You and I play a game, we roll two dice and both pick a number before rolling them. The one closer to the sum wins. The distribution is non-uniform, with 40 % chance of getting a sum of 12 and a uniform distribution for all the other numbers. What is your strategy? Go first or second?
I mean, 12 is the most likely one so for sure you want today 12, but your opponent will say 12 as well and the game ends in a tie. If you don't want to tie and you want to win then you should go second and see what your opponent does and say a sum which is higher than the one of your opponent. However, knowing this, your opponent will say 12 and be satisfied with a tie, otherwise he would lose.
We have $11$ possible numbers to get from two dice and so as there is a $40\%$ chance of getting a $12$ the other $10$ combinations "share" the remaining $60\%$ and so all other combinations have a $6\%$ of being rolled.
Notice if you guess $x$ your opponent will only ever guess $x-1$ or $x+1$. Why?
Because guessing anything else is strictly dominated by one of the other two.
Consider I guessed $8.$ If you guess $3$ you will win if a $2,3,4$ or $5$ is rolled . However if you guessed $7$ then you would win on all of these combinations as before but also now you win if a $6$ or $7$ is rolled.
Let us work our way down:
Guessing $12$ seems bad because as we noted before they will guess $11$ and win $60\%$ of the time and you only win $40\%$.
Guessing $11$ is better, if they guess $12$ you win $60\%$ and if they guess $10$ you will win if an $11$ or $12$ is rolled so $46\%$ of the time. Your opponent will naturally try to minimize this so they will guess $10$.
Hence guessing $11$ results in winrate of $46\%$
Guessing $10$ seems great! If they guess $11$ as we saw before you win $100-46 = 54\%$ of the time. If instead, they guess $9$ you will win if a $10,11 $ or $12$ is rolled. And so win $6+6+40 = 52\%$ of the time! They will minimise your winrate and so will guess $9$ leaving you with $52\%$ chance to win.
Is this the best you can do? Yes. If you guessed any lower than $10$ they will just guess one above you and they will win on at least $10's,11's$ and $12's$ > $52\%$
Another way to do this would have been to look at the CDF and find the "halfway point" where the dice are roughly equal to land above and below and start searching around there.