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introduction game theory was developed for the purpose of analyzing competitive situations involving conflicting interests in other words game theory is used for decision making under conflicting situations where there ...

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                                              Introduction 
                                              Game theory was developed for the purpose of analyzing competitive situations involving 
                                              conflicting  interests.  In  other  words,  game  theory  is  used  for  decision  making  under 
                                              conflicting situations where there are one or more opponents (i.e., players). For example, 
                                              chess, poker, etc., are the games which have the characteristics of a competition and are 
                                              played according to definite rules. Game theory provides solutions to such games, assuming 
                                              that each of the players wants to maximize his profits and minimize his losses.  
                                              What are the underlying assumptions of game theory?  
                                                       1)  There are finite numbers of competitors (players). 
                                                       2)  The players act reasonably. 
                                                       3)  Every player strives to maximize gains and minimize losses. 
                                                       4)  Each player has finite number of possible courses of action. 
                                                       5)  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. 
                                                       6)  The pay-off is fixed and predetermined. 
                                                       7)  The pay-offs must represent utilities.  
                                              Pay-off Function 
                                              Let [aij] be any pay off matrix of order m X n. Then the pay-off function or mathematical 
                                                                                                                                                                                                                                                         m          n
                                              expectation of a game which is denoted by E (p,q) and defined as  E(p,q)                                                                                                                                                   aij piqj  
                                                                                                                                                                                                                                                        i  1       j  1
                                              Where p, q are the mixed strategies for A and B respectively. 
                                              Value of the Game 
                                              It refers to the expected outcome per play, when players follow their optimal strategy. It is 
                                                                                                                                                                    a .a                  a .a
                                              generally denoted by V and defined byv                                                                                   11       22           12       21             . 
                                                                                                                                                           (a             a ) (a                          a )
                                                                                                                                                                11            22               12            21
                                              In the subsequent sections of this chapter, we provide several trivial illustrations. It should 
                                              be noted that we make no pretense about the realism of these illustrations. 
                                              Fair Game: 
                                              If the value of the game is zero i.e. there is no loss or gain for any player, the game is 
                                              called fair game. 
                                              Pure Strategy 
                                              The simplest type of game is one where the best strategies for both players are pure 
                                              strategies. This is the case if and only if, the pay-off matrix contains a saddle point. To 
                                              illustrate, consider the following pay-off matrix concerning zero sum two person game. 
                                               
                                               
                                               
                                               
                                              DJ                                                                                                                                                                                                       Page 1 
                                               
                   
                  Example-01: For what value of  , the game with the following pay-off matrix is strictly 
                  determinable? 
                                                                     Player-A 
                                                          5            2 
                                             -1                        -8 
                                             -2           3               
                  Solution: We ignoring the value of   and determine v and vof the matrix by computing the 
                  minimums and column maximums. For this we have the matrix below. 
                                                                  Player-A 
                                                          5            2 
                                             -1                        -8 
                                             -2           3               
                  We have maximin for A (v)=2 and minimax for B (v ) =-1.The strategy for A is III and  for 
                  B    is  I.    If  the  game  is  to  be  determinable  with  its  value  v,  then  v  should  be  such 
                  that 1 v 2. 
                  Thus for a strictly determinable game,  1   v  2 
                  Example-02: What is the optimal plan for both the players? 
                                       Player-B              
                                    I     II   III    IV 
                  Player-A    I    -2     0    0      5      
                              II    4     2    1      3      
                              III 
                                   -4    -3    0      -2 
                                    5     3    -4     2      
                              IV 
                  Solution: We use the maximin (minimax) principle to analyze the game. 
                                           Player-B              Minimum 
                                       I     II    III    IV 
                  Player-A      I     -2     0     0       5         -2 
                                II    4      2     1       3         1 
                                III   -4     -3    0      -2         -4 
                                                            
                                      5      3     -4      2         -6 
                                IV 
                   Maximum            5      3     1       5           
                  DJ                                                                           Page 2 
                   
                            
                           Select minimum from the maximum of columns. ie. Minimax = 1 
                           Player A will choose II strategy, which yields the maximum payoff of 1.  
                            
                           Select maximum from the minimum of rows. ie.  Maximin = 1 
                           similarly, player B will choose III strategy.  
                           Since the maximin = the minimax, therefore, = 1.  
                           The optimal strategies for both players are: Player A must select II strategy and player B 
                           must select III strategy. The value of game is 1, which indicates that player A will gain 1 
                           unit and player B will sacrifice 1 unit. 
                           Mixed Strategy                                                                                                  WBUT-08 
                           In  situations  where  a  saddle  point  does  not  exist,  the  maximin  (minimax)  principle  for 
                           solving a game problem breaks down. The concept is illustrated with the help of following 
                           example. 
                           A mixed strategy game can be solved by following methods:  
                                1.  Algebraic Method 
                                2.  Calculus Method 
                                3.   Linear Programming Method  
                           Dominance                                                                                                CS-511/2003 
                           The principle of dominance states that if one strategy of a player dominates over the other 
                           strategy in all conditions then the later strategy can be ignored. A strategy dominates over 
                           the other only if it is preferable over other in all conditions. The concept of dominance is 
                           especially useful for the evaluation of two-person zero-sum games where a saddle point 
                           does not exist. 
                           Rule: 
                            
                                1.  If all the elements of a column (say ith column) are greater than or equal to the 
                                     corresponding elements of any other column (say jth column), then the ith column is 
                                     dominated by the jth column and can be deleted from the matrix. 
                                      
                                2.  If all the elements of a row (say ith row) are less than or equal to the corresponding 
                                     elements of any other row (say jth row), then the ith row is dominated by the jth row 
                                     and can be deleted from the matrix. 
                                      
                           Advantages & Limitations of Game Theory 
                           Advantages 
                                1.  Game theory gives insight into several less-known aspects, which arise in situations 
                                     of  conflicting interests. For example, it describes and explains the phenomena of 
                                     bargaining and coalition-formation. 
                                2.  Game theory develops a framework for analyzing decision making in such situations 
                                     where interdependence of firms is considered. 
                                3.  At  least  in  two-person  zero-sum  games,  game  theory  outlines  a  scientific 
                                     quantitative technique that can be used by players to arrive at an optimal strategy. 
                           Limitations 
                           DJ                                                                                                                  Page 3 
                            
                              
                                   1.  The assumption that players have the knowledge about their own pay-offs and pay-
                                         offs of others is not practical. 
                                   2.  The techniques of solving games involving mixed strategies particularly in case of 
                                         large pay-off matrix are very complicated. 
                                   3.    All the competitive problems cannot be analyzed with the help of game theory. 
                             Example-05: Find the value of 2X2 game algebraically by using mixed strategies? 
                                                                     B 
                                                               A     2                3           
                                                                             4        1
                             Solution: The problem has no saddle point. It can be solved by  mixed strategies. Let 
                             p=(p ,p )  and  q=(q ,q )  with    p p                                       p ,p > 0  and  q1+q2 =1,  q ,q >0  be  the 
                                     1    2                       1   2                    1+ 2=1nad         1   2                                         1   2
                             probabilities for the player A and B respectively. Assuming the existence of the value of the 
                             game, considering from the A’s point of view, 
                             E (p)=2p +3p =2p +3(1-p ) 
                               1            1       2       1            1
                             E (p)=4p - p = 4p  -(1-p ) 
                               2            1      2        1          1
                             To  determine  the  optimal  values  of  p ,p ,  we  have  2p +3(1-p )=  4q   -(1-q ).  Solving 
                                                                                               1   2                         1           1           1           1
                                 *                   *
                             p1 =5/6 and p2 =1/6. 
                              
                             Again for B’s point of view 
                             E (q)=2q +4q =2q +4(1-q ) 
                               1            1       2       1            1
                             E (q)=4q - q = 4q  -(1-q ) 
                               2            1      2        1          1
                             To  determine  the  optimal  values  of  p ,p ,  we  have  2q +4(1-q )=  4q   -(1-q ).  Solving 
                                                                                               1   2                         1           1           1           1
                                 *                   *                                                                     *        *
                             q1 =2/3 and q2 =1/3. And the value of the game (v) =2q1 +3q2 =7/3. 
                                                                                     *                             *
                             Hence the optimal strategies are p =(5/6,1/6) and q (2/3,1/3) and v=7/3. 
                                                                                *    *          5 2              1 2              5 1              1 1 7
                             Again it can be verified that E(p ,q )=2                           6 3 4 6 3 3 6 6 1 6 3 3. 
                              
                              
                              
                              
                              
                              
                              
                             Exercises 
                              
                             01. Solve the following 2X3 game graphically                                                    MB/205/07 
                                                                                     B 
                                                               A       1`     3      11         
                                                                      8       5        2
                              
                             02. Explain the following in terms of game theory:                                MB/205/08 
                                  a)  Pay-off matrix    b) Pure strategy     
                                  c)  Mixed strategy   d) Saddle point 
                                   e) Value of a rectangular game 
                              
                              
                              
                             DJ                                                                                                                              Page 4 
                              
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...Introduction game theory was developed for the purpose of analyzing competitive situations involving conflicting interests in other words is used decision making under where there are one or more opponents i e players example chess poker etc games which have characteristics a competition and played according to definite rules provides solutions such assuming that each wants maximize his profits minimize losses what underlying assumptions finite numbers competitors act reasonably every player strives gains has number possible courses action choices assumed be made simultaneously so no knows opponent s choice until he decided own course pay off fixed predetermined offs must represent utilities function let any matrix order m x n then mathematical expectation denoted by p q defined as aij piqj j mixed strategies b respectively value it refers expected outcome per play when follow their optimal strategy generally v byv subsequent sections this chapter we provide several trivial illustratio...

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