Game Development Reference
In-Depth Information
in strategic games: best response, Nash equilibrium, dominated strategies
and mixed strategies and to clarify the relation between these concepts.
In the first part we consider the case of games with complete information .
In the second part we discuss strategic games with incomplete information ,
by introducing first the basics of the theory of mechanism design that deals
with ways of preventing strategic behaviour , i.e., manipulations aiming at
maximising one's profit. We focus on the concepts, examples and results, and
leave simple proofs as exercises.
1.2 Basic concepts
Assume a set
of players, where n> 1. A strategic game (or
non-cooperative game ) for n players, written as ( S 1 ,...,S n ,p 1 ,...,p n ),
consists of
{
1 ,...,n
}
a non-empty (possibly infinite) set S i of strategies ,
a payoff function p i : S 1 ×
...
×
S n R
,
for each player i .
We study strategic games under the following basic assumptions:
players choose their strategies simultaneously ; subsequently each player
receives a payoff from the resulting joint strategy,
each player is rational , which means that his objective is to maximise his
payoff,
players have common knowledge of the game and of each others' ratio-
nality. 1
Here are three classic examples of strategic two-player games to which we
shall return in a moment. We represent such games in the form of a bimatrix,
the entries of which are the corresponding payoffs to the row and column
players.
Prisoner's Dilemma
CD
C
2 , 2
0 , 3
D
3 , 0
1 , 1
1 Intuitively, common knowledge of some fact means that everybody knows it, everybody knows
that everybody knows it, etc.

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