How can a board/card game be considered to be only moderately based on chance?

Question

I'm not sure if this is the right forum to ask this, but I guess it's related to statistics... My intuition is that, if a game has any element where luck is involved (e.g. rolling a dice or dealing shuffled cards), then that ultimately renders the game being entirely based on luck/chance. And yet, different games have different 'chance' ratings. For example, Monopoly's chance rating is high while Catan's rating is low/moderate. I don't know if my question makes sense but hopefully someone understands what I'm trying to get at.

Answer 1

Suppose that I have a bag containing 100 balls. 99 of them are red and one is blue. In the game, you choose a colour - one of red or blue. I randomly pick a ball from the bag, and if it matches your chosen colour you win, otherwise you lose. This is probabilistic, but you are very likely to win if you choose red.

On the other hand, if the bag had equal numbers of red and blue, then whatever you choose, your results are entirely up to chance.

Answer 2

Consider the following two games:

Game 1: A quizmaster asks a question. When you answer the question correctly, you earn 100 points (pounds/euros/whatever), otherwise you lose 100 points. After each question, a coin is flipped. If it is heads, you earn 1 point, if it is tails, you loose 1 point.

Game 2: A quizmaster asks a question. When you answer the question correctly, you earn 1 point (pounds/euros/whatever), otherwise you lose 1 point. After each question, a coin is flipped. If it is heads, you earn 100 points, if it is tails, you loose 100 points.

Both games have some element of luck (the coin flip) and some element of skill/tactics (the questions). However, one of the games is much more luck-based than the other.

Similarly, Catan is based quite a bit on skill/tactics (placing your first village on a tile bordering only on 2s and 12s is bad, and placing it on a tile bordering 6s and 8s is good. Having a good distribution over all resources is usually good. etc. Then when you get resources, there is some tactic in what you spend it on, and there's social tactic in making people like you and not put the robber-thingie near you) whereas in Monopoly you just buy whatever your luckily land on and have not very much to say.

Answer 3

If you wanted to put it "formally", suppose you have $n$ players. For each choice of strategies $\bar{S} = (S_1, \ldots, S_n)$, the $i$-th player has a chance $p_i(\bar{S})$ to win, depending on the factor of luck.

One way of formalizing the "luckyness" is the following. We suppose the game is symmetric just for simplicity (as all the games you quoted are). Fix the stragies $S_2, \ldots, S_n$ and consider the random variable (where strategies have the same weight) $$X: S \mapsto p_1(S, S_2 \ldots, S_n)$$ The more this variable is similar to a constant, the more the game is random: no matter which strategy you choose, you always have the same probability. On the other hand, the more is polarized, the more you have to choose wisely. A cheap way to measure this is to set

$$ R = (\textrm{Rationality}) = \frac{ \textrm{Var}(X)}{E(X)^2}$$ That is the normalized variance of $X$. If $R$ is zero then the distribution is constant and the game is not rational at all; if for example we have only $1$ strategy out of $N$ realizing victory with probability $1$ and the other never realising victory, we have $R \sim N^2$, because the game is very rational.

Hope you enjoyed!

EDIT: there is something I swept under the carpet: here $R$ depends on the strategies $S_2, \ldots, S_n$ of the other players. This makes sense: if the other strategies are stupid, even if the game is supposed to be a lot based on chance, it could end up being very rational what strategy to take. For example in Monopoly if the other players never buy or build anything, your play will not depend much on chance.

If we make the further assumption that other players will try to make your play as less rational as they can, so that you have less choice to make a good strategy, then we should set

$$ R = \inf_{S_2, \ldots, S_n} R(S_2, \ldots, S_n)$$

Answer 4

One example to consider: imagine a chess variant where the only difference was that White's first move was chosen at random. This is clearly an element of luck that might put White at a disadvantage.

Nevertheless, we wouldn't call this a game of pure luck. It's closer to being a game of pure skill, because if an amateur repeatedly played a world chess champion at this game, it is unlikely the amateur would ever win.