Game Theory Nita H. Shah Ravi M. Gor Hardik Soni

1. Game Theory

2. Syllabus
 Introduction
 Two person Zero-sum Games
 Maximin and Minimax Principles
 Mixed Strategies, Expected Pay-Off
 Solution of 2 x 2 Mixed Strategy Game
 Solution of 2 x 2 Mixed Strategy Game by the method of oddments
 Dominance Principle
 Graphical method for solving a 2 x n game

3. Introduction
 There are two opposite parties with conflicting interests, and the action of one depends
upon the action which the opponent takes.
 The outcome of the situation is controlled by the decisions of all the parties involved is
termed a competitive situation.
Example:
 All sort of activities, in games, sports, business, economic, political, social and military
fields.

4. Conflict over time:
 The generic term used to characterize these situations is games.
 Participants -> Competitors is “ The success of one is usually at the expense of the other”.
 Each participant selects and executes those strategies.
 He / She believes will results in his “Winning the game”.
 The participants make use of deductive and inductive logic in determining a strategy for
winning.

5.  Theory of games started in 20th century.
 “John Von Neumann and Morgenstern” in 1944 published their
“Theory of games and economic behavior”.
 Von Neumann’s approach “The minimax principle”.
 It involves the fundamental idea of the minimization of maximum loss.
 A competitive situation is called a game.

6. Four properties:
 Players  There are a finite number of competitors
 Game involving n persons is called n person game.
 Game interests of only two parties may be in conflict is called two person game.
 A list of finite or infinite number of possible course of action each called a strategy, is
available to each player.
 The list need not be the same for each player.
 The strategy of a player is the pre-determined rule by which a player decides his course of
action from his own list during the game.

7. Two types of strategies
 Pure strategy  One knows in advance all strategies, out of which he always selects
only one particular strategy.
 This choice is made irrespective of the strategy others may choose.
 Mixed Strategy  A Player decides in advance his courses of action in accordance
with some fixed probability distribution.

8. Advantages
 The mixed strategy over the pure strategy is that a player has only finite choices of pure
strategies but he has infinite number of mixed strategies.
 A play is played when each player chooses one of his courses of action.
 The choices are assumed to be made simultaneously, so that no player knows his opponent’s
choice until he has decided his own course of action.

9.  Pay-Off  Every play is a combination of courses of action is associated with an outcome,
which determines a set of gains, one to each player.
 A loss is considered as a negative gain.
 Thus, after each player of the game, one player pays to other an amount determined by the
courses of action chosen.
 A pay-off matrix is a table which shows how payments should be made at the end of a play
or the game.

10. Two person zero-sum games
 A game with two players in which the gain of one player is equal to
the loss of the other.

11. Zero-sum game:
 A game in which the gains of one player are the losses of the other players.
 Example:
 The algebraic sum of gains to all players after a play is bound to be zero.

12. Example 18.1
Player B
Strategy B1 Strategy B2
Player A Strategy A1 Player A Wins 2 Player A Wins 3
Strategy A2 Player B Wins 1 Player B Wins 2

13. Solution
Player B
B1 B2
Player A A1 2 3
A2 -1 -2

14.  A wins the game only by playing his strategy A1  A1 plays all the time.
 B realizes that A will play strategy A1 all the time and in an effort to minimize A’s
gains, plays his strategy B1.
 The solution to the game is A1, B1.
 A wins 2 points and B loses 2 points each time the game is played.
 The value of the game is A is 2 and B is -2.

15. Optimal Strategies
 It provides the best situation in the game in the sense that it provides maximal pay-off to the players.

16. Value of the game
 The expected outcome per play when the players follow their optimal strategy.
 It is generally denoted by V.
 A game, whose value is zero, is called a fair game.
 By solving a game, to find the best strategies for both the players and the value of the game.

17. Maximin and minimax principles
 A player lists his worst possible outcomes and then he chooses that strategy which corresponds to
the best of these worst outcomes.
 The best strategies for each player on the basis of the maximin and minimax criterion of optimality.
 MAXIMIN = MAXIMUM (ROW, MINIMUM)
 MINIMAX = MINIMUM (COLUMN, MAXIMUM)

18. Saddle-point situation
 It guarantees that neither of the player is tempted to select a better strategy.
 Pay-off to A > max(i) min(j) aij
 Pay-off to B < min(j) max(i) aij
 If the maximin value for a player is equal to the minimax value for another player is
 max(i) min(j) aij = min(j) max(i) aij
 The game is said to have a saddle point and the corresponding strategies are the optimal strategies.
 The amount of pay-off at the saddle point is the value of the game V
 max(i) min(j) aij = min(j) max(i) aij = V
 It can be recognized because it corresponds to both the smallest numerical value in its row and the
largest numerical value in its column.

19.  If a game has a saddle point, then the pure strategies corresponding to the saddle
point are the optimal strategies and the number at that point is the value of the
game.