game theory. brief history of game theory 1913 - e. zermelo provided the first theorem of game...
Post on 15-Dec-2015
219 Views
Preview:
TRANSCRIPT
Game theory
Brief History of Game Theory
• 1913 - E. Zermelo provided the first theorem of
game theory asserts that chess is strictly determined
• 1928 - John von Neumann proved the minimax
theorem
• 1944 - John von Neumann / Oskar Morgenstern’s
wrote "Theory of Games and Economic Behavior”
• 1950-1953, John Nash describes Nash equilibrium
• 1972 - John Maynard Smith wrote “Game Theory
and The Evolution of Fighting”
What is Game Theory?
Game theory is a study of how to mathematically
determine the best strategy for given conditions
in order to optimize the outcome.
Basic Terms or Terminology
• Player: The competitors in the game are known as
players. A player may be individual or group of
individuals, or an organization.
• Strategy: A strategy for a player is defined as a set
of rules or alternative courses of action that he
should adopt. A strategy may be of two type: Pure
Strategy and Mixed Strategy.
• Pure Strategy:- If the players select the same
strategy each time, then it is referred to as pure-
strategy. In this case each player knows exactly
Basic Terms or Terminology…
what the other player is going to do, the objective of
the players is to maximize gains or minimize losses.
• Mixed Strategy:-When the players use a
combination of strategies and each player always
kept guessing as to which course of action is to be
selected by the other player at a particular occasion
then this is known as mixed strategy. Thus there is a
probabilistic situation and objective of the player is
to maximize expected gains or to minimize expected
losses.
Basic Terms or Terminology…• Optimum strategy: A course of action or play which
puts the player in the most preferred position,
irrespective of the strategy of his competitor, is called
an optimum strategy.
• Payoff: Each combination of a course of action and
an event is associated with a payoff, which measures
the net benefit to the decision maker the accrues
(accumulate) from a given combination of decision
alternatives and events. ( conditional profit values or
conditional economic consequences )
Basic Terms or Terminology…
• Value of the Game: It is the expected payoff of
play when all the players of the game follow their
optimum strategies. The game is called fair if the
value of the game is zero and unfair, if it is non-
zero.
• Payoff Matrix: When the players select their
particular strategies, the payoffs (gains or losses)
can be represented in the form of a matrix called the
payoff matrix.
Two Person Zero-Sum game
• When there are there are two competitors playing a game, it
is called a two-person game. In case the number of
competitors exceeds two, say n, then the game is termed as
a n-person game.
• Zero-sum means that the algebraic sum of gains and losses
of all the players is zero. Zero-sum game with two players
are called two-person zero sum games. In it the loss (gain)
of one player is exactly equal to the gain (loss) of the other.
• If the sum of losses or gains is not equal to zero, then the
game is of non-zero sum game.
Two Person Zero-Sum game…
• Let player A has m strategies A1, A2, ...,Am and
players B has n strategies B1,B2,…,Bm.
• Assumptions:
Each player has his choices from amongst the pure
strategies.
Player A is always the gainer and player B is
always the loser. (i.e. all payoffs assumed in terms
of player A). Let aij be the payoff which player A
gains from player B if player A chooses strategy Ai
and B chooses Bj. Then the payoff matrix of A is:
mnm3m2m1
3n333231
2n232221
1n131211
n321
a ..... a a a
a ..... a a a
a ..... a a a
a ..... a a a
A Player
B B B B
BPlayer
The payoff matrix for players B is -aij
Two Person Zero-Sum game…
• Example:- Consider a two person coin tossing
game. Each player tosses an unbiased coin
simultaneously. Player B pays Rs. 7 to A, if (H,H)
occurs and Rs. 4, if (T,T) occurs; otherwise player A
pays Rs. 3 to B.
(This two-person game is zero-sum game)
Pure strategies for players are H and T.
The maximin-minimax Principle
• For player A, minimum value in each row
represents the least gain (payoff) to him if he
chooses his particular strategy. Then the player A
selects the strategy that maximizes his minimum
gains. This choice of player A is called the maximin
principle and corresponding gain is called maximin
value of the game.
• On the other hand, player B likes to minimize his
losses. The maximum value of each column
represents the maximum loss to him if he chooses
The maximin-minimax Principle…his particular strategy. These are written in the
matrix by column maxima. He will then select the
strategy that minimize his maximum losses. This
choice of player B is called the minimax principle,
and the corresponding loss is the minimax value of
the game.
• If the maximin value equals the minimax value, then
the game is said to have a saddle point
(equilibrium) and the corresponding strategies are
called optimum strategies. The amount of payoff at an
equilibrium point is known as the value of the game.
The maximin-minimax Principle…Example: Consider a two-person zero sum game
with a 3X2 payoff matrix for player A.
4 6 A
6 8 A A Player
2 9 A
B B
BPlayer
3
2
1
21
Pure strategies for player A (SA) = A1, A2, A3
Pure strategies for player B (SB) = B1, B2
The maximin-minimax Principle…Example:…….
Suppose player A starts the game ( he knows fully
that whatever strategy he adopts, B will select that
particular counter strategy which will minimize the
payoff to A)
If A selects A1 then B will reply by selecting B2, as
this corresponds to the minimum payoff to A in in
row-1 corresponding to A1.
Similarly if A chooses A2, he may gain 8 or 6
depending upon B’s strategy.
Games without saddle points —Mixed Strategy Games
• It is not necessary that there is always a saddle point
exist.
• In such cases to solve games, both the players must
determine an optimal mixture of strategies to find a
saddle point.
• The optimal strategy mixture for each player may be determined by assigning to each strategy its probability of being chosen. The strategies so determined are called mixed strategies because they are probabilistic combination of available choices of strategies
Mixed Strategy Games… • The value of game obtained by the use of mixed
strategies represents which least player A can expect
to win and the least which player B can lose. The
expected payoff to a player in a game with arbitrary
payoff matrix (aij)of order m*n is defined as:
m
1i
n
1jjiji qapq)E(p,
where p and q denote the mixed strategies for players A and B respectively
Maximin-Minimax Xriterion• Consider a m*n game without any saddle point, i.e.
strategies are mixed. Let p1, p2, ….,pm be the
probabilities with which player A will play his
moves A1, A2, ….,Am respectively; and let q1, q2,
…., qn be the probabilities with which player A will
play his moves B1, B2, …., Bn respectively.
• Also, p1+p2+ ….+pm=1 and q1+ q2+ ….+qn =1
• Pi >=0 and qj >=0
• The expected payoff function for player A is :
m
1i
n
1jjiji qapq)E(p,
• For player A (maximin-minimax criterion)
strategyjth his uses Bwhen
A, gain to expected th denotes a p min. Here,
a p ,.......,a p ,a p min. max.
a p min. max. q)E(p, min. max.v
ij
m
1ii
j
in
m
1iii2
m
1iii1
m
1ii
jp
ij
m
1ii
jpqp
• For player B (maximin-minimax criterion)
strategy.ith his usesA
whenB, of loss expected thedenotes a q max. Here,
a q,.......,a q,a q max. min.
a q max. min. q)E(p, max.min.
ij
n
1jj
i
jm,
n
1jjj2,
n
1jjj1,
n
1jj
iq
ij
n
1jj
iqpq
v
The relationship holds good in general and when pi and qj
correspond to the optimal strategies the relation holds in ‘equality’
sense and the expected value for both the players becomes equal to the
optimum expected value of the game.
vv
• For any 2x2 two-person zero-sum game without any saddle
point having the payoff matrix for player
• the optimum mixed strategies
• and determined by
a a A A Player
a a A
B B
BPlayer
22212
12111
21
q q
B B S and
p p
A A S
21
21
B
21
21
A
1211
2122
2
1
1211
2122
2
1
aa
aa
q
q ,
aa
aa
p
p
)a(aaa
aaaa v
bygiven isA togame
theof valueThe 1.qq and 1pp where
21122211
12212211
212 1
Example-1
• For the game with the following payoff matrix,
determine the optimum strategies and the value of
the game:
1qq,1pp
;q q
B B S and
p p
A A S
4 3
1 5A
B
2121
21
21
B
21
21
A
17/5 vis game theof value theand
2/5 3/5
B BS,
4/5 1/5
A AS
:are B andA for strategy optimum Hence
17/5. down to )q,E(p keep
can B 3/5,q because 17/5, than more of sure bet can'A
17/5.atleast isn expectatio his that ensuresA 1/5,p If
5/17)5/3)(5/1(5
435p
)q)(1p4(1)q(11.p)qp3(1q5p q)E(p,
thenfunction, payoff expected thedenotes q)E(p, If
21
A
21
A
11
1
1
11
1111
11111111
qp
qpq
Example-2
• Consider a modified form of matching biased coins
game problem. The matching player is paid Rs. 8 if
the two coins turn both heads and Rs. 1 if the coins
turns both tails. The non-matching player is paid Rs.
3 when the two coins do not match. Given the
choice of being the matching or non-matching
player, which one would you choose and what
would be your strategy?
• The payoff matrix for the matching player is given by
1 3- T Palyer Matching
3- 8 H
T H
player matching-Non
The payoff matrix does not posses any saddle point.
The players will used mixed strategies. Optimum
mixed strategies for matching player and non-
matching player are:
15
4
)33(18
)3(1q
)a(aaa
aaq
15
4
)33(18
)3(1p
)a(aaa
aap
121122211
12221
121122211
21221
The expected value of the game is :
15
1
)33(118
)3)(3(1*8
)a(aaa
aaaav
21122211
21122211
Graphic Solution of 2xn and mx2 games
• Consider the following 2xn game:
2n232221
1n131211
n321
a ..... a a a
a ..... a a aA Player
B B B B
BPlayer
• It is assumed that the game does not have a saddle point.
• Let the optimum mixed strategy for A be given by
1pp, p p
A A S 21
21
21
A
• The expected payoff of A when he plays SA against
B’s pure strategies B1, B2,…Bn is given by:
22n11n1nn
222112122
221111111
papa)(pE B
.
.
.
papa)(pE B
papa)(pE B
E(p) Payoff Expected sA' Strategies Pure sB'
• According to maxmin criterion for mixed strategy
games, player A should select the value of p1 & p2
so as to maximize his min. expected payoffs.
• This may be done by plotting the expected payoff
lines Ej(p1)=(a1j-a2j) p1 + a2j (j=1,2,…,n)
• The highest point on the lower envelope of these
lines will give max. of the min. expected payoffs to
player to player A as also the max. value of pi.
• The two lines passing through maximin point
identify the two critical moves of B, which
combined with two of A, yield a 2x2 matrix that can
• This may be done by plotting the expected payoff
lines Ej(p1)=(a1j-a2j) p1 + a2j (j=1,2,…,n)
• The highest point on the lower envelope of these
lines will give max. of the min. expected payoffs to
player to player A as also the max. value of pi.
• The two lines passing through maximin point
identify the two critical moves of B, which
combined with two of A, yield a 2x2 matrix that can
be used to determine the optimum strategies of the
two players, for the original game, using the results
of mixed strategies (without saddle point).
• Similarly, in mx2 games, the upper envelope of the straight
lines corresponding to B’s expected payoffs will give
expected payoff to player B and the lowest point on this
then gives the minimum expected payoff (minimax value)
and the optimum value of q1.
• EXAMPLE:
Solve the following game graphically:
2 3 0 1 A A Palyer
2- 0 1 2 A
B B B B
BPlayer
2
1
4321
• Since maximin.# minimax. (no saddle point), so players
have to choose mixed strategies, Let the player A plays the
mixed strategy SA against B
1pp ;p p
A A S 21
21
21
A
Then A’s expected payoffs against B’s pure moves are given by
2p4)(pE B
3p3)(pE B
p)(pE B
1p)(pE B
E(p) Payoff Expected sA' Strategies Pure sB'
1144
1133
1122
1111
• These expected payoff equations are plotted as
functions of p1 as shown in figure which shows the
payoffs of each column represented as points on two
vertical axis 1 and 2, unit distance apart.
• Thus line B1 joins the first payoff element 2 in the
first column represented by +2 on the axis 2, and the
second payoff element 1 in the first column
represented by +1 on axis 1.
• Similarly, lines B2, B3, b4 joins the corresponding
representation of payoff elements in the 2nd, 3rd and
4th columns.
• Since the player A wishes to maximize his
minimum expected payoff we consider the highest
point of intersection H on the lover envelope of the
A’s expected payoff equations. This point H
represents the maximin expected value of the game
for A.
• The lines B2 & B4, passing through H, define the
two relevant moves B2 & B4 that alone B needs to
play.
• The solution of 2x4 game, boils down to the simpler
game with 2x2 payoff matrix:
2 0 A A Palyer
2- 1 A
B B
BPlayer
2
1
42
5
1q-1q ,
5
4
)2(21
)2(2q
)a(aaa
aaq
5
3p-1p ,
5
2
)2(21
02p
)a(aaa
aap
thenB, andA for strategy optimum thebe
q q
B B S and
p p
A A S if Now
24 221122211
12222
12 121122211
21221
42
21
B
21
21
A
5
2
2)(021
2)x0(1x2
)a(aaa
aaaa v
is game theof valueexpected The
1/5 0 4/5 0
B B B B S and
3/5 2/5
A A S
are B andA for strategies optimum theNow,
21122211
21122211
4321
B
21
A
Dominance Property
• It is observer that one of the pure strategies of either
player is always inferior to at least one of the
remaining ones.
• The superior strategies are said to dominate the
inferior ones.
• Clearly, a player would have no incentive to use
inferior strategies which are dominated by the
superior ones. In such cases of dominance, we can
reduce the size of the payoff matrix by deleting
those strategies which are dominated by the others.
Dominance Property…
• Thus if each element in one row, say kth of the
payoff matrix is less than or equal to the
corresponding elements in some other row, say rth,
then player A will never choose kth strategy.
• The value of the game and the non-zero choice of
probabilities remain unchanged even after the
deletion of kth row from the payoff matrix. In such
a case the kth strategy is said to be dominated by the
rth one.
• General rules for dominance are:
Dominance Property…
• If all the elements of a row, say kth, are less than or
equal to the corresponding elements of any other
row, say rth, then kth row is dominated by rth row.
• If all the elements of a column, say kth are greater
then or equal to the corresponding elements of any
other column, say, rth, then kth column is
dominated by the rth column.
• Dominated rows or columns may be deleted to
reduce the size of the matrix, as the optimal
strategies will remain unaffected.
Dominance Property…
The modified dominance property: The dominance
property is not always based on the superiority of
pure strategies only. A given strategy can also be
said to be dominated if it is inferior to an average of
two or more other pure strategies only. More
generally, if some convex linear combination of
some rows dominates the ith row, then ith row will
be deleted. Similar arguments follow for columns.
Example
• Two firms are competing for business under the
condition so that one firm’s gain is another firm’s
loss. Firm A’s payoff matrix is
10 14 16 Adv.)(High A
15 12 13 Adv.) (MediumA A Firm
2- 05 10 Adv.) (NoA
Adv.)(High B Adv.) (Medium B Avd.) (No B
B Firm
top related