# Definitional ambiguity for weakly dominated strategy with beliefs

I have been reading some game theory and I have come across the following belief-based definitions. As is recommended, it is instructive to prove the equivalence between definition 3 and the usual 'rival strategy profile' based definition of strictly dominant strategy (similarly definition 6 and the usual 'rival strategy profile' - based weakly dominant strategy). I don't have a problem in doing that. But the source I am reading has a pedagogical gap it seems as it doesn't provide definition 4. Can somebody provide that?

Note: Script U denotes the expected utility over all $$s_{-i} \in S_{-i}$$

Definition 1: strictly dominated strategy with belief $$\mu_{i}$$

A strategy $$t_i \in S_i$$, for player $$i$$, is strictly dominated by strategy $$s_i \in S_i$$ with respect to belief $$\mu_i \in \Delta (S_{-i})$$ if

$$\mathscr{U}_i(s_i,\mu_i) > \mathscr{U}(t_i, \mu_i)$$

We write, $$s_i \succ_{\mu_i} t_i$$

Definition 2: Strictly dominated strategy

A strategy $$t_i \in S_i$$, for player $$i$$, is strictly dominated by strategy $$s_i \in S_i$$ if

$$\mathscr{U}_i(s_i,\mu_i) > \mathscr{U}_i(t_i, \mu_i) \space \forall \mu_i \in \Delta (S_{-i})$$

Definition 3: Strictly dominant strategy

A strategy $$s_i \in S_i$$, for player $$i$$, is a strictly dominant strategy if it strictly dominates strategy $$t_i \in S_i$$ for all $$t_i \in S_i$$

Definition 4 Weakly dominated strategy with belief $$mu_i$$

I think this can't be defined? Please confirm.

Definition 5 Weakly dominated strategy

A strategy $$t_i \in S_i$$, for player $$i$$, is weakly dominated by strategy $$s_i \in S_i$$ if

$$\mathscr{U}_i(s_i,\mu_i) \ge \mathscr{U}_i(t_i, \mu_i) \space \forall \mu_i \in \Delta (S_{-i})$$

with strict inequality for at least one $$\hat{\mu_i} \in \Delta (S_{-i})$$

Definition 6 Weakly Dominant Strategy

A strategy $$s_i \in S_i$$, for player $$i$$, is weakly dominant strategy if it weakly dominates strategy $$t_i \in S_i$$ for all $$t_i \in S_i$$

• Who uses Definition 1? Commented Aug 10 at 23:10
• It may be helpful to provide a reference to "the source" you're reading. The mainstream textbooks typically define strategic dominance in terms of, well, strategies; whereas beliefs are used to define the closely related concept of rationalizability. Commented Aug 12 at 20:56