OPERATIONS RESEARCHGAME THEORY
Ing. Daniel Orlando Rivera Royero
What is Game Theory ?
What is Game Theory?
• Game theory is the study of multi-personal
decision problem, " conflict of interest" .
• It helps to understand the phenomena
observed when agents interact.
• We will only consider non-cooperative
games.
• We will emphasize the effect of the
payments.
Game Theory
Historical Development1713 James
Waldegraveproposed a
minimax solution of the card game
"le Her"1838 Cournot duopoly performs an
analysis based on game theory 1
913 Ernest Zermelo
shows that in every finite zero-sum game with
perfect information
between two people, there is at least one strategy for one player with which you can not
lose
1928 Von Neumann
's Minimax theorem shows 1
944 John von
Neumann and Oskar
Morgenstern published the book " Theory of Games and
Economic Behavior”
1950-1
960 First use of
models of game theory in economics and first checks the predictions of
game theory in experimental economics.
Desarrollo Histórico
19
94 John C. Harsanyi , John
F. Nash and ReinhardSelten received the
Nobel Prize in economics for his
pioneering analysis of equilibria in the theory
of non- cooperative games.
20
07 Leonid Hurwicz , Eric
Maskin and Roger Myerson receive the
Nobel Prize in Economics for having laid the foundations of
mechanism design theory .
Desarrollo Histórico
• A set of N Players (2 Players)
• Each player chooses an action 𝑎𝑘 of
𝐴𝑘 possible actions.
• The player k gets a payment
Assumptions
Player 2
A B CP
layer
1
A 51 51.5 52.5
B 50.5 57.5 54
C 52 53 53.5
Game Theory Componets
• The players are rational maximize
profits.
• The players are intelligent They use
their knowledge or experience (they can
deduce and infer).
Assumptions
• Games complete information: Payment
information and action is published.
• Games of complete and imperfect
information : games aren’t sequential.
Game Types
Definition
There is public information of rationality
and each player knows that each player is
rational, and every player knows that each
player knows that each player is rational ...
Infinitum .
Solution methods
Games between 2 people, the strategies of
each of the opponents can be:
• Pure strategies
• Mixed strategies
Solution methods
Average profit per game during many plays.
It corresponds to the minimum value that will
win Player A, always that play intelligently,
no matter how plays the Player B.
V* = Value Play
Value Play
Find how each competitor must combine
their strategies independently as play your
opponent to guarantee himself at least V * .
Solution a Game
1. Each competitor seeks:
• (max(min E(profit))
• (min(max E(losses))
2. Rationalization.
Establish strategies that are rational for the
player.
Solution a Game
Is the point that Player A and Player B always
choose.
This point matches the strategy that maximizes the
minimum profit in Player A, and the strategy that
minimizes the maximum loss in Player B.
NOTE. It is the lowest of the row and the highest
value of the row column.
Saddle point
Player 2
A B CP
layer
1
A 51 51.5 52.5
B 50.5 57.5 54
C 52 53 53.5
Saddle point
• Identify , if any, is theSaddle point.
• If there the Saddlepoint, what would bethe value of the gameafter a number ofmoves ?
Player B
B1 B2
Pla
yer
A A1 -5 4
A2 -4 -8
Player B
B1 B2
Pla
yer
AA1 2 1
A2 -3 -4
A3 -5 -6
Saddle point
• Domain rule for the row: every value of thedominant row must be greater than orequal to the corresponding value in thedominated row.
• Rule domain for columns : each value ofthe dominant column must be less than orequal to the corresponding value of thedominated column.
Rationalization
Find key strategies in
the following situations:
Player B
B1 B2
Pla
yer A
A1 2 6
A2 3 1
Player B
B1 B2
Pla
yer A
A1 -4 -2
A2 -6 -3
Player B
B1 B2 B3 B4
Pla
yer A
A1 -4 -6 2 4
A2 -6 -3 1 2
Rationalization
Steps.
• Find Saddle Point.
• If (Saddle Point Exists){
• Define=> Pure Strategies (Players A & B)
• } else {
• Rationalize
• Define=> Mixed Strategies (Players A & B)
• }
Strategies A % Market A Strategies B % Market B Difference
a1 47 b1 53 -6
a2 54,5 b1 45,5 9
a3 53 b1 47 6
a1 56 b2 44 12
a2 48,5 b2 51,5 -3
a3 47 b3 53 -6
a1 48,5 b3 51,5 -3
a2 56 b3 44 12
a3 54,5 b3 45,5 9
Example MARKET GAME
PLAYER 2
B1 B2 B3
PL
AY
ER
1
A147, 53 56, 44
48.5,
51.5
A2
54.5,
45.5
48.5,
51.556, 44
A353, 47 47, 53
54.5,
45.5
Example
B1 B2 B3
A1 -6 12 -3
A2 9 -3 12
A3 6 -6 9
Example
Are there key strategies ?
Does saddle point ?
B1 B2
A1 -6 12
A2 9 -3
Example
Reduced matrix
Whenever a game has no saddle point,
game theory, each player can assign a
probability distribution for the set of
strategies .
Mixed strategies
Play intelligently is:
Find a combination of strategies indifferent of
how your opponent plays.
Otherwise, your opponent might try to take
advantage of the way you are playing.
Mixed strategies
B1 y B2 1-y
A1 x -6 12
A2 1-x 9 -3
Example
Example
Player A
−6𝑥 + 9 1 − 𝑥 = 12𝑥 − 3 1 − 𝑥𝑥 = 0.4
Player B
−6𝑦 + 12 1 − 𝑦 = 9𝑦 − 3 1 − 𝑦𝑦 = 0.5
Example
B1 0.5 B2 0.5
A1 0.4 -6 12
A2 0.6 9 -3
𝑉 = −6 × 0.4 × 0.5 + 12 × 0.4 × 0.5 + 9 × 0.6 × 0.5 − 3 × 0.6 × 0.5
𝑉 = 3
Question 1
What is the
meaning of Value
Play?
Pregunta 2
What is the saddle
point ?
Question 2
What is a fair game
?
Exercise 1
Exercise 2