Mathematics Magazine 83, pp. 100-110.

**ISSN/ISBN:** 0025-570X
**DOI:** 10.4169/002557010X482862

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**Abstract:** The Multiplication Game is a two-person game in which each player chooses
a positive integer without knowledge of the other players number. The two numbers are
then multiplied together and the first digit of the product determines the winner. Rather
than analyzing this game directly, we consider a closely related game in which the players
choose positive real numbers between 1 and 10, multiply them together, and move the
decimal point, if necessary, so that the result is between 1 and 10. The mixed strategies are
probability distributions on this interval, and it is shown that for both players it is optimal to
choose their numbers from the Benford distribution. Furthermore, this strategy is optimal
for any winning set, and the probability of winning is the Benford measure of the players
winning set. Using these results we prove that the original game in which the players choose
integers has a well-defined value and that strategies exist that are arbitrarily close to optimal.
Finally, we consider generalizations of the game in which players choose elements from a
compact topological group and show that choosing them according to Haar measure is an
optimal strategy.

**Bibtex:**

```
@article {MR2649322,
AUTHOR = {Morrison, Kent E.},
TITLE = {The multiplication game},
JOURNAL = {Math. Mag.},
FJOURNAL = {Mathematics Magazine},
VOLUME = {83},
YEAR = {2010},
NUMBER = {2},
PAGES = {100--110},
ISSN = {0025-570X},
CODEN = {MAMGA8},
MRCLASS = {91A60},
MRNUMBER = {2649322 (2011b:91084)},
DOI = {10.4169/002557010X482862},
URL = {http://dx.doi.org/10.4169/002557010X482862},
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Probability Theory