Normal-Form Dominance Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Normal-Form Dominance Game

• Players: Player 1 and Player 2.
• Strategies: Player 1 chooses from $\{\textit{A},\textit{B},\textit{C},\textit{D}\}$; Player 2 chooses from $\{\textit{X},\textit{Y},\textit{W},\textit{Z}\}$.
• Rules: Players move simultaneously; iterated deletion removes dominated strategies.

Payoff Matrix

Derivation (Iterative Deletion of Dominated Strategies)

Step 1 (Player 1)

• $D$ is strictly dominated by $C$ since $2>1$, $3>2$, $4>3$, and $2>1$ across columns $X,Y,W,Z$.
• Reduced matrix after deleting $D$:

Step 2 (Player 2)

• After deleting $D$, $Z$ is strictly dominated by $W$ since (for rows $A,B,C$) $3>2$, $2>0$, and $2>1$.
• Reduced matrix after deleting $Z$:

Step 3 (Player 1)

• $C$ is strictly dominated by $B$ since $3>2$, $4>3$, and $5>4$ across $X,Y,W$.
• Reduced matrix after deleting $C$:

Step 4 (Player 2)

• After deleting $C$, $W$ is strictly dominated by $Y$ since $4>3$ (row $A$) and $3>2$ (row $B$).
• Reduced matrix after deleting $W$:

Step 5 (Player 1)

• $B$ is strictly dominated by $A$ since $4>3$ (col $X$) and $5>4$ (col $Y$).
• Reduced matrix after deleting $B$:

Step 6 (Player 2)

• After deleting $B$, $Y$ is strictly dominated by $X$ since $5>4$ (row $A$).
• Reduced matrix after deleting $Y$:

• Only $(A,X)$ survives the full elimination path.

Nash Equilibrium

Result:

The unique Nash equilibrium in pure strategies is:

Insights

Insight:

Iterated deletion captures iterated rationality.

• Once a strategy is strictly worse regardless of beliefs, it can be removed.
• This can create new dominated strategies in the reduced game.