Here’s a card game: You lay out the A,2,3 of Spades, Diamonds, Clubs in random order on the table face up. So that’s 9 cards in total. There are two players and they take turns picking up cards from the table, one at a time. The winner is the first to collect a triplet where a triplet is any one of the following sets of three:
- Three cards of the same suit
- Three cards of the same value
- Ace of Spaces, 2 of Diamonds, 3 of Clubs
- Ace of Clubs, 2 of Diamonds, 3 of Spades
Got it? Ok, this game can be solved and the solution is that with best play the result is a draw, neither player can collect a triplet. See if you can figure out why. (Drew Fudenberg got it almost immediately [spoiler.]) Answer and more discussion are after the jump.
The game is isomorphic to Tic-Tac-Toe. The 9 cards represent a 3×3 matrix where the value of the card is the column and the suit is the row and we put Diamonds in the center row. To collect a triplet then is equivalent to getting three in a row in Tic-Tac-Toe.
I’ll bet that you would have no trouble saving a draw in Tic Tac Toe but, before you saw the isomorphism, might actually have lost a game or two of this version. (Win bets at parties: beat your friends at this game, then bet them you can beat them at Tic Tac Toe.) I don’t know of any theory of “bounded rationality” in game theory that would predict differences in play between this game and Tic Tac Toe.
I think that having subjects play, unbenknownst to them, several games that are isomorphic to another could be the basis for some interesting experiments. Keeping the same basic strategic elements but changing the framing could allow you to isolate differences in levels of sophistication that are different than the usual “cognitive hierachies.” For example Tic Tac Toe emphasizes geometry (trivially) whereas the variant emphasizes more abstract thinking.