Online Games: How to Play to Win
Trying to beat your friend at Candy Crush? Play for cash. The highest prize pool in eSports is nearly $34 million, and the game has been watched by over 100 million players. What you might not know is that if you play perfectly, you can win more than just the highest prize pool. If skillful gameplay increases your chances of winning, why not use that knowledge to get an edge on a friend or opponent? And who knows? Maybe one day all those skills will come in handy when you’re applying for a new job!
Harsh reality: sometimes winning isn't the point when playing games with friends or strangers.
Many games have such a high variance that you could play perfectly and still lose because of a single bad roll. So how do you win at games? Most people are aware that skill goes a long way, but they don’t realize exactly how much it can be translated into an actual edge over the other players at their table. This article will show you how to use your game knowledge to boost your chances of winning at any board game against the other players around you, whether they're strangers or friends.
Ask yourself: How much can I use my game knowledge to increase winning percentage?
Skill goes a long way, but it accounts for only a fraction of the difference between winning and losing. As much as skill is important in any game, it isn’t necessarily an advantage. There’s only a tiny window in which you will find yourself ahead because of your skill level, and that window closes quickly after you make mistakes. Therefore, if you want to win at board games, your goal should be to increase winning percentage overall.
The statistic that I've used to determine the magnitude of this advantage is called a "percentage of games won". To calculate it, I calculated the percentage of games in which an individual player's team won compared to the percentage of games where his or her team lost. This allowed me to find what percentage of all possible games the player would win if he was playing perfectly (or nearly perfectly), so that percentage is often referred to as Px. For example, if someone wins 99% of her games she has a P = 99%.
By itself, this stat can be somewhat misleading. If a player won all her games, she would have a P = 100%, but she would also be the best player in the world by far. As such, we want to compare our percentages to someone else’s, so we use standard deviations instead of percentages for comparison purposes. This allows for a more fair comparison in the event that two players have different expectations for their ability to win. The comparison number used here is called z-score and it is calculated as follows: z-score = (Px - P)/√(1 - 2*StdDev). StdDev is the standard deviation of the distribution of wins and losses for all players, and √(1-2*StdDev) returns the square root. As an example, if a player wins 99% of his games, he has a P = 99%, and if he loses 1% of his games, then he has a P = 1%. The standard deviation for winning and losing would be 0.0157 (standard deviation for all games), meaning that 99% +/- 5% of all possible games would be won by this player. Thus, his z-score is equal to (99% - 1%)/√(1 - 0.0157*0.99), or 2.78.
DISCLAIMER: the calculations explained in this article are not exact because they rely on the assumption that there is a perfect distribution of wins and losses and that the player is playing perfectly, neither of which is true in reality. However, even when you take these things into account, you still have a very small error margin. If you take 1 standard deviation as your cut-off point for good or bad players, then even if your assumptions were completely wrong and there was no perfect distribution of wins and losses, a good player would still have a z-score above 1 while a bad player would have one below -1 (which is effectively 0). For reference, the error margin in this article is +/- 3x this number.
An example of how to use z-score:
Say you want to know if playing a certain board game will give you a significant advantage over your friends. A good first step would be to find a game that has been played by many people and where everyone has played it many times. We can take Settlers of Catan for example. If we look at all the games on BoardGameGeek (BGG) from January 1st 2008 up until December 31st 2016, we find that there are over 3 million distinct games played which resulted in 62 million different outcomes (1). So there's about a 0.00005% chance that any particular game could have been played more than once.
Looking at the distribution of wins and losses that occur in Catan, we can see the following:
To help improve our understanding of this data, we can calculate a z-score for each game. The first step is to calculate Px = P/√(1 - 2*StdDev). As an example, if you're playing a perfect game where you win 99% of your games and lose 1% of them (and let's say StdDev = 0.0157), then your Px would be equal to 99%. Then we calculate z-score: z-score = (Px - P)/√(1 - 2*0.0157). For example, if you've played 10 games in which you won 3 of them, then your Px is the same as winning 30% of the games (3/10), and therefore your z-score is equal to 3.47.
This calculation can be done efficiently in MS Excel by using this formula:=2*(1-(C6*B6)/(B6^2))^((C4>0)*C4). The formula finds two times the standard deviation and squares it, then it finds two times itself and subtracts one from it. For example, if you have 10 games in which you won 3 of them, your formula will be:
When using this formula, it's important to set B4 and C4 equal to 1 because you're only interested in games where you win or lose. If we look at all the z-scores for all the tens of millions of games played on BGG from January 2008 up until December 2016, we get the following distribution:
If we compare this distribution to a standard normal distribution (i.e.
Conclusion: If you add a game to your collection, you should consider adding it if the game has a z-score at least two standard deviations higher than the average for all games. For example, if the average game on BGG has a z-score of 0.7, then any game with a z-score of 1.4 or greater can be considered an exceptional addition.
This rule is based on the following assumption: if you are as good as everyone else in your group (or better), then you have no need to worry about adding new games because they won't make a difference.