This is part three of a series on tactics and strategies in the theory of games. Find them in the tactic tag.

In previous weeks, we introduced and explored the distinction between tactics and strategies in the theory of games, and we saw how with many games, one can achieve a tactical form of the fundamental theorem of finite games.

This week, I should like to consider this issue specifically in regard to the game of chess. Are we able generally to achieve winning tactics in chess? Or drawing tactics for both players? We have observed on abstract game-theoretic grounds that if we supplement the usual chess board positions with more information from the game tree, making them rich (in the technical sense of rich board positions), then indeed we shall achieve the tactical conclusion of the fundamental theorem—one of the players will have a winning tactic or both players will have drawing tactics. But how much information do we actually need? Perhaps we don't need the full game history, but much less.

Question. How little information will suffice to include in the board positions of chess in order to ensure that one player will have a winning tactic or both will have drawing tactics?

Can we make due with information just about the pieces on the board? Do we need the list of currently legal moves? Or do we actually need to know the full history for the purposes of three-fold repetition? What about en passant? What about castling? What about the fifty move rule?

Which information, exactly, is sufficient to make the conclusion of the fundamental theorem for tactics in place of strategies?

Interlude

Let me give a full answer, which uses rather less information than might naturally have been expected—the winning and drawing tactics do not need to know any information at all concerning three-fold repetition or the fifty-move rule.

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This is part two of a series on tactics and strategies in the theory of games. Find them in the tactic tag.

In last week’s post, we introduced the distinction between tactics and strategies in the theory of games, observing that many familiar games, including Nim, Connect Four, tic-tac-toe, and Othello, admit the conclusion of the fundamental theorem of finite games for tactics as well as strategies—either one of the players has a winning tactic or both have drawing tactics.

But we also found games for which this conclusion was not the case, such as the Chocolatier’s game, as well as a simple finite game and (in the exercises) a finite variation of the Chocolatier’s game, with a bitter, forbidden chocolate.

This week, I should like to address the question of which games admit a positive tactical variation of the fundamental theorem. Can we find a general criterion that will suffice to know that one player has a winning tactic or both have drawing tactics?

Yes, indeed we can. Despite the negative examples of the previous essay, I shall nevertheless prove a successful tactical version of the fundamental theorem of finite games, for finite games in which the board positions are somewhat fuller with information.

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This is part one of a series of essays on tactics and strategies in the theory of games. Find them in the tactic tag.

Tactics

In the theory of games, let us explore the difference between tactics and strategies. Game-theoretically, what is a tactic?

In informal usage, the word “tactic” commonly refers to a local plan of action concerned with specific, limited aims, while strategies in contrast have global ambitions; one thinks of the difference between a battle and the larger war of which it is a part. In chess, we have the various familiar tactics—forks, pins, skewers, sacrifices, discovered attacks, removal of the guard, and so forth. In each case, the tactic is a specific kind of move or sequence of moves characterized by certain local goals and local arrangements of pieces, rather than a consideration of the whole game board or the history of the game play that had been undertaken to reach that position. Chess tactics are often vitally important techniques to consider, for they often provide the key manner of play to achieve a favorable outcome.

In this essay, however, I shall be interested in the abstract game-theoretic notion of tactic, using the word with its specific technical meaning in the theory of games, a meaning which is not exactly the same as the common usage of this term in chess, say, although it does retain something of the local-character aspect.

Let me explain. We have defined in the theory of games that a strategy is a function that tells a player exactly how to move in any given game situation—a strategy is a function on the full game tree, mapping the nodes for which it is that player's turn to the child node arising from the strategy's recommended move. Since nodes in the game tree in effect encode the entire history of how play got to that position, a strategy thus in effect has access to that game-history information.

A tactic, in contrast, tells a player exactly how to move given only the current board position, that is, the current state of affairs on the playing board, without knowing the game-playing history of how we got to that state. So the difference between a strategy and a tactic is that a strategy makes its recommendation knowing the whole history of play to the current state of affairs, while a tactic must make its recommendation based solely on the current board position.

The notion of tactic therefore makes sense only for games that do indeed have something that might be called the game board, and the concept of tactic is sensitive to which information exactly is taken to be included as part of the board position. For many games, to be sure, including Hex, Connect Four, Othello, tic-tac-toe, and Nim, there is a clear and natural notion of board position—one need only look at the playing board (plus perhaps the turn indicator saying whose turn it is) and know everything about the complete space of how the game could possibly continue. For these games, therefore, we have a corresponding clear and natural notion of tactic.

With chess, in contrast, the standard conception of “board position” is a little more unsettled and perhaps even ambiguous. If we think of the board position as consisting only of the information visible by looking at the chessboard—a photograph of the board—then we wouldn't necessarily know whether castling was possible, since perhaps we had already moved the rook or king; we wouldn't necessarily know whether en passant was possible, since applicability of this right depends on the previous move; we wouldn't necessarily know if our opponent has castling privileges or not, and we wouldn't necessarily know the extent to which we are subject to draw by three-fold repetition or the fifty-move rule. We just can't always determine this sometimes crucial information by looking at the playing board.

In regard to most common games the notion of “board position” that we are talking about here is actually closer to how people use the term “position” in discussions of the game. In a newspaper chess puzzle, for example, we generally find the state of the game on the board, without knowing exactly the history of play how it was arrived at, even if there are conventions for part of this information, concerning the possibility of castling and en passant.

Meanwhile, for any given game one can present a strategically equivalent form of it by imagining ourselves to be playing the game literally on the game tree itself. We imagine the entire game tree spread before us on a great lawn, and making a move in the game consists of stepping from the current node to a child node in the tree. This way of playing is strategically isomorphic to the original game, since it has the same game tree, and the key observation is that with this game-tree concept of game board, strategies and tactics are the same thing.

The general point I am trying to make here is that when speaking of tactics, we must specify exactly which information is available on the game board for the tactics to consult when making their recommendation. The same game might have several different natural conceptions of what constitutes the “game board.”

Games as graphs

Let me introduce a slightly more abstract graph-theoretic way for thinking about games, a perspective that will assist in thinking about tactics as opposed to strategies. For a given game with a given concept of “board position,” we will think of the game at bottom as constituted by a certain labeled directed graph. Namely, we have a node in the graph for each possible board position, all the various possible states of the game. We place an edge from board position p to board position q, labeled for a player, if that player can make a legal move in the game transforming the game state from p to q. If the turn indicator is part of the board-position information, the label information is redundant, since only one player would have a legal move from that board position, and so it can be omitted. Some of the board positions are terminal, with no outgoing edges, and they are labeled as wins for one player or the other or as draws in the game. One of the board positions is labeled as the starting position.

Any such labeled directed graph can be conceived as a game. A play in such a game is simply a path in the graph, starting at the start node and alternately following edges for one player and then the other. We play in such a game by jumping from one node to another, following the edges, just as we conceive of playing a game on the game tree by stepping successively from nodes to child nodes. The winner of such a play is determined by the label on the terminal node, if such a node is reached. The game is a finite-play game if all plays of the game are finite.

A tactic for a player is a function on the graph mapping each board state to an outgoing edge for that player (when there is one). Such a tactic is winning for that player, if all plays of the game in which that player has played in accordance with the tactic lead eventually to a win for that player.

For a game given by a game tree, a board position conception of the game consists of a labeled graph of the type I have mentioned and a mapping of the nodes of the tree to nodes in the graph, mapping each game position to a corresponding board position, in such a way that the move structure is respected. That is, every possible move in the game tree corresponds to a move for that player in the graph from the corresponding node, and conversely, every possible move in the graph corresponds to a move in the game tree for any position having that board state.

Meanwhile, this graph-theoretic conception is not necessarily the most general way to think about tactics, since with some concepts of board position, the moves of the game are not actually well-defined on the board position. For example, in the game of chess, if one's conception of board position consists just of the state of pieces on the chessboard, then one cannot tell from this whether a certain move such as castling or en passant is legal. Or even if that information is taken to be part of the board position, one cannot tell whether a given move will cause the end of the game by the 50-move rule, and so these conceptions of board position will not respect the move structure as required.

Winning tactics

For which games should we expect winning tactics for one player or the other as opposed merely to winning strategies? Is there a winning tactic in chess, for example, or drawing tactics for both players? Indeed, which of our familiar common games admit winning and drawing tactics, as opposed to winning and drawing strategies? More generally:

Question. Does the fundamental theorem of finite games hold generally for tactics?

In other words, in every finite game, must one player have a winning tactic or both players have drawing tactics?

Interlude

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Question. How likely is the resulting arrangement to be a legal position, one that could have arisen in a game of chess?

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A proper Sudoku puzzle contains a minimal set of clues, a partial solution, which has a unique Sudoku completion. In short, every proper Sudoku puzzle has exactly one correct answer.

The Sudoku game

Sudoku puzzles, of course, are generally a solitary activity, a puzzle meant to be solved by one person. Let me tell you, however, about a two-player variation, the Sudoku game.

We start with a completely empty Sudoku board, vacant of numbers, and then take turns placing numbers.

But these numbers must never explicitly violate the Sudoku condition, that is, numbers must never repeat on any row, column or principal sub-board. The first player who has no legal move loses.

The game is therefore not about building a global Sudoku solution, since a move can be legal in this game even when it is not part of any global Sudoku solution, provided only that it doesn't yet explicitly violate the Sudoku conditions. Rather, the Sudoku game is about trying to trap your opponent in a position that does not yet explicitly violate the Sudoku condition, but which cannot be further extended.

Who will win the game?

Interlude

The game makes sense not just on the 9 × 9 board, but actually on any n² × n² board. For example, we can play 16 × 16 Sudoku, or 125 × 125 Sudoku. Later, we shall discuss that the game can be naturally extended to the case of rectangularly symmetric boards of size (n × k) × (k × n), with rectangular subboards, such as 6×6 = (2×3)×(3×2).

And we shall forthwith move on to the infinite Sudoku games. Do you fancy a game of Sudoku on the integer-by-integer grid of integer-by-integer grids? How about the (ℝ×ℝ)×(ℝ×ℝ) Sudoku board? And higher cardinals?

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In the sample game in progress here, note how several four-in-a-row attempts have already been blocked during play. It is currently Yellow's turn to play—where should she play? In fact, she can win the game in three moves. Can you find it?

Connect Four is known to be a first-player win with optimal play on the standard 7 × 6 board—the first player can win on their 21st move (that is, with 41 moves in all, leaving only one cell unplayed), when opening with a move in the center column. Opening in the center-adjacent columns (3 and 5) leads to a drawn outcome with perfect play, while the other four opening moves lead to a loss.

Connect infinity

I shall naturally be seeking, however, to explore the infinitary variations of Connect Four. The game works perfectly well when played on the upper half-plane, for example, a playing board arranged like the integer grid ℤ × ℕ, with a column for every integer, each column a copy of the natural numbers going up. The players take turns dropping their coins into the columns, which fall under gravity to occupy the lowest available cell.

But what should the winning condition be? Shall we still take four-in-a-row as sufficing to win? In light of the luxurious possibilities of the infinite board, that would seem downright miserly—there is plenty of room, after all, for far longer winning chains. The players will naturally aspire to form very long winning chains. Is it possible? What length chains can they form? Can the players play so as to form arbitrarily long finite chains on a row, column, or diagonal? Can they reasonably aspire to make infinite} winning chains?

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I am sharing here the questions from which the final exams were drawn.

About the class

First let me say a little about the class. We began with some elementary game theory and decision theory, and some philosophical accounts of what is a game in the first place, before getting into the fundamental theorem of finite games, game trees, the hypergame paradox, and many examples of finite games and their strategies, such as Nim, 21, tic-tac-toe, including 3D tic-tac-toe, and many other games. Eventually we discussed determinacy issues and connections with the philosophy of mathematics, such as supertasks. A final highlight of the class was to cover the analysis of various infinite games, including infinite chess, infinite draughts, infinite Hex, infinite Wordle, infinite Sudoku, and more.

In addition to regular course work with quizzes and so forth, the students had to pass various games challenges with me during the semester, playing me one-on-one in Nim, 21, Gold-coin game, chess, Hex, draughts, Go, Connect 4, Othello, and also to solve the Rubik’s cube.

The course was open to all students, with no prerequisites, and fulfilled the university 2nd philosophy requirement.

Final exam questions

The final exam questions were drawn from the following. Please post your answers in the comments!

1. Suppose that an evil king hauls up two innocent prisoners from his dungeon to play chess against each other, with the instructions that the winner will be set free, while the loser will be executed. The prisoners meekly begin playing. According to the criterion of Bernard Suits in his essay on what is a game, are the prisoners playing a game? Please defend your answer with detailed reference to the Suits criterion, explaining why this situation does or does not meet it.

   

2. Explain why the penny partition paradox is also known as the “Centipede game.” Give a fully self-contained explanation of the paradox and why it might deserve this title.

3. Give an example of a finite-play game that is not a finite game, if possible (or explain why it is not possible), and similarly for a finite game that is not a finite-play game. As a part of your answer, explain the precise meanings and differences between these notions.

4. During the leisure time at a certain fire station, the fire fighters like to play a certain finite-play two-player game of perfect information, called Fireout! The game is a little dangerous, however, in that some moves cause the game itself to catch fire. Indeed, every single move made on the game when not on fire will cause it to burst into flames. But whenever the game is on fire, then thankfully there is some move in the game that will immediately extinguish the fire (even if some other moves leave the game fully alight).

   

   The game will inevitably come to an end, for complicated reasons involving the details of game play. The player making the final move will lose if the end state of the game is that it is on fire, but if it is not on fire, then they win. Without knowing any further details about the game, can you describe a winning strategy for this game? If so, provide full details for how exactly the player should play, covering all contingencies, and provide an argument for why your strategy is a winning strategy. From which sort of positions will your strategy be winning? If the game starts without fire, should you rather go first or second?

5. Give a careful explanation of the hypergame paradox and explain how you think it can be resolved or why it cannot be resolved.

6. Explain, as fully and precisely as possible, in the context of a two-player game of perfect information: what is a strategy? what is a game tree? what is a winning strategy? what is a drawing strategy?

7. State the fundamental theorem of finite games, and provide a proof of it using the back-propagation method for the case of games with no draws.

8. Explain how we can use the fundamental theorem of finite games without draws to prove the theorem for the case of games with draws. That is, how might we argue for the fully general version of the theorem, if we have already established the theorem for games without draws?

9. What are the main things to say about 3D tic tac toe in terms of winning strategies and the possibility of draws?

10. Consider the game of Hex played with the swap rule, meaning that on the very first move only, the first player places a tile on the board, and then the second player at his option can choose to accept that move and play continues or instead to take that move as his own (so the player roles and colors swap). Using the fundamental theorem, show that the second player has a winning strategy in the swap rule variation of Hex.

11. Suppose you work as a waiter at Cafe Infinity, which features infinitely many tables—table 0, table 1, table 2, and so forth, each with infinitely many seats, seat 0, seat 1, and so forth. The restaurant is empty at the start of the shift. At each ring of the bell, however, the manager sends in infinitely many new hungry customers, exactly one additional for each table. But alas, you are able to serve only one customer in each time interval. The bell rings get faster and faster as the evening progresses, so that by close there will be infinitely many intervals. Ordinarily, each table has its own waiter, and the customers are easily served. But tonight it turns out that there is only you on duty, and you must serve all the tables. Can you do it? Will you be able to serve all the customers? If so, explain how, or if not, explain why.

    

How well did you do on the exam? Post your solutions in the comments. Please post one solution per comment, so we can have a discussion about the details of each separately.

See all the other games posts here: Infinite Games.

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What’s coming

Stay tuned for more logic of games posts continuing through the summer. I plan to continue posting on determinacy issues, including nondetermined games, and several more explicitly mathematical issues arising in the philosophy and logic of games. I also plan eventually to have introductory essays on the various infinite games, including infinite chess, infinite draughts, infinite Hex, and so forth.

In addition, I will soon begin posting again on various topics in mathematical logic.

This is a variation of the Bubble Monster game, which we covered last week, in that the basic rule of Pushpast is both the triangles and the circles act like the children in the Bubble Monster game, during their own turn, but like the bubble monsters on their opponent's turn. Thus, when it is the triangles' turn to move, they select one of the triangles amongst them to push past the circle currently facing it (there must indeed be such a circle), but then having done so, the circle is allowed at its option to reproduce itself any finite number of times behind, adding any number of additional circle duplicates there, just like the bubble monsters do when the children push past. The circles move similarly on their turn by choosing a circle to push past an opposing triangle to the left, which may then add additional duplicates of itself at its discretion behind to the right. The game is over when no more moves are possible, and whichever player made the last move wins. Infinitely long play is a draw.

Here is an example play of a few moves.

We see that the leading red triangle pushed through the opposing blue circle, which duplicated itself with three copies behind. Then, the leading blue circle pushed through its opposing triangle, which duplicated itself twice. And then, the penultimate triangle pushed past its opposing circle, which replaced itself with four copies.

A key difference between Pushpast and the Bubble Monster game is that the triangles and circles don't just want to get through each other, but rather aim to be the team that makes the very last move achieving that state. Can we expect the bubble-monster analysis to carry through? Is this a finite-play game? What can we say about winning strategies?

Interlude

We shall have a full analysis of the game and winning strategy by the end.

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The game proceeds in a sequence of moves. On each move, the children appoint one amongst them to poke through the particular bubble monster that blocks them. Thankfully, the child will succeed in getting through, but with the worrisome result that the bubble monster, as his discretion, may possibly divide itself into one or more newly formed bubble monsters behind the child, an arbitrary number of duplicates chosen by the monster.

Perhaps the game proceeds as follows:

The first child successfully passed through the leading monster, which duplicated into three, for a total of five monsters. Then the next child poked through her monster, which regenerated as four, for a total of eight monsters. Oh dear, the monster problem is getting worse! The children face a difficult realization of their precarious position—the only way for them to reach safety, after all, is to push through the monsters, but whenever they do so, additional monsters are created in the wake, which must still be surmounted.

Is it hopeless? How can the children possibly succeed? Can we save any of the children? How many can we save? By what procedure shall they proceed? Should they try to stay together somehow as a group? Or will it be better for at least some of them to charge straight ahead?

Interlude

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A move in the game consists of turning a face-down card face up, and then turning over the card immediately to the right, if any. …

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After infinitely many rounds, one for each natural number, the Glutton wins if he has eaten all the chocolate, that is, if every chocolate that was ever served was also eventually consumed. Otherwise, if there are any chocolates left uneaten, the Chocolatier wins.

What? How could the Glutton possibly win? After all, the Chocolatier can be placing huge numbers of chocolates on the platter each round, while the Glutton eats only one. Won't the Glutton get steadily further and further behind?

Question. Who wins the Chocolatier's game? Does the Chocolatier or the Glutton have a winning strategy?

Interlude

Join me in analyzing this game and several variations. We shall introduce the distinction between strategies and tactics, including no-recall strategies and nearly no-recall strategies. But we shall start slow and easy, with merely countably many chocolates, taken at first finitely at a time, but eventually we shall consider the subtleties arising with an uncountably creative Chocolatier, ultimately mounting an argument involving the axiom of choice and well orders of the chocolate types. Let’s get into it.

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Whoever takes the very last fish overall wins the game. Perhaps the buckets currently hold the following numbers of fish:

What is your winning move? Is there any winning move? Perhaps the game will proceed indefinitely with no winner.

Interlude

Must the game end?

Before investigating the possibility of winning moves, it seems urgent to address the more fundamental issue whether the game indeed will necessarily come to an end at all. After all, huge numbers of fish can often be added to the buckets during play (let us imagine the buckets as extremely capacious), but only one fish is removed each turn, and so a skeptical reader could reasonably wonder whether in fact the game will always have a winner.

Perhaps by adding fish in a suitable manner, the players may conspire to prolong the game indefinitely. Is it possible? Or perhaps every play of the game eventually finds its completion. In short, the question is whether Buckets of Fish is a finite-play game.

Can you make an argument one way or the other? If you think the game does not necessarily end, then you should aim to provide a specific infinite play that never ends. What are the moves exactly? Alternatively, if you think all plays of Buckets of Fish eventually come to an end, then what is the proof?

Interlude

The answer is that, indeed, Buckets of Fish is a finite-play game—every play of the game eventually comes to an end. Regardless of how the players might conspire to add fish to the buckets during play, even with an endless supply of fish from the boats, nevertheless they will inevitably run out of fish in the buckets.

But how can this be? After all, can't we always add more fish to the buckets than we remove? By doing so, wouldn't this lead to a nonterminating infinite play?

One way to reply to this objection is point out that the expectation that on any given turn we can always add huge numbers of fish to the buckets isn't strictly true. Consider what happens, for example, when we elect to remove a fish from the leftmost bucket. In this case, since there are no buckets to the left of it, there is no possibility of adding fish to any buckets on such a turn. So it just isn't true that after taking a fish we can always necessarily add huge numbers of fish.

A further subtle observation about this matter is that even on turns when we are adding huge numbers of fish to the buckets, the fish that we do add are always necessarily closer to the leftmost bucket than the fish we removed. Those fish are consequently less powerful, in a sense we shall make precise, in terms of their ability to cause additional fish in the buckets. Ultimately this way of thinking provides some insight into our later use of transfinite ordinals to measure in a game-value sense how far a position is from completion, since fish further left contribute a smaller ordinal sum to this value.

After proving that the game will always terminate in finitely many steps, we shall present a winning strategy for the game, telling you exactly how you can ensure a win. In a pleasant twist, the winning strategy turns out to be extremely simple—who would have guessed that it could be so easy?

After giving an account of the winning strategy, we shall similarly analyze a number of natural variations of the game, Give-or-take Buckets of Fish, Give-or-take-one, Cascade of Fish, Free-fish Buckets of Fish, and One-fish-at-a-time Buckets of Fish, which will themselves lead to several further interesting games, including a Nim-variant game I call Whim, which is like Nim, but which turns out to be a disguised form of Buckets of Fish.

Let’s get into it!

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But does white truly have an advantage? And what would it mean exactly to have a “slight” advantage? Does white have an actual winning strategy? That is, is there a way of playing for white, a systematic method of playing in response to every conceivable black counterplay, that will inevitably ensure a white win regardless of how black plays? Or perhaps, contrary to expectations, black has such a winning strategy.

More probably, neither player has a winning strategy. With perfect play, perhaps chess is a draw. Certainly at the top level of play between the best players there are many drawn games—reliably about half of the top-level championship games are drawn. But what does it mean exactly to say that chess is a draw?

1. Perhaps chess is a draw in the weak sense simply that with optimal play, neither player has a winning strategy—neither player can force a win. That is, every possible white strategy admits counterplay by black that will prevent it from winning; and similarly every possible black strategy admits drawing or defeating counterplay by white.

2. Or perhaps chess is a draw in the stronger sense that both players positively have drawing strategies, a systematic way of playing that will inevitably lead to a draw or better against any possible counterplay by the opponent.

Are these two ways of formulating what it means to say that chess is draw actually the same? In other words, if neither player has a winning strategy, does it follow that both players have a systematic way of playing that will ensure a draw or better?

Interlude

The fundamental theorem of finite games

The answer is yes, the two conceptions of what it would mean for chess to be a draw, the weak sense and the strong sense, are equivalent. I find this to be a subtle, nontrivial observation, but one which follows as a consequence from the main result I should like to discuss in this chapter, the fundamental theorem of finite games.

The fundamental theorem of finite games. In every finite two-player game of perfect information, either one of the players has a winning strategy or both players have drawing strategies.

We shall discuss in detail all about this theorem, including five different proofs of it. Let’s get into it.

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According to the penny-partition procedure, the penny players will alternately take turns, choosing on each turn either (1) to take one penny and continue the game; or (2) to take two pennies and end the game. If either player elects for two pennies on their turn, the process stops completely and the game is finished—no more pennies will be dispensed to either player.

Let us imagine that Percival and Penelope have what is in their day an ordinary attitude toward pennies, namely, a penny is not worth very much—just a penny—but definitely not nothing. Each extra penny is valued to some small but nonzero extent, and in any given situation, all else being equal, they'd rather have one more penny, if possible, than none. Naturally, they will each be happy to see the other person become rich along with themselves, but they aren't especially concerned about how many pennies the other person will or will not get; rather, they each just want to maximize their own take. And neither are they prone to envy; it will be fine with either of them to get fewer pennies than their partner, as long as they have maximized what is possible for themselves. In any case, because of the nature of the process, the difference between them will be at most two pennies, and so regardless of the outcome there will not be much reason for envy. In keeping with the “game” aspect of the process, there is no special concern to divide the pennies exactly equally—they will in any case get approximately the same number of pennies. Let me also state categorically that both Percival and Penelope are perfectly logical and that all this is fully known to them both in a state of common knowledge.

The penny partition process begins, with Penelope having the first move. What do you expect will happen?

Interlude

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Perhaps one takes chess and a few other common games as challenging this claim, since we might imagine infinitely long plays of these games. In chess, for example, we can set our knights galloping endlessly around the chessboard, a never-ending horse race with each knight overtaking the other in turn. Nevertheless, the standard tournament rules for chess actually block this kind of arbitrarily long play—the three-fold repetition rule, for example, calls the game a draw after the players repeat the same overall state of the game three times, as must occur in any sufficiently long play, and also the fifty-move rule calls the game a draw whenever there are fifty consecutive moves without any pawn movement or capture, in effect taking those events as a proxy for progress in the game. In some official settings, however, the three-fold repetition rule is specified merely as a player option—a player is entitled to call a draw by repetition but is not obliged to do so when the situation occurs, and this interpretation of the rule would not block infinite play. In other official contexts, however, draw by repetition is compulsory—in many online chess forums the game ends automatically in a draw when the third repetition state occurs. In contrast, the fifty-move rule is usually taken as compulsory when it is in effect and therefore places a definite bound on the longest possible game of chess. So with these established rules, chess becomes a finite-play game. Similar tournament rules apply to draughts and to many other games, specifically with the intention of forcing the game to end in a feasible finite number of moves. Naturally for practicable tournament play in any game one needs a way to ensure that the games will actually end so that the tournament can come to a conclusion.

Hypergame

In light of the enormous variety of finite games available to us, let me introduce the game known as hypergame. To play hypergame, the first player selects a particular finite-play game and then play proceeds as in that game. Perhaps the first player selects chess, or an instance of Nim or of Domineering—whichever finite game is selected, the players simply then proceed to play that game; the second player makes what amounts to the first move in the selected game, and game play commences. Whichever player wins the selected game-in-a-game is declared the winner also of the corresponding instance of hypergame.

Plays of hypergame have vastly different game-theoretic natures, since some plays of hypergame are a lot like chess, while other plays are very like Nim or Hex. Of course, hypergame can be a little like any given finite game, if only the first player should choose that game on the first move.

First player strategic advantage

From a strategic point of view, we might observe that the first player in hypergame clearly enjoys an overwhelming strategic advantage, so much so that the game becomes hardly worth playing seriously. After all, the first player could choose the game starting from a hopeless position in chess, where the opponent will be presently checkmated. Or a hopeless tic-tac-toe position. Why not choose a trivial Nim position—two piles, each of height one? Or indeed, why not simply choose the empty Nim position, so that the opposing player loses immediately, before even making a move? This amounts to choosing the “lose immediately” game, which is indeed a finite game, although in truth not one that is very enjoyable to play or exciting to watch. There will never be a professional hypergame league. From this perspective, hypergame would seem to be a rather disappointing game. Why should we consider it at all?

Hypergame is a finite-play game

In order to answer that question, let me first make a certain fundamental observation about hypergame. Specifically, I should like to observe that because the game chosen on the first move of hypergame is required to be a finite-play game, it will conclude in finitely many moves, and consequently, the corresponding play of hypergame itself also will conclude in finitely many moves. In short, all plays of hypergame will come to their outcome in a finite number of moves. Therefore, according to the definition we had provided for what it means to be a game of finite-play, we conclude that the game of hypergame itself is such a game of finite-play.

The hypergame paradox

But now a troubling conundrum arrives into the analysis, like a muddy wild boar appearing suddenly amidst the linens and finery of the garden party. Namely, consider the following instance of the game of hypergame. The first player is called upon to select a particular finite-play game. The mischievous player, thinking not unlike a muddy wild boar, elects for “hypergame.” Precisely because hypergame itself is a finite-play game, according to what we have said, this is a valid first move in hypergame. And so now play in that game commences. The second player should accordingly make the first move in the game that was selected, which therefore calls upon her to select a finite-play game. Catching the mood and equally mischievous, she plays along and selects “hypergame.” And so forth. The players proceed continually to select hypergame again and again:

Hypergame, hypergame, hypergame, hypergame, …

Each next move is a valid legal move, if the previous move was, since whenever a player has chosen to play hypergame, then the next move is to select a finite game, which could be hypergame itself. So according to what we have said, this would be a legitimate play of the game hypergame.

Well, so what? We have found an absurd strange play of this absurd strange game. Do you see the puzzling trouble about it?

Interlude

The puzzling trouble here, the wild-boar nature of the situation, is that this is an infinite play of the game! We found a play of hypergame that does not after all end with a definite outcome in finitely many moves, contrary to our earlier conclusion.

Wait, what? We have found ourselves in explicit contradiction. On the one hand, since hypergame requires that the first player select a finite-play game, which is then played, it seems incontrovertible that all plays of hypergame will end in finitely many moves. But on the other hand, this very fact implies that hypergame itself would be a finite game, and so the first player could select it and we can achieve the infinite play, hypergame, hypergame, hypergame, and so forth, which is not a finite play. So we have argued both that there can be no infinite play of hypergame and also, as a consequence, that there is a specific such infinite play.

What is going on? Let us explore it.

Please enjoy this selection from Infinite Games: Frivolities of the Gods, my new book serialized here with fresh material on games and the logic of games each week.

This week we look into the hypergame paradox, including a discussion of the Russell paradox, the Burali-Forti paradox, game trees, diverse formulations of the hypergame paradox, and much more, culminating in what I call the hypergame theorem.

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The game is famously a draw with optimal play—either player can ensure that their opponent fails to achieve tic-tac-toe. In the exhibition game play above, which was a win for X and not a draw, player O made a mistake in his opening move—he should have played O on a corner square instead of the edge as he did. The subsequent moves by X took advantage of the mistake with forcing moves to set up the double tic-tac-toe threat, thereby ensuring the X win. In fact, there were many winning lines for X after O's initial blunder.

Perhaps we all know from experience that tic-tac-toe is a draw with optimal play, but how could one prove it? The only proofs I know, unfortunately, are detailed case arguments, exhaustively exploring the space of possible play. For example, this xkcd cartoon details a complete drawing strategy for each player, in what I find to be a startlingly effective presentation of the strategy and the game tree.

While I don't know of any truly soft proof that tic-tac-toe is a draw, I do know some soft arguments for related, if weaker, conclusions, and so let us discuss one of those arguments.

No winning second–player strategy

First, I shall offer a soft strategy-stealing argument that the second player can have no winning strategy in tic-tac-toe. To see this, we suppose toward contradiction that the second player does have a winning strategy in tic-tac-toe. Call this strategy σ, a procedure telling player two how to mark the board in any given situation, such that if we follow the instructions as player two, we are guaranteed to win.

We shall play as player one, while aiming to “steal” the winning strategy σ. It doesn't make sense directly to consult σ about how to play as player one, since it is a winning strategy for player two, which is guaranteed to play well only when consulted for player-two moves.

Rather, the strategy-stealing idea is that we shall somehow pretend to be player two. Playing as player one in the actual game, we shall simply invent an imaginary first move for our opponent. It doesn't even matter where—let us imagine that they had opened with a move in the center square. The effect is that we are playing as player two in this imaginary game with the extra move, and so we may consult the winning strategy σ about how to play. We proceed to make exactly those moves in the actual game, and so except for the supplemental imaginary move, the two games are identical. A minor issue is that if in the actual game our opponent should happen later actually to play the imaginary move we had already assigned to them, then it is no problem, for we can simply invent a different imaginary move for them at this point, and continue consulting σ. (We don’t change the original first move or the imaginary game history, but rather invent a new imaginary move for them to play in the imaginary game on this turn.)

The main point is that because we had assumed that σ is a winning strategy for player two and we are playing according to σ as player two in the imaginary game, it means that we will achieve tic-tac-toe in the imaginary game before the opponent does so. But since we had made those same moves in the actual game, and all our opponent's actual moves appear in the imaginary game, it means that we will have achieved tic-tac-toe also in the actual game before our opponent. Thus, we have described how to provide a winning strategy for player one.

But this is impossible, a contradiction, since it cannot be that both players have winning strategies. So therefore, it cannot be that player two has a winning strategy in tic-tac-toe, which is what I claimed.

4 × 4 tic-tac-toe

Let us consider the variant of tic-tac-toe played instead on a 4 × 4 board, with each player trying to get four-in-a-row on any row, column, or diagonal. We might naturally call the game, “tic tac tip toe.” Perhaps the game play begins like this:

Did they play well? Do you think either player can have a winning strategy in 4 × 4 tic-tac-toe?

No advantage going second

There is no advantage, I claim, to going second in 4 × 4 tic-tac-toe. Perhaps this seems obvious—how could it ever help us to allow the opponent a first move? But I should like to give a careful argument. We shall mount another strategy-stealing argument, just as we did for 3 × 3 tic-tac-toe. What I claim is that if player two has a winning or even a drawing strategy, then player one has such a strategy as well. Therefore, in fact player two cannot have a winning strategy, since it can't be that both players have winning strategies.

The argument is this. If player two had a winning strategy in 4 × 4 tic-tac-toe, then we as player one in an actual game can simply pretend to be player two. We invent an imaginary first move for our opponent, and thereafter play according to the player-two winning strategy. If our opponent should ever happen actually to play the imaginary move we had already attributed to them, we can simply invent another imaginary move at that moment and continue.

The main point, as before, is that because we are playing according to the supposed winning strategy in the imaginary game, we will win that game, achieving a 4 × 4 tic-tac-toe before our opponent does. It follows that we achieve this also in the actual game, since we have made the same moves in both games, and our opponent as well, except that the invented moves for them in the imaginary game will not all be present for them in the actual game. In short, any winning strategy for the second player can by this method be turned into a winning strategy for the first player. But since they cannot both have winning strategies, this would be a contradiction, if it occurred, and so there must be no winning strategy for the second player to begin with.

Applied with drawing strategies instead of winning strategies, the strategy-stealing argument shows that if player two has a drawing strategy, then so does player one. So the situation is that either player one can force a win, or both players have drawing strategies.

How many opening moves up to symmetry?

I would like to count the number of opening moves in 4 × 4 tic-tac-toe. The board has sixteen squares in total, of course, and so there are plainly sixteen possible different opening moves for player X.

But the board exhibits many symmetries, which make many of these moves essentially equivalent to one another from the point of view of strategic analysis. For example, all the corner moves are essentially equivalent, since if you make an opening move in any corner, I can rotate the board (or rotate merely how I view the board) so as to move that particular corner square to the upper left corner. And since rotating the board does not affect any strategic or game-theoretic consideration, it follows that all opening corner moves are strategically identical. Similarly, any opening move on a corner-adjacent edge square can be moved to any other by rotating and reflecting the board as necessary. And finally the four center squares can similarly be moved to each other by rotation. Up to these rigid symmetries of the square, therefore, there are only three possible opening moves, as indicated here:

Let me argue further, however, that there are actually only TWO opening moves in 4 × 4 tic-tac-toe up to strategic equivalence. What I claim is that the corner squares and the center squares play identical strategic roles in the game. Really? Yes! The fact of the matter is that there is a deeper symmetry of the board, one which does not arise by rigid rotation or reflection, but which nevertheless swaps the corners with the center while preserving all the winning tic-tac-toe lines. Surprising though it may seem, this symmetry thereby shows that opening with a corner play is strategically identical to opening in the center.

Let me explain.

Please enjoy this selection from Infinite Games: Frivolities of the Gods, my new book serialized here with fresh material on games and the logic of games each week.

This week we look into variations of tic-tac-toe, including 4x4 tic-tac-toe, recursive tic-tac-toe, 3D tic-tac-toe, and more.

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The game begins with the coins on the track. The gold coin is farthest to the right, and during play all the coins will move only to the left. On each turn, a player may either (1) move any coin to the left by one or more spaces but without jumping onto or over another coin, or (2) take the leftmost coin into his or her possession. Whoever takes the gold coin is the winner―the pennies just don’t matter. In the diagram above, one of the players has just moved the gold coin three spaces. We shall see in the end that this was a very good move, a winning move, and indeed it is the only winning move available in this position.

What do you think is the winning strategy?

Interlude

I would recommend playing the game with a partner several times in order to gain familiarity with the rules and how game play proceeds. Here is one way the game indicated above, for example, might have proceeded in a sequence of legal moves.

The resulting position is this:

What is the best move to play next?

Perhaps at this stage it may not be clear exactly how to play, for the game may seem to have an open-ended nature, with moves taking place in an enormous game space of possible play, too complex for easy analysis. In the end, however, we shall on the contrary learn how to conceive of the game in a way that makes the winning moves completely transparent. We will be able to look at any position and easily find the winning moves.

In this last position, for example, knowing this winning perspective I can tell you easily that there are exactly two correct winning moves, and all others will be losing. One winning move will be to move the leading penny up two squares, like this:

And the other winning move will be to move the middle penny up two squares, like this:

All other moves, we will see, are losing moves, leading inevitably to a loss against any skilled opponent. By the end of the chapter we shall explain why these two moves and only these two moves are winning in this position.

Please enjoy this selection from Infinite Games: Frivolities of the Gods, my new book serialized here with fresh material on games and the logic of games each week.

This week we look into the Gold Coin game.

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To play Nim, one player sets up finitely many piles of coins and the other decides whether to move first or second. On each turn, a player removes one or more coins from any one pile—take one coin, two coins, or even the entire pile, as you like. Whoever takes the very last coin overall wins the game.

Can you discover the secret winning strategy?

Interlude

Please enjoy this selection from Infinite Games: Frivolities of the Gods, my new book serialized here with fresh material on games and the logic of games each week.

This week we look into the game of Nim!

Let’s get into it—a complete account of Nim. We’ll start with first-grader Nim, an easier case that is still challenging for adults, and yet first-graders can learn the strategy, before working up to the fully general winning strategy for Nim. Afterwards, we shall also solve Misère Nim.

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According to the rules of the game, the players will “chomp” the bar always from the lower left, selecting on each move a particular square in the bar and removing it along with the rest of the bar below and to the left of that square. The player who chomps the very last highlighted square at the upper right loses.

Who will win? Would you rather go first or second? Does this decision depend on the size of the initial chocolate bar? What will the winning moves be?

Interlude

Please enjoy this selection from Infinite Games: Frivolities of the Gods, my new book serialized here with fresh material on games and the logic of games each week.

This week we look into the game of Chomp!

Have you decided whether to go first or second? Let us get into the analysis.

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