Decisiontree&game theory

Decision making and
Game Theory

 Decisions are made at every level of management to ensure organizational or business goals are achieved.
 Decision-making involves the selection of a course of action from among two or more possible alternatives in order to
arrive at a solution for a given problem”.
 Steps to Good Decision Making
 1.Define problem and influencing factors.
 2. Establish decision criteria.
 3. Select decision-making tool (model).
 4. Identify and evaluate alternatives using decision-making tool (model).
 5. Select best alternative.
 6. Implement decision.
 7. Evaluate the outcome.

Terminologies
 Decision Maker: The decision maker is charged with responsibility of making
the decision.
 The acts or decision: The alternative decisions or strategies that are available
to decision maker.
 Events: Occurrences which affect the achievement of the objectives, these
are also called as outcomes or state of nature.
 Pay off table: It is the outcome in quantitative form if the decision maker
adopts a particular strategy under a particular state of nature

Types of Decision Making
 1) Decision making under certainty
The outcome of a decision alternative is known (i.e., there is only one state of
nature.)
 2) Decision making under risk
The outcome of a decision alternative is not known, but its probability is known.
 3) Decision making under uncertainty
The outcome of a decision alternative is not known, and even its probability is
not known.

 Decision under Certainty:
 The decisions may be taken when the problems are under certainty i.e., where a complete
knowledge about the nature of future conditions is known. For example, we know that if we toss an
unbiased coin, one of two equally likely outcomes (i.e., either head or tail) occur, and the
probability of each outcome is predetermined.
 Decision making under uncertainty. In this case, each course of action has several possible
consequences and the decision maker does not know the probability of each of them. It is therefore
a scenario poor in information, as opposed to the previous case. The decision is complicated
because past experiences do not make it possible to predict the future and there are many
uncontrollable variables.
 Decision making under risk. This scenario presents an intermediate situation between the two
two previous ones: each alternative, strategy or course of action has several possible
consequences, but the decision maker knows the probability of each of them. Although the choice
choice will not be as easy as in the case of decisions under certainty, it will be possible to apply a
a decision-making model that facilitates it.

Decision Making Under Uncertainty
 1.Maximax Criterion /optimism
 2. Maximin Criterion/ pessimism
 3. Minimax Criterion /regret
 4. Laplace Criterion (Equally likelihood/Rationality/Bayes)
 5. Hurwicz alpha Criterion ( Rationality or Realism)

Criterion 1: Maximax (Optimistic)
 An adventurous and aggressive decision maker may choose the act that would
result in the maximum payoff possible
 Step 1 - Pick maximum payoff of each alternative.
 Step 2 - Pick maximum of those maximums in Step 1; its corresponding
alternative is the decision.
 This is viewed as an optimistic approach, “Best of bests”.

States of Nature Decisions
A1 A2 A3
S1 220 180 100
S2 160 190 180
S3 140 170 200
maximum 220 190 200
Max(220,190,200)=220 is the maximax i.e maximum of
maximums
Ex1:
What is the decision taken by the decision maker for the following payoff matrix
under maximax criterion?
States of Nature Decisions
A1 A2 A3
S1 220 180 100
S2 160 190 180
S3 140 170 200

(Payoffs) State of Nature
Action Oil Dry
Drill for oil 700 -100
Sell the land 90 90
State of Nature Maxim
um
payoff
Action Oil Dry
Drill for oil 700 -100 700
Sell the land 90 90 90
Ex2: What is the decision taken by the decision maker for the following payoff matrix
under maximax criterion?
Max(700,90)=700,Hence the decision taken by decision maker is
to drill for oil

Criterion 2: Maximin (Pessimistic)
 This is also called Waldian criterion. This criterion of decision making stands
for choice between alternative courses of action assuming pessimistic view of
nature (payoff matrix)
• Step 1 - Pick minimum payoff of each alternative
• Step 2 - Pick the maximum of those minimums in Step 1, its corresponding
alternative is the decision
This is viewed as a pessimistic approach, “Best of worsts”

State of Nature Decision
A1 A2 A3
S1 -1 4 10
S2 3 -2 6
S3 18 14 -3
minimum -1 -2 -3
Maximum(-1,-2,-3)=-1 i.e. maximum of minimum decisions,
The decision taken by decision maker is A1
under maximin criterion?
A1 A2 A3
S1 -1 4 10
S2 3 -2 6
S3 18 14 -3

 For each action, find minimum payoff over all states of nature
 Then choose the action with the maximum of these minimum payoffs
State of Nature Minim
um
payoff
Action Oil Dry
Drill for oil 700 -100 -100
Action Oil Dry
Sell the land 90 90
Ex2:
under maximin criterion?
Max(-100,90)=90,Hence the decision chosen by decision maker is to
sell the land

Criterion 3: Minimax decision criterion
 Application of the minimax criterion requires a table of losses or table of
regret instead of gains. Regret is amount you give up due to not picking the
best alternative in a given state of nature. Regret = Opportunity cost =
Opportunity loss (cost matrix)
• Step 1 - Construct a ‘regret table’,
• Step 2 - Pick maximum regret of each row in regret table,
• Step 3 - Pick minimum of those maximums in Step 2, its corresponding
alternative is the decision

State of Nature Decisions
A1 A2 A3
S1 0 4 10
S2 3 0 6
S3 18 14 0
maximum 18 14 10
Minimum(18,14,10)=10 i.e.)minimum of maximum
decisions,Decision made by decision maker is A3
Ex1: What is the decision taken by the decision maker for the following Regret matrix
under minimax criterion?
State of Nature Decisions
A1 A2 A3
S1 0 4 10
S2 3 0 6
S3 18 14 0

 For each action, find maximum regret over all states of nature
 Then choose the action with the minimum of these maximum regrets
(Regrets) State of Nature Max
Regret
Action Oil Dry
Drill for oil 700 -100 700
Action Oil Dry
Sell the land 90 90
Ex2: What is the decision taken by the decision maker for the following Regret matrix
under minimax criterion?
Min(700,90)=90,Hence the decision taken is to sell the land

Criterion 4: Equally Likely or Laplace
criterion
 The decision maker makes a simple assumption that each state of nature is
equally likely to occur & compute average payoff for each. Choose decision
with highest average payoff. Also known as Laplace criterion
• Step 1 - Calculate the average payoff for each alternative.
• Step 2 - The alternative with highest average if the decision.

Events Acts
A1 A2 A3
E1 20 12 25
E2 25 15 30
E3 30 20 22
All the three outcomes has equal probability of occurrence i.e.) 1/3=0.34,find
expected payoff for each alternative or acts
A1->0.34(20+25+30)=25.5
A2->0.34(12+15+20)=15.98
A3->0.34(25+30+22)=26.18
A3 has maximum payoff, hence it is the optimal choice.
under laplace criterion?

Criterion 5: Hurwicz (Realism)
 This method is a combination of Maximin and Maximax criterion. Also known as criterion of
rationality. neither too optimistic nor too pessimistic
• Step 1 - Calculate Hurwicz value for each alternative
• Step 2 - Pick the alternative of largest Hurwicz value as the decision.
𝐻𝑢𝑟𝑤𝑖𝑐𝑧 𝑣𝑎𝑙𝑢𝑒 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 = 𝑚𝑎𝑥 𝛼 + (𝑚𝑖𝑛)(1 − 𝛼)
where 𝛼 is called coefficient of realism

Events Acts
A1 A2 A3
E1 20 12 25
E2 25 15 30
E3 30 20 22
Let ᾱ=0.6
For A1, maximum payoff=30,minimum payoff=20,Hence decision D1=(30*0.6)+20*(1-0.6)=26
For A2,D2=(20*0.6)+12*(1-0.6)=16.8
For A3,D3=(30*0.6)+22*(1-0.6)=26.8
D3 is maximum, hence the optimal decision is A3
under Hurwicz criterion?

 Ex2:The following payoff matrix gives the payoff of different strategies S1,S2,S3 against
conditions N1,N2,N3 and N4
Conditions strategies
S1 S2 S3
N1 4000 20000 20000
N2 -100 5000 15000
N3 6000 400 -2000
N4 18000 0 1000
Indicate the decision under 1)pessimistic 2)optimistic 3)Equal probability
1)Pessimistic : max((-100,0,-2000))=0,S2 is the strategy
2)Optimistic : max(18000,20000,20000)=20000,either S2 or s3 can be chosen
3)Equal probability value:
For S1:1/4(400-100+6000+18000)=6975
For S2:1/4(20000+5000+400+0)=6350
For s3:1/4(20000+15000-2000+1000)=8500, Hence S3 is the optimal decision

Decision under risk
 Risk implies a degree of uncertainty and an inability to fully control the outcomes or consequences of
such an action. Risk or the elimination of risk is an effort that managers employ. However, in some
instances the elimination of one risk may increase some other risks. Effective handling of a risk
requires its assessment and its subsequent impact on the decision process. The decision process
allows the decision-maker to evaluate alternative strategies prior to making any decision. The process
is as follows:
1. The problem is defined and all feasible alternatives are considered. The possible outcomes for each
alternative are evaluated.
2. Outcomes are discussed based on their monetary payoffs or net gain in reference to assets or time.
3. Various uncertainties are quantified in terms of probabilities.
4. The quality of the optimal strategy depends upon the quality of the judgments. The decision-maker
should identify and examine the sensitivity of the optimal strategy with respect to the crucial factors.

Expected Monetary Value (EMV)
 To calculate the EMV in project risk management:
1. Assign a probability of occurrence for the risk.
2. Assign monetary value of the impact of the risk when it occurs.
3. Multiply Step 1 and Step 2.

 Ex1:A1,A2,A3 are the acts and S1,S2,S3 are the state of nature for the following payoff matrix,
Also known that p(S1)=0.5, P(S2)=0.4 and P(S3)=0.1, what is the decision taken by decision
maker?
State of nature Decision
A1 A2 A3
S1 30 25 22
S2 20 35 20
S3 40 30 35
EMV for A1=0.5*30+0.4*20+0.1*40=15+8+4=27
EMV for A2=0.5*25+0.4*35+0.1*30=12.5+14+3=29.5
EMV for A3=0.5*22+0.4*20+0.1*35=11+8+3.5=22.5
Highest EMV is for A3,hence the decision chosen is A3

 Ex2: Given the following pay off of three acts A1,A2 and A3 and states of nature S1,S2 and S3,
the probabilities of state of nature are respectively 0.1,0.7,0.2.calculate and tabulate the EMV
and conclude which of the acts can be chosen as the best.
States of Nature Acts
A1 A2 A3
S1 25 -10 -125
S2 200 440 400
S3 650 740 750
Solution: Tabulating the EMV, EMV is maximum for A2,hence A2 is chosen
act
Act A1
Prob*payoff
Act A2
Prob*payoff
Act A3
Prob*payoff
0.1*25=2.5 0.1*-10=-1 0.1*-125=-12.5
0.7*200=280 0.7*440=308 0.7*400=280
0.2*650=130 0.2*740=148 0.2*750=150
EMV for A1 =412.5 EMV for A2=455 EMV for A3=417.5

DECISION MAKING Decision Making Under RiskCalculate Expected Values (Ei)
Well Drilling Example-Decision Making Under Risk
N1:Dry Hole N2 :Small Well N3:Big Well
P1=0.6 P2=0.3 P3=0.1
A1:Don’t Drill $0 $0 $0
A2:Drill Alone $-500,000 $300,000 $9,300,000
Alternative
State of Nature / Probability
A3:Farm Out $0 $125,000 $1,250,000
Expected
Value
E1=0.6*0+0.3*0+ 0.1*0
$0
E2=0.6*(-500,000)+0.3*(300,000)+ 0.1*(9,300,000)
$720,000
E3=0.6*0+0.3*(125,000)+ 0.1*(1,250,000)
$162,000
$720,000
A2 is the solution if you are willing to risk $500,000

n

j=1
(pjOij)
Ei= E1=0.999*(-200)+0.001*(-200) E1=$-200
E2=0.999*0+0.001*(-100,000) E2=$-100
Example of Decision Making Under Risk
Not Fire
in your house
P1 =0.999 P2=0.001
Insure house $-200 $-200
Do not Insure house 0 $-100,000
Alternatives
Fire
in your house
Would you insure your house or not?

Decision trees
 A Decision tree is graphical representation of the decision process indicating
decisions alternatives,states of nature, probabilities attached to the states of
nature and conditional benefits and losses.
 It consist of network of nodes and branches.
• Root Node: A root node compiles the whole sample, it is then divided into multiple
sets which comprise of homogeneous variables.
• Decision Node: That sub-node which diverges into further possibilities, can be
denoted as a decision node.
• Terminal Node: The final node showing the outcome which cannot be categorized
any further, is termed as a value or terminal node.
• Branch: A branch denotes the various alternatives available with the decision tree
maker.
• Splitting: The division of the available option (depicted by a node or sub-node) into
multiple sub-nodes is termed as splitting.

Steps in Decision tree analysis
 identity all the options you have to complete your project.
 Predict potential outcomes i.e evaluate the results that each option will bring
 Analyse the results:determine the different outcomes are.
 Optimise your decisions: determining which option will best fit your project

 Ex1:Consider the following payoff table
States of Nature Probability Alternatives
A1 A2
S1 0.6 4000 10000
S2 0.4 2000 -5000
EVM of decision alternative A1=4000*0.6+2000*0.4=3200
EMV of decision alternative A2=10000*0.6-5000*0.4=4000,alternative A2 yields higher EMV,hence A2 is
the best decision

 Ex2:Construct the decision tree for following
Expand Maintain status quo Sell now
Good
competit
ion
Probability=0.7 800000 1300000 320000
Poor
competit
ion
Probability=0.3 500000 -150000 320000
Solution:EMV for Expand=800000*0.7+500000*0.3=710000
EMV for Status quo=1300000*0.7+150000*0.3=865000
EMV for sell=320000*0.7+320000*0.3=320000
Status quo gives maximum EMV hence is the best option

DECISION MAKING Decision Making Under RiskCalculate Expected Values (Ei)
Decision Trees
No Fire:
Fire:
No Fire:
Fire:
Decision
node Ai
Chance
node Nj
Outcome
(Oij)
Probability
(Pj)
Expected Value
Ei
(-200) (0.999)
x =
x = (-199.8)
(-200) (0.001)
x = (-0.2)
(0) (0.999)
x = (0)
(-100,000) (0.001)
x = (-100)
+
+
= $-200
=$-100
Mathematical solution is identical,
visual representation is different

Advantages of Decision trees
 Easy to read and interpret
 One of the advantages of decision trees is that their outputs are easy to
without even requiring statistical knowledge.
• Important insights can be generated based on experts describing a situation
(its alternatives, probabilities, and costs) and their preferences for outcomes.
• Help determine worst, best and expected values for different scenarios.
• Can be combined with other decision techniques.
• Depicts Most Suitable Project/Solution: It is an effective means of picking out
the most appropriate project or solution after examining all the possibilities.

Game Theory
 Theory of games is decision theory with general features of competitive
situations. It is helpful when two or more individuals have conflicting
objectives when trying to make decision.
 Properties of a Game
 There are finite numbers of competitors called ‘players’
 Each player has a finite number of possible courses of action called
‘strategies’
 All the strategies and their effects are known to the players but player does
not know which strategy is to be chosen
 Strategy can be classified as: 1. Pure strategy 2. Mixed Strategy

Basic Terms used in game theory
 Player –Competitor(individual or group or organization)
 Strategy – Alternate course of action(choices)
 Pure strategy – Using same strategy each time (deterministic)
 Mixed strategy – Using the course of action depending on some fixed probability.
 Optimum strategy – The choice that puts the player in the most preferred position
irrespective of his competitors strategy.

Two person zero – sum game
 Definition: Only 2 persons are involved in the game and the gain made by
one player is equal to the loss of the other.
 As the name implies, these games involve only two players .They are called
zero-sum games because one player wins whatever the other one loses, so
that the sum of their net winnings is zero.
 In general, a two-person game is characterized by
The strategies of player 1.
The strategies of player 2.
The pay-off table.

 Terms used
 Pay off matrix: The representation of gains and losses resulting from different actions of the
competitors is represented in the form of a matrix
 Payoff is the outcome of playing the game..
 Value of game: It is the expected outcome of the player when all the players of the game
follow their optimum strategy.
 Fair game: Value of the game is zero.
 Total payoff to the two players at the end of each play is zero

Formulation of Two person zero – sum game
payoff table
a11 a12 ……… a1n
a21 a22 …........ a2n
.
.
am1 am2 ………. amn
B1 B2 ……… Bn
A1
A2
.
.
Am

Formulation of Two person zero – sum game
 A1,A2,…..,Am are the strategies of player A ,row designates the course of actions
available for A
 B1,B2,…...,Bn are the strategies of player B, column designates the course of actions
available for B
 aij is the payoff to player A (by B) when the player A plays strategy Ai and B plays Bj
(aij is –ve means B got |aij| from A)

 Ex1.Consider the game of the odds and evens. This game consists of two players A,B,
each player simultaneously showing either of one finger or two fingers. If the number
of fingers matches, so that the total number for both players is even, then the player
taking evens (say A) wins Rs.1 from B (the player taking odds). Else, if the number
does not match, A pays Rs.1 to B. Thus the payoff matrix to player A is the following
table:

Optimum Solution
 A game can be solved by using the following three methods,
based on the nature of the problem.
 Saddle point concept/Max-min and Min max principle
 Dominance rule
 Graphical method.

Min- Max and Max-Min principle
 Max –Min : A row(winning) payer will select the maximum out
of the minimum gains.
 Min- Max : A column(loosing) player will always try to minimize
his maximum losses.
 Saddle point: If the max-min and min-max values are same then
the game has a saddle point and is the intersection point of both
the values.
 A game is fair if maximin value=minimax value=0
 A game is said to be determinable if maximin=minimax≠0

8 6 2 8
8 9 4 (SP) 5
7 5 3 5
B1 B2 B3 B4
A1
A2
A3
Max
Row min
Col 8 9 4 8
2
4
3
min max
max min

Solution
 The solution of the game is based on the principle of securing the
best of the worst for each player. If the player A plays strategy 1,
then whatever strategy B plays, A will get at least 2.
 Similarly, if A plays strategy 2, then whatever B plays, will get at
least 4. and if A plays strategy 3, then he will get at least 3
whatever B plays.
 Thus to maximize his minimum returns, he should play strategy
2.

Solution (cont..)
 Now if B plays strategy 1, then whatever A plays, he will lose a
maximum of 8. Similarly for strategies 2,3,4. (These are the
maximum of the respective columns). Thus to minimize this
maximum loss, B should play strategy 3.
 and 4 = max (row minima) = min (column maxima) is called the
value of the game.
 4 is called the saddle-point.
6/19/2021 45

 Ex 2:Solve the game whose payoff is given by
 solution
B1 B2 B3 Row minima
A1 1 3 4 1
A2 0 -4 -3 -4
A3 1 5 -1 -1
Column
maxima
1 5 4
Maximin=max(1,-4,-1)=1
Minimax=min(1,5,4)=1
Minimax=maximin, Hence saddle point exists i.e 1 and the optimal
strategy is (A1,B1)
B1 B2 B3
A1 1 3 4
A2 0 -4 -3
A3 1 5 -1

 Ex3: Determine which of the following two person zero sum games are strictly
determinable and fair. Given the optimum strategy for each player in the case
of strictly determinable games.
B1 B2
A1 -5 2
A2 -7 -4
Solution:
B1 B2 Row
minima
A1 -5 2 -5
A2 -7 -4 -7
Column maxima -5 2
Maximin=max(-5,-7)=-5
Minimax=min(-5,7)=-5
Maximin=minimax=-5,saddle point exist at(A1,B1)and is not equal to 0,hence game is
strictly determinable

 Ex4: Determine which of the following two person zero sum games are strictly
determinable and fair. Given the optimum strategy for each player in the case of
strictly determinable games.
B1 B2
A1 0 2
A2 -1 4
Solution
B1 B2 Row
minima
A1 0 2 0
A2 -1 4 -1
Column maxima 0 4
Maximin=max(0,-1)=-0
Minimax=min(0,4)=-0
Maximin=minimax=0,saddle point exist at(A1,B1)and is equal to 0,hence game is fair

Games without saddle point(mixed strategy)
 No pure strategy or no saddle point exists.
 The optimal mix for each player may be determined by assigning each
strategy a probability of it being chosen. Thus these mixed
strategies are probabilistic combinations of available better
strategies and these games hence called Probabilistic games.
 The probabilistic mixed strategy games without saddle points are
commonly solved by any of the following methods
 Analytical Method
 Graphical Method
 Simplex Method
6/19/2021 49

Analytical Method
 A 2x2 game without saddle point can be solved using following formula.
 The optimum mixed strategies is given by Sa and Sb,where
A1 A2
P1 P2
SA=
B1 B2
q1 q2
Sb=
Sb=
Where p1,p2,q1,q2 are as follows and V
is the value of the game
B1 B2
A1 a11 a12
A2 a21 a22

 Ex1:Solve the following game and determine its value
Player Y
strategy1 Strategy
2
Row
minima
Player
X
strategy
1
4 1 1
Strategy
2
2 3 2
Column
maxima
4 3
Maximin=max(1,2)=2
Minimax=min(4,3)=3
Maximin ≠ minimax, saddle
point does not exist, hence
apply mixed strategy

Conclusion: Out of 4 trials player X will use first strategy once and
second strategy thrice
Out of 2 trials player Y will use first strategy once and second
strategy once

 Ex2:Two player A and B match coins. If the coins match, then A wins two units
of value, if the coin do not match, then B win 2 units of value. Determine the
optimum strategies for the players and the value of the game.
 Solution: The pay off matrix for player A
Since maxmin=-2 and minmax=2. i.e maxmin is not equal
to minmax, therefore there is no unique saddle point.
Games with no saddle point should be solved using mixed
strategy

 Ex3:Consider a modified form of " matching biased coins" game problem. The
matching player is paid Rs. 8.00 if the two coins turn both heads and Rs. 1.00
if the coins turn both tails. The non-matching player is paid Rs. 3.00 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?
 Solution: The pay off matrix for matching player
Since maxmin=-3 and minmax=1. i.e maxmin is not equal to minmax, therefore there is no
unique saddle point. Games with no saddle point should be solved using mixed strategy.

Dominance Rule
 Definition: A strategy is dominated by a second strategy if the
second strategy is always at least as good (and sometimes better)
regardless of what the opponent does. Such a dominated strategy
can be eliminated from further consideration.
dominance method Steps (Rule)
Step-1: 1. If all the elements of Column-i are greater than or equal to the corresponding
elements of any other Column-j, then the Column-i is dominated by the Column-j and
it is removed from the matrix.
eg. If Column-2 ≥ Column-4, then remove Column-2
Step-2: 1. If all the elements of a Row-i are less than or equal to the corresponding elements
of any other Row-j, then the Row-i is dominated by the Row-j and it is removed from
the matrix.
eg. If Row-3 ≤ Row-4, then remove Row-3
Step-3: Again repeat Step-1 & Step-2, if any Row or Column is dominated, otherwise stop the
procedure.

 Ex1: Solve the game given below in Table after reducing it to 2 × 2 game
Solution: Reduce the matrix by using the dominance property. In the given matrix for player A, all the elements in
Row 3 are less than the adjacent elements of Row 2. Strategy 3 will not be selected by player A, because it gives
less profit for player A. Row 3 is dominated by Row 2. Hence delete Row 3, as shown in table.

 For Player B, Column 3 is dominated by column 1 (Here the dominance is opposite because
Player B selects the minimum loss). Hence delete Column 3. We get the reduced 2 × 2 matrix as
shown below in table.
Now, solve the 2 × 2 matrix, using the maximin criteria as shown below in table.
There is no saddle point and the game has a
mixed strategy.

 Ex2:Solve the following game
ayer APlayer B B1 B2 B3 B4
A1 3 5 4 2
A2 5 6 2 4
A3 2 1 4 0
A4 3 3 5 2
Solution:
Use Dominance rule to reduce the size of the payoff matrix to 2X2
Row-3 ≤ Row-4, so remove Row-3

 Column-2 ≥ Column-4, so remove Column-2
Column-1 ≥ Column-3, so remove Column-1

 Row-1 ≤ Row-3, so remove Row-1
Check for the saddle point
B3 B4 Row
minima
A2 2 4 2
A4 5 2 2
Column maxima 5 4
Maximin is not equal to minimax,2≠4,Hence no saddle point exist apply mixed
strategy formula to find p1,p2 and q1,q2

Player A A1 A2 A3 A4
0 3/5 0 2/5
Player B B1 B2 B3 B4
0 0 2/5 3/5
The optimal strategy for the players is

 Ex3:Find Solution of game theory problem using dominance method
layer APlayer B B1 B2 B3
A1 1 7 2
A2 6 2 7
A3 5 1 6
Solution:Use Dominance rule to reduce the size of the payoff matrix
row-3 ≤ row-2, so remove row-3
column-3 ≥ column-1, so remove column-3

 Check for saddle point
B1 B2 Row
minima
A1 1 7 1
A2 6 2 2
Column maxima 6 7
maximin≠minimax,2≠6,saddle point does not exist hence use mixed strategy
formulae
P1=(2-6)/(3-13)=-4/-10=2/5
P2=1-p1=1-(2/5)=3/5
Q1=(2-7)/)3-13)=-5/-10=1/2
Q2=1-q1=1-(1/2)=1/2
Hence the optimal strategy is
Player A A1 A2 A3
2/5 3/5 0
Player B B1 B2 B3
1/2 1/2 0

Decisiontree&game theory

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à Decisiontree&game theory

Similaire à Decisiontree&game theory (20)

Plus de DevaKumari Vijay

Plus de DevaKumari Vijay (20)

Dernier

Dernier (20)