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I am skeptical that there will be such an argument. The reason is that the winning Nim strategy is essentially unique. Namely, a Nim position is balanced if the binary digits of the heap sizes have al …

The other answers are truly excellent and have settled the intended question. For a bit of fun, however, allow me to mention the following paradoxical solution. Namely, with a certain precise and rea …

I'll start things off by observing that this is what is known as an open game, since if player 1 wins, then the winning condition is satisfied after finitely many moves. It follows by the Gale-Stewart …

There are numerous proofs of what I call the fundamental theorem of finite games. Theorem. (Fundamental theorem of finite games) In any finite two-player game of perfect information, one of the pla …

Here is one possible answer. An essential feature of any board game, in the way I am thinking about it, is that there are only finitely many game states that are realizable on the board. This fact, co …

Here is a complete winning strategy for the Justice game. One wins the Justice game simply by following the usual Nim strategy, with all the same winning positions and moves (except if the position is …

No, the negative a game is simply the game in which the player's roles are swapped, hereditarily. You can see this in the definition you provided $$-G=\{ -G^R\mid -G^L\}$$ since the left options in $- …

Here is an amusing concrete non-determined game, under the assumption that the dependent choice principle fails. Assume DC fails. This means that there is a set $X$ and a binary relation $R$ on $X$, …

(My argument is somewhat easier if you consider games where the players play $0$s and $1$s, so that the payoff set is in Cantor space $2^\omega$, and we use the usual coin-flipping probability measure …

My wife and I have a standing agreement where I pick up our son Horatio from school and she picks up our daughter Hypatia. One day, because I knew I would be near Hypatia's school, it was convenient …

This doesn't answer the question that was asked, but rather an alternative kind of infinite Hex, played on the infinite hexagonal lattice board as shown below. I am posting it because people intereste …

This question arises from what I find interesting in the recently asked question What is a chess piece mathematically? My answer to that question was that mathematically, game pieces are in general e …

For even $n$, I claim that nobody has a winning strategy, and therefore both players have drawing strategies. To see this, observe first that by the fundamental theorem of finite games, we know that …

I don't know the answer to your question, but perhaps we might gain insight from the following Interview with Jason Simmons, a professional rock/paper/scissors player, which appeared a few years ago o …

To prove that this is a metric, consider the following theorem. Theorem. If the second player can survive for $n$ steps in the $(\Gamma_1,\Gamma_2)$ game, and for $m$ steps in the $(\Gamma_2,\Gamma_3 …