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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