I am confused between the difference of what a solutions that a Nash equilibrium method provides versus an IESDS method. Let me explain through an example:
We have player 1 (P1) with strategies, $s_1$ = $\{ X, Y,Z \}$ and player 2 (P2) with strategies, $s_2$ = $\{ A, B, C \}$
$$
\begin{array}{c|lcr}
& \text{A} & \text{B} & \text{C} \\
\hline
X & 0,2 & 1,0 & 0,0 \\
Y & 0,1 & 0,0 & 1,1 \\
Z & 0,1 & 0,1 & 0,1
\end{array}
$$
Now, using the method of Nash equilbrium, for every column, which is strategy for P2, we identify the highest payoff for P1. Let's Underline these payoffs. This gives:
$$
\begin{array}{c|lcr}
& \text{A} & \text{B} & \text{C} \\
\hline
X & \underline{0,2} & \underline{1,0} & 0,0 \\
Y & \underline{0,1} & 0,0 & \underline{1,1} \\
Z & \underline{0,1} & 0,1 & 0,1
\end{array}
$$
The second step of this method is, for every row, which is a strategy for P1, identify the highest payoff for P2. Let's Over-line these payoffs that satisfy this condition. This yields:
$$
\begin{array}{c|lcr}
& \text{A} & \text{B} & \text{C} \\
\hline
X & \overline{\underline{0,2}} & \underline{1,0} & 0,0 \\
Y & \overline{\underline{0,1}} & 0,0 & \overline{\underline{1,1}} \\
Z & \overline{\underline{0,1}} & \overline{0,1} & \overline{0,1}
\end{array}
$$
All entries that are left with both an underline and an overline are Nash equilibria, correct? So we have four in this case: $(X,A),(Y,A),(Z,A),(Y,C)$.
I am confused because if we go with the IESDS (by weak dominance), then the only strategy that remains is $(Y,A)$, so are they both right?
Best Answer
For the record: this game has the four Nash equilibria in pure strategies that you have found above. However, contrary to your statement above, under IEWDS (iterated elimination of weakly dominated strategies) three of them survive: $(X,A),(Y,A),(Z,A)$.
It is generally known that IESDS never eliminates NE, while IEWDS may rule out some NE. The game presented in your question illustrates this fact, with the additional twist that the eliminated NE achieves the (unique) first-best for Player~1. So, here IEWDS eliminates the equilibrium uniquely preferred by one player.
This is why we say (tongue-in-cheek) that IEWDS should be applied with caution.