CGD Podcast Episode 18: Single/Multi-player and 50% Win Rates

cgdplogo_twitterToday I talked about how and why games work best with a 50% winrate (even single player games). That’s because learning in games is extremely hard due to their inherently complex and ambiguous nature. Getting a loss when you had a 10% chance to win doesn’t necessarily tell you much about your choices in that match. In order to learn, you must compare yourself to yourself.

In addition, I talked a lot about why single-player is considered a strange thing for strategy games.


  • Simon

    Not sure I’m convinced on your points about other player input necessarily being ‘random’ – that rings false for me. 1. Because I’m a pedant and it’s not actually true; 2. Because it seems to suggest that people’s intentions can’t be read even in principle, even though reading people’s intentions (to a reasonable non-perfect level of accuracy) seems to be necessary to make sense out of any social situation; and 3. Because in a game worth playing there are such things as ‘good moves’ and you can take it an axiom that an opponent will be trying to do them, which gives you some ‘objective’ insight into what might happen.

  • Jereshroom

    -Perhaps we evolved to like 50/50 odds so much *because* it is what lets us learn best. “Because it feels good”, “because it’s fair”, and “because we get more accurate feedback” might be basically the same thing, the same way “because I was hungry” is nearly the same as “because my body needed energy”.

    -(Similar to what Simon said) Even if reading opponents through facial expressions and such may not be effective (though perhaps it is), I’ve definitely had success figuring out what my opponent would do based on their play-style and skill level (eg. “she usually attacks even when she shouldn’t”) and exploited that. That isn’t randomness.

  • Van

    The only case in which your opponent’s behavior is deterministic (meaning not random) is if you could predict their actions with 100% accuracy. Saying “usually she does something”, implies that you can’t get it 100% right. Rather it implies that “she does something x% of the time”, which is the textbook definition of randomness. Does it give you some predictability? Of course it does. But, once again, unless you have absolutely 100% predictability, your opponent’s behavior remains uncertain, non-deterministic, random.

  • Van

    Being able to predict something “to a reasonable NON-PERFECT level of accuracy” is the very definition of randomness. I struggle to think how you can be a pedant and contradict yourself so obviously about something that we’ve all learned in school, no offense.

  • Jereshroom

    If we define randomness as anything that isn’t certain, then basically everything is random, including the outcome of a game of chess. A more useful definition for game design that Keith Burgun has used is “information that enters the game state which is not supposed to ever be predictable”*. Since figuring out your opponents future actions is part of any game with a decent amount of interactivity, one’s opponents’ actions are not random.

    * (also, if you look at other definitions for randomness, I think you’ll find that calling a player’s actions random is a big stretch)

  • Jereshroom

    That’s not the definition I’ve seen used, and certainly not in the context of game design. Keith Burgun,, and Wikipedia are all against you. Surrender now! (Also, I can’t find your definition anywhere on the internet, even with the “non-perfect” replaced with an asterisk)

  • kreylix

    Not “pretty much it” – I count way more single-player games than you’re thinking of: Isn’t Farmville, Candy Crush Saga/Soda, Clash of Clans, and many others of that ilk single-player games? Sure, the server your data is on is shared with other players, and sure there’s some quasi-limited interaction, but its not interactive play between players. (you could describe as single-player with fake multi-player background) Also, almost all computer games (PC, Apple, Tandy, etc.) in the 80s single-player games. Slots, Pinball and Coin-ops mostly were. Tetris, Carmen Sandiego, Might & Magic, Ultima Underworld, etc., Mario games, all single-player. Also, also, solitaire card games were very popular until the 80s. Couldn’t we say, instead, that single-player dominated in the 80s and 90s (until internet games really got going, like WoW).

  • Van

    I’ve already read that article. I have much respect for Keith Burgun, regarding his philosophical and aesthetic insights into game design. However, researching mathematical concepts by reading philosophical articles on gaming sites is the opposite of productive, even more so when you’ve already received a formal education on the matter. The definition you quoted isn’t wrong as it hints at what randomness is, but you still got the wrong idea. My previous post is guilty of the same thing. Formal definitions don’t just hint at stuff and they don’t allow you to get the wrong idea. For that reason I suggest you do your future research by studying math.

    A more formal definition of randomness, without getting into lengthy mathematical proofs (which, let’s be honest, won’t be useful to you at this point), is to say that a repeated occurrence is random when the outcomes of INDIVIDUAL instances of it can’t be predicted.

    A random occurrence can still have long term predictability. We can, for example, approximate its probability distribution through Bayesian inference or we can even know it in advance (as is the case with a die roll). That being said, humans have very obvious biases (or fixed weights to their randomness), which makes us predictable in long runs, but individual occurrences are still unpredictable and therefore random. You even said it yourself “she USUALLY does something”… Think about what you mean by “usually”.

  • Those things you listed aren’t really games in the way I’m talking about it. They’re not competitive contests of decision making, they’re more like gambling toy-apps (as far as I know). They’re just a totally different animal. Very few things are like single player competitive strategy games.

  • Jereshroom

    Okay, for a mathematics definition you’re right. But that’s why different fields use different definitions of words. “Organic” means something very different in the food labeling law than it does in chemistry. In game design, Burgun’s, Wikipedia’s, and the dictionary’s definitions — the lack of an understandable pattern — are far more helpful than a definition that could just be replaced with the word “uncertain”.
    With your definition, the outcome of ANY game is random, even single-player games with what I would consider completely deterministic mechanics. Because you don’t know in advance how well you will do.

  • Van

    The definition I gave you does not contradict Burgun’s or Wikipedia’s, nor does it refer to something else. It’s the same thing with more accurate wording. It most certainly applies to all kinds of RNG in games.
    But that’s beside the point. We can use any definition you like and your self-contradiction would remain. The definition isn’t the problem, your misconception of it is.

    The best way to realize this is if you try to explain your point of view, rather than me throwing any more arguments your way. So I ask you – in your first post what do you mean by “she USUALLY does something”? What about the times that are not usual, what do you call those and how are they different from the times when you fail to predict the outcome of a die roll?

  • Van

    P.S. – who knows, maybe you’re not contradicting yourself, rather you’re just referring to something else entirely?

  • Jereshroom

    Bloons Tower Defense 1.
    Perhaps a bit solvable, but it has a low execution cap, no grinding or anything, no randomness, and a clear win-loss state.
    (Unless it’s a puzzle, due to lack of input randomness. Do single-player games need randomness to not be puzzles? Or is having many “solutions” enough?)

  • Jereshroom

    By “usually”, I mean that there are variables I couldn’t possibly predict (psychology stuff) that affect her decisions as a player. Which I agree is similar to a die, where there are variables I couldn’t possibly predict (physics stuff) that affect how the die rolls.
    So okay, I get what you’re saying now, it is the same in many ways.

    However, I still don’t think defining randomness this way is helpful for game design. It is too broad a definition. It means that computer chess even against a highly-predictable AI is still random, because the game tree itself is too complex to fully understand even when our opponent is. It means that skee-ball is random because we can’t completely understand physics and our own bodies. It means Mario (without access to the code) is random because we can’t be expected to know the exact hit-boxes and velocities of everything.

    I think we need to limit our definition of randomness to systems that are not meant to even be even partially solved. Here is a quote from later in the same article I linked to above

    “A rolling die is a closed system of its own that really has nothing to do with the greater game system.

    This is distinct from other kinds of ‘unpredictable’ or ‘uncertain’
    events. In chess, for example, players have some limit to the number of
    turns they can look ahead. Beyond that point, the events that occur are
    indeed unpredictable for that player. However, players can and do learn
    to look further and further down the possibility tree as they get better
    at the game. Part of the skill of chess is being able to explore that
    ever-increasing possibility space and come out with more predictive

    The same applies to the physics of Golf, the minds of Go players, the execution of Street Fighter moves, and the deck choices of Hearthstone opponents. You can use skill to improve your odds and/or predictive ability.

    Improving your predictive skill for dice, on the other hand, is both nigh-impossible and cheating.

    Note: My definition does have some slight strangeness. For instance, I would consider card drawing without replacement to be very slightly non-random, as card-counting takes a bit of skill. (Many people will forget/fail to recalculate the odds of a poker deck every time a card is revealed.)

  • Van

    “It means that computer chess even against a highly-predictable AI is still random” – the outcome of such a game might be considered random, but it doesn’t follow that the game itself is arbitrary.

    I think you’re a bit fuzzy on the difference between uncertainty and randomness and I think you’re referring to uncertainty, more so than randomness. For example, in statistics uncertainty refers to having incomplete knowledge of a probability distribution (like not knowing all of the sides of a die or not knowing hit boxes as you mentioned), while randomness refers to individual occurrences being unpredictable (like not being able to predict a die on a roll to roll basis). The presence of uncertainty leads to random results (for obvious reasons) and that’s why sometimes randomness and uncertainty might be used interchangeably, but statisticians know to keep the difference in mind.

    Perhaps what you’re getting at isn’t that human behavior in games functions differently than RNG, but rather that the two play completely different roles. Rather than redefining randomness, maybe the roles of RNG and human behavior need new terms.

  • Jereshroom

    So would “chess is a game with randomness” be true, false, or incoherent?

  • Van

    Depends on what you mean by that, so I’d say incoherent.
    Chess doesn’t have inherent randomness, but it does have uncertainty.
    In game theory games are classified based on many factors. One
    classification is by cause of uncertainty. In chess the
    cause of uncertainty is its large state space. In Poker the cause is RNG
    and hidden information. Another way to classify chess is to say that it’s a deterministic game, meaning that for every player input, the game can make a transition to only one other state and also that the game state changes only based on player input.

  • Jereshroom

    So would you consider opponents to be basically the same as semi-random AI added to the game whenever you play?

  • Van

    I don’t know what you mean by “semi-random” or by “basically the same”, but I would say that in chess you can play in such a way that your opponent’s decisions don’t matter.
    I get what you’re getting at, but I think you’re extrapolating Keith Burgun’s point in a wrong way.

  • Jake Forbes

    What role do you think there is for formalized handicapping as a way to better find that 50% win rate, or should it only be a logistical problem of matchmaking? Would you say a player learns better when an opponent is at an objective advantage or disadvantage but statistically has an even chance at winning when skill is factored in, as opposed to a “fair” game where one player is 60% likely to win? Go is the only game I can think of where handicapping is taken for granted, or maybe Golf, but there are other examples like Mario Kart or Left 4 Dead where the game “cheats” to keep the playing field more perpetually close. I know those aren’t good examples of what you’d define a game (except for Go) but there is no reason, for example, that a mismatched Agricola game couldn’t be better balanced by giving the novice a free wood or two at the start of th game.

  • The problem with handicapping is that you’re playing a different game. If I’m playing “Go with 5 stones already down”, that’s kind of a different game than “Go”. To me, that’s not really optimal. Go was designed to be played with 0 stones already down. That was the optimal configuration for the game, so we’re making it non-zero amount worse with the handicap. With that said, I think if you can’t use matchmaking for whatever reason, it’s often preferable to use handicapping instead of just letting it be some 80/20 winrate business.

    Another note: games like Go which are sort of not-very-designed probably can take the abuse more than other more carefully balanced games. With Go you can actually scale the board and play it at 9×9, 13×13, or probably 25×25 if you wanted – it’s flexible. This is a bad sign for the game in general but also means that the damage incurred by handicaps is smaller.

  • Pelle Nilsson

    The most recent podcast reminded me of this one, so I thought it was worth to go back and make the comment I did not bother to post immediately, but now that I signed up to disqus anyway.

    I do not like games to win 50/50, as a player or designer. I have played many (and designed a few) solitaire boardgames. You should check out the very active community for those on bgg (the 1-player guild). I am sure almost everyone there would agree with me that 95/5 is a better ratio (95 % losses that is). But for me at least this is the same with digital strategy games. I want to feel when I start playing that the game is very difficult, and then work hard to eventually beat the game, and when I reach that point I am generally satisfied with it and will probably move on to a new game. I want to feel the progress in being slightly closer to winning each time, but never be awarded with a YOU WON screen, until I really win, and not by lowering the difficulty level (myself, or the game doing it automatically). This is unlike a multiplayer game where I can always keep improving and find a better opponent. Yes, in a digital game you can simulate that by adjusting the difficulty-level automatically, but I still prefer the game to always require me to improve to have a chance of winning at all, always look at (the current difficulty-level) and think “wow, this can never be possible, I need to try many different strategies to win!”. I do not see any need to compare this to multiplayer, because multiplayer is much different (in many ways; it is not, ideally, at all like single-player plus AI, that many seem to think).

    Of course the more casual a game is the more it is true that you want to be nice to the player. Wanting games to feel like big obstacles to eventually overcome is probably a characteristic of more hardcore gamers, but that does not mean that it is not something you need to consider, and that means that 50/50 is not always the best answer. Other than that I rather liked this episode really.

  • It seems like you have a different idea about what 50% win rate means. 50% means when you’re playing YOUR BEST, you can win but just barely. The 50% is determined by placing you against an “opponent” of basically identical strength as yourself. Imagine you’re a great chess player. Would you consider it an “easy” / “casual” game if we could clone you and have you play against a copy of yourself? I don’t think so at all. I think it would be very, very hard.

    50% win does not mean “you win most of the time”. Imagine playing your absolute HARDEST and still losing half of the time. This isn’t “casual”. I hope that you also heard the other reasons why a 50% win rate specifically is ideal that I mentioned in the podcast.

  • Pelle Nilsson

    Cloning myself to play myself sounds boring, yes. I want a challenge. I want to start a new game, feel like it is a huge (HUGE) challenge. To work hard to overcome that challenge. And then to conquer it and feel good about how I perform. I will not feel like that if I play against my clone, because I will be able to win 50 % of the time if I just play well enough, which sounds way too easy to be interesting in a solitaire game (although ideal for a two-player game). So I want to lose badly, then lose slightly less badly, until I eventually reach a point of skill where the game becomes easy and I win every time, because I do not like the outcome of games to be very random. At that point I can move on to play another game. (And as I commented on your latest podcast, that other game could be a more difficult or just different scenario that is part of the same game as that I just managed to win, because that is essentially the same as moving on to a new game, as long as it is interesting as a new challenge and just not more of the same).

    I heard your reasoning. I do not remember the details unfortunately, but I remember that you were very quick to conclusions at times and that I thought it sounded like you were ignoring possibilities that were at least equally likely to the one you chose to prove your point. I will actually have a second listening now just to be able to say something more specific, because I find it interesting when theory contradicts reality, and I am certain the issue here is that the theory is missing some bits. I have read enough on various solitaire-game forums to know for sure that my way of looking at this is common, and I do not see any kind of 50/50 no matter how you define it. But I will try harder to figure it out. 🙂

  • Pelle Nilsson

    Yup, five minutes into the podcast is where you introduce a false statement that then makes further reasoning (even if well-reasoned) give you the wrong result. You assume that the crucial element to learning is to win or lose. You then somehow use that as a fact that lose or win is what allows progress, which is even more wrong. There are many, many ways of learning without winning, and being far from 50/50 is actually not blurring this at all. I never ever got close to win Nethack for instance but I feel how I improve every time I try to play, because it is obvious how much further down into the dungeons I can go. There are many different ways of measuring progress than win/lose. If I play a wargame I can often see easily if I did better or worse than last time by looking at my losses or how fast I manage to progress across the map. And I can use that feedback to improve until I figure out how to play the game well and can win each time. It is simply not correct at all that 50/50 makes it easier to evaluate your progress and you reason around a lot in circles trying to prove it.

    On the other hand your discussion early in this podcast about how a real single-player game is different from a two-player game is excellent and something I have thought a lot about and tried to convince other about, but never had much success. Hope you reach out to more, because I think that is an important idea (and completely unrelated to 50/50 or not). I googled for my own old posts and for instance found a few from 4 years ago that I still think are quite good for explaining this (if you care to read), for instance this post: Incidentally earlier in that thread I reason a bit about why I think balanced games are important for two-player games, but not for one-player games, and I still agree with my old self. 🙂

  • No, you have to work HARDER THAN YOU CAN to beat yourself. I would assume that you and your clone will both be playing basically at the peak possible level. Somehow, you have to find a way to play BETTER THAN your peak possible level to beat your clone. This is what you’re not getting. Playing against yourself would not be boring at all, and it would be kind of the perfect challenge. The question is “can you do better than you can do?” If you’re saying “of course, OBVIOUSLY I can do better than I can do”, then I don’t know, we might have deeper English language barriers between us or something.

  • Pelle Nilsson

    Nope. No need to question my language skills, even if my grammar might not be perfectly correct at all times. I understand you quite clearly. I do not disagree that it would require all my effort to win 50 % against my clone. What I disagree about, and have probably an explanation for in my other recent post, is your reasoning to reach the wrong conclusion that this 50 % would be a good level, and I also explained how I would like that level, or rather learning-curve, to look like.

    Perhaps one issue here, judging from your intro, is that you are not yourself very familiar with one-player games in general? When you try to reason to some conclusion, in particular when it seems like you already made up your mind about something that sounds good from the beginning (eg “50 % win ratio is ideal” … very catchy design rule!), it might be difficult to not make logical shortcuts (like the bad assumptions about 50/50 being ideal for learning, just repeated over and over rather than proven, that then results in your loop in your “proof”)?

  • Conor

    So in your model a player makes a series of decisions, then gets a win or a loss to contextualize those decisions. A win should inform the player that their series of decisions were good and a loss should inform the player that their series of decisions were poor. If you have a high probability of winning, say 100%, the game is not doing a good job of teaching you because it can give no information about your decisions. Irrespective of your decisions, you were going to win. Similarly for a 100% probability to lose, losing gives you no information about your decisions. If your probability to win increases to 10%, winning or losing now gives you some information about your moves but not a huge amount. The closer you get to a 50% win-rate, the more information winning and losing gives you about your decisions. This idea of “information gain” from a random variable has been formalized in statistics as Shannon entropy and following that theory lead us to some interesting conclusions.

    We can model the outcome of a match for a player as a random variable X with outcomes either win or loss, and some probability distribution p(x). The entropy of X is defined as H(X)=-sum{ p(x)log(p(x)) }, or in English as the negative sum over all outcomes of the probability of an outcome times the log base 2 of the probability of that outcome. If the win-loss ratio for a game is 100:0, the entropy would be H(X) = -1*( 0*log(0)+1*log(1) ) = -1*( 0*1+1*0 ) = 0, which fits our intuitive definition before, there 0 bits of entropy. If we change the win-loss ratio to 10:90 then the entropy would be H(X)= -1*( 0.1*log(0.1)+0.9*log(0.9) )=0.47 bits of entropy, which is greater which is as expected. If we make it so as the win-loss is 50:50 then there is exactly H(X)= -1*( 0.5*log(0.5)+0.5*log(0.5) ) =-1*( -0.5+-0.5 )=1 bit of entropy. From this we can see entropy matches our intuitive understanding of “information gain” upon a win or loss. So if we agree that entropy and the intuitive idea of “amount of information winning gives to the player” that means 50% is the best win-rate right?

    Well looking at the definition of entropy, we can actually quite easily increase our entropy to greater than 1. We can do this by introducing more possible outcomes. If we have a random variable Y with four outcomes all equally likely we have H(Y) = -1*( 0.25*log(0.25) +0.25*log(0.25) +0.25*log(0.25) +0.25*log(0.25) ) = 2. This would suggest that if we used some other random variable with a greater variety of outcomes we could teach our players faster and more accurately than we ever could with a binary win/loss. An example of this kind of random variable could be a position in a racing game, you could come 1st, 2nd, 3rd, or 4th, all equally likely for a particular game. Where playing with a 10% win-rate won’t distinguish good play very well compared to a 50% win-rate, playing with a 50% win-rate wont distinguish good play as accurately as a 4 place ranking system. The even probability four place ranking system conveys more information to the player about the quality of their decisions.

    Intuitively I think this also holds up. A player might play the game twice and lose both times and the win/loss method wouldn’t be able to distinguish between the two and would rate the players series of decisions equally bad. A ranked system might be able to tell them that the first set of decisions landed them in 4th place but the second set of decisions was actually slightly better and put them in 3rd, thus contextualizing their decisions better, teaching them faster.

    This theory also becomes interesting when you extrapolate it further, instead of 4 places, why not 8? why not 30? Each increase in the number of possible final states gives you an increase in entropy and therefore more information to contextualize play. At a large enough number of end states every possible series of decisions can be ranked accurately against one another, thus giving the best possible information to the player and maximizing the speed of their learning.