Matrix with a dominant strategy

Problem

Construct a $2\times 2$ payoff matrix in which player G1 has a strictly dominant strategy and G2 does not. Identify the Nash equilibrium.

Solution

A strategy is strictly dominant for G1 if it gives a greater payoff whatever G2 does. It is enough to choose the numbers so that row $A_1$ always beats row $A_2$. An example (payoff $(G1, G2)$): $\begin{array}{c|cc} & B_1 & B_2 \\ \hline A_1 & (3,\,1) & (4,\,0) \\ A_2 & (1,\,0) & (2,\,1) \end{array}$ For G1: against $B_1$ it is better to play $A_1$ ($3>1$), against $B_2$ it is still better to play $A_1$ ($4>2$). Hence $A_1$ dominates $A_2$. G2, on the other hand, has no dominance: against $A_1$ it prefers $B_1$ ($1>0$), against $A_2$ it prefers $B_2$ ($1>0$). Since G1 certainly plays $A_1$, G2 responds with its best response against $A_1$, that is $B_1$. The Nash equilibrium is $\boxed{(A_1,\ B_1)}$

Links

Topics: Probabilita
Concepts: Equilibrio di nash · Strategia dominante
Skills: Modellizzare
Exercise type: Problema modellizzazione