Game Development Reference
In-Depth Information
Although, economists usually interpret awareness as 'being able to con-
ceive about an event or a proposition', there are other possible meanings
for this concept. For example, awareness may also be interpreted as 'un-
derstanding the primitive concepts in an event or proposition', or as 'being
able to determine if an event occurred or not', or as 'being able to compute
the consequences of some fact' [Fagin and Halpern, 1988]. If we interpret
'lack of awareness' as 'unable to compute' (note that this interpretation is
closely related to the discussion of the previous section!), then awareness of
unawareness becomes even more significant. Consider a chess game. Although
all players understand in principle all the moves that can be made, they are
certainly not aware of all consequences of all moves. A more accurate repre-
sentation of chess would model this computational unawareness explicitly.
We provide such a representation.
Roughly speaking, we capture the fact that player i is aware that, at a
node h in the game tree, there is a move that j can make she ( i ) is not aware
by having i 's subjective representation of the game include a 'virtual' move
for j at node h . Since i might have only an incomplete understanding of what
can happen after this move, i simply describes what she believes will be the
game after the virtual move, to the extent that she can. In particular, if she
has no idea what will happen after the virtual move, then she can describe
her beliefs regarding the payoffs of the game. Thus, our representation can
be viewed as a generalisation of how chess programs analyse chess games.
They explore the game tree up to a certain point, and then evaluate the
board position at that point. We can think of the payoffs following a virtual
move by j in i 's subjective representation of a chess game as describing the
evaluation of the board from i 's point of view. This seems like a much more
reasonable representation of the game than the standard complete game tree!
All the definitions of games with awareness can be generalised to accom-
modate awareness of unawareness. In particular, we can define a generalised
Nash equilibrium as before, and once again show that every game with
awareness (now including awareness of unawareness) has a generalised Nash
equilibrium [Halpern and Rego, 2006].
There has been a great deal of work recently on modelling unawareness
in games. The first papers on the topic were by Feinberg [2004, 2005]. My
work with Rego [2006] was the first to consider awareness in extensive games,
modelling how awareness changed over time. There has been a recent flurry on
the topic in the economics literature; see, for example, [Heifetz et al., 2006b,
Li, 2006a,b, Ozbay, 2007]. Closely related is work on logics that include
awareness. This work started in the computer science literature [Fagin and
Halpern, 1988], but more recently, the bulk of the work has appeared in the
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