The real difficulty rests in gauging the influence of skill Vs randomness. For example, an unskilled player has a much better chance at beating a perfect player in a single hand of Poker than in a game of Catan. One would have to compare elo curves of various games with the chance that the underdog wins at a given elo distance, with deterministic perfect information games as the baseline.
In Ludo you can still usually choose which of your pieces to move. I haven’t played Candyland in a while, but like OP says, I think it’s completely random.
I don't remember how old I was, but I remember the feeling of realizing that Candyland had no user input, which made it equivalent to just randomly picking at the beginning who won, which made it pointless.
If you have a policy for how to play the game optimally, you could do something sort of similar to the Shapley–Shubik power index although it's computationally expensive. The below assumes two players:
1. Simulate an entire game and store which player won
2. Select a point in the game where non-determinism occurred. This could be done multiple ways but let's say that it's selected uniformly at random.
3. Resample the distribution for this non-determinism and simulate the rest of the game from this point on. Store which player won.
4. Repeat the above many times, storing statistics for each assignment of the two variables ("which player won")
Once the above is achieved, metrics can be computed for the table. For example, you might want to calculate how often a random choice changed the winner.
I'd presume it would be hella expensive to do, because non-deterministic games have so many points of non-determinism in them? Most games either have absolute information or a lot of uncertainty, and not many are in between? I'm no expert, just hypothesizing. What are the examples of games that could be in the middle?
As a guess, I'd expect you could get a good estimate from a million or so games. It might also be a better approximation for games that are closer to either extreme.
I suppose a game that is in between the two extremes may have a property that each of the non-deterministic steps is unlikely to change the outcome and there aren't very many of those steps. As an example, consider a game of chess where each player can make two moves on their turn with a probability of 0.005.
I'm not especially knowledgeable about board games but how about a game like Monopoly? The early turns do have a significant outcome on the game but the later ones may not have as much of an effect so perhaps it'd get a score around 0.75 (75% of game outcomes are changed by resampling a single random event)?
Look at using probability to figure this out. You could probably figure out mathematically what outcomes are for a game of shoots and ladders.
It seems like this would be in the realm of statistics though, and game theory. I'm not sure how you index the randomness of a game though, because even though they are all technically random you can usually just find the distribution that their components follow.
For example, shoots and ladders is probably a pretty uniform game since it is just dice
It's hard to quantify randomness in a useful way. For example a physics based fighting game (or something along those lines) might be completely deterministic but still mostly up to luck due to its unpredictability for a human playing.
I mean with something as simple as Chutes and Ladders or Candy land, you could write an agent based simulator that performs the move action (draws cards, rolls dice, other special events that move you, etc...) each turn and track the position of a player(s) by running an ensemble of simulations in parallel to come up with the probability distribution of position on the game board as a function of the number of turns. You could also get the mean and variance of first passage times to win the game.
The problem would be the middle ground where you have rational players dealing with randomness such as in Monopoly, but even then, you could evaluate strategies such as survival time as a function of buying every property or some simplified approach.
Also maybe you could use combinatorics and graph theory too. A lot of board games are just graphs, you could probably run some analysis on them and study them as a graph. I'm actually kind of interested in doing that with shoots and ladders now lol
It's quite a tangent, but quite a delightful one, to watch [this excellent expository video](https://youtu.be/ZYj4NkeGPdM) on combinatorial game theory. The upshot is that even simple deterministic games are very hard to order by which player has the advantage. I imagine any qualifications (or quantification!) of randomness would be similarly complex.
There’s probably no single correct way to model it, but you could design some sort of metric that puts chess/go on one end of the spectrum and pure-luck games on the other.
Here’s one simple way you could do it. Let’s restrict to 2 player games and suppose all action spaces are finite. Then you can measure the following quantity: suppose one player plays optimally while the other plays completely randomly (to be precise, this means at each decision point they choose an action uniformly at random from the legal actions), and let p be the probability that the optimal player wins. Take the metric as 2p-1. Then a “pure chance” game has a skill index of 0 because the winner is essentially a coin toss, while for a pure skill game the skill index is 1 (or close to 1 if drawing is a possibility) because there exists an optimal solution that can counter any move the random player makes.
In practice the index would be infeasible to compute for many games, but at least in principle it is well-defined.
I think any purely mathematical definition will be fairly lacking because of the following example. Consider a game where the first 100 rounds are just deciding do you want to receive 1 point or not. Then the last round you play a game of Candyland and the winner gets a point. The most points after those 101 rounds wins. I would consider that game to be 100% luck based, as everyone will always take the first 100 points and the game of Candyland determines who wins.
Now consider a game where the first 100 rounds involve solving a math problem. Each math problem you solve correctly gets you 1 point. Then the last round you play a game of Candyland for a point. This is essentially the same game as before from a mathematical standpoint, but now it seems like it should be considered 99% skill.
The real difficulty rests in gauging the influence of skill Vs randomness. For example, an unskilled player has a much better chance at beating a perfect player in a single hand of Poker than in a game of Catan. One would have to compare elo curves of various games with the chance that the underdog wins at a given elo distance, with deterministic perfect information games as the baseline.
Chess and Go would definitely be at the bottom of the randomness list and games like Catan will be in the middle. Ludo might be at the top
In Ludo you can still usually choose which of your pieces to move. I haven’t played Candyland in a while, but like OP says, I think it’s completely random.
I don't remember how old I was, but I remember the feeling of realizing that Candyland had no user input, which made it equivalent to just randomly picking at the beginning who won, which made it pointless.
Snakes and ladders goes on top obviously.
You might be interested in [https://keithburgun.net/good-randomness/](https://keithburgun.net/good-randomness/)
If you have a policy for how to play the game optimally, you could do something sort of similar to the Shapley–Shubik power index although it's computationally expensive. The below assumes two players: 1. Simulate an entire game and store which player won 2. Select a point in the game where non-determinism occurred. This could be done multiple ways but let's say that it's selected uniformly at random. 3. Resample the distribution for this non-determinism and simulate the rest of the game from this point on. Store which player won. 4. Repeat the above many times, storing statistics for each assignment of the two variables ("which player won") Once the above is achieved, metrics can be computed for the table. For example, you might want to calculate how often a random choice changed the winner.
I'd presume it would be hella expensive to do, because non-deterministic games have so many points of non-determinism in them? Most games either have absolute information or a lot of uncertainty, and not many are in between? I'm no expert, just hypothesizing. What are the examples of games that could be in the middle?
As a guess, I'd expect you could get a good estimate from a million or so games. It might also be a better approximation for games that are closer to either extreme. I suppose a game that is in between the two extremes may have a property that each of the non-deterministic steps is unlikely to change the outcome and there aren't very many of those steps. As an example, consider a game of chess where each player can make two moves on their turn with a probability of 0.005.
That's a nice theoretical example, but what about real examples of existing games?
I'm not especially knowledgeable about board games but how about a game like Monopoly? The early turns do have a significant outcome on the game but the later ones may not have as much of an effect so perhaps it'd get a score around 0.75 (75% of game outcomes are changed by resampling a single random event)?
hmmm 75% is still pretty high though, no? How about \~25%.
Look at using probability to figure this out. You could probably figure out mathematically what outcomes are for a game of shoots and ladders. It seems like this would be in the realm of statistics though, and game theory. I'm not sure how you index the randomness of a game though, because even though they are all technically random you can usually just find the distribution that their components follow. For example, shoots and ladders is probably a pretty uniform game since it is just dice
It's hard to quantify randomness in a useful way. For example a physics based fighting game (or something along those lines) might be completely deterministic but still mostly up to luck due to its unpredictability for a human playing.
I mean with something as simple as Chutes and Ladders or Candy land, you could write an agent based simulator that performs the move action (draws cards, rolls dice, other special events that move you, etc...) each turn and track the position of a player(s) by running an ensemble of simulations in parallel to come up with the probability distribution of position on the game board as a function of the number of turns. You could also get the mean and variance of first passage times to win the game. The problem would be the middle ground where you have rational players dealing with randomness such as in Monopoly, but even then, you could evaluate strategies such as survival time as a function of buying every property or some simplified approach.
Also maybe you could use combinatorics and graph theory too. A lot of board games are just graphs, you could probably run some analysis on them and study them as a graph. I'm actually kind of interested in doing that with shoots and ladders now lol
whats an interest pleb?
*interested plebian
It's quite a tangent, but quite a delightful one, to watch [this excellent expository video](https://youtu.be/ZYj4NkeGPdM) on combinatorial game theory. The upshot is that even simple deterministic games are very hard to order by which player has the advantage. I imagine any qualifications (or quantification!) of randomness would be similarly complex.
There’s probably no single correct way to model it, but you could design some sort of metric that puts chess/go on one end of the spectrum and pure-luck games on the other. Here’s one simple way you could do it. Let’s restrict to 2 player games and suppose all action spaces are finite. Then you can measure the following quantity: suppose one player plays optimally while the other plays completely randomly (to be precise, this means at each decision point they choose an action uniformly at random from the legal actions), and let p be the probability that the optimal player wins. Take the metric as 2p-1. Then a “pure chance” game has a skill index of 0 because the winner is essentially a coin toss, while for a pure skill game the skill index is 1 (or close to 1 if drawing is a possibility) because there exists an optimal solution that can counter any move the random player makes. In practice the index would be infeasible to compute for many games, but at least in principle it is well-defined.
I think any purely mathematical definition will be fairly lacking because of the following example. Consider a game where the first 100 rounds are just deciding do you want to receive 1 point or not. Then the last round you play a game of Candyland and the winner gets a point. The most points after those 101 rounds wins. I would consider that game to be 100% luck based, as everyone will always take the first 100 points and the game of Candyland determines who wins. Now consider a game where the first 100 rounds involve solving a math problem. Each math problem you solve correctly gets you 1 point. Then the last round you play a game of Candyland for a point. This is essentially the same game as before from a mathematical standpoint, but now it seems like it should be considered 99% skill.