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# Two persons zero sum game

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### Two persons zero sum game

1. 1. Yasir Hashmi 01
2. 2. Game theory
3. 3. CLASSIFICATION OF GAMES • Zero Sum Games: Most instances involve repetitive solution. The winner receives the entire amount of the payoff which is contributed by the loser. • Non-Zero Sum Games: The gains of one player differ from the losses of the other. Some other parties in the environment may share in the gain or losses.
4. 4. CLASSIFICATION OF GAMES • Two-Person Game: A game with 2 number of players. • N-Person Game: A game with N number of players, where >2.
5. 5. CLASSIFICATION OF GAMES • Pure-Strategy Game: A game in which the best strategy for each player is to play one strategy throughout the game. • Mixed-Strategy Game: A game in which each player employs different strategies at different times in the game.
6. 6. PAY OFF: • It is the sum of gains and losses from the game that are available to the players. • If in a game sum of the gains to one player is exactly equal to the sum of losses to another player, so that the sum of the gains and losses equals zero then the game is said to be a zero-sum game.
7. 7. PAY OFF: • There are also games in which the sum of the player’s’ gains and losses does not equal zero, and these games are denoted as non-zero-sum games.
8. 8. STRATEGY: • The strategy for a player is the set of alternative courses of action that he will take for every payoff (outcome) that might arise.
9. 9. STRATEGY: Strategy may be of two types: (a) Pure strategy If the players select the same strategy each time, then it is referred as pure strategy. In this case each player knows exactly what the opponent is going to do and the objective of the players is to maximize gains or to minimize losses.
10. 10. STRATEGY: (b) Mixed Strategy When the players use a combination of strategies with some fixed probabilities and each player 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.
11. 11. TWO PERSON ZERO SUM GAME: A game which involves only two players, say player A and player B, and where the gains made by one equals the loss incurred by the other is called a two person zero sum game.
12. 12. TWO PERSON ZERO SUM GAME: For example: If two chess players agree that at the end of the game the loser would pay 50Rs to the winner then it would mean that the sum of the gains and losses equals zero. So it is a two person zero sum game.
13. 13. PAY OFF MATRIX: • If Player A has m strategies represented as A1, A2, --- , Am and player B has n strategies represented by B1, B2,--- ,Bn. • Then the total number of possible outcomes is m x n. • Here it is assumed that each player knows not only his own list of possible courses of action but also those of his opponent.
14. 14. PAY OFF MATRIX: • It is assumed that player A is always a gainer whereas player B a loser. Let 푎푖푗 be the payoff which player A gains from player B if player A chooses strategy 푖 and player B chooses strategy 푗, then the pay off matrix is: Player B`s strategies 푩ퟏ 푩ퟐ 푩ퟑ 푩풏 Player A`s strategies 푨ퟏ 풂ퟏퟏ 풂ퟏퟐ 풂ퟏퟑ 풂ퟏ풏 푨ퟐ 풂ퟐퟏ 풂ퟐퟐ 풂ퟐퟑ 풂ퟐ풏 푨ퟑ 풂ퟑퟏ 풂ퟑퟐ 풂ퟑퟑ 풂ퟑ풏 푨풎 풂풎ퟏ 풂풎ퟐ 풂풎ퟑ 풂풎풏
15. 15. PAY OFF MATRIX: • By convention, the rows of the payoff Matrix denote player A’s strategies and the columns denote player B’s strategies. • Since player A is assumed to be the gainer always so he wishes to gain a payoff 푎푖푗 as large as possible and B tries to minimize the same.
16. 16. METHODS OF SOLVING 2-PERSON ZERO-SUM GAMES: 1. In case of Pure Strategy game, maximizing player arrives at optimal strategy on the basis of maximin criterion and minimizing player’s strategy is based on minimax criterion. 2. For no saddle point, we try to reduce the size of game using dominance rules.
17. 17. SOLUTION OF PURE STRATEGY GAMES: Following methods are used for solving 2 person zero sum games : 1. In a two person game if saddle point exists it is solved using pure strategies but in case of no saddle point, mixed strategies decide the results. 2. The game is solved when maximin value equals minimax value. This value is reffered as the value of game.
18. 18. EXAMPLE: Firm B Row B1 B2 B3 minimum Firm A A1 2 18 4 2 A2 16 10 8 8 Column maximum 16 18 8 As shown, The value of game is 8. The following steps are followed: 1. Find maximin value: a) Find minimum value in each row denoting minimum possible game from each strategy of A. b) Maximum value is the maximum of these minimum values.
19. 19. EXAMPLE: Firm B Row B1 B2 B3 minimum Firm A A1 2 18 4 2 A2 16 10 8 8 Column maximum 16 18 8 2. Find minimax value: a) Find maximum value in each column denoting minimum possible loss from each strategy of B. b) Minimax value is minimum of these maximum values.
20. 20. EXAMPLE: Firm B Row B1 B2 B3 minimum Firm A A1 2 18 4 2 A2 16 10 8 8 Column maximum 16 18 8 3. Find saddle point: a) At the right of each row, write the row minimum and underline the largest of them. b) At the bottom of each column, write the column maximum and underline the smallest of them
21. 21. EXAMPLE: Firm B Row B1 B2 B3 minimum Firm A A1 2 18 4 2 A2 16 10 8 8 Column maximum 16 18 8 At the right If these two elements are equal, the corresponding cell is the saddle point and the value is value of the game.