On Game Theory, Part III

Breaking the Equilibria

If you read part II.5, the case study, I term-dropped ‘equilibria.’  Here’s what the deal with that is:

Suppose we have a game that we are in the process of Knowing.  (We know all the rules, outcomes, and possible decisions, but have yet to analyze them and figure out a dominant strategy.  The game is currently Unknown, at least to us, but is Knowable.)

John Nash says that there will be a Nash Equilibrium in this game: a dominant strategy.  (And he’s a smart guy, having won a nobel prize and getting played by Russel Crowe and all.)

For a given outcome, neither player will want to change their decision, given the other player’s decision.  Yeah, its wordy.

Let’s revist the Prisoner’s Dilemma.  (The whole cooperation/competition thing.)  If I have cooperated, no matter what you did, given a chance to betray you, I’m better off taking it.  Likewise, if I’ve betrayed you, it’s a bad idea for me to switch to cooperating with you, even if I know for certain you’re cooperating with me.

That’s called a Pure Equilibrium.  There is a single dominant, obvious strategy of a single choice and only a single choice; any deviation from this strategy causes one to be screwed.  I posit that any game with a Pure Equilibrium will not be fun.  Figuring out that Equilibrium may be enjoyable as a mental exercise, but the application will be an exercise in rote performance.

Mixed Equilibria are more interesting.  That’s when there’s a dominant strategy, but it doesn’t consist of a single option.  Paper Rock Scissors is a good example.  Choosing randomly is the soundest strategy not because it’s always the best matchup against no matter what you face (consider: if your opponnent always picks rock, you should always pick paper), but because it gives the highest constant degree of success no matter what you face (consider: if you always pick rock, you’re hosed if someone who favors paper comes along.)  I posit that any game with a Known Mixed Equilibrium will not be fun.

I can combine these two premises into the following theory:

Any game with an identifiable dominant strategy (i.e. Nash Equilibrium) will not be fun.

(Remember my definition of Known and Unknown back in part II?  That whole dominant strategy part is a classic definition of a Known game.  This means that any Known Game, played by people who know it, cannot be fun!  Yikes!)

Analysis of said game can be tons of fun if you’re into that kind of thing, but once the analysis is all said and done, the honeymoon is over.

(I suspect that many players who suffer from ‘analysis paralysis’ and overanalyze their moves are having fun, however they are in the process of analyzing the game as they play and gradually changing it from Unknown to Known on a subconscious level.)

Remember all that stuff I said back in Part II about keeping your games Unknown?


With the exception of increasing uncertainty, all the others simply make the game more complex.  Therein lies the last hope of fun for all gamers, whether they prefer rpgs, cards, boards, or videos: we are human, and we are imperfect.

We cannot always calculate the dominant strategy for a game.  This is true the more complex the game is, and the less time we have to analyze it.  Only by pushing the game out of the rational mind’s calculatory zone can we give the creative mind a chance to shine and imagine.

Consider the card game, collectible or otherwise: the act of shuffling exposes the players to constantly changing conditions.  Each individual instance of play has its own unforseen starting conditions and continuing circumstances.  One cannot analyze the entire field of decisions in advance, and with each new card draw, the field changes anew.  With card games, I am forever venturing into the Unknown.


8 thoughts on “On Game Theory, Part III

  1. Rahvin says:

    There’s a corallary to this you may want to explore regarding risk analysis in game design.

    Most designers (including myself) seem to try to sidestep the issue of equilibria by assigning a certain risk level to each given dominant strategy. This different strategies can prevail to differing degrees, depending on how much risk you’re willing to take.

    The problem with this, is that most players seem to settle on a certain “acceptable” risk level for themselves, often regardless of what game they are playing. Thus, an attempt to make diverse dominant strategies becomes an excersise in interpreting the opponent’s risk level and after that the whole concept of risk analysis can get thrown out the window. Because there is no analysis anymore; it’s essentially pre-analyzed and if the game has no other way of differing dominant strategies, it will not be fun unless you can routinely play against new players. (An advantage of CCGs over RPGs.)

    Whenever you try to make an “Unknown Game” you are essentially saying that it is “Unknown” to the players at the time of game design, but you also have to explore rather player-choices can be easily interpreted, predicted, and “Known” this side-stepping one advantage of your game.

    Burning Wheel suffers from this problem. It has a paper-scissors-rock style combat scripting system, but most players (including Luke) seem to settle on their preferred combat strategy and thus the Unknown game-within-a-game becomes Known.

  2. Rahvin says:

    Regarding your comments on shuffling cards:

    Yes and no.

    Most RPGs that offer differing strategies, allow a player to do one of two things:

    1) Come up with different ways of achieving the same end result.

    2) Leave the end result open in interpretation based on the method of resolution.

    A card game on the other hand, at any given moment in which you have your new hand, actually does something else:

    3) Opens up new options and new tactics to change the game field, in a way that will hopefully be helpful to you later.

    No RPG I know of has this number 3.

    In a card game, there isn’t really a “I want to do this, but what WAY do I want to do this — except certain CCGs at the strategic level of play, pre-game). Yet every RPG seems to believe this to be the answer.

    In a card game you can’t get to your end result from where you are, but you can select a new end result now that will hopefully escalate the situation in a way favorable to you. Let’s call this the End of Turn Result.

    If you want to capture this element in an RPG, a way has to be made to include an End of Turn Result — a point where a player has achieved something meaninful, to the best of his ability, to the exclusion of other End Turn Results he could have chosen.

    For example, consider the superhero situation of the difficult choice of rescuing the girl or saving the city.

    In an RPG, there might be different ways to accomplish these tasks, or the player might have to choose between them, but who cares? It’s a fictional girl and a fictional city.

    In a card game, each of these choices could be their own cards with their own game effects in the future. Thus, it matters more to the Gamist player. Each is a worthy End of Turn Result in and of itself and the player won’t feel like he wasted his turn or any of mental faculties on pointless analysis.

    Most games apply strategy and tactics to result HOW TO GET THERE, I say the THERE has to be worth going to. That’s an advantage board games and card games have over RPGs. The only real “end of turn result” you have in an RPG is that you get to “win” the adventure and start a new one, which some players don’t really like doing anyway.

    Experience points and similiar reward mechanisms don’t qualify as an “End of Turn” condition because they’re not tied to elements within the game — likely you’ll earn the reward either way. Magic Items in D&D, however, qualify. If you score a magic item in D&D it is a meaningful tangible game effect linked to a story effect (hopefully) chosen by the player.

    I hope I stayed on topic through that…

  3. Willow says:

    There’s alot of good stuff here, Rahvin. Too much that can be justifiably replied to in the space of one comment, but here’s a quickie:

    Alot of modern rpgs do have a more realized reward system where you won’t just get rewards either way. I’ll make sure to have an essay on Shattered Vista’s rewards system in the future.

  4. Willow says:

    “3) Opens up new options and new tactics to change the game field, in a way that will hopefully be helpful to you later.

    No RPG I know of has this number 3.”

    See Escalate in Dogs in the Vinyard (so awesome, its one of my atmoic maneuvers!)

  5. Rahvin says:

    Ha, my absolute favorite thing about Dogs in the Vineyard!!! I’ve been looking at Bloodlines to see how I can add a similiar Escelate feature in, but I just can’t!!! It’s not cool enough!!! 🙂 It’s so worth having escelate though, that I might just scratch the whole conflict system and start from scratch…

  6. So, this is a response to both this op and the last part of the game theory, and its just a bit of a freestyle on games in general…just in case any of it seems interesting to you.
    You said at one point that chess bores you, because if you know more about it than the opponent then you just get thrashed. Firstly, I agree, i hate playing chess with my friends because I just get thrashed…however…occasionally I win…and that feels great…to luckily/intuitively come up with a winning strategy. Still, it makes sense to play against players your own ability, when this is the case you both know the strategies but there’s still unknowability, and strange strategies that add to it. There’s the story of how Deep Blue beat Gary Kasparov by making a move that looked idiotic, but Kasparav thought it was some genius move that a poor human couldn’t understand, and resigned soon after in desperation. Psychology plays a part in all things (this relates to someone elses post in the other thread about a group that don’t bother with unrecoverable items, psychology and personality can be all important).
    Anwyay, strictly speaking Chess is only a known game at the opening and the end game (it can be ‘known’ by the very best computers, but only so many moves ahead). In the mid game it is conisdered a ‘nightmare’ game that has too many possible routes to be analysed.
    This relates to my current (board) game of choice, Go, which is totally nightmare..computers can’t beat moderately good human players…there’s just far too many possible moves to play (I read somewhere that strictly speaking there are more possible games of go than there are particles in the known universe…check it on wiki..which I think is where I read that). Anyway, I’m still beginning, but am struggling to find players as good as or better than me..cause I have to keep on teaching people to play first, but this is off topic. Anyway, i think the important thing here is that systems should be complicated enough to understand in a meaningful way and simple enough to be hard to figure out the ‘Nash Equilibria’. if it sounds as if I’m getting my adjectives mixed up bear in mind the number of rules in chess, and the fact that go only has about six rules (there’s a few more about scoring that vary from country to country…house rules etc) to learn…yet is played in a ‘big enough space’ (we’re talking phase space here if you know what I mean by that) that it seems like anything is possible (particles). In fact a couple of simple strategies can guide you (which I learnt online and is why I can beat my friends when I teach them) that help…but knowing when to apply them and exactly how they work is what you need to know to play well. Basically they’re a bit like common sense…and part of the joy of the game is gained from the ‘proverb’ I found on the internet, mis-quoted here:
    Go is a game played with black and white stones on a carved grid, but those who win, see a Battlefield and two armies.
    A lot of tactics can be analogised from warlike scenarios and strategies: for example keeping supply lines to the front…isolated stones can die very quicly. If you apply enough imagination you can actually start to see trenches and borders appearing on the board.

    Anyway…i’ve gone MASSIVELY off topic now, sorry about that. I just wanted to say that complicating a game can sometimes be about simplifying the rules as well as adding new elements. Sorry if this is all irrelevant and incoherent, but I figured it might be interesting to you.

  7. Willow says:

    I’ve heard very good things about Go, and played a few times.

    “Anyway, i think the important thing here is that systems should be complicated enough to understand in a meaningful way and simple enough to be hard to figure out the ‘Nash Equilibria’”

    I know what you mean, but you’ve got it backwards. The elegant simplicity of Go makes it structurally easy to grok; the massive number of options makes it impossible to determine a strictly best strategy.

  8. I know…I just suddenly got the Terry Gilliam ‘Clever enough for Kids and Exciting enough for adults’ thing in my head and wanted to try it out. I think I meant complicated enough to have verisimilitude…but still…its about simplicity..not complication. After all…life on Earth and the structure of the universe is (theoretically) based on a number of very simple rules…and look how complicated that is

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