I have invented a very interesting card game. All the cards from 2 to 10 (in four colours) are divided evenly between the two players (the deck is shuffled before dealing the cards, of course). Now the players put cards face-down on the table and when both cards are there, they flip them. If the cards are the same, they both score a point. Else, they determine who scores a point following those rules:
3 beats 2
4 beats 3
2 beats 4
Then
6 beats 5
7 beats 6
5 beats 7
And
9 beats 8
10 beats 9
8 beats 10
Plus, finally
5,6,7 beat 2,3,4
8,9,10 beat 5,6,7
2,3,4 beat 8,9,10
Then the cards are discarded. The game goes on until the players have no cards left in their hands. The task is to
- Find a deal in which one player has a winning strategy (if there is one).
- Calculate the odds for a player to win a given deal in polynomial time.
The players can look into their hands and make choices. They also (if they remember the discarded cards) know their opponent's cards.
I've called this game "ideal", because each time I played the score seemed not to depend on the deal, although the deal was random and the cards were different every time.