Is chess a zero sum game?

Question

"Chess is a zero sum game" is a claim you see often, even on this forum.

Game theorists will tell you that it is just obvious. Either the game is a draw, nobody wins or loses, or one side wins and the other loses. A win is equal and opposite to a loss and a draw is worthless. QED.

Economic trade is a classic example of a non-zero sum game. In fact it is a win-win game. If that weren't the case then there would be no trade. Nobody would trade freely if they knew they were going to lose in advance.

Answer 1

Edit: let me add something because this answer seems to be misunderstood. FIDE ratings are based on game scores (see the Mathematics of Elo Ratings. It was created by Arpad Elo as a method for calculating the relative skill levels of players in zero-sum games such as chess. A player's Elo rating is represented by a number which may change depending on the outcome of rated games played. After every game, the winning player takes points from the losing one. So, if chess is a zero-sum game, so are the FIDE ratings!

The question then is: are FIDE ratings a zero-sum game because of pairings of players with different K-factors? A question I reinterpreted in a more provocative way: is chess empirically a zero-sum game?

As stated in Brian Towers' answer:

Not necessarily. My k FIDE factor is 20. If your's is also 20 then, yes, my loss will match your gain, but if you are a junior or still haven't played 30 games (I think) then your k factor will be 40 and you will gain double what I gain. Similarly if you have ever been over 2400 then your k factor will be 10 and your gain will only be half my loss.

In the May 2020 FIDE list of the standard rating (see here), I observe a distribution of K-values for 2,700 players rated with

• a Fide rating in April and May,
• at least one game recorded in the list of of May 2020.

The distribution is as follows:

     K      Freq.   Percent
    10        21     0.78   
    20     1,704    63.11
    40       975    36.11   
    Total  2,700   100.00

Then, I plotted the estimated probability density function of the difference between their rating in April and May (given that the formula for updating a player's rating is based on the expected and the actual scores of the games played). As expected, the difference is centered on zero because the points lost by some players are won by others.

However, the sum of points gained is greater (23,144) than the sum of points lost (-21,564), which represents an average gain of +0.59 per game. However, this average is not statistically different from 0! Conclusion: we cannot reject the hypothesis that chess is (empirically) a zero-sum game!

One caveat: I did not observe the results and matches of these players; I assumed that the points won by some players in this list are lost by others in the same list.

This aside, I like very much the point about the experience gained and the sharing of ideas during and after a game, which suggests that "chess is anything but zero sum"!

Answer 2

When I said chess is a zero sum game in that answer, I wasn't referring to anything involving ratings. Obviously if we include ratings then chess isn't strictly a zero sum game, since the gains and losses aren't always balanced out. But different rating systems are arbitrary, and aren't part of the game of chess itself.

Your question should be "Is the FIDE rating system a zero sum game?"

Answer 3

Chess is a zero sum game. There is 1 point available and it is divided over the players, what one gains the other loses. It is a simple concept with a simple answer. Tournament rules mention some very rare cases where the result is 0-0 or 0.5 - 0, but in my opinion those are "outside the game". That a mobile phone may ring is not really a matter for the game theoretical analysis of the game of chess.

Chess games played in tournaments are not zero sum, but the tournament as a whole is. There are many situations where e.g. neither player wins anything in case of a draw, but a win by one of the sides puts that player in a prize-winning position. Not zero-sum. But a chess tournament, viewed as a multi-player game, is not the same game as chess. And the tournament as a whole usually has a fixed prize pool and is thus again zero sum.

Ratings are not a game at all. They are a method to estimate player strength by some third party, not a system of "points" that you "win". I could have my own private system of ratings that I never tell you about, based only on published chess results. I don't think you can be considered to be playing a game if you don't even necessarily know of its existence. When Jeff Sonas computed what ratings would have been for players before 1970, where those players retroactively playing the ratings game? That is absurd. Ratings are not a game.

Answer 4

The main reason why people often claim chess in a zero-sum game is simple: it is!

Considerations about ratings, the joy of playing, the scoring system for a certain type of tournament... are not an intrinsic part of the game of chess. The utility function for chess only considers three values: win > draw > loss. The value for the first player determines the value for the second player. When player 1 sees their utility function maximized, player 2 sees it minimized and vice-versa.

Please note that the distinction between "sum-0" and "sum-1" is irrelevant here since chess strategy would be exactly the same if the scoring system were "0.5, 0, -0.5" rather than "+1, +0.5, 0". In contexts like business this distinction is important because you have the "0-monetary-value choice" of not participating. But once you are in a game of chess, the only way to opt-out is to lose

Chess if therefore a zero-sum game.. Your "chess career" may not be, though.

Answer 5

This answer ignores the question of ratings, instead looking at a setting like a tournament, league, or similar.

Here, a single game of chess is indeed zero sum, under the typical scoring rules (1pt for a win, 0.5 for a draw, 0 for a loss) and assuming that the goal of players is to maximize points earned. That this is "actually 1-sum, not 0-sum" does not go against this assertion: Since the number of games each participant/team gets to play is fixed, one might as well score (+0.5 / 0 / -0.5) without changing the resulting outcomes at all. (That is why typically, the interesting property of a game is really being constant-sum, and not necessarily zero sum. The former are sometimes referred to as "rescaled zero-sum games".) If the goal is not to maximize points earned, but rather e.g. to place as high as possible in a given tournament, it is well possible that a match is not zero sum; say, if both players would need a win to move up a rank, and losing or drawing would not make a difference by that metric.

It is important to point out that the zero-sum nature of a match does not arise from the fact "Either the game is a draw, nobody wins or loses, or one side wins and the other loses." alone. In football ("soccer"), the same is true, but games are often scored 3 for a win, 1 for a tie, 0 for a loss. Thus, a match is not 0-sum (or constant sum): Two teams would e.g. rather trade wins than tie twice. Two possible aims of the three points for a win rule are to incentivize more aggressive, arguably more interesting-to-watch play, and curb potential for collusion.

Answer 6

I've seen a lot of answers here discussing wins and loses, piece trades, and even ratings. Here is my take on chess being a "zero-sum game":

There are exactly two players, so any move, any trade, any event in-game, affects both players in the opposite manner. It is theoretically impossible for there to be a move that is net positive for both players, or a move that is net negative for both players. If players trade queens, this is not net negative for both players. It may be negative in varying degrees, but the net result is simply how it affects each players' odds of winning, and since there are only two players, if player 1's odds go from 60% to 65%, then player 2's odds decrease by 5% as well.

Answer 7

A game of chess, considered as an abstract game, is a zero sum game. A game of chess, considered as a way to try advance one's rating may or may not be a zero-sum game; it depends on the rating system. A game of chess, considered as a human activity, is not a zero-sum game, as both players may gain pleasure from it, whether or not one loses.

Answer 8

A different take on the question "Is the FIDE rating system a zero sum game?": one could define a player's scaled ELO rating with the formula

scaled_rating = ELO_score / k. 

By the definition of ELO scores, this quantity is zero-sum: if I play a game against you, then your scaled rating change plus my scaled rating change is exactly zero.

There are two exceptions, because of how the factor k is defined:

• When you turn 18 or play your 30th game, your scaled rating doubles.
• When you first pass 2400, your scaled rating doubles.

These are always positive variations, and when a new players gets a FIDE ELO score it is positive. So technically chess is a positive-sum game: the sum of the scaled ratings of all players in the world (living and dead) is an increasing quantity.

However, the events that make this sum increase are probably not what you imagined. (In particular, there are at most 3 in the life of each player).

Answer 9

The claim that chess is a zero-sum game is clearly nonsense, apart from in the most theoretic way. Just looking at the scoring system gives the game away -

Win = 1 point
Draw = 0.5 points
Loss = 0 points

Clearly it is a one-sum game :-).

OK, joking aside, what about rating? Surely if we play a game and the game is rated then what one side gains in rating points will be matched by the other side's loss in rating points? Rating-wise it is zero sum.

Not necessarily. My k FIDE factor is 20. If your's is also 20 then, yes, my loss will match your gain, but if you are a junior or still haven't played 30 games (I think) then your k factor will be 40 and you will gain double what I gain. Similarly if you have ever been over 2400 then your k factor will be 10 and your gain will only be half my loss.

Putting these purely numerical factors aside it is clear that it is mostly a non-zero sum game. There is a FIDE rule which says that for a game to count as being played for rating purposes each side must have made at least one move. The event which lead to this rule was a weekend tournament in the 1970's when Tony Miles (England's first OTB GM) and Stewart Reuben (later an IA and long time member of FIDE's Rules Commission) agreed a zero-move draw. This guaranteed Miles first prize and gave Reuben a share of second prize. For them it was win-win, at least until the organizers saw what they had done, disqualified them and asked for the money back!

Whenever I play a serious/rated game against a much stronger player it is clear that even if (when!) I lose I don't actually lose anything real and in fact I gain. I gain the experience of having my ideas tested by a much stronger player. If there is time after the game and my opponent agrees to a postmortem then I also gain a free lesson.

Clearly chess is anything but zero sum.

Answer 10

The problem is whether to add or subtract a draw. Subtract 1/2 from 1/2 and you get 0, but add them and a draw is a "win-win" and as a popular site says "a wild game that ended peacefully," which is cold comfort if both spiders are dead, or both prisoners in the classic zero sum prisoner's dilemma end up in jail!! There is stochastic resulting bias if at a game state complexity of 10^123 (log of branching factor 35^80 plies) results in chance-like choices at the edge of chaos.

Answer 11

I would like to bring in the point that it is wrong to consider chess a game being played for the result. Both sides have to play trying to overcome or withstand the other side to play chess, but that does not mean that we play chess in order to win. Nor does it mean that what is- intrinsically- to be won in the practice of chess is deducible from the score.

Just a few examples: Quite obviously, a good game is more of a win-win game than a game played poorly by either side.

If you try to become a better fighter, losing or winning is not the key factor by which you value your gain.

It is intrinsic to chess that it is played by beings with vision and agency. It is hard to give engines this credit. Even though they exploit-apply vision and agency, we do not see them as beings- they need not be beings to do what they do. I'd say engines, computers can not play a game of chess, yet (it is not what they are set out to do, yet), and therefore they can not win, draw or lose a game, while for us playing them, the result is 0, 1/2 or 1 (in order of frequency).

The establishment of the rules of mate and stalemate can be seen as a win-win woven into the fabric of chess: the defender gets more credit in form of more points, the attacker gets more credit for making the difference against the harder odds. You could argue the other way round that it was far too easy to achieve stalemate, and it were lose-lose: The attacker frustrated without the resource of a stalemate win, the defender bored and inexact with stalemate draws just too easy to bring home. (It's the first one, isn't it)?

The zero-sum thing is importantly giving shape to the game, we have to play to mate or save the kings. But it is also unimportant in as far as it does not reflect lots of things counting for us, like, for example, the love for wood.

Answer 12

The only times chess is not a zero-sum game are in situations described by a famous Tal quote: "You must take your opponent into a deep dark forest where 2+2=5, and the path leading out is only wide enough for one." In that case chess becomes, after simplifying the equation, "+1" sum game.