Game Development Reference

In-Depth Information

the game that
i
believes to be the true game when the history is
h
, and

I
consists of the set of histories in Γ
h
he currently considers possible. For

example, in the examples described in Figures 8.2 and 8.3, taking Γ
m
to be the

augmented game in Figure 8.1, we have

(Γ
m
,

)=(Γ
A
,I
), where
I
is the

F

(Γ
A
,

information set labelled
A.
1 in Figure 8.2, and

F

unaware,across
A

)=

(Γ
B
,

). There are a number of consistency conditions that have

to be satisfied by the function

{

across
A
}

F

; the details can be found in [Halpern and

Rego, 2006].

The standard notion of Nash equilibrium consists of a profile of strategies,

one for each player. Our generalisation consists of a profile of strategies, one

for each pair (
i,
Γ
), where Γ
is a game that agent
i
considers to be the true

game in some situation. Intuitively, the strategy for a player
i
at Γ
is the

strategy
i
would play in situations where
i
believes that the true game is Γ
.

To understand why we may need to consider different strategies consider,

for example, the game of Figure 8.1.
B
would play differently depending on

whether or not he was aware of down
B
. Roughly speaking, a profile
σ
of

strategies, one for each pair (
i,
Γ
), is a
generalised Nash equilibrium
if

σ
i,
Γ
is a best response for player
i
if the true game is Γ
, given the strategies

σ
j,
Γ
being used by the other players in Γ
. As shown by [Halpern and Rego,

2006], every game with awareness has a generalised Nash equilibrium.

A standard extensive-form game Γ can be viewed as a special case of a

game with awareness, by taking Γ
m
=Γ,

(Γ
m
,h
)=(Γ
m
,I
),

where
I
is the information set that contains
h
. Intuitively, Γ corresponds to

the game of awareness where it is common knowledge that Γ is being played.

We call this the
canonical representation
of Γ as a game with awareness.

It is not hard to show that a strategy profile
σ
is a Nash equilibrium of Γ iff

it is a generalised Nash equilibrium of the canonical representation of Γ as a

game with awareness. Thus, generalised Nash equilibrium can be viewed as

a generalisation of standard Nash equilibrium.

Up to now, I have considered only games where players are not aware of

their lack of awareness. But in some games, a player might be aware that

there are moves that another player (or even she herself) might be able

to make, although she is not aware of what they are. Such awareness of

unawareness can be quite relevant in practice. For example, in a war setting,

even if one side cannot conceive of a new technology available to the enemy,

they might believe that there is some move available to the enemy without

understanding what that particular move is. This, in turn, may encourage

peace overtures. To take another example, an agent might delay making a

decision because she considers it possible that she might learn about more

possible moves, even if she is not aware of what these moves are.

Γ
m

G

=

{

}

, and

F

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