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

Question

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?

Answer 1

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.

ADDENDUM, addressing common objections to this answer:

• 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.

Answer 2

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).

Answer 3

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.

Answer 4

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!

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

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

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

Answer 11

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:

Breaking Symmetry (combinatorial games)
Explaining Diversity: Symmetry-Breaking in Complementarity Games
Symmetry-Breaking in two-player games via strategic substitutes and diagonal nonconcavity: A synthesis
Excerpt from Modern Principles of Economic Mechanics Vol. 1

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.)

Answer 12

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

Answer 13

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).

Answer 14

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.
Tie breaker ... ad infinitum.

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

Answer 15

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.

Answer 16

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.