<p>I agree with Herr, the payoff matrix looks right. Also, there are no strictly dominated strategies because a strictly dominated strategy cannot be a best response for any possible belief. However, If any player believes that the other player is choosing 19, then every strategy (both pure and mixed) is a best response.</p>

CC BY-SA 4.0
2019-05-08T19:49:01.087Z
2019-05-08T19:49:01.087Z
Regio
<p><strong>Q1</strong></p>

<p>Your table seems to be correct. Here is a quick Python implementation for generating the payoffs:</p>

<pre><code>def payoff_calculator(x, y):
    if x+y &lt; 14:
        return (x+1,y+1)
    else:
        if x==y:
            return (x,y)
        else:
            return (y,20-x) if x &lt; y else (20-y,x)

primes = [2,3,5,7,11,13,17,19]
payoffs = [[payoff_calculator(i,j) for i in primes] for j in primes]

for row in payoffs:
    print(*row)
</code></pre>

<p><strong>Result</strong></p>

<pre><code>(3, 3) (4, 3) (6, 3) (8, 3) (12, 3) (18, 13) (18, 17) (18, 19)
(3, 4) (4, 4) (6, 4) (8, 4) (17, 11) (17, 13) (17, 17) (17, 19)
(3, 6) (4, 6) (6, 6) (8, 6) (15, 11) (15, 13) (15, 17) (15, 19)
(3, 8) (4, 8) (6, 8) (7, 7) (13, 11) (13, 13) (13, 17) (13, 19)
(3, 12) (11, 17) (11, 15) (11, 13) (11, 11) (9, 13) (9, 17) (9, 19)
(13, 18) (13, 17) (13, 15) (13, 13) (13, 9) (13, 13) (7, 17) (7, 19)
(17, 18) (17, 17) (17, 15) (17, 13) (17, 9) (17, 7) (17, 17) (3, 19)
(19, 18) (19, 17) (19, 15) (19, 13) (19, 9) (19, 7) (19, 3) (19, 19)
</code></pre>

<p><strong>Q2</strong></p>

<p>You are right, there are no strictly dominated strategies here. This is because each action is a best response to some opponent action. For example, 2 is a best response to opponent moves 13, 17, and 19, 3 is a best response to 11, 13, 17, and 19, and so on. Similarly, there is no strictly dominant strategy. For example, 19 is a best response to 2, 3, 5, 7, and 19, but not to 11, 13, or 17.</p>

CC BY-SA 4.0
2019-06-12T11:26:26.113Z
2019-06-12T11:59:27.273Z
2019-06-12T11:59:27.273Z
Frederic Schneider
<p>Two players simultaneously announce a prime number less than 20.Denoting <span class="math-container">$p_{i}$</span> the number announced by the player <span class="math-container">$i$</span>, the payoffs are:</p>

<p>-If <span class="math-container">$p_{1}+p_{2}&lt;14$</span> each player receives as payment the value of <span class="math-container">$p_{i}+1$</span>.</p>

<p>-If <span class="math-container">$p_{1}+p_{2} \geq{14}$</span> and <span class="math-container">$p_{i}&lt;p_{j}$</span>, player <span class="math-container">$i$</span> receives <span class="math-container">$p_{j}$</span> and the player <span class="math-container">$j$</span> receive <span class="math-container">$20-p_{i}$</span>.</p>

<p>-If <span class="math-container">$p_{1}+p_{2} \geq{14}$</span> and <span class="math-container">$p_{i}=p_{j}$</span>, each player receives a payoff <span class="math-container">$p_{i}$</span>.</p>

<p>Q1:My game in normal form is correct?</p>

<p><span class="math-container">\begin{array}{|c|c|c|c|}
\hline
 &amp; 2 &amp; 3&amp;5&amp;7&amp;11&amp;13&amp;17&amp;19 \\ \hline
2&amp;3,3&amp; 3,4&amp;3,6 &amp; 3,8&amp;3,12&amp;13,18&amp;17,18&amp;19,18\\ \hline
3 &amp; 4,3 &amp; 4,4&amp; 4,5&amp;4,8&amp;11,17&amp;13,17&amp;17,17&amp;19,17\\ \hline
 5&amp; 6,3 &amp; 6,4&amp; 6,6&amp;6,8&amp;11,15&amp;13,15&amp;17,15&amp;19,15\\ \hline 
7&amp; 8,3 &amp; 8,4&amp; 8,6&amp;7,7&amp;11,13&amp;13,13&amp;17,13&amp;19,13\\ \hline 
11&amp; 12,3 &amp; 17,11&amp; 15,11&amp;13,11&amp;11,11&amp;13,9&amp;17,9&amp;19,9\\ \hline 
13&amp; 18,13 &amp; 17,13&amp; 15,13&amp;13,13&amp;9,13&amp;13,13&amp;17,7&amp;19,7\\ \hline 
17&amp; 18,17 &amp; 17,17&amp; 15,17&amp;13,17&amp;9,17&amp;7,17&amp;17,17&amp;19,3\\ \hline 
19&amp; 18,19 &amp; 17,19&amp; 15,19&amp;13,19&amp;9,19&amp;7,19&amp;3,19&amp;19,19\\ \hline 
\end{array}</span></p>

<p>Q2:There are strictly dominated strategies?
For the table, I think not. Is correct?</p>

<p>Thanks!</p>

CC BY-SA 4.0
2019-05-08T15:26:35.057Z
The game looks correct (I only checked a few cells though). Based on the matrix, there is no strictly dominated strategy, mainly because of the last row/column.
Herr K.
<game-theory><dominant-strategy>
Are there strictly dominated strategies?
Franciscolli
# Are there strictly dominated strategies?

(Asked by: username_0 on 2019-05-08)

Two players simultaneously announce a prime number less than 20.Denoting $p\_{i}$ the number announced by the player $i$, the payoffs are:

-If $p\_{1}+p\_{2}<14$ each player receives as payment the value of $p\_{i}+1$.

-If $p\_{1}+p\_{2} \geq{14}$ and $p\_{i}<p\_{j}$, player $i$ receives $p\_{j}$ and the player $j$ receive $20-p\_{i}$.

-If $p\_{1}+p\_{2} \geq{14}$ and $p\_{i}=p\_{j}$, each player receives a payoff $p\_{i}$.

Q1:My game in normal form is correct?

\begin{array}{|c|c|c|c|}
\hline
& 2 & 3&5&7&11&13&17&19 \\ \hline
2&3,3& 3,4&3,6 & 3,8&3,12&13,18&17,18&19,18\\ \hline
3 & 4,3 & 4,4& 4,5&4,8&11,17&13,17&17,17&19,17\\ \hline
5& 6,3 & 6,4& 6,6&6,8&11,15&13,15&17,15&19,15\\ \hline
7& 8,3 & 8,4& 8,6&7,7&11,13&13,13&17,13&19,13\\ \hline
11& 12,3 & 17,11& 15,11&13,11&11,11&13,9&17,9&19,9\\ \hline
13& 18,13 & 17,13& 15,13&13,13&9,13&13,13&17,7&19,7\\ \hline
17& 18,17 & 17,17& 15,17&13,17&9,17&7,17&17,17&19,3\\ \hline
19& 18,19 & 17,19& 15,19&13,19&9,19&7,19&3,19&19,19\\ \hline
\end{array}

Q2:There are strictly dominated strategies?
For the table, I think not. Is correct?

Thanks!

**Tags:** game-theory, dominant-strategy

> **Comments:**
> **username_1**: The game looks correct (I only checked a few cells though). Based on the matrix, there is no strictly dominated strategy, mainly because of the last row/column.  

---

## 2 answers

### Answer by username_2 on 2019-05-08. Score: 1

I agree with username_1, the payoff matrix looks right. Also, there are no strictly dominated strategies because a strictly dominated strategy cannot be a best response for any possible belief. However, If any player believes that the other player is choosing 19, then every strategy (both pure and mixed) is a best response.


---

### Answer by username_3 on 2019-06-12. Score: 1

**Q1**

Your table seems to be correct. Here is a quick Python implementation for generating the payoffs:

```
def payoff_calculator(x, y):
    if x+y < 14:
        return (x+1,y+1)
    else:
        if x==y:
            return (x,y)
        else:
            return (y,20-x) if x < y else (20-y,x)

primes = [2,3,5,7,11,13,17,19]
payoffs = [[payoff_calculator(i,j) for i in primes] for j in primes]

for row in payoffs:
    print(*row)

```

**Result**

```
(3, 3) (4, 3) (6, 3) (8, 3) (12, 3) (18, 13) (18, 17) (18, 19)
(3, 4) (4, 4) (6, 4) (8, 4) (17, 11) (17, 13) (17, 17) (17, 19)
(3, 6) (4, 6) (6, 6) (8, 6) (15, 11) (15, 13) (15, 17) (15, 19)
(3, 8) (4, 8) (6, 8) (7, 7) (13, 11) (13, 13) (13, 17) (13, 19)
(3, 12) (11, 17) (11, 15) (11, 13) (11, 11) (9, 13) (9, 17) (9, 19)
(13, 18) (13, 17) (13, 15) (13, 13) (13, 9) (13, 13) (7, 17) (7, 19)
(17, 18) (17, 17) (17, 15) (17, 13) (17, 9) (17, 7) (17, 17) (3, 19)
(19, 18) (19, 17) (19, 15) (19, 13) (19, 9) (19, 7) (19, 3) (19, 19)

```

**Q2**

You are right, there are no strictly dominated strategies here. This is because each action is a best response to some opponent action. For example, 2 is a best response to opponent moves 13, 17, and 19, 3 is a best response to 11, 13, 17, and 19, and so on. Similarly, there is no strictly dominant strategy. For example, 19 is a best response to 2, 3, 5, 7, and 19, but not to 11, 13, or 17.


---

