Game Development Reference

In-Depth Information

In Apt et al. [2008] only strict preferences were considered and so defined

finite games with parametrised preferences were compared with the concept

of
CP-nets
(Conditional Preference nets), a formalism used for representing

conditional and qualitative preferences, see, e.g., Boutilier et al. [2004].

Next, in Roux et al. [2008]
conversion/preference games
are intro-

duced. Such a game for
n
players consists of a set
S
of
situations
and for

each player
i
a
preference relation

i
on
S
and a
conversion relation

→
i
on
S
. The definition is very general and no conditions are placed on the

preference and conversion relations. These games are used to formalise gene

regulation networks and some aspects of security.

Finally, let us mention another generalisation of strategic games, called

graphical games
, introduced by Kearns et al. [2001]. These games stress

the locality in taking a decision. In a graphical game the payoff of each player

depends only on the strategies of its neighbours in a given in advance graph

structure over the set of players. Formally, such a game for
n
players with the

corresponding strategy sets
S
1
,...,S
n
is defined by assuming a neighbour

function
N
that given a player
i
yields its set of neighbours
N
(
i
). The payoff

for player
i
is then a function
p
i
from
×
j∈N
(
i
)
∪{i}
S
j
to R.

In all mentioned variants it is straightforward to define the notion of a

Nash equilibrium. For example, in the conversion/preferences games it is

defined as a situation
s
such that for all players
i
,if
s →
i
s
, then
s
i
s
.

However, other introduced notions can be defined only for some variants.

In particular, Pareto e
ciency cannot be defined for strategic games with

parametrised preferences since it requires a comparison of two arbitrary

joint strategies. In turn, the notions of dominance cannot be defined for the

conversion/preferences games, since they require the concept of a strategy

for a player.

Various results concerning finite strategic games, for instance the IESDS

Theorem 1.3, carry over directly to the strategic games as defined in Osborne

and Rubinstein [1994] or in Apt et al. [2008]. On the other hand, in the

variants of strategic games that rely on the notion of a preference we cannot

consider mixed strategies, since the outcomes of playing different strategies

by a player cannot be aggregated.

1.7 Mechanism design

Mechanism design is one of the important areas of economics. The 2007 Nobel

Prize in Economics went to three economists who laid its foundations. To

quote from
The Economist
[2007], mechanism design deals with the problem

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