### s 1 s 2 > s 1 s 1 > s 2 s 2 > s 2 s 1

```Rational Choice
Sociology
Lecture 5
Game theory I: the concept and
classification of games
The Concept of Game I
Game theory is the branch of RCT that analyzes
the problem of rational choice in strategic
situation; or: theory of strategically rational
action
 Strategic situation is the situation of choice
involving at least two rational actors making
interdependence
 Interdependence means that outcomes of
alternative choices of each actor depend on
what other actors choose
 Game is analytical model of a strategic situation

The Concept of Game II
Actor2
(Columns)
Action 1
Action 2
C1
r11
r21
C2
r12
r22
Actor 1
(Rows)
Action
1
Action
2
Action
1
r11
r12
Action
2
r21
r22
The Concept of Game III

The parametric situations can be considered as involving the
game against nature (including people behaving like nature)

Strategic actions can be considered as the subclass of the social
actions in the sense of Max Weber. Social action: action oriented
to the actions of other actors. Strategic action is instrumentally
rational social action in the strategic situation
The social actions of at least two actors constitute a game (or
can be described as game) if following conditions are satisfied:
Each actor is instrumentally rational (maximizes utility or
expected utility): this means that her preferences are consistent
(satisfy axioms discussed above)
Each actor knows (1): assumption of symmetry
Each actor knows that other actors know that she knows (2): =
assumption of the common knowledge of rationality
From the same information each actor makes the same
conclusions (no misperceptions or misunderstandings): =
assumption of common alignment of beliefs

(1)
(2)
(3)
(4)
The Concept of Game IV
The description of a game includes the specification of:
 Actor set N {1,2,.., n}, including at least two players
 Strategy set S {1,2, …, m}, including at least two strategies
 Outcome set O. The size of this set depends on how many
players there are and how many strategy choices each of
them has.
E.g. if there are two players and each of them has 2 strategies,
there are 2×2= 4 outcomes; if there are two players, one of
them has 2, another 3 strategies, there are 2×3=6 outcomes;
if there are 3 players, one of them has 2, another 4, and the
third one 6 strategies, there are 2×4×6=48 outcomes etc.
Names of the outcomes can be built using as elements the
names of strategies that cause them

Utility functions U that for each player and for each outcome
ascribe the utility index u (ordinal or cardinal)
The Concept of Game V
Actor 2
s1
Actor 1
s1
s2
3
4
s2
s1s1 3
1 s1s2
4
s2s1 1
2 s2s2
2
Actor 1: s2s1 > s1s1 > s2s2 >s1 s2
4
3
2
1
Actor 2: s1s2 > s1s1 > s2s2 > s2s1
4
3
2
1
The Concept of Game VI
Actor 2
s1
Actor 1
s1
s2
20
22
s2
s1s1
s2s1
-4
-55
3 s1s2
-2
14 s2s2
-30
Actor 1: s2s1 > s1s1 > s2s2 >s1 s2
22(4) 20(3) 14(2) 3(1)
Actor 2: s1s2 > s1s1 > s2s2 > s2s1
-2(4) -4(3) -30(2) -55(1)
The Concept of Game: an Example
(Prisoner’s Dilemma)
Bill
Do not confess
John
(cooperate)
Do not confess
(cooperate)
Confess
(defect)
-2 s1s1 -2
-1
s2s1 -10
Confess
(defect)
-10 s1s2
-8 s2s2
-1
-8
The police have caught two suspected drug dealers, John and Bill. They are now sitting in two
separate cells in the police station and cannot communicate. The prosecutor tells them that they
have one hour to decide confess their crimes or deny the charges. The prosecutor has enough
evidence to prove the charge to sentence them for 2 years for a well-documented offences.
However, to sentence them for drug trafficking the prosecutor should have the confession of at
least one of them. The legal situation is as follows: if both prisoners confess they will get 8 years
each. However, if one confesses and the other does not, then the prisoner who confesses will be
rewarded and get away with just 1 year in prison, whereas the other will get 10 years.
Taxonomy (classification) of games


Two-person and N-person (N>2) (How many players)
Cooperative and non-cooperative games
(in cooperative games players can agree on binding contracts that force them to respect
whatever they have agreed)



Non-iterated (one-shot) and iterated games
Simultaneous-move versus sequential-move games
Games with perfect information versus games with
imperfect information
(sequential game is a game with perfect information if players have full information about the
strategies played by the other players in earlier rounds; if they have only some information about
their previous moves, the sequential game is with imperfect information)

Constant (including zero) sum games versus nonconstant sum
Two-Person Constant-Sum Games
A game is constant-sum if for all outcomes the
sum of payoffs for all players is constant.
 A special case of constant-sum game is zerosum game where this sum is zero.
 All constant-sum games sometimes are called
zero-sum, because each constant-sum game can
be transformed into a zero-sum game by
subtracting from each payoff half of the sum of
the payoffs

Two-person zero-sum game
s1
s2
s3
0
s1
s2
s3
s1
-0,5XX 0,5ZZ
1,5 -1,5Z
2,5 -2,5Z
-1, 5X 1,5
0
1,5 -1,5Z
-2,5X 2,5
s1
2
3
4
1
5
s2
2
3
1
4
2,5 2,5
s2
-0,5 0,5Z
s3
0
5
4
1
0
s3
-2,5X 2,5
5
0
Two-Person Zero-Sum Games




Zero-sum games are models of so-called antagonistic
conflicts (e.g. war). For each outcome, one player can
gain only as much the other player loose.
Zero-sum games is the oldest province of the game
theory. They were investigated in an exhaustive way in
the book J. von Neumann and O. Morgenstern Theory of
Games and Economic Behavior (1944)
In this book, the authors proved that in the zero-sum
games the choice according to minimax(maximin) rule is
rational (utility-maximizing) for all players
A pair of strategies are in equilibrium if the
outcome determined by the strategies equals the
minimal value of the row and the maximal value
of the column (minimax condition rule)
Two-Person Zero-Sum Games
s1
s2
s3
s1
9
-9Z
8
-8 Z
7XX
s2
7
-7
-5 X
5
6
s3
4
-4
1
-1
-2 X
-7 ZZ
-6
2
Classification of Games (limited to three dimensions
of their differences) I
Two-Person and N-Person games (1A, 1B)

Cooperative and non-cooperative games (2ά, 2β)

Zero-sum and nonzero-sum games (3i, 3ii)
From combinatorial point of view, eight combinations of
attributes (types of games) are possible:
(1) 1A, 2ά, 3i (two-person cooperative zero-sum)
(2)
1A, 2ά, 3ii (two-person cooperative nonzero-sum)
(3) 1A, 2β, 3i (two-person non-cooperative zero-sum)

(4) 1A, 2β, 3ii (two-person non-cooperative
nonzero-sum)
(5)
(6)
(7)
(8)
1B, 2ά, 3i (N-person cooperative zero-sum)
1B, 2ά,3ii (N-person cooperative nonzero-sum)
1B, 2β, 3i (N-person non-cooperative zero-sum)
1B, 2β, 3ii (N-person non-cooperative nonzero-sum)
Classification of Games (limited to three
dimensions of their differences) II
(1)
1A, 2ά, 3i (two-person cooperative zero-sum)
This combination is inconsistent (contradictory): in antagonistic
conflict between two players, there is no ground for cooperation
However, if there are at least 3 players and the situation is zero-sum,
it is possible for some of them (two or more) to get payoffs at the
expense of others, so there is ground for cooperation: coalition of
two (or more) against the other(s)
Generally, the N-person (N>2) game theory is the analytical tool of
the coalition theory. For other applications of game theory, the
knowledge of 2-person games is sufficient. In Coleman’s book, at
some places 3-person games are used.
Among them (4) 1A, 2β, 3ii (non-cooperative nonzero-sum) games
are most interesting and widely applicable for the construction of
models in social theory and research
```