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What is the law of diminishing marginal utility?

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In its most general form, the "law" of diminishing marginal utility states that, in the absence of "tipping points", as increasing amounts of a good or of a service are consumed, past some point of consumption the utility (usefulness) of successive increases drops. This is follows from an assumption that economic actors are rational, and therefore put each available amount to the best possible use, so that (on the assumption that there is no tipping point) the next available amount must then go to a less important use.

In mainstream economics, it is often assumed that utility can be quantified. In that case, the marginal utility would be an actual arithmetic difference. To get the marginal utility of the nth unit of a good or service, one could subtract the total utility without that unit from the total utility with that unit.

In that case, the "law" of diminishing marginal utility would imply that these arithmetic differences became ever smaller quantities. Here's a purely hypothetical example:

Quantity of Good . Total Utility . Marginal Utility
... ... ... ... 0 ... ... ... ... ... ... ... 0 ... ... ... ... ... ... undefined
... ... ... ... 1 ... ... ... ... ... ... . 10 ... ... ... ... ... ... 10
... ... ... ... 2 ... ... ... ... ... ... . 19... ... ... ... ... ... ... 9
... ... ... ... 3 ... ... ... ... ... ... . 27... ... ... ... ... ... ... 8
... ... ... ... 4 ... ... ... ... ... ... . 34... ... ... ... ... ... ... 7
... ... ... ... 5 ... ... ... ... ... ... . 40... ... ... ... ... ... ... 6
... ... ... ... 6 ... ... ... ... ... ... . 45... ... ... ... ... ... ... 5

If marginal utility diminishes asymptotically, as in this hypothetical sequence

MU = 16 , 8 , 4 , 2 , 1 , .5 , .25 , .125 , ...

then total utility may always grow (though its growth may become very slow). On the other hand, when there can be "too much of a good thing", so that a good becomes a bad when there's too much of it, then marginal utility can become negative, and total utility can actually shrink with additional units. Continuing the first hypothetical example:

Quantity of Good . Total Utility . Marginal Utility
... ... ... ... 5 ... ... ... ... ... ... ... 40... ... ... ... ... ... ... 6
... ... ... ... 6 ... ... ... ... ... ... ... 45... ... ... ... ... ... ... 5
... ... ... ... 7 ... ... ... ... ... ... ... 49... ... ... ... ... ... ... 4
... ... ... ... 8 ... ... ... ... ... ... ... 52... ... ... ... ... ... ... 3
... ... ... ... 9 ... ... ... ... ... ... ... 54... ... ... ... ... ... ... 2
... ... ... . 10 ... ... ... ... ... ... ... 55... ... ... ... ... ... ... 1
... ... ... . 11 ... ... ... ... ... ... ... 55... ... ... ... ... ... ... 0
... ... ... . 12 ... ... ... ... ... ... ... 54... ... ... ... ... ... -1
... ... ... . 13 ... ... ... ... ... ... ... 52... ... ... ... ... .... -2
... ... ... . 14 ... ... ... ... ... ... ... 49... ... ... ... ... ... -3
Law states that for any good or service, the marginal utility of that good or service decreases as the quantity of the good increases, ceteris paribus. In other words, total utility increases more and more slowly as the quantity consumed increases.
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