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MOBK014-01 MOBK014-MacKenzie-v1.cls February 20, 2006 17:35
6 GAME THEORY FOR WIRELESS ENGINEERS
also a substantial body of literature on cooperative games, in which players form coalitions
and negotiate via an unspecified mechanism to allocate resources. Cooperative game theory
encompasses concepts such as the Nash bargaining solution and the Shapely value. Although
there are certainly profitable applications of both theories and deep connections between the
two, the amateur game theorist would be well advised to mix the two theories with extreme
caution. Mixing the two theories can easily lead to results that are incomprehensible, at best,
and utter nonsense, at worst.
Perhaps the mostcommon mistake among legitimate applications ofgame theory, though,
is a failure to clearly define the game and the setting in which it is played. Who are the players of
the game? What actions are available to the players? What are the players’ objectives? Does an
equilibrium exist for the given game? Is it unique? Is there a dynamic process by which players
update their strategies? If so, what is it and is it guaranteed to converge?
Finally, there are two general “philosophies” on applying game theory, and it is extremely
important to consider which philosophy you are using. In many cases, researchers who have not
clearly thought through their reasoning will jump back and forth between the two philosophies.
This is bound to create holes in their arguments, as the philosophies are mutually exclusive.
The first philosophy might be called a direct application of game theory. In this philoso-
phy, the users’ actual preferences are said to be modeled by the players’ utility functions. What
makes the application of this philosophy difficult is that it is very difficult to extract preferences
from users. Furthermore, nailing down a single explicit utility function that represents the
preferences of all users is not possible, since different users will have different preferences. It is
unlikely that even a parameterized utility function can represent the preferences of every user
except in very simple (perhaps contrived) situations. Even if one could postulate the existence
of such a function, it would take considerable effort to prove that such a formulation was valid
for the majority of users in the system.
Thus, if one wants to claim that the utility functions in a game represent actual users’
preferences, then one must operate with a bare minimum collection of assumptions about the
users’ preferences. Typical assumptions might include an assumption that users prefer higher
signal-to-interference-and-noise ratios (SINRs) to lower SINRs and that users prefer to spend
less power rather than more to obtain a given SINR. This approach can produce beautiful
results. It is, however, a very challenging task to execute well.
The second philosophy might be called an “engineering” application of game theory.
This philosophy assumes that the engineer is capable of programming the devices in the system
to behave as if they are maximizing a chosen utility function. Since the engineer chooses the
utility function, it can be whatever the engineer desires. The burden in this philosophy lies
in explaining why a particular utility function has been chosen and why game theory is being
employed at all. Though perhaps easier than demonstrating that a particular utility function
represents the preferences of all users, this may still be a difficult challenge to meet.