Even though you are now more sure which machine is better (IE decreases from 8% to 1%), it comes at a cost. The reason is that by playing each machine 250 times instead of 100 times at the start, you are playing an additional 150 games on the weaker machine, which decreases your average winnings. With a much lower IE, why didn’t our average winnings increase? However, our average winnings decreased to $963. Surely it is much harder for Manchester United to lose three games in a row? With 250 games for each machine, the IE is now reduced to 1%. To lower the IE, we can extend the above method to play each machine 250 times at first, before deciding which is better. But since we choose the correct machine most of the time, on average we will still receive a good amount. In 8% of the time, Machine 2 will surpass Machine 1 in pay-outs using our setup, and we will be choosing the wrong machine to play the remaining games. In fact, the identification error (IE) is 8%. This is akin to Manchester United losing two premier league games in a row to inferior opponents – unlikely and unexpected, but it has happened before. Since the chances of winning for both machines are quite close to each other, this is not surprising.
In the top-right scenario however, Machine 2 actually had a higher win rate. To better illustrate this method, I ran four simulations of the setup, playing the game 100 times each on Machine 1 (solid line) and Machine 2 (dotted line).Īs you would expect, in most cases Machine 1 is paying out more money. On the other hand, if received insider information about which machine has the higher pay-out, and you play it for all 2,000 games, you will receive $1,000 on average. As a comparison, randomly playing the machines for all 2,000 games will give $900 on average. This turns out to be a pretty good strategy. Thereafter, for the next 1,800 games, we play just that machine to maximize our winnings. We simply play each machine 100 times, check how much we have won from each machine so far, and then decide that the machine giving you more money is the better machine. Most people would be able to think of this intuitive method to find out which machine is better. You may also have two experimental drugs that treat the same disease but with different chances of success. For example, you may need to choose between two different kinds of ads to put on your website, but you are not sure which ad is more appealing to your customers and will thus give a better click-through rate in the long run. This simple setup can surprisingly be applied to many different situations.
What is the best way to play if you want to maximize your winnings?
The catch is, these chances are unknown to you, and so you do not know which machine will give you a higher chance of winning.
For machine 1, this chance is 50%, and for machine 2 it is slightly lower at 40%. For an individual machine, the chances of winning are the same every time you play it, and it is not affected by the results of your previous plays. When you play a game, you either win (get $1) or lose (get nothing). At each time you can pull the arm of either slot machines, but not both, and we assume you can play this game 2,000 times. You have two slot machines / one-armed bandits (the number of machines can be changed). If we make some assumptions, we can view the problem of choosing penalty takers as a Multi-armed Bandit (MAB) problem, and we will learn whether Van Gaal’s policy is optimal. The player that missed will go down to the bottom of the list, and will be back at the top after the four other players have missed their penalties. While most football teams appoint one or two players to take penalty kicks in all matches, Van Gaal has a list of five players that rotates whenever a player misses a penalty. Manchester United, under Dutch coach Louis van Gaal, has an unconventional policy on which player takes a penalty during a match.
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