# Are there strictly dominated strategies?

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 $$p_{i}, player $$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!

• 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. May 8 '19 at 15:26

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.

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.