Applicability of Game Theory

[Mike Holmes]

Game theory has no problem with different players placing different values on different aspects of the outcome. But it requires all that information -- that is, the payoff for every player for every possible combination of individual moves -- to be already known going in. It then says that each player has an optimum strategy for how to allocate their moves to maximize their own value over repeated plays, and if the game is really really simple you might even be able to calculate what that strategy is.

First, this is incorrect. You're working from Von Neumann's model and ignoring the additions to it by people like Nash (yes, the guy from A Beautiful Mind). Nash's Equilibrim theory and other work is what enables Game Theory to be applied to things like RPGs. These theories propose games that are more "real life" in that the players can have only paritial intelligence. Further, often in RPGs players do have perfect intelligence (Sorcerer advocates this, for example).

In any case, it is in fact of great benefit that game theory doesn't tell us the winning moves for complex games. For we don't want to create games like Tic-tac-toe where the outcome is a forgone conclusion. It's precisely the fact that we can determine that there are optimum strategies and not what they are that is valuable. That we can see that players will strategize effectively, but not what the best strategy is. See TROS combat for an excellent example of this. Every element of the design is Jake subconciously applying these principles.

Why not discuss them in an explicit manner so as to avoid common pitfalls?

It's a great irony. Game Theory was intended to discover winning strategies for real life situations. As it turns out, by focusing on games, it happnens to better create a theory that gives us a way to create non-real-life situations with strategies that we know are interesting, but unsolved.

For example, Nash points out how The Prisoner's Dilemma is actually flawed as originally proposed. Because he finds the equivalent of the Metagame within the actual execution of it (the game is not set in a vaccuum). Hence we see that the Prisoner's Dilemma can be used as a game mechanic in a larger environment, and predict some of the possible strategies.

It's by predicting what a game mechanic will cause that we analyze their effects on an RPG. This is what Game Theory is all about.

At least that's how I see it. How does it seem to everyone else, pro or con?

Mike

[deadpanbob]

Mike,

This seems like precisely the type of applied theory that's needed for RPGs.  I must confess that my reading of Game theory to date hasn't sparked a clear connection with RPGs and their mechanics.

When you study Game Theory, do you see direct parallels and ways to apply GT to RPG design directly?

If so, that would be a wonderful thing to discuss.

Cheers,


Jason

[Mike Holmes]

When you study Game Theory, do you see direct parallels and ways to apply GT to RPG design directly?

Consantly. For example, the Equilibrium theory is interesting in it's application because most games want there to be an Equilibrium point. Essentially this is "Game Balance". Game Theory can tell you if your game has such a point. In fact, without knowing much of the math, you can determine that one exists or does not, which is all that's needed.

This is an elementary association, however. And as I keep saying, I've only scratched the surface. Which does mean that my amature understanding might be misapplying the theory. So Walt might be right. I only wish that an expert on the subject might come in and comment. (If I sound like I'm floundering a bit it's because I am not anything like an expert).

Til then, however, I'm satisfied to abuse the theory in an attempt to see if anything good comes of it. :-)


Note, to reiterate, what GNS does is say, essentially that there are three currencies sought, and each is provided by making decisions in one of three ways. Thus, for Gamists, the currency is not neccessarily "victory points" or some such hard exchange. It can be, instead, "moments of satisfactory challenge". For Simulationists, it might be "moments of immersion". We can't be sure what the actual form of the currency is, but how it's attained is possible to ascertain.

So by calculating what sort of behaviours are supported mechanically using Game Theory, we determine what GNS mode (amongst other things) is supported.

For example, an intuitive use of Game Theory to discern whether a game was Gamist, might look at whether or not the tactical challenges are in fact "balanced" in terms of PvP, and PvG. This is simply looking for an Equilibrium point, and can be done mathematically if neccessary.

It's neccessary to use equilibrium in this case because RPGs are not, usually, zero-sum games (in which case we could use the Min-max theorem from Von Neumann's work to determine "balance"; I think this would work for Rune, for example).

Now, I can look at Rune and guess that it's balanced; that the designers have made it so by whatever method. So where's the usefulness if it can be done intuitively? Well, sometimes you can't, just looking at something determine these things. Oft times this results in a change to a rule that makes the effect obvious and observable. But if you're aware of Game Theory principles, you can sometimes keep an otherwise difficult rule (without having to resort solely to playtesting which can be problematic itself).

Mike

[Wormwood]

Mike,

It occured to me after some of our discussions earlier that much of my Ur-game analysis in my Speculative Physics column borders on an attempt to perform game theoretic analysis on RPGs.

The basic idea of an Ur-game, was a game that describes in a discrete way some model for player interaction with the game. Essentially placing this model on top of the game provides a game which Game Theory can directly be applied to. I decided to call this an Ur-game, since the term meta-game is taken.

One interesting implication of this is a basic model (in the article Desk Juggling: Designing Players and the Statistical Metagame ) suggests that the ideal chance of victory for a gamist player is approximately 62%, with just one Player, and the chance descending to a limit near 50% as the number of Players increases.

Now admittedly this model is based on a major simplification, but I find it generates a useful rule of thumb.

I've also been trying to decide how to develop a sequence of Ur-games to simulate parts of the GNS model (not precisely the same as the theory, but certainly related). As of yet I'm trying to determine how to handle what I've been referring to as decision classes, and the hierarchies of their use by players.

In short, I strongly support the idea of game theoretic analysis of RPGs, I believe it's a fairly poorly developed area, and hope some further interest is spurred by this discussion.

  -Mendel S.

[epweissengruber]

It's nice to see you applying Game Theory to RPG situations.  But could you insert more operational definitions into your posts?  That way your posts could be taken up and employed by gamers/designers/GMs who do not have much specialized knowledge of that field.  

Frex:

When you study Game Theory, do you see direct parallels and ways to apply GT to RPG design directly?

Consantly. For example, the Equilibrium theory is interesting in it's application because most games want there to be an Equilibrium point.
Mike

Don't like to confess to innumeracy but --- what is Equilibrium (in its strict Game Theory definition)?

[Mike Holmes]

I knew that I was going to get trapped into this. But it's my own fault for brining it up.

From this page:
http://www.ags.uci.edu/~jalex/egt.html

The Nash bargaining game
The Nash bargaining game is a two-player noncooperative game where two players attempt to divide a good, say a cake, between them.  Each player requests an amount of the cake. If their requests are compatible, each player receives the amount requested; if not, each player receives nothing. The simplest form of the Nash bargaining game assumes the utility function for each player to be a linear function of the amount of cake they get. (We may assume the utility functions of the players are equal since utility functions are determined only up to a nonnegative multiplicative constant and a constant term.)
According to traditional game theory, an infinite number of Nash equilibria exist for this game. Given any request, the corresponding strategy of the equilibrium pair simply requests the remainder of the cake.  If the first person did not request the entire cake for herself, we have a strict Nash equilibrium.  If the first player did request the whole cake, the equilibrium is not a strict Nash equilibrium since the second player receives the same amount regardless of what she demands. (If she makes her equilibrium demand of 0, then player 2 receives nothing. However, if player 2 makes any nonzero demand, she will still receive the same amount, namely nothing, because any nonzero demand will push the total sum of demands greater than the amount of cake available.)  If both players act to maximize expected utility, traditional game theory dictates each should demand half. Intuitively, this appears not only as the rational thing to do (`rational' meaning maximizing personal expected utility), but also as the "fair" thing to do.

This is a classic example of something that intuitively gets inserted into designs.

This is much less intuitive:

The ultimatum game
The ultimatum game is another a two-player noncooperative game where two players attempt to divide a good, again, say a cake, between them.  However, we assume that one player (the proposer) has sole possession of the cake and offers a certain amount of the cake to the second player (the receiver), keeping the rest for himself. The second player has only two choices: take the offer or leave it. If player two takes the offer, each player receives the amount of cake due. If player two chooses to leave it, each player receives nothing.
Compared to the Nash bargaining game, the ultimatum game has a significantly larger strategy space.  Each strategy has two components, prescribing what demand the player will make as a proposer and what demands the player will accept as a receiver.  If the cake divides into N pieces and we forbid purely altruistic behavior (demanding nothing) and completely greedy behavior (demanding everything) the game has 2^(N-1)*(N-1) possible strategies. Most treatments of the ultimatum game consider only a small subset of the possible strategies.

According to von~Neumann-Morgenstern game theory, if the good can divide into infinitely many pieces, an infinite number of Nash equilibria exist.  When talking about the ultimatum game, though, it proves fruitful to use another solution concept, that of subgame perfection. We say an equilibrium is subgame perfect if the strategies present in that equilibrium are also in equilibrium when restricted to any subgame.  Consider a population of players where all make fair offers (half of the cake) and only accept fair offers, a strategy typically called "Fairman." Although this strategy is a Nash
equilibrium (no player can do better by changing her strategy), it is not subgame perfect: in a mixed population containing players of all strategies, Fairman does not do as well as the strategy which makes a fair offer but accepts any offer. Consequently, if one thinks a credible equilibrium of a game must be subgame perfect, the number of credible equilibria shrink.  If players act to maximize expected utility, then proposers should demand the entire cake minus epsilon (if the cake is infinitely divisible) or N-1 pieces (if the cake has N pieces). Receivers, on the other hand, should accept any nonzero offer.

These sorts of implications work out so well in actual play that I can't recommend them enough. And there's ton's more where this came from.


BTW, I don't mean to imply that Game Theory is the only workable methodology for determining behavior. Obviously Behavioral Psychology can be used, and in fact we've had some great threads on reward frequency based on that. IOW, there are probably a great number of ways to look at how to design. It just happens that Game Theory is one that I think is potentially of great use.

Mike

[Jeffrey Miller]

Mike,

This seems like precisely the type of applied theory that's needed for RPGs.  I must confess that my reading of Game theory to date hasn't sparked a clear connection with RPGs and their mechanics.

Hmm.. can we work in (non)self-adjusting systems theory into this, as well?

-j-

[Mike Holmes]

Hmm.. can we work in (non)self-adjusting systems theory into this, as well?

Sounds like a new thread. But why not?

Mike

[epweissengruber]

Yes -- posting a nice condensed definition goes a long way to establishing common ground.

And, yes, it can serve as the start of a whole new thread.

I think I will be opening a website or blog that documents a lot of the ideas I am working with concerning play and games -- but I will be coming from a philosophical and literary angle, so I don't know if I will produce much of direct use to this forum, but some might like to check from time to time, just to see what is going on.