Reason 1

A pure strategy that is undominated by other pure strategies may be dominated by a mixed strategy.

Consider the Nole - Grace game again:

 Grace's Laces U M1 M2 D Nole's Soles U 1, 0 4, 2 2, 4 3, 1 M 2, 4 2, 0 2, 2 2, 1 D 4, 2 1, 4 2, 0 3, 1

There is no dominant pure strategy for either Nole or Grace.

If Nole plays U with probability p = 1/2 and D with probability 1-p = 1/2 then when Grace plays U Nole gets 2.5 ( = 1x1/2 + 4x1/2).  When Grace plays M1 then Nole gets 2.5 with his mixed strategy. When Grace plays M2, then Nole gets 2 with his mixed strategy.  When Grace plays D then Nole will get 3 by playing his mixed strategy. Put the result of the mixed strategy in a new row of the table:

 Grace's Laces U M1 M2 D Nole's Soles U 1, 0 4, 2 2, 4 3, 1 M 2, 4 2, 0 2, 2 2, 1 D 4, 2 1, 4 2, 0 3, 1 N's mix 2.5, * 2.5, * 2, * 3, *

So the mixed strategy dominates M for Nole. In fact, for any Pr(U) = p < 2/3 the mixed strategy (U, D, p) dominates M for Nole.

If Nole plays U with probability p1 and D with probability p2 and M with probability 1-p1-p2 then this mixed strategy will dominate M as long as .You should be able to prove this.

If Grace plays U, M1, and M2 with probabilities 1/3, 1/3, 1/3 then this mixed strategy beats her pure strategy of playing D.

What is the message here?

When we consider mixed strategies for Nole then MN is dominated so it can be stricken from the table.  Similarly for Grace, consideration of mixed strategies allows us to strike DG from the table.  Making these changes:

 Grace's Laces U M1 M2 D Nole's Soles U 1, 0 4, 2 2, 4 3, 1 M 2, 4 2, 0 2, 2 2, 1 D 4, 2 1, 4 2, 0 3, 1

Grace now sees that M1 dominates U for her.  After striking UG, Nole sees that U dominates D for him. Finally, Grace sees that M2 dominates M1. The solution to the game is {UN, M2G}, which we had seen earlier was the Nash equilibrium.