Game theory and Cournot and Bertrand equilibriumFind all equilibria of this game: Row/Column L M R ...

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Game theory and Cournot and Bertrand equilibriumFind all equilibria of this game: Row/Column L M R  U (6,6) (2,9) (2,7)  D (4,8) (4,5) (5,6)        
	
Document Preview: Autumn 2012 December 17, 2012 Question 1 Find all equilibria of this game: Row/Column L M R U (6,6) (2,9) (2,7) D (4,8) (4,5) (5,6) Question 2 The two commanders, i = 1, 2, of opposing armies choose simultaneously the quantity qi of resources that they want to commit to a battle. The choice qi has to satisfy: 0 = qi = 1. The battle is won by the commander who chooses to commit the larger quantity of resources. If both commanders commit the same resources, then each commander has a chance of 0.5 of being the winner. The winner wins a prize of value v > 0, and keeps his resources, so that his von Neumann Morgenstern utility is equal to v. The loser wins no prize, and loses all his resources, so that his von Neumann Morgenstern utility is equal to -"qi. Part a Show that there is no Nash equilibrium in pure strategies of this game in which q1 =q2

Robert · Accepted Answer

1) We select largest no in columns from first element and largest element in row form 
second element if they occur at same place it is equilibrium for pure strategy 
2
( 6 ,6) (2, 9 ) (2,7)
1
(4, 8) ( 4 ,5) (5,6)
player
player
 Which is not at the same place so we have to search for 
mixed strategy Nash equilibrium. 
Now by definition a pair of mixed strategies 
 * *,n mX S Y S  is a Nash equilibrium if 
* * *
1 1, ,              nE X Y E X Y X S          
And 
* * *
2 2, ,              mE X Y E X Y Y S          
Where Sn and Sm are mixed strategies for player 1 and Player 2 respectively. 
By dominance rule we see that strategy ( 2,7) for player 2 is always dominated by 
strategy (2,9). 
So we get reduced game as 
2
(6,6) (2,9)
1
(4,8) (5,

Game theory and Cournot and Bertrand equilibrium Find all equilibria of this game: Row/Column L M R U (6,6) (2,9) (2,7) D (4,8) (4,5) (5,6) Document Preview: Autumn 2012 December 17, 2012 Question 1...

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