Game Development Reference

In-Depth Information

enemy is unaware of (and thus being able to make moves that the enemy is

unaware of) can be critical; in financial markets, some investors may not be

aware of certain investment strategies (complicated hedging strategies, for

example, or tax-avoidance strategies).

To understand the impact of adding the possibility of unawareness to the

analysis of games, consider the game shown in Figure 8.1 (this example, and

all the discussion in this section, is taken from [Halpern and Rego, 2006]).

One Nash equilibrium of this game has
A
playing across
A
and
B
playing

down
B
. However, suppose that
A
is not aware that
B
can play down
B
.In

that case, if
A
is rational,
A
will play down
A
. Although
A
would play across
A

if
A
knew that
B
were going to play down
B
,
A
cannot even contemplate this

possibility, let alone know it. Therefore, Nash equilibrium does not seem to

be the appropriate solution concept here.

A

across
A

B

across
B

(0,2)

down
A

down
B

(1,1)

(2,3)

Figure 8.1 A simple game

To find an appropriate analogue of Nash equilibrium in games where players

may be unaware of some possible moves, we must first find an appropriate

representation for such games. The first step in doing so is to explicitly

represent what players are aware of at each node. We do this by using what

we call an
augmented game
.

Recall that an
extensive game
is described by a game tree. Each node

in the tree describes a partial history of the game - the sequence of moves

that led to that node. Associated with each node is the player that moves at

that node. Some nodes where a player
i
moves are grouped together into an

information set for player
i
. Intuitively, if player
i
is at some node in an

information set
I
, then
i
does not know which node of
I
describes the true

situation; thus, at all nodes in
I
,
i
must make the same move. An augmented

game is an extensive game with one more feature: associated with each node

in the game tree where player
i
moves is the
level of awareness
of player

i
- the set of histories that player
i
is aware of. (The formal definition of an

augmented game can be found in [Halpern and Rego, 2006].)

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