Does Unpredictable Four have an optimal solution?

Question

The rules of Unpredictable Four are quite simple. One player (the crazy) tries to be unpredictable, while still achieving a goal -- and the other player (the psychic) tries to predict them. However, my simulation shows that if the crazy plays purely randomly -- they are trivially defeated (with an expected return of about 6%).

I have been unable to think of anything coming close to an optimal solution, for either player, wherein if the opponent knows your strategy they can not counter. However, wikipedia tells me all such games should have a nash equilibrium. Could someone shed some light on how I can find an optimal solution for a game such as this?

Answer 1

I wanted to comment but I'm a bit short of reputation points. Interesting game here. A few points here.

1. Crazy playing randomly is unlikely to be optimal because crazy is hoping to make a run of 4 in a row. A complete specification of a strategy is a probability over stars/moon given every possible history of play. So after the first move, a strategy would have to indicate how crazy plays after playing stars/moon assuming psychic passed. By the last move the strategy has to consider all possible histories for 9 moves--this might be a large space!

2. Psychic at least has a 50 percent chance of winning by guessing on the first move. Psychic can also pass until crazy makes 3 in a row and then have a 50/50 chance.

3. Although such games do have solutions, it is far from trivial to solve them unfortunately (see comment about strategy space in 1.). Here's another game: I am thinking of a number 1, 2, ..., N. You guess a number and I tell you "higher" or "lower" until you get the number. You pay me $1 for each guess. Even though this game sounds simple, it is far from trivial to solve! http://link.springer.com/chapter/10.1007%2F978-3-642-25280-8_10

Hope this helps. There might be some ways to simplify strategy, I will let you know if I come up with anything.