theorem
admitted
Dominant Strategy Profile is a Nash Equilibrium
If every player \(i\) plays a weakly dominant strategy \(s_i\), then the resulting profile \(\sigma = (s_i)_{i \in I}\) is a Nash equilibrium.
Proof
A weakly dominant strategy weakly dominates every alternative. In particular, for any player \(i\) and any deviation \(s'_i\), we have \(u_i(\sigma[i \mapsto s'_i]) \le u_i(\sigma)\). This means each player is best responding, so \(\sigma\) is a Nash equilibrium. \(\square\)
References
- [MSZ, Chapter 2] Maschler, Solan, and Zamir, Game Theory. Relationship between dominant strategies and Nash equilibrium.