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My first remark is that if you allow players to pick their own theories, but only allow consistent theories to win, then you will not be able to compute the winner of the contest. The reason is that t …

The game of infinite Hex, proceeding from an arbitrary position, is a good example with all the features you seek. The game was the subject of my Oxford student Davide Leonessi's masters MFoCS dissert …

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 …

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 $- …

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 …

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 …

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 …

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers. The ques …

Allow me to mention that since the players in effect adopt the roles of the quantifiers $\forall$ and $\exists$, as Bob has a winning strategy just in case for every move for Alice, there is a reply b …

I like this question a lot. It provides an interesting way of talking about some of the ideas connected with the maximality principle and the modal logic of forcing. Let me make several observations. …

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 …

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 …

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 …

This amounts to the Gale-Stewart theorem showing that open games are determined. The issue of draws can be easily finessed, as I explain below. Specifically, a game of perfect information is open for …

Unless I have misunderstood, here is a counterexample. Let $A$ be the $2\times 2$ identity matrix. This has your Pareto property on $x-y$, but if $x=[{a\atop b}]$, then $Ax=x$, and there is no way to …