Review1 of Liber De Ludo Aleae (Book on Games of Chance) by ...

The values distributed upon suspension <strong>of</strong> the game, conform to the expected values.
The argument is an (early) example <strong>of</strong> the application <strong>of</strong> a “minimax” principle. Both players wish to
maximize the amount they would receive, and minimize their losses. The “motivation” is illustrated by
the following “pay<strong>of</strong>f matrix”, with the expected proceeds for player A indicated under the relevant
circumstances.
Player A
Rolls a favourable
number
Does not roll a
favourable number
Player A would like to maximize the row minimums, while player B wishes to minimize the column
maximums. Both would settle on 48 (for player A).
After discussing his theory relating to the equitable distribution <strong>of</strong> the wager amount, Pascal presents the
following rule :
“…the value (by which I mean only the value <strong>of</strong> the opponent’s money) <strong>of</strong> the last game <strong>of</strong> two is double
that <strong>of</strong> the last game <strong>of</strong> three and four times the last game <strong>of</strong> four and eight times the last game <strong>of</strong> five,
etc.”
Using the recursive procedure applied by Pascal in the case <strong>of</strong> a game <strong>of</strong> three, with a game <strong>of</strong> four, we
would proceed as follows:
State <strong>of</strong> Game
Agreement to
settle the
wager
Division if player A's
number turns up next
10
Player B
No agreement
to settle the
wager
Division if player B's
number turns up next
Row
Minimum
48 64 48
48 32 32
Column Maximum 48 64
Division if players
agree to suspend the
game
(3,2) {64,0} {32,32} {48,16}
(3,1) {64,0} {48,16}* {56,8}
*this corresponds to
the state (3,2) shown
above
(3,0) {64,0} {56,8}**
**this corresponds to
the state (3,1) shown
above
{60,4}