Game Theory
Game theory, branch of applied mathematics that provides tools for analyzing
situations in which parties, called players, make decisions that are
interdependent. This interdependence causes each player to consider the other
player's possible decisions, or strategies, in formulating strategy.
ZERO SUM GAME
In a game if the algebraic sum of payments to all players to zero the game is
called zero sum game. In a zero sum game the play does not add a single money
to the total wealth of all players, it mearly results in a new distribution of initial
money among them.
Two persons zero sum game
Zero sum games with two players are called two persons zero sum game or
rectangular game. In this case the gain of one player is exactly equal to the loss of
the other. The basic assumption in a two person zero sum game are
a) There are exactly two players with opposite interest
b) The number of strategies available to each player is finite(maynot be
common)
c) For each specific strategies selected by a player ,there results a payoff
d) The amount won by one player is exactly equal to the amount lost by the
other
Pay off matrix
In a two person zero sum game, the resulting gain can be represented by a matrix
called the payoff matrix or gain matrix. Consider a game with two players A and B
in which player A has m strategies .Then the pay off matrix of A is denoted by
, B 1 2 . . . . . . . . . j……………. n
A 1
a11 a12 a1j a1n
2
a21 a22 a2j a2n
3
a31 a32 a3j a3n
:
i ai1 ai2 aij ain
:
M am1 am2 amj amn
Example
Consider a two person zero sum game of tossing a coin. Let A and B be
two players. Each player tosses an unbiased coin twice successively.
Player B pays Rupees 7 to A if {H,H} occurs and Rupees to 4 to A if {T,T}
occurs. On the other hand A pays rupees 7 to B if {H, T} occurs and
Rupees 4 to B if {T, H} occurs. Then the payoff matrix A & B as follows.
Payoff matrix of A
H B T
H 7 -7
A
-4 4
T
Game theory, branch of applied mathematics that provides tools for analyzing
situations in which parties, called players, make decisions that are
interdependent. This interdependence causes each player to consider the other
player's possible decisions, or strategies, in formulating strategy.
ZERO SUM GAME
In a game if the algebraic sum of payments to all players to zero the game is
called zero sum game. In a zero sum game the play does not add a single money
to the total wealth of all players, it mearly results in a new distribution of initial
money among them.
Two persons zero sum game
Zero sum games with two players are called two persons zero sum game or
rectangular game. In this case the gain of one player is exactly equal to the loss of
the other. The basic assumption in a two person zero sum game are
a) There are exactly two players with opposite interest
b) The number of strategies available to each player is finite(maynot be
common)
c) For each specific strategies selected by a player ,there results a payoff
d) The amount won by one player is exactly equal to the amount lost by the
other
Pay off matrix
In a two person zero sum game, the resulting gain can be represented by a matrix
called the payoff matrix or gain matrix. Consider a game with two players A and B
in which player A has m strategies .Then the pay off matrix of A is denoted by
, B 1 2 . . . . . . . . . j……………. n
A 1
a11 a12 a1j a1n
2
a21 a22 a2j a2n
3
a31 a32 a3j a3n
:
i ai1 ai2 aij ain
:
M am1 am2 amj amn
Example
Consider a two person zero sum game of tossing a coin. Let A and B be
two players. Each player tosses an unbiased coin twice successively.
Player B pays Rupees 7 to A if {H,H} occurs and Rupees to 4 to A if {T,T}
occurs. On the other hand A pays rupees 7 to B if {H, T} occurs and
Rupees 4 to B if {T, H} occurs. Then the payoff matrix A & B as follows.
Payoff matrix of A
H B T
H 7 -7
A
-4 4
T
