# Are strict dominance solvable games weakly dominance solvable?

Okay consider a game $G$ if a strategy $s_i$ has the following property we call $s_i$ the strictly dominant strategy

$$u_i(s_i,s_{-i})>u_i(s_i',s_{-i}) \\ \forall s_{-i} \ \forall s_i' \epsilon S'_i$$

Where $s_{i}$ indicates the strategies of players other then $i$ in the game and $S'_i$ is the set for strategies of player $i$ except the specific strategy $s_i$

Now let's look at the definition of a weakly dominant strategy

if a strategy $s_i$ has the following property we call $s_i$ the weakly dominant strategy

$$u_i(s_i,s_{-i})≥u_i(s_i',s_{-i}) \\ \forall s_{-i} \ \forall s_i' \epsilon S'_i \ and \\ \exists s_i' \epsilon S'_i \ such \ that: \ u_i(s_i,s_{-i})>u_i(s_i',s_{-i})$$

Okay I believe from these two definitions we can derive that any strictly dominant strategy $s_i$ is also a weakly dominant strategy

Definition of strict dominance solvable is as follows :

A strict dominance solvable game is a game where the equilibrium outcome is strict dominance equilibrium. A weakly dominance solvable games is a game where the equilibrium outcome is weakly dominance equilibrium.

So It seems like then a strictly dominance solvable game is always a weakly dominance solvable game. Am I wrong?

You're right. Let $s^*=(s_1^*,\dots,s_N^*)$ be the equilibrium of a strictly dominance solvable game. Then by definition, $$u_i(s_i^*,s_{-i})>u_i(s_i,s_{-i})$$ for all $i$, all $s_i\ne s_i^*$ and all $s_{-i}$. This implies that $$u_i(s_i^*,s_{-i})\ge u_i(s_i,s_{-i})$$ for all $i$, all $s_i\ne s_i^*$, all $s_{-i}$ and with strict inequality for at least some $s_i$ (in fact, for all $s_i\ne s_i^*$). This makes $s^*$ an equilibrium satisfying the weak dominance solvability criterion.
• An example: Both players have to choose between strategy $A$ and $B$. If your opponent choses $A$ your payoff is 0, otherwise it is 1. In these games both $A$ and $B$ are weakly dominant strategies. You can eliminate either one and have whatever you want as a solution: All strategy profiles are weakly dominant equilibria. – Giskard Feb 21 '17 at 20:20