Quantum Tic-Tac-Toe

Stupidest Classical Game

Once a game of Quantum Tic-Tac-Toe is over, the collapsed moves imply a classical game. It is usually the case, that if the quantum game was strategically sound, the classical game was not. This month's challenge is to find a reasonably good quantum game that yields a really really bad classical game - the stupidest classical game you can find. There is no right answer, but there are a number of interesting attacks on creating such a game from scratch, and several different measures of stupidity.

The main reason for this challenge is to emphasize the Ascertainity Principle. In physics, the Ascertainity Principle states that given only classical data it is usually possible to determine whether the measured system was classical-like or quantum-like. Similarly, given only the classical Tic-Tac-Toe moves, it is usually possible from strategic considerations to determine whether the game was played under classical or quantum rules.

It is remarkable that we have been able to "ascertain" that we exist in a quantum realty, not a classical one, given the counterintuitive nature of quantum mechanics and the lack of experiential evidence at the scale of typical human activities. As it is, this realization is only a hundred years old.

Be sure to include your name and email address! We want to give credit to the person that sends the best solution.

Last month, we looked at how rapidly the game tree branches in Quantum Tic-Tac-Toe, and considered how to keep the number of branches small. The obvious way is to "prune early, prune often." The extremum of that strategy is for player O to match X's every move, forcing a collapse every pair of moves. Can X win if O does that?

Consider this example game:

Move		X		O
1		1 - 6		6 < 1
3		3 - 8		8 < 3
5		9 - 4		4 < 9
7		7 - 2		2 < 7
9		5 < 5

X plays in squares 1 and 6, O plays in the same squares, and X chooses that O really moved in square 6 (the direction of the arrow shows which ghost mark becomes real), giving himself a real mark in square 1. He continues and she replies in kind, until X has all four corners. The last move of the game is self-collapsing (since there is only one square for both ghost moves to go into), and X has the center, and two points.

Of course, this example is too obvious, and O isn't going to fall for it. She could play move eight in squares 2 and 5 instead of 2 and 7; this almost lets her win (take a look at the ensemble of real games), but X can still win by a half-point (what's his move?). If she sees where the game is going a bit earlier, O could play move six in squares 4 and 5, which gives her a shot at winning by a half-point. X can defeat her still, but it's more difficult.

The cause of X's troubles is greed; he loses by wanting the two-point win. His better choice would be squares 2 and 4 on move five; this guarantees him a 1-point win, either on move six (if O continues her "mimic" strategy), or on move seven (if she abandons it). So, if O pursues the "mimic" strategy for even two moves, she loses. It is an open question whether she can win if she collapses X's first move.