Game Theory Basics – 2 Saddle Point

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#Operations #Research #Math #Statistics #Game #Theory #Saddle #Point #Minimax #Maximin #Value # #FreeStudy

Game Theory Basics:

Two-person, zero-sum game
A game with only two players (player A and player B) is called a ‘two-person, zero-sum game’, if the losses of one player are equivalent to the gains of the other so that the sum of their net gains is zero.
Two-person, zero-sum games are also called rectangular games as these are usually represented by a payoff matrix in a rectangular form.

Number of activities
The activities may be finite or infinite.

Payoff
The quantitative measure of satisfaction a person gets at the end of each play is called a payoff

Payoff matrix
Suppose the player A has ‘m’ activities and the player B has ‘n’ activities. Then a payoff matrix can be formed by adopting the following rules
 Row designations for each matrix are the activities available to player A
 Column designations for each matrix are the activities available to player B
 Cell entry Vij is the payment to player A in A’s payoff matrix when A chooses the activity i and B chooses the activity j.
 With a zero-sum, two-person game, the cell entry in the player B’s payoff matrix will be negative of the corresponding cell entry Vij in the player A’s payoff matrix so that sum of payoff matrices for player A and player B is ultimately zero.

Value of the game
Value of the game is the maximum guaranteed game to player A (maximizing player) if both the players uses their best strategies. It is generally denoted by ‘V’ and it is unique.

Saddle point
A saddle point of a matrix is the position of such an element in the payoff matrix, which is minimum in its row and the maximum in its column.
Procedure to find the saddle point
 Select the minimum element of each row of the payoff matrix. Write them in a new column besides the matrix and mark them with circles wherever they are in the matrix. From the column of the minimum values, find out the maximum value and mark it with circle. This value is known as “Maximin” value.
 Select the maximum element of each column of the payoff matrix. Write them in a new row below the matrix and mark them with squares wherever they are in the matrix. From the row of the maximum values, find out the minimum value and mark it with square. This value is known as “Minimax” value.
 If their appears an element in the payoff matrix with a circle and a square together then that position is called saddle point and the element is the value of the game. In other words, if the “Minimax” value and the “Maximin” value are the same, then it is the saddle point.

Solution of games with saddle point
To obtain a solution of a game with a saddle point, it is feasible to find out
 Best strategy for player A (i.e. the strategy with “Maximin” value)
 Best strategy for player B (i.e. the strategy with “Minimax” value)
 The value of the game
The best strategies for player A and B will be those which correspond to the row and column respectively through the saddle point.

* If Maximin value = Minimax value = V, then the game is strictly determinable, otherwise not

* If Maximin value = Minimax value = V = 0, then the game is ‘FAIR’, otherwise it is not fair.

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32 comments

  1. Little Monk 21 June, 2020 at 11:14 Reply

    Very good. But you did not say what if the minimax and maximin were not the same? what would that mean? Why Game point = 0 is fair etc.
    Thanks

  2. Sk Samirul Islam 21 June, 2020 at 11:14 Reply

    player B

    player A 10 81 32 43 93

    59 63 39 69 73

    71 20 5 27 84

    34 14 44 44 69

    Dear Sir, Please help me to solve this GAME, I am unable to do this from last 10 days…….
    Please help me…..

  3. s.waqar Hasan 21 June, 2020 at 11:14 Reply

    any one have answer of this question:
    . Two firms dominate the market for surgical sutures and compete aggressively with respect to research and development. The following payoff table depicts the profit implications of their different R&D strategies. a. Suppose that no communication is possible between the firms; each must choose its R&D strategy independently of the other. What actions will the firms take, and what is the outcome? b. If the firms can communicate before setting their R&D strategies, what outcome will occur? Explain.

    Firm B’s R&D Spending Low Medium High Low 8, 11 6, 12 5, 14 Firm A’s R&D Medium 12, 9 8, 10 6, 8 Spending High 11, 6 10, 8 4, 6

  4. Ayeman Ibna Hasan 21 June, 2020 at 11:14 Reply

    sir, what should i do, if the maximin and minimax value are not equal but there exists the same value like in
    Row minimum – 2, 8, 4
    Column maxima- 6, 10, 2
    Here, the maximin is 8
    And minimax is 2
    But in between maximin and minimax there exists 2, can i take this 2 as value of the game?
    Or should i use mixed strategy?
    Thank you.

  5. debashmita guha roy 21 June, 2020 at 11:14 Reply

    If anyone among minimax;maxmini and value of game is 0 then we can say that the game is fair??????

  6. Sreemana Bhattacharyya 21 June, 2020 at 11:14 Reply

    Sir please explain the game theory in the logic of set theory… In our syllabus we study the game theory in respect of set theory… Bt I can't understand those theories… please sir help me…

  7. Prachi Garg 21 June, 2020 at 11:14 Reply

    hello sir, i want to thank you for all these lectures. it was my mcom final year and i studied all chapters of operational research from your videos and i attempted all questions with 100% accuracy. sir u simplified this subject so much. your teaching method is very effective and simple. thankyou very much

  8. Poulami Karmakar 21 June, 2020 at 11:14 Reply

    Please suggest some good books on game theory which has a lot of information on game theory… Please sir suggest books … Which gives me a Cristal clean concept of game theory

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