play. This conclusion is difficult to accept primarily because of bias towards risk-aversion, and the widespread, correct judgement that money in reality does not possess a linear utility. In fact, money usually possesses diminishing utility, and is sometimes modelled as proportional to the base-10 logarithm of wealth rather than to wealth itself. Applying this principle, the amount of utility derived from the game can be calculated as that given by precisely $4. However, the paradox reemerges for a variant of the game which has infinite utility: the pot starts at $100 and is multiplied by 10 to the power of 2 to the power of the number of heads for each additional heads. A like intuition might possibly suggest that this second game cannot, as a matter of certainty, be worth more than $10 septillion, and is virtually certain to be worth closer to $100,000; in any case, here too intuition does not consider it to be of infinite value.

The reason for this further discrepancy is the widespread and, again, correct judgement that there exists some huge amount of wealth after which absolutely no more utility is possible from gaining money. Suppose, for example, this amount is $75 trillion, approximately world GDP. For the original game, this brings the monetary value down to $47.06 floored, and the utility-adjusted value down to $3.99 floored. For the variant game, this brings the monetary value down to $9,375,012,502,550 (about $9.375 trillion), and the utility-adjusted value down to $54,247.86 floored. Clearly, the utility-and-maximum-wealth-adjusted values can provide reasonable answers given real human preferences.

I could be wrong because I am just working with borrowed knowledge, and I do not like the conclusion, but for the case of starting with n I am getting a regularised expectation of:

]]>The divergent expectation is making me go crazy

]]><will write back when free>

]]>Perhaps that is the expectation, and the banker profits after all xD

]]>What happens if you start with $n instead of $2?

]]>I have heard of a solution where dollar value is replaced by utility, or risk-aversion, and so the higher rewards that are much less likely become exponentially less valuable and the expectation is a bit over $4. Can you please explain what is involved in this resolution?

]]>But doesn't that conclusion bother you? Would you give up your house to be the player in this game, where your chance of earning your house's value or better is worse than 250,000 to 1?

]]>Let's call the bankroller A and the player B.

I agree, the expected win for B is infinite. A doesn't ever get a chance to 'win' so eventually B must win.

Conclusion. Don't be A.

Bob

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