# Is it possible to design a two player game of skill with absolutely no luck?

I really do mean absolutely no luck. This would disqualify even games like Chess as White has an inherent advantage, winning between 52 and 56 percent of all games. A good test for a game with absolutely no luck is:

• If player 1 plays again as player 2 and vice-versa, both playing to the best of their abilities, then the second match would be played out exactly the same as the first match.

Is it even possible to design such a game? If so then what would the rules of such a game entail? If not then why?

• If we're going to expand the definition of luck to include first-player advantage, such a game would have to have simultaneous turns. However, such games lend themselves to mixed strategies which seem like they would also be covered by generalized luck. May 31, 2017 at 3:49
• Rock Paper Scissors? May 31, 2017 at 8:11
• You say game of skill but it seems you are actually referring to games of strategy. They're NOT the same thing. Most direct competitions (e.g. archery) are games of pure skill with no luck. They don't have *strategy*—just skill. May 31, 2017 at 11:16
• @Wildcard, though real-life sports have the annoying property that seemingly external conditions may affect the results (at least in theory). Any outdoor sport suffers from luck effects due to weather conditions changing (gusts of wind just as you're releasing the arrow or kicking the ball). In any tournament, the match schedule may give one player/team more rest time than the other or just plain have the game take place at a time more comfortable for one (being sleepy in the morning and a match at dawn). The last one might of course be counted as part of skill. May 31, 2017 at 11:58
• @Aadit, "the second match would be played out exactly the same as the first match." -- would it be? If player A won the first match, then B has no reason to play an identical match, but exactly the opposite, they have nothing to lose by doing something completely different. May 31, 2017 at 12:00

By Zermelo's Theorem, every 2 player game has one of the following:

1. Chance
2. Hidden information (this also includes simultaneous moves)
3. Either first or second player has a strategy that will guarantee them the win
4. Both players can guarantee a draw (or force the game to go on forever)

Any game that has (1) is clearly out.

Any game that has (2) is also out if you view chance as our way of modeling information that is unknown. After all, the same rules of probability work whether it's unknown because it will be chosen soon by the roll of a die, or is unknown because an opponent has already chosen something but isn't telling us.

Any game that has (3) is out by your definition, which is reasonable because presumably you choose who goes first by chance, and that determines the winner.

So your only choice is to go for (4): games that under perfect play end in draws (or are infinite). For example, tic-tac-toe can be thought of as fitting your requirements because under perfect play every game is always a draw, no chance involved.

Note that we can't always distinguish games in category (3) and (4) - for example, though you talk about the "inherent advantage" of first in chess, it's entirely possible that this advantage is merely an artifact of our incomplete understanding of perfect play, and that two perfect players might always play to a draw (perhaps even in the same way each time). You can also convert any game with (3) into a game with (4) by wrapping two rounds together in a single game where first player alternates, and declaring the overall game a draw unless one player wins both times. So while chess may or may not fit your definition, two-round-chess does.

• The English definition of "game" is broader and fuzzier than the "extensive-form" definition I am implicitly using to apply Zermelo's Theorem. If you wish to consider other types of games, including timed contests of prowess and trivia, read other answers.

• My analysis of (2) is certainly controversial - I consider rock-paper-scissors a game containing luck but you don't have to. For more on different views of chance, read about Bayesian vs frequentist schools of thought.

• I consider simultaneous moves under hidden information because you could imagine that players alternate but the first player's move is hidden until the second player moves.

• Note that I didn't quite follow the test you provide for "absolutely no luck" because it seems overly restrictive - a game like tic-tac-toe (or two-round tic-tac-toe, perhaps) has absolutely no luck, yet since there are multiple equally good ways of playing to a draw, the game may not be played the same way each time. May 31, 2017 at 3:55
• It's worth noting that the world championship games are played with equal numbers of whites and blacks for each player, and I believe the Swiss system is deterministic, so there's not strictly speaking any chance in who gets to be white or black in a tournament. And it's generally thought (though not proven) that chess solves to a draw. May 31, 2017 at 4:12
• What if you replaced the coin-flip for player order with an arm-wrestle? Would that count as skill rather than chance? May 31, 2017 at 8:55
• @Wildcard In a simultaneous game, whether you have made a good decision depends on what decision your opponent makes. Consider rock-paper-scissors. Throwing rock was only the correct decision if your opponent throws scissors, which you have no control over. Compare chess, where the correct decision depends only on the current board state, which you have full knowledge of. May 31, 2017 at 21:39
• @Wildcard I'll amend my statement to say Set is not a strategic game. Prowess-based games are certainly a thing, but they're not very interesting to analyze. However, Basque chess does not have simultaneous turns. It is two separate sequential games being played simultaneously. The two boards do not affect each other, so there is no meaningful simultaneity to the turns. May 31, 2017 at 23:42

I find it a bit odd to include the first move advantage in chess as an element of luck -- it assumes that luck is involved in choosing the color, but that choice isn't usually seen as part of the game, more of the tournament organization.

But there are a few ways to fix it.

The most obvious and usual is to play multiple matches, with alternating colors. There is probably still some luck involved, because it may be an advantage to start first or second. World Championship matches use this format, and they switch around the order of who goes first in each 2-game block half way during the match (so it goes A B A B A B B A B A B A ).

Another is to simply accept some number for the built-in advantage to white (say 54%) and instead of always giving 1 point for a win, adjust the scores for winning as black and winning as white so that the expected scores are equal. The math is left as an exercise for the reader and there will be some issues in practice with e.g. Swiss tournaments where people get subtly different scores.

But there is an easier way: simply play two (or another even number) of games simultaneously, one with each color, each with their own clock. By symmetry, there is no advantage for either player, and that was the only luck involved.

This is called Basque chess (video of two players playing it, article about some exhibition Basque chess matches involving some top players).

• If white has an advantage, and you alternate colors, wouldn't you just expect perfect players to win whenever they're white, lose when they're black, and the whole match will be a draw? You need to play an odd number of games to be sure to get a winner, and then you have the problem of which player gets an extra chance to be white. May 31, 2017 at 22:52
• @Barmar There's nothing in the question that says the game must always have a winner. Jun 1, 2017 at 5:34
• @Bamar: they can also just draw all the games, chess is very likely to be a draw with perfect play. But that is fine. Jun 1, 2017 at 7:44
• @Barmar That's exactly what should happen, according to the OP's stipulations. If two players have identical skill, and one still loses, it must have been due to randomness, first-player advantage, or something else unrelated to skill. In a game with no luck and no first player advantage, equally skilled players should always finish in a draw. According to the OP's stipulations, a "no luck" game can't have a winning strategy. If it does, it comes down to luck as to which player is in the correct seat to implement that strategy, which contradicts the premise. Jun 1, 2017 at 13:31
• @Barmar, the problem of two equally skilled sided always ending in a draft is removed when one player is better right? If one player is more skilled, he can win both games of chess, playing as white and black, so then the score is 2-0 and the more skilled one wins. Only with equal skill it should be 1-1 (twice draw, giving 0.5 points) every time.
– user20863
Jun 1, 2017 at 14:37

Consider the game of go, where Black starts, but White gets a number of points (the komi) to balance the advantage of Black's first move. Now, of course this may still not lead to an absolutely balanced game, but the complexity of the game pretty much swamps the remaining difference. From a purely theoretical standpoint the game is still a win for either Black or White (depending on the amount of komi), but as @BenjaminCosman stated in their answer, any two-player game of pure skill has a deterministic result anyway.

If you can't agree on the correct komi, one option is a bidding system like the pie-rule used in Hex: let one player decide the komi, and then the other chooses which color to play.

If that still doesn't satisfy your needs, have the players play an even number of games in a row, with alternating colours, and declare the player with more wins as the winner. (That of course works in chess, too.)

In go, you could also consider playing just two games, and take the total score difference of the two games as the final result. (i.e. if the first game ends with A winning by +2.5, and the second with B winning by +1.5, then A is the final winner, this is harder to do in chess.) However, that does change the strategy of the game, since usually the margin of winning does not matter.

• Go is a finite deterministic game (basically, there's a rooted oriented graph for the moves, each move is an edge, each position is a vertex), so it either has a winning strategy for one or the other player, or has a draw strategy for both players. So assuming the players are Oraculum A and Oraculum B, go does not qualify. And nothing else than assuming they are Oracula makes sense to me (or I may have misunderstood the question).
– yo'
Jun 5, 2017 at 16:41
• @yo', Sure. That also rules out all variants of chess, as mentioned in other answers. But to me, a "game of skill" between two entities with power great enough to know the optimal play does not make any sense. In a smaller scale, it doesn't make sense to call tic-tac-toe a game of skill between two adults familiar with it. Jun 5, 2017 at 19:28
• I think the issue may lie in the subjectivity of non-triviality in regard to games. Games typically designated as non-trivial are those which are intractable to the strongest human player, and more lately, defy computational solution, but to the average 5 year old child, tic-tac-toe is decidedly non-trivial. :) Aug 17, 2017 at 18:53
• @yo' You can't draw in Go because of the half point komi. Or I think you hear some say a quarter of a point in chinese scoring rules. Aug 31, 2017 at 14:56
• @snulty: While komi is often an odd multiple of half a point, in order to avoid jigo (a draw), this is not universal. To make the game fair, komi would have to exactly cancel out the amount by which Black would win given perfect play to maximise the score; that means it would be a whole number for almost all rule sets. With this komi perfect play would then lead to a draw. (Of course there is an archaic version of Ing rules in which the result can include all sorts of odd fractions, so that theoretically correct komi for those rules would be fractional.) Dec 17, 2018 at 0:11

The card game "Set" may qualify. (Your requirements are awfully stringent but also not precisely defined, so you may disagree on a technicality—but I believe it fits.)

It's described as the "Family game of visual perception," which is a good description. There are no turns. Twelve cards are dealt, and as soon as you see a set you call out "set" and then take the set you saw. Three cards are dealt to replace the three you took, and play continues.

There can be (will be) variations from game to game, obviously, as there are over a thousand distinct sets and several million million possible "boards" of 12 cards. But in any given game, both players have exactly the same information. I wouldn't count the starting board arrangement as "luck" as it doesn't give either player any advantage.

The point you could reasonably question is whether perception counts as a skill. I fully believe it does. It's certainly not susceptible to turn-based mathematical strategy, though!

This game is of the second type listed in this answer.

A sample of a set "board," with notes on the rules:

• Ricochet Robots springs to mind from this. The game basically consists of the players staring at a board, and calling out the numbers of moves they require to solve the puzzle currently presented, with the lowest number of course getting to present their solution and winning a point. May 31, 2017 at 11:48
• In theory people with the same amount of "skill" would say SET at exactely the same time, something not covered by the rules of the game.
– user20863
Jun 1, 2017 at 15:50
• Similarly, the game Spot It could also fit this criteria. No turns, cards in hand are secret but each will have exactly one symbol matching the card in play (no hand or card is "better" or "worse" than any other). It's all about your skill at parsing sets of symbols quickly and locating matches.
– bta
Jun 1, 2017 at 22:42
• @bta, that was the ultimate nerd snipe as I'm now knee-deep in the math of the game Spot It. :) Jun 2, 2017 at 0:07
• But isn't there luck in which card or symbols you happen to look to first? Jun 22, 2017 at 18:24

For perfect symmetry, you need both players to have the same opportunity to make moves. This suggests that either:

1. Players play the game twice, rotating the first player position between them, with the final score of the game being the score differential between the two games.
2. Players play a game in which moves are played simultaneously. Either their moves do not directly interact with each other (in which case they are essentially solving the same puzzle and trying to get the better score), or there are rules preventing their moves from interfering with each other (for example, maybe if pieces are close enough together they can't move them any closer), or there are rules that resolve what happens when their moves interfere (for example, pieces which move into the same space on the same turn are removed from the board).

You may consider the simultaneous action in 2 to be a form of luck or hidden information, but it's done in a way that two players of equal skill would always make the same moves against each other, so that seems to meet the requirement. Of course, as pointed out, any perfectly symmetrical game played by players of equal skill is guaranteed to end in a draw, so it's only interesting if the game allows you to distinguish between quite fine differences in player skill - tic-tac-toe is very poor in this regard, since everyone above a fairly low skill level is able to force a draw easily; by comparison, in a game like Go even a small difference in skill level can result in a very deterministic result.

• An example of (1) is the way competitive bridge is played. A team of 4 players plays the same deal of cards twice - once as North-South, and once as East-West. This takes the element of chance out; you truly measure "given the hand you are dealt, how well does your team play it?" - with the final score being the difference between the two games. I suppose that is not a "two player game" though... May 31, 2017 at 16:25
• @Floris This is not a "symmetrical" situation. Since top-level card players are skilled in remembering the cards that have been played, on the second play of the hand each player has (potentially) perfect knowledge of all the cards held by all the other players for the whole duration of the hand (I'm assuming that the situation where exactly the same cards are dealt to one of the players in two separate games is too unlikely to be relevant). That was not the case for the first play of the hand. May 31, 2017 at 20:01
• @alephzero sorry if I didn't describe this clearly. A team is four people, called NESW. NS of team 1 play EW of team 2. The score is recorded. The cards (same deal) are then given to EW of team 1 and NS of team 2 (who did NOT see game 1). So they have NO prior knowledge of the cards. Only relative skills of the four players on each team determine the outcome as they are facing the exact same set of cards and their decisions will determine their outcome. May 31, 2017 at 20:08
• Duplicate bridge removes a lot of the luck that would otherwise be inherent, but there is still plenty left. For example, to make a contract may involve choosing between two 50% chances: you can make a play that wins when West holds the king of spades, or a play that wins when West holds the king of diamonds, but no play wins in both cases. When I play the hand I may choose to hope that West has the king of spades, and at the other table they hope for the king of diamonds. We both played the hand with equal skill, but one of us guesses right and scores better. Jun 2, 2017 at 17:33

# Yes, 2 games of chess!

After the first game players switch sides and play again. The slight advantage is completely removed.

The premise in the middle of the OP is incorrect and irrelevant.

• Agreed. Tic-tac-toe would work as well. Jun 1, 2017 at 5:35

A game that may fit your constraints: Goofspiel, also known as GOPS - Game of Pure Strategy, for 2 or more players.

Goofspiel is played using cards from a standard deck of cards, and is typically a two-player game, although more players are possible. Each suit is ranked A (low), 2, ..., 10, J, Q, K (high).

1. One suit is singled out as the "prizes"; each of the remaining suits becomes a hand for one player, with one suit discarded if there are only two players, or taken from additional decks if there are four or more. The prizes are shuffled and placed between the players with one card turned up.

2. Play proceeds in a series of rounds. The players make "closed bids" for the top (face up) prize by selecting a card from their hand (keeping their choice secret from their opponent). Once these cards are selected, they are simultaneously revealed, and the player making the highest bid takes the competition card. Rules for ties in the bidding vary, possibilities including the competition card being discarded, or its value split between the tied players (possibly resulting in fractional scores). Some play that the current prize may "roll over" to the next round, so that two or more cards are competed for at once with a single bid card.

3. The cards used for bidding are discarded, and play continues with a new upturned prize card.

4. After 13 rounds, there are no remaining cards and the game ends. Typically, players earn points equal to sum of the ranks of cards won (i.e. Ace is worth one point, 2 is two points, etc., Jack being worth 11, Queen 12, and King worth 13 points). Players may agree upon other scoring schemes.

• This is a really interesting suggestion. Even though the game utilizes random order of the prize cards, there is perfect information for any given decision. Even though the sequence of future prize card is unknown, all players have access to the same statistical information as each prize card is revealed. I can see advantage for any player, or luck being a factor in this game. It seems to definitely qualify! Aug 17, 2017 at 19:10
• Goofspiel/GOPS has hidden information, which is excluded as per point 2 of Benjamin Cosman's answer. Mar 29, 2018 at 16:12

Each player makes a tower of stacked bricks. Their brick supplies and building areas are separate and identical. The builder of the first tower to topple loses. If both towers are standing after a predetermined amount of time, the builder of the tallest tower wins.

• A large truck rumbles by outside, causing your slightly taller tower to fall over just before time is up. Was that bad luck or lack of skill on your part to withstand unexpected vibration? May 31, 2017 at 23:47
• @Michael That's a foul, and requires you to start over. Tournaments are not played near roads. Jun 1, 2017 at 6:49

No. Whatever you do to exclude luck from your game, either player may choose to overrule you by making their own choice of moves random. A player who chooses to move randomly (or even uses a random factor to advise their moves when unsure) has introduced chance back into the game, and may win or lose by luck.

• Not necessarily. The ability to read your opponent, and the ability to outsmart your opponent, can both be seen as components of skill and/or strategy. In this case, the player decides to make unreadable moves which confuse their opponent through sheer unpredictability, at the expense of their own ability to strategise and plan. Knowing when to discard strategy and act solely at random is a skill-based decision, which makes such confusion-oriented gameplay a valid tactic for outwitting an opponent (it's a mind game, basically, which causes them to doubt their ability to read you), May 31, 2017 at 19:14
• and a valid tactical choice as long as it falls within the game's rules. It may introduce chance into one side, but this doesn't necessarily cause the game as a whole to fall back on chance (no human can be truly random, so this strategy can and will backfire as soon as your opponent learns to read your "randomness"). May 31, 2017 at 19:14
• @JustinTime My point is that it is possible for acting randomly to cause an unskilled player to play perfectly, hence "getting lucky". You argue that games with a random player can still involve strategy, and I agree; but the question is whether a game can be found with no luck whatsoever, and in that context, your argument is irrelevant. May 31, 2017 at 19:47
• My argument is also that even a "random" player will become readable over time, due to humans not being truly random unless they use an exterior source of randomness. May 31, 2017 at 20:35
• I agree with this. Richard Garfield (creator of MTG) has a great lecture on chance in games where he says basically the same thing, using chess as an example. He shows that chess does in fact have luck because a player making random moves could theoretically make the right set of moves to beat the best players in the world. Jun 1, 2017 at 2:54

I believe the game of Connect 6 was designed to eliminate the advantage/disadvantage of going first or second:

According to Professor Wu, the handicap of black's only being able to play one stone on the first turn means that the game is comparatively fair

One way that a game can be made entirely skill- or strategy-based is by meeting the following conditions:

Examples are provided, in the context of a card game. In this fictional card game, each player has their own deck, and draws a card at the start of their turn. On their turn, they are allowed to play one card face-up, and set one card face-down. On their opponent's turn, they are allowed to activate one face-down card. A full turn consists of each player getting a turn; for any full turn N, these are turn N1 and N2; player 1 is the player that goes first (and thus gets 11), and player 2 is the one that goes second (and thus gets 12).

1. Barring player actions, both players will have identical game states.

• Both players will have the same resources become available over the course of the game (e.g., both players use the same deck).
• These resources will become available in a predetermined order, with no chance involved (e.g., both players must stack their decks in the same order).
• Both players will start with the same number and type of resources available (e.g., both players start with the exact same hand).

This guarantees that:

1. Both players have the ability to make the same moves (e.g., if player 1 can play card X on turn 31, player 2 can play card X on turn 32).
2. Chance is removed from hidden information, as all possibilities are known; the unknown changes from "what do I have to expect" (e.g., which deck is the opponent using) to "which of their resources is this" (e.g., which card out of the cards both players have drawn is it).

This allows players to infer hidden information using their deductive skills, as their opponent's game state can be inferred from their own (e.g., if player 2 sets a card face-down, then player 1 can look at which cards they've drawn (as they know player 2 has drawn the same cards), and eliminate the cards player 2 has already used, to determine all possibilities for that card; they can then logically determine which of these possibilities it's likely to be).

2. Turn advantage will be removed. This can be done in multiple ways.

• Both players can take turns simultaneously; each player plans out their turn, and reveals their actions once both have confirmed that they're done planning (e.g., both players set the cards they plan to use on turn N face-down, and tap their deck when they're done preparing; once both players tap their deck, they both flip the cards they set that turn face-up).
• All games can be a two-game match, and a player must win both games to win the match; this guarantees that each player will have first-turn advantage, eliminating its effect on the game (e.g., to beat player 1 in a match, player 2 must win both when they go first, and when player 1 goes first).
• Whichver player has the advantage receives a handicap that compensates for it (e.g., on turn 11, since player 1 gets to draw before player 2 does, they can only play one card instead of the normal two; player 1 will thus have more options available, but player 2 will have more options in play).
• Going first and going second have distinct advantages; player 1 chooses which of these they get, and by extension, whether they go first or second (e.g., the player that goes first might be able to play 2 cards face-up once during the game, while the player that goes second might be able to set 2 cards face-down once during the game; player 1 decides which of these advantages they want, which also decides turn order).
• Player 1 has the option of giving up their turn advantage; if they don't, player 2 receives an advantage (e.g., player 1 chooses whether they want to draw on turn 11; if they choose to draw, then player 2 can play a card from their hand during turn 11).

Out of these options, the first two are the most viable ways to eliminate chance. While the others might work, they are much harder to balance, because the full implications of turn advantage are often unknown during the game's design stage and preliminary testing; they will only truly become known once the pros start milking them for all they're worth (e.g., the advantage of drawing first might be more than enough to compensate for only being able to play one card on your first turn, or only being able to play 1 card might be crippling even if you do draw first).

A lot depends on your definition of a game for purposes of this question. If you're avoiding chance playing any part then the game must come down to a measure of some skill or trait (or combination thereof). Any such game, being perfectly symmetrical and deterministic, would then proceed to end the same between any two players every game without exception (assuming both players played to the best of their abilities).

For an example I've just invented a game, I call it Who is Taller: Aggregate Edition Two players get their height measured by the same means over a period of time (to account for shifts over the day) and then their heights are compared. No matter which is player 1 or player 2 the results are the same. Obviously this game is probably pretty terrible, my sales are sitting at zero currently, but it meets the criteria you've put forth.

Any game like this would make repeated plays meaningless unless the skill or trait is something a player could improve meaningfully either from playing the game itself or externally. At which point you've discovered all sorts of games involving strength (arm wrestling), dexterity (darts), pattern finding & reflexes (Set, as mentioned in Wildcard's answer), ...etc. Though obviously only one of those examples is actually a board or card game, there are certainly a number of board style games that involve taking physical actions especially involving dexterity, timing, or balance and it mainly depends where you want to draw your line in the sand on what counts and what doesn't.

Of course to truly replicate the starting state in a skill/trait measuring game as mentioned above with the players swapped you'd need to somehow go back in time to run the second test as one player could gain more skill from the playing of the first game than the other, thus shifting the result; but assuming you could do so the result should be the same every time.

The common thread though is that in most of these you're looking at some kind of well structured competition instead of a more traditional game, as the goal is not to allow multiple players to make interesting decisions and try to come out on top with chance of success determined by skill, but rather to see if your abilities are greater when measured in a particular way than that of another.

Although your categorization of what constitutes luck is non-standard, I understand perfectly the point you are making regarding first player advantage.

It is possible, and I designed a set of games, [M], which, for certain types of symmetrical configurations, eliminates turn-based advantage, and relies on symmetry breaking.

CONTEXT

For context, symmetry-breaking is a phenomena observed in nature, but, like everything in nature, is related to combinatorics. Most of the thinking on this subject in relation to games will be found in Combinatorial Game Theory (CGT), and combinatorics is at the root of all games discussed on this Stack.

Here are a few articles I found on the subject, to demonstrate I am not alone in pursuing this idea in relation to Game Theory and CGT:

The underlying problem with combinatorial games (all games are combinatorial;) is that the underlying mathematics are unforgiving: a set of mechanics is either too imbalanced, always resulting in a win for the advantaged player, or too balanced, always resulting in stalemate.

In deterministic games (those which involve no elements of chance such as card shuffling or random number generation, commonly dice) complexity becomes a balancing factor. Chess may have a built-in advantage for the starting player, but has sufficient complexity that the second player may still engineer a win. Complexity partly subjective in that it relates to the tractability of a problem. Games such as chess, which are intractable to the strongest human players, are regarded as non-trivial, but for a 5 year-old, tic-tac-toe is distinctly non-trivial. Tic-tac-toe contains a non-symmetric element in the odd-order (3x3) gameboard, but nevertheless always results in a draw if the game has become tractable to both players. Unlike 8x8 chess, the less symmetrical tic-tac-toe provides no inherent advantage to the starting player where the game is tractable to the participants

SYMMETRY BREAKING IN EVEN-ORDER [M]

In the set of games [M] restricted to 2 players, symmetric grid configurations where n is even n^2(n^2), n^3(n^3), ..., n^2(m^2), ..., allow the second player to draw a stalemate by simply mirroring the placements of the starting player. This is because in such configuration there are an even number of turns and board positions.

However, if the starting player is perceived by the second player to have made a sub-optimal placement, the second player may change their mirroring strategy an attempt to engineer a victory.

In this model, there is no inherent advantage for the starting player. Neither is there an advantage for the second player where the configuration is non-trivial because, while the onus is on the starting player to place optimally, where the gametree is intractable, there is no guarantee that the second player's assessment of a move as suboptimal is correct.

Thus, even-order [M] is solved to infinity in that the second player can always draw a stalemate, it is not solved in the context of objective optimality of a given move over the course of the game, particularly in that the [M] placement constraints, drawn from Sudoku, allow for unpredictable patterns to emerge than can yield advantage if exploited. ([M] has similar emergent complexity to Chess and Go, with less rules and no special conditions.)

This principle may be extended to [M] with any even number of players (2,4,6,...), and becomes more interesting in that only even players (P2, P4, P6, ...) can break the board symmetry, where odd players (P1, P3, P5, ...) do not have this option.

Where is really gets interesting is in even-order [M] with an odd number of players, where number of players is a factor of total board positions, for instance as 3P|2^2(3^2) which yields 36 board positions and 12 placements for each of the three players.

In a configuration like 3P|2^2(3^2), the initiative for symmetry breaking cycles between players, still always on an even-numbered turn, but now all players cycle between odd and even numbered turns:

01: P1
02: P2 (initiative)
03: P3
04: P1 (initiative)
05: P2
06: P3 (initiative)
07: P1
08: P2 (initiative)
09: P3
10: P1 (initiative)
11: P2
12: P3 (initiative)

You'll notice that over 12 turns, 1/3 of the game, each player has the initiative an equal number of times. (This configuration is distinct from even-order configuration with an even number of players because the game cannot result in a stalemate.)

• I'd love to know more about symmetry breaking. It seems quite relevant to the concept of initiative in role-playing games. Aug 15, 2017 at 21:29
• @AaditMShah Initiative in this context is distinct, in that it involves a method of ordering turns (player actions). By contrast, symmetry-breaking relates to partisan strategy in a Combinatorial Game Theory framework, typically in 2-player games, where the second player can mirror the actions of the first player until some point of divergence, based on perception of advantage. That said, I've been thinking about symmetry-breaking in > 2P games, and initiative would certainly interesting to explore in that model. Aug 16, 2017 at 20:30
• @AaditMShah I updated the answer to provide more context on symmetry breaking as related to games, and provided a concrete example in the set of games [M], which I designed, and which are unique in relation to all other previous games. I linked the [M] rules in the above answer, but you can also find simplified rules here: mclassgames.com/how-to-play Aug 16, 2017 at 23:11
• Is there somewhere I could learn how to play [M] interactively? Just reading the rules is confusing. Aug 17, 2017 at 4:28
• @AaditMShah My team and I are planning to go into open Beta sometime in September, but if you have an android or iOS device, can add you to the list for the closed Beta (pst me at dukezhou@mclassgames.com) We're not set up for even-order configurations atm, but you can definitely get a sense of how the [M]echanics work with the 9x9 "classic" Sudoku configuration. Aug 17, 2017 at 18:20

I am going to criticize your definition of Luck.

Yes, it's possible. Take the game Anomia

This game requires players to quickly flip over cards and name items from subjects.

For example, if the card 'Small Dog Breed' was flipped, someone could say "poodle" and take the card for a point.

If your definition of a game without luck is what you're going for, then here is how each game would go:
card flip. "Monkey".
Both players: Marmoset.
Tie breaker.
card flip. "Tree".
Both players: aspen.

But clearly, luck is inherent in this game, with the knowledge of the players, and the cards which were turned over.

There is a common mechanic that I've always called the 'cake cutting mechanic' although that doesn't seem to be the common name.

lets say player A goes first. Player A then make a move within the rules of the game. Player B then has a choice. They can make a move back and then the game continues as normal. Alternatively they can say they like Player A's move so much they want it and both players swap sides and Player B now has Player A's opening Move and Player A then takes a turn in response to there opening move and from this point players stay with the pieces they have.

So if you perceived white in Chess as having an advantage opening with Kings Pawn forward 2 then the Black player just says 'I want to be white now' and turn the board around.

This is rule is used in Ponte Del Diavolo for example.

• I think you're referring to the "pie rule". It's a great suggestion, but I don't think it constitutes the elimination of advantage (which in the question is regarded as an element of luck) if the game is intractable, because neither player really knows if the first choice by the first player is truly optimal. Aug 17, 2017 at 19:24

Yes, and it's already been done. It's called InstaGib over LAN. Latency is neglible, weapons always kill when they hit and have 100% accuracy. Everyone has the same amount of life, the same gun, infinite ammo.

You both start at the same time, and you can eliminate the random factor of spawns by placing only 2 spawnpoints.

One for Red, one for Blue. Here's an example. https://www.youtube.com/watch?v=SuRJNn-R1C4

edit: monoRed is right. Excuse my noobness.

• This is the board games stack exchange, so video game answers probably aren't what the asker was looking for. May 31, 2017 at 14:13
• While true, I'd argue that anything that can be done in such a video game could be replicated in a board or card game, such as by having simultaneous resolution with equal information combined with some kind of dexterity task involved in playing to simulate players who are simply faster at processing information and reacting. Which would seem to count except in the case that not having perfect information might be considered having chance involved. Jun 1, 2017 at 1:36