I'm starting to play around with some properties of combinatorial games, and I am having problems formalizing an argument based around the game of polite chocolate.
There is an $n \times m$ grid of chocolates, where each chocolate is labelled $c_{i,j}$ for being in row $i$, column $j$. When a given player takes the chocolate $c_{i,j}$, that player must take all chocolates $c_{k,l}$ where $k \le i$ and $l \ge j.$ The idea being, given I take a chocolate, I must take all chocolates in the upper-right square from from the chocolate. The person who takes the chocolate $c_{n,1}$ loses.
Take this game simply with two players, where each player takes turns making moves in the game. I have a feeling that if player 1 has a winning strategy in the case of the $n \times r$ board ($n$ columns, $r$ rows of chocolate) when $n \gt 1 \wedge r \gt 1,$ and this is because if player $2$ has a winning strategy, player $1$ can steal this given strategy. However, I am having difficulties formalizing this argument. Any assistance would be greatly appreciated.
We show that Player 1 has a winning strategy except in the case of the $1\times 1$ chocolate bar. The game ends in no more than $mn$ moves, and ties are not possible, so one of the players has a winning strategy. We show it is Player 1, by using a strategy-stealing argument.
Suppose to the contrary that Player 2 had a winning strategy. Then Player 2 would have a winning response to Player $1$ taking the top right square on her first move. But the result of that response can also be achieved in a first move by Player 1.