I think what you say might be true. I once found a dumbed down version which goes like this: (A simplified version of Mia [0]) On your turn, you secretly roll

I think what you say might be true. I once found a dumbed down version which goes like this: (A simplified version of Mia [0]) On your turn, you secretly roll a die and then make a statement about what number you rolled. The next player may then call your bluff. If you lied, you lose, otherwise they lose. Or they may roll again, but must claim to have rolled something higher than whatever you rolled. If you don't lose, you win. Suppose you have to explain your (mixed, i.e. involving randomness) strategy to other players before playing. This is a simulation of the realistic situation of playing many times and other players learning your strategy. Let us examine the situation where you are given a 4 from the previous player and you decided to roll. (You would have to prove that this can be a correct move, but what follows is in principle true in any situation where you roll.) If you roll a 5 or 6, you say so. But if you roll a lower number, you have to claim a 5 or 6 regardless. If you announce a 6, you lose instantly, because the next player has no chance of rolling higher, so they must call your bluff. Thus in a single game the correct move is to announce a 5. But since the other players know your strategy, you can increase you chances of them believing you have a 5 if you sometimes claim a 6 and thereby forefeit the game. It is an easy calculation to see that this strategy yields more wins than always claiming a 5 in this situation. I think this is analogue to the poker example: Play a clearly bad move and thus gain an advantage by being unpredictable. In my example it is not so hard to see that the advantage can in principle outweigh the cost of playing bad moves. [0] https://en.wikipedia.org/wiki/Mia_(game) https://en.wikipedia.org/wiki/Mia_(game)