h2H
THE FORMAL DEFINITIONS
The fundamenta l notion s o f a zero-su m game ,
mixed an d pur e strategies , an d expecte d payof f ar e denned .
It i s tim e t o mak e som e forma l definitions . Fo r purel y pedagogica l reason s w e
begin wit h game s i n whic h eac h o f the player s ha s onl y tw o options , leavin g th e
more genera l cas e for th e en d o f the section .
Taking ou r cu e fro m th e Penny-matchin g an d Bombing-sortie s game s o f th e
previous section , w e define a 2 x 2 zero-sum game a s a squar e arra y o f 2 x 2 = 4
numbers. Thus , th e mathematica l representatio n o f th e Penny-matchin g gam e
is the arra y
1
- 1
- 1
1
and th e mathematical representatio n o f the Bombing-sortie s gam e i s the arra y
80%
90%
100%
50%
Just a s i t i s convenient t o represen t th e additio n o f 8 oranges t o 6 oranges b y
the abstrac t equatio n 8 + 6 = 14, we shall ignor e th e actua l detail s o f the game s
in mos t o f th e subsequen t discussio n an d simpl y dea l wit h array s o f unitles s
numbers. Thus , th e mos t genera l 2 x 2 zero-su m gam e ha s th e for m
a
c
b
d
13
http://dx.doi.org/10.1090/mawrld/013/02
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