#game skill

There are at least two ways to look at this.

The first "dogma"

is more straightforward: compare someone who makes choices randomly with someone who makes informed decisions. Let's say four people sit down to play Monopoly, and three of them choose at random when to mortgage properties, when to build houses, and when to accept a trade. And let's say the fourth, while not a perfect player, follows a few basic rules:

1. Never accept a trade that gives an opponent a Monopoly, unless that trade also gives you a more valuable Monopoly, or a less valuable Monopoly and at least double the combined value differential of the properties involved in cash.

2. Always accept trades and sometimes offer trades that give you a Monopoly, but don't give your opponent a Monopoly, even if you are giving away more net value than you are receiving, up to $500.

3. Never mortgage unless you have to.

4. Build houses up until the point that landing on no single square on the board will force you to sell a house, and no further.

This fourth player won't be a Monopoly superstar, but they do at least have some heuristic for approaching the game. My bet is that the fourth player will win 95% of the time. Even if we modify the rules to have the random players never accept or offer trades, I bet the fourth player will have significantly more than a 25% chance of winning. If we wanted to get technical (which we do, but we'd have to program Monopoly from the top down first, which would take a while, oh well), then we could make a formula out of this pretty easily.

First, play 100 games between one heuristic player and three random players. Then take the odds of the heuristic player winning. Next, take four random players, and take the odds of one specific random player winning.

So, to make up some bullshit notation:

O(H | H, R, R, R) - O(R1 | R1, R, R, R) = S(H, R)

Or, odds of H wins given H plays with three Rs, minus the odds of R1 wins given R1 plays with three other Rs, equals the skill of H with reference to R.

Or, made more englishly-readable still, we take the odds of a player using an okay heuristic winning against three random opponents, and subtract the odds of a random player winning against three random opponents. We should get the differential of how much better your chances of winning are from having serviceable (but not excellent) skill.

Let's say that H does win 95% of the time. If four random players duke it out, then any given one should win 25% of the time. So the value of S(H, R) with respect to Monopoly would be 0.7. Your odds are 70% better of winning a four-player game against randos if you yourself are not a rando.

Probably the equation could be made more general by multiplying the result by a factor to do with number of players. That way, you could compare 2-player games like chess with more commonly 4-player games like Monopoly, or n-player games like Yahtzee.

Probably this formula could be made better. If I think of ways to improve it, I'll keep you posted. If you think of ways to improve it (if you have background in statistics, it will probably be easy), you should keep me posted, please.

Anyway.

All this to say, if you are playing Monopoly with one player that understands the game's principles pretty well, and a bunch of other players that don't, luck is barely a factor.

But that doesn't mean Monopoly is off the hook.

Here's the second "dogma." Let's say we use a similar formula, but this time we're comparing players with an ok heuristic to one perfect player. A perfect Monopoly player is possible, though I'm not sure it would be within a computer's power to efficiently simulate. Probably it wouldn't. A perfect player would calculate all possible die rolls and always make the best decision. It would only offer or accept trades that increased its odds of winning. It would always build the number of houses or mortgage the correct properties to increase its odds of winning.

We call the player that uses an okay heuristic H. Let's call the perfect player P. For some games, simulating a perfect player is feasible (any solved game, including Connect Four and Checkers, which I am Capitalizing for Consistency). For Monopoly it isn't, so if we actually wanted to run this experiment we'd have to just give the Perfect player a really, really good heuristic, and consult with Monopoly experts and advanced statistics or something.

Anyway, our formula would be:

O(P | P, H, H, H) - O(H1 | H1, H, H, H) = S(P, H)

So we'd find the odds of a perfect player beating a bunch of players with an ok heuristic, and subtract the odds of a player with an ok heuristic playing a bunch of other players with an ok heuristic.

As long as all four players have the same strategy, the odds of winning should be 25% (assuming order of play is randomized too, which in standard Monopoly rules it is). So we'd take the odds of a perfect player's victory, and reduce it by 25%.

I'm betting perfect play against players with an alright heuristic would only yield a victory rate of about 40%. If that. And for a sufficiently good heuristic, like a complicated one used by a top human player, it might even be as low as 30%, or 35%. These numbers are being conjured from the void, and may have little bearing on reality. But I do have at least some basis for them.

The main decisions in Monopoly have to do with when to build houses, when to leave jail, and when to trade. A perfect player will occasionally make nuanced decisions that are confusing to a player that's not a quantum superintelligence. But it still won't accept dumb trades, or mortgage properties for no reason. In most cases, it'll probably behave the same as just a fairly skilled player, because so rarely in Monopoly do you actually make important, non-obvious choices. If you can build houses, you should. It's only a question when you're low enough on cash that you might have to sell them soon. So while the perfect player will up its odds of winning in those persnickety border cases, it probably won't be by much.

I'm betting the value of S(P, H) would be somewhere below 0.2, and below 0.1 for a really solid heuristic (better than the one I outlined). If I'm right, then Monopoly has a really low skill floor - you can be sufficiently bad that luck doesn't matter (and if you're not willing to run some basic calculations or core Monopoly values when thinking about a trade, you will be sufficiently bad), and you will almost always lose. But there isn't a very high skill ceiling - above a level of fairly attainable skill, your odds of winning get diminishing returns, bad. You can think much more deeply about Monopoly than your fellow players, and only gain a few percentage points toward your odds of winning, provided everybody at least understands some fundamental and sorta nuanced principles of the game (Dark Greens are usually really bad, Oranges are amazing, waiting in jail is a solid choice in the late game, etc.)

Further Thoughts

If I had a copy of Monopoly on Python, or the amphetaminesque (no, I don't take amphetamines) drive to create one (I think it'd be legal, since I own the game), these would be some really fun tests to run, rather than just thought experiments. It'd be really hard to get good data, though, for comparing high skill levels to top skill levels, or deciding what definitely qualified as better and worse heuristics (you could go by results, but you might get into nasty rock-paper-scissors patterns - it's good to trade more vs. enemies that don't develop too much, but that doesn't mean that accepting more trades is good across the board).

You could make this way more fun by creating prototypes of various skill levels and play styles, and running their odds in groups of different makeup. Ie. figure out what happens when one conservative player, two vindictive players, and a total sap play each other 100 times. Or when one person who makes only great trades, two people who make any old trade that gets them a Monopoly, and one player who refuses to trade at all play 100 times. The works.