<p>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?</p>
<p>Note: Script U denotes the expected utility over all  <span class="math-container">$s_{-i} \in S_{-i}$</span></p>
<p><strong>Definition 1: strictly dominated strategy with belief <span class="math-container">$\mu_{i}$</span></strong></p>
<p>A strategy <span class="math-container">$t_i \in S_i$</span>, for player <span class="math-container">$i$</span>, is strictly dominated by strategy <span class="math-container">$s_i \in S_i $</span> with respect to belief <span class="math-container">$\mu_i \in \Delta (S_{-i})$</span> if</p>
<p><span class="math-container">$\mathscr{U}_i(s_i,\mu_i) &gt; \mathscr{U}(t_i, \mu_i)$</span></p>
<p>We write, <span class="math-container">$s_i \succ_{\mu_i} t_i$</span></p>
<p><strong>Definition 2: Strictly dominated strategy</strong></p>
<p>A strategy <span class="math-container">$t_i \in S_i$</span>, for player <span class="math-container">$i$</span>, is strictly dominated by strategy <span class="math-container">$s_i \in S_i $</span> if</p>
<p><span class="math-container">$\mathscr{U}_i(s_i,\mu_i) &gt; \mathscr{U}_i(t_i, \mu_i) \space \forall \mu_i \in \Delta (S_{-i})$</span></p>
<p><strong>Definition 3: Strictly dominant strategy</strong></p>
<p>A strategy <span class="math-container">$s_i \in S_i$</span>, for player <span class="math-container">$i$</span>, is a strictly dominant strategy if it strictly dominates strategy <span class="math-container">$t_i \in S_i $</span> for all <span class="math-container">$t_i \in S_i $</span></p>
<p><strong>Definition 4 Weakly dominated strategy with belief <span class="math-container">$mu_i$</span></strong></p>
<p>I think this can't be defined? Please confirm.</p>
<p><strong>Definition 5 Weakly dominated strategy</strong></p>
<p>A strategy <span class="math-container">$t_i \in S_i$</span>, for player <span class="math-container">$i$</span>, is weakly dominated by strategy <span class="math-container">$s_i \in S_i $</span> if</p>
<p><span class="math-container">$\mathscr{U}_i(s_i,\mu_i) \ge \mathscr{U}_i(t_i, \mu_i) \space \forall \mu_i \in \Delta (S_{-i})$</span></p>
<p>with strict inequality for at least one <span class="math-container">$ \hat{\mu_i} \in \Delta (S_{-i}) $</span></p>
<p><strong>Definition 6 Weakly Dominant Strategy</strong></p>
<p>A strategy <span class="math-container">$s_i \in S_i$</span>, for player <span class="math-container">$i$</span>, is weakly dominant strategy if it weakly dominates strategy <span class="math-container">$t_i \in S_i $</span> for all <span class="math-container">$t_i \in S_i$</span></p>

CC BY-SA 4.0
2024-08-10T23:10:48.520Z
Who uses Definition 1?
Michael Greinecker
CC BY-SA 4.0
2024-08-12T20:56:13.170Z
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.
Herr K.
<microeconomics><game-theory><dominant-strategy>
Definitional ambiguity for weakly dominated strategy with beliefs
Karan Kumar
# Definitional ambiguity for weakly dominated strategy with beliefs

(Asked by: username_0 on 2024-08-10)

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 $t\_i \in S\_i$

**Tags:** microeconomics, game-theory, dominant-strategy
> **Comments:**
> **username_1**: Who uses Definition 1?  
> **username_2**: 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.  

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## 0 answers

