AMS 335/ECO 355 Game Theory
Fall 2010
Chapter 1 Games of Complete Information
Zhen Xu
Page
1
of
5
Static Game
: each player simultaneously chooses a strategy
1
Complete information
: each player’s payoff function is common knowledge among
all the players. Unlike the
perfect information
, players do not necessarily have the
knowledge of the actions inside the game.
1.1
Normal Form games and Nash Equilibrium
Definition
An n-player
normal form game
specifies the players’ strategy spaces
ሺS
ଵ
,ڮ,S
୬
ሻ
and their payoff functions
U
ଵ
,ڮ,U
୬
. We denote this game by
Gൌ
ሼS
ଵ
୬
; U
ଵ
୬
ሽ
.
The n-player normal-form game specifies:
1)
The players are numbered from 1,2,
…
,n and an arbitrary player
i
2)
The strategy set available to player
i
: S
i
. Let s
i
denote an arbitrary element of this
set, i.e.
s
୧
אS
୧
3)
The payoff received by player i is the result of each combination of strategies, i.e.
u
୧
ൌU
୧
ሺs
ଵ
,s
ଶ
,ڮ,s
୬
ሻ
Table 1 a two-player normal form game
Player 2
s
21
s
22
…
s
2q
Player
1
s
11
U
1
(s
11,
s
21
), U
2
(s
11,
s
21
)U
1
(s
11,
s
22
), U
2
(s
11,
s
22
)
…
U
1
(s
11,
s
2q
), U
2
(s
11,
s
2q
)
s
12
U
1
(s
12,
s
21
), U
2
(s
12,
s
21
1
(s
12,
s
22
), U
2
(s
12,
s
22
)
…
U
1
(s
12,
s
2q
), U
2
(s
12,
s
2q
)
s
1p
U
1
(s
1p,
s
21
), U
2
(s
1p,
s
21
1
(s
1p,
s
22
), U
2
(s
1p,
s
22
)
…
U
1
(s
1p,
s
2q
), U
2
(s
1p,
s
2q
)
S
1
=( s
11
, s
12
,
…
, s
1p
), S
2
=( s
21
, s
22
,
…
, s
1q
)
Take s
1i
∈
S
2
and s
2j
∈
S
2
, then u
1
=U
1
(s
1i
, s
2j
) and u
2
=U
2
(s
1i
, s
2j
).
For a two-player normal form game, the above table is common knowledge to the two
players in this game.
Definition
In the normal form game
G ൌ ሼS
ଵ
୬
ଵ
୬
ሽ
, let
s
୧
and
s
୧
ᇱ
be
feasible strategies for player
i
(i.e.
s
୧
୧
and
s
୧
ᇱ
୧
). Strategy
s
୧
ᇱ
is
strictly
dominated
by strategy
s
୧
if for each feasible combination of the other players’
strategies, the payoff of player i from playing
s
୧
ᇱ
is less than the payoff from playing
s
୧
:
U
୧
ሺs
ଵ
୧ିଵ
୧
ᇱ
୧ାଵ
୬
ሻ൏U
୧
ሺs
ଵ
୧ିଵ
୧
୧ାଵ
୬
ሻ
for each
ሺs
ଵ
୧ିଵ
୧ାଵ
୬
ሻ
that can be constructed from the other players’
strategy spaces
S
ଵ
୧ିଵ
,S
୧ାଵ
୬
The above equation can be simplified into
U
୧
ሺs
୧
ᇱ
ି୧
୧
ሺs
୧
ି୧
ሻ
The subscript
–i
is known as the players other than
i
1
Do not restrict to simultaneous move. The players know each others’ chosen strategies at the same time. They do