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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