GAMES AND DECISIONS
Lecture 4
Bimatrix Games / Non-Constant-Sum Games
The most famous example of bimatrix game:
Prisoner’s dilemma
Two prisoners are isolated and charged being jointly
responsible for a crime. They have two strategies: to
deny or confess.
Deny
Player1…….Confess
Player 2
Deny
Confess
-2,-2
-10,-1
-1,-10
-6,-6
Nash equilibrium is (-6,-6). This games shows that
the pursuit of the individual best interests can lead
to outcomes that are bad for all of them.
From economic point of view, Nash equilibrium is
not Pareto efficient. There is an outcome that would
make all players better off.
One possible solution: penalization of the strategy
“confess”. Assume that both prisoners belong to a
gang (organized crime).
Battle of Sexes
Husband (player 1) and wife (player 2) want to meet in
the evening. They both forget their decisions from the
morning and have no cellular phones (coordination is not
possible).
But they both know their preferences:
 Husband prefers to go to the football match. Wife
prefers to go shopping.
 Husband and wife will have a pleasure from
spending afternoon together (+1), and (+1) if one
does what he/she prefers.
 If they stay alone, utility is 0, not dependent on their
choices.
Matrix for husband
football
Husband……..shopping
Wife
football shopping
2
0
0
1
Matrix for wife
football
Husband……..shopping
Wife
football shopping
1
0
0
2
Bimatrix game
football
Husband……..shopping
Wife
football shopping
2, 1
0, 0
0, 0
1 ,2
There two Nash equilibria in pure strategies, and one
in mixed strategies.
The negotiation (cooperation) may be possible in
this situation.
Game of Chicken
There are two boys driving their cars.
Stay
Player1…….Swerve
Player 2
Stay
Swerve
-100,-100
+1,-1
-1,+1
0, 0
Two Nash equilibria in pure strategies, one NE in
mixed strategies.
Is it possible to win by some trick?
Game with dominated Nash Equilibrium
1
Player1…….2
1
7, 9
-2, 0
Player 2
2
-2, 1
6, 4
Game without Nash Equilibrium in pure strategies
1
Player1…….2
1
3, 5
4, 1
Player 2
2
2, -1
-2, 5
Nash Equilibria in bimatrix games:
-
unique NE in pure strategies.
more NE, one NE dominates other NE.
more NE (pure or mixed).
one NE in mixed strategies.
Note: any matrix game can be represented as a
bimatrix game.
John Nash:
Equivalence Theorem: there is at least one NE in
bimatrix game.
Theory of bimatrix games:
- noncooperative theory / cooperative theory
There are two alternatives of cooperative games:
- with transferable payoffs (you can cooperate and
you pay compensation),
- with non-transferable payoffs (you can cooperate,
you cannot compensation)