Eliminating Dominated Strategies

First a definition:

A dominated strategy:

An example: In a political primary Ms. McCain must decide whether to move to the left or to the right in her policy rhetoric.  Ms. Bradley must decide how to mold her publicity.  The choices are to head up to the high road, keep on the middle course, or drop down to the depths of low brow rhetoric.  The payoff in terms of additional delegates in the state is given by

For Bradley the strategy 'Down' is dominated by either of the other two strategies, so she should never play that.  McCain can see that as well, so she only needs to consider the first two rows of her Left and Right strategies.  Noting that Bradley won't play Down, McCain sees that Left dominates Right.  Following this reasoning, Bradley sees that given McCain's commitment to Left, it is best for her to play Up.   The solution to this game is (Up, Left).

We found the solution to this game through the use of Iterated Elimination of Dominated Strategies.  In order for this to work the players have to know that their opponents are rational and will not play dominated strategies.

Dominance solvable:

• Over how many layers of elimination can we count on the players behaving rationally and making no mistakes?
  

  		Player 2
  		Left	Center	Right
  Player 1	Top	4,5	1,6	5,6
  	Middle	3,5	2,5	5,4
  	Down	2,5	2,0	7,0

  Right is dominated by Center, at which point Down is dominated by Middle. In the remaining 2x2 game Player 2 will pick Center and Player 1 will pick Middle.

  By using IEDS, Player 2 risks getting zero if her opponent isn't behaving rationally. Should 2 guarantee herself 5 and play Left, or play the IEDS solution?
  

• Does order of elimination matter?
  

  		Player 2
  		Left	Right
  Player 1	Top	0,0	0,1
  	Bottom	1,0	0,0

  If the dominated strategies are eliminated simultaneously then the solution is Bottom - Right.  But if we first take away Left, the Player 1 is indifferent between her two choices.
  

• Are all games dominance solvable?
  

  		McCain
  		Left	Middle	Right
  Bush	Up	4,3	6,7	0,4
  	Down	5,5	5,-1	-4,-2

McCain must decide whether to move his policy positions to the Left, the Middle or to the Right. Bush must decide whether to stoop Down to attack ads or move Up to issue ads. The payoffs are the delegates they will pick up going into the convention. Is this game dominance solvable? Does it have an equilibrium?

Let's go to a longer example, the Bertrand model of duopoly.