How to judge whether NE involves a Weakly Dominated Strategy?

Question

Consider the following normal form game:

(i) Find all Nash Equilibria in pure strategies.

I know how to find it. The answer is $\mathrm{NE}:\{C ; Y\},\{D ; Z\}$.

(ii) Based on the Pareto Dominance Criterion and Weak Dominance Criterion, can you make a good prediction in this game? Explain.

Solution: Neither NE involves a Weakly Dominated Strategy, so Weak Dominance Criterion does not apply. Neither NE yields a weakly higher payoff for both players, so Pareto Dominance does not apply. We cannot make a good prediction in this game based on NE, Pareto Dominance, and Weak Dominance.

Question: I don't quite understand part (ii). How to answer part (ii)? How to see whether the NE involves a Weakly Dominated Strategy? and whether NE involves Pareto Dominance?

Thank you.

Answer 1

You are right that neither the Pareto dominance nor Weak dominance criteria apply to refine the set of Nash equilibria.

To make this clearer, notice by iterated elimination of strictly dominated strategies, the rationalisable actions are $C,D,Y,Z$. Since the set of actions played in Nash are a subset of rationalisable actions, your game collapses to $$\begin{array}{c|cc} \text{P1} \backslash \text{P2} & Y & Z \\ \hline \text{C} & (4,3) & (0,0) \\ \text{D} & (0,0) & (3,4) \\ \end{array} $$

Therefore, the game reduces to the standard coordination game "Battle of the Sexes". I hope it is clear in this simplified game that neither the Weak dominance nor Pareto dominance criteria apply.

Therefore, we cannot well predict which equilibria will arise. This presents an example where the Nash solution concept is insufficient, even when supplemented by equilibrium refinements such as Weak and Pareto dominance.

A possible resolution is to introduce a correlation device to coordinate the players, giving rise to the concept of a correlated equilibrium.