January 19, 2001



EXAMPLE WHERE ITERATED DOMINANCE DOES NOT LEAD TO THE UNIQUE NASH

EQUILIBRIUM





Conjecture 1: If a game has a unique Nash equilibrium in pure strategies,

then that equilibrium can be reached by iterated dominance.



Counterexample;



                 		        COLUMN:

                                   

           		       LEFT	CENTER		RIGHT                                                    

                                                                  

 		    UP  	50,1	 1,35		1,1



ROW: 		MIDDLE		1,35	 50,1		1,1



		  DOWN          1,1	 1,1	       10,10*                                                   

                                                                  

                                                                  

    The starred combination, DOWN, RIGHT, is Nash. 



Consider ROW. Is any strategy weakly dominant? 

  UP is best, for 50,  if COL picks LEFT.

 MIDDLE is best, for 50,  if  COL picks CENTER.

 DOWN is best, for 10,  if COL picks RIGHT. 



Consider COL. Is any strategy weakly dominant?

LEFT is best, for 35, if ROW picks MIDDLE.

CENTER IS best, for 35, if ROW picks UP.

RIGHT is best, for 10, if ROW picks  DOWN. 



Thus, no strategy is dominated, and iterated dominance gets us nowhere. 





There  is, to be sure, more than one equilibrium, but there is only one

equilibrium in pure strategies. I think there are two other equilibria.

One  mixes UP, MIDDLE and LEFT, CENTER. The other mixes all three

strategies for each player. (But I haven't checked these.)



That suggests the conjecture: 





Conjecture 2: If a game has a unique Nash equilibrium, which is in   pure

strategies, then that equilibrium can be reached by iterated dominance.





Proof idea: ...



                                                              

                     Two concepts of dominance: 



1. Strategy X never does better than Y. (the standard definition)



2. Strategy X is not the best response to ANY pure  strategy of the other

player.

                                                                  

                       An interesting point: a strategy might be best

against a mix of two strategies of the other player, though it is not a

best response to any single pure strategy. That willb e true whenever a

straegy is dominated according to definition 2 but not by defintion 1.