More formally, the probability of winning ${\displaystyle n}$ dollars is ${\displaystyle {\tfrac {n}{2n}}}$ and so the overall expected value of playing the game (assuming that the house has unlimited resources and will allow the game to continue until a flip comes up tails) is given by:

 ${\displaystyle \sum _{1}^{\infty }{\tfrac {1}{2}}}$ 5(d)

If the amount of money that our gambler should be willing to pay to play a game is constrained only by the demand that it be less than the expected return from the game, then this suggests that she should pay any finite amount of money for a chance to play the game just once. That seems very strange. While there are a number of solutions to this problem, the one of most immediate interest to us was proposed in Bernoulli (1738).[1] Bernoulli suggested that we ought to think of utility gained from the receipt of a quantity of some good (in this case money) as being inversely proportional to the quantity of that same good already possessed. He justifies this by pointing out that

The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount[2]

Bernoulli’s original suggestion of this fairly straightforward (albeit still non-linear) relationship between wealth and utility has been refined and expanded by a number of thinkers.[3] The

1. Translation by Sommer (1954).
2. Op. cit., pp. 158-159
3. The principle of diminishing marginal utility was developed by a number of economists over the course of several decades, and continues to be refined to this day. See, for example, Menger (1950), Bohm-Bawerk (1955), and McCulloch (1977). While the originators of this principle (particularly Menger and Bohm-Bawerk) were associated with the Austrian school of economics, diminishing marginal utility has found its way into more mainstream neoclassical economic theories (Kahneman and Deaton, 2010).

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