Game Development Reference
In-Depth Information
We use the player's awareness level as a way of keeping track of how
the player's awareness changes over time. For example, perhaps A playing
across A will result in B becoming aware of the possibility of playing down B .
In financial settings, one effect of players using certain investment strategies
is that other players become aware of the possibility of using that strategy.
Strategic thinking in such games must take this possibility into account. We
would model this possibility by having some probability of B 's awareness
level changing.
For example, suppose that in the game shown in Figure 8.1
players A and B are aware of all histories of the game;
player A is uncertain as to whether player B is aware of run across A ,
down B
and believes that B is unaware of it with probability p ; and
the type of player B that is aware of the run across A ,down B is aware
that player A is aware of all histories, and B knows that A is uncertain
about his ( B 's) awareness level and knows the probability p .
Because A and B are actually aware of all histories of the underlying game,
from the point of view of the modeller, the augmented game is essentially
identical to the game described in Figure 8.1, with the awareness level of both
players A and B consisting of all histories of the underlying game. However,
when A moves at the node labelled A in the modeller's game, she believes
that the actual augmented game is Γ A , as described in Figure 8.2. In Γ A ,
nature's initial move captures A 's uncertainty about B 's awareness level. At
the information set labelled A. 1, A is aware of all the runs of the underlying
game. Moreover, at this information set, A believes that the true game is Γ A .
At the node labelled B. 1, B is aware of all the runs of the underlying
game and believes that the true game is the modeller's game; but at the
node labelled B. 2, B is not aware that he can play down B , and so believes
that the true game is the augmented game Γ B described in Figure 8.3. At
the nodes labelled A. 3 and B. 3 in the game Γ B , neither A nor B is aware of
the move down B . Moreover, both players think the true game is Γ B .
As this example should make clear, to model a game with possibly unaware
players, we need to consider, not just one augmented game, but a collection
of them. Moreover, we need to describe, at each history in an augmented
game, which augmented game the player playing at that history believes is
the actual augmented game being played.
To capture these intuitions, starting with an underlying extensive-form
game Γ, we define a game with awareness based on Γ to be a tuple
Γ =(
, Γ m ,
), where
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