Author: Shruti Mistry

We all may already know the game of penalty kicks, there are two players, the striker and the goalie. The striker must shoot either left or right to score a goal and the goalie needs to dive left or right to stop the goal. Suppose that the goalie always manages to stop the ball and the kicker is not always accurate in his kicks, he is accurate in kicking on left and can score a goal, however sometimes he misses and hits the right side with a probability of X. This occurs even if the goalie guesses incorrectly. The payoff matrix for this looks like:

Let Y axis be the goalie and X axis be the striker.

How does penalty kicks play out in reality?

That is the basic model of penalty kicks, but let’s delve deep into the reality of penalty kicks.

Real life penalty kicks are different in two ways, one that kicking to the right and kicking to the left is not always the same. For example, a player that is right footed can be more accurate in shooting towards the left rather than shooting towards the right. This tells us that if he hits towards the right he is more likely to miss the shot, as the goalie becomes more likely to stop the ball on the right.

The second way it differs is that the players can choose to shoot towards the middle, and likewise the goalie can also choose to defend the middle.

We know that a striker’s probability of scoring is equal no matter the direction he shoots in, because suppose it the probability was not equal. This means that if the striker is continuously kicking towards the right, then the goalie is also likely to dive towards the right, then there is a chance of exploitation, where the striker can suddenly kick left, which can result in a higher scoring percentage. To avoid this exploitation, the striker’s probability of scoring is equal. Just like the striker, the goalie also has an equal probability of defending where if one of the sides is more likely to be scored in then someone has a higher advantage than others.

This is data that shows the percentage of shots scored from the strikers. From looking at the Total column and row for both striker and goalie we notice that they all have an equal probability where the difference isn’t too significant.

Then the main question that remains is: How can we improve our chances of scoring?

Suppose you are very accurate in shooting to the right side but not as accurate when shooting to the left side, so you improve your accuracy towards the left side. From intuition, you think that you will now shoot more often to the left side. However this is incorrect because you must consider the weakness of the goalie, shooting towards the left side can decrease your chances of getting a goal, because it may be the side that is better defended by the goalie.

Hence, with this mixed strategy approach we can improve our chances of scoring.

References:

https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/

If you would have said your future goal was to be a “Youtuber” two decades ago, everyone would have thought you have lost your mind. However nowadays, “about 3 in 10 American children want to be a Youtuber, and lesser preferences include being a teacher, astronaut, and a musician. [1]” Being an established Youtuber is not easy, Youtube has over a billion users, totaling 228575 distinct user channels, 400249 user recommendations, and 400249 Youtube recommendations [2]. Youtube remains to be competitive by utilizing its click-through-rate with its accurate video recommendation algorithms. These numbers are indeed daunting for some of us who would want to start a youtube channel in hopes of earning money.  Then, the common question of all YouTubers is “How can I get more views?”

In the study by Yonghyun Ro, Han Lee, and Dennis Wan, nodes in the network represent a youtube video. Two nodes are connected with a directed edge if one video recommends the other. The researchers study the technique of PageRank, which allows them to observe a pattern for a highly viewed video for the categories. The PageRank algorithm works as so: each video on youtube is given a PageRank, which represents the number of other videos it connects to via edges. The more outgoing edges it has, the more links that particular video has, and hence it will have a higher PageRank. However, there is a slight problem to this, it becomes very easy for others to forge the PageRank of a video so that their PageRank is higher. This introduces the Random Surfer Model, which randomly picks a video to visit, and goes to the videos that it’s linked to and then picks another video randomly, so on. It keeps a score of how many times a video has been visited, those videos that have more links are visited more frequently. This brings up another problem: what if all videos are not connected via a link? Then it limits us to only visit that specific cluster of videos and disregard the rest of the videos. This is solved by occasionally resetting the random surfing by its damping factor so that it does not disregard any of the videos.

The researchers analyzed the total PageRank score of Youtube around 750,000, where each video has its own PageRank score. Youtube allows each video to have about 20 recommended videos. Then, if video A is one of the recommended videos of video B, there is a directed edge from B to A. As understood from the PageRank algorithm, if a video has a higher PageRank score means that the video is related to many other videos in the network. Thus, it will have a larger influence than other videos within the network and will be recommended to a larger percentage of the users.

Above is the data that compares the PageRank of a video to its views and its video length. The graphs have x-axis as the PageRank score in log10 base and y-axis is the percentile of each video feature (views and length).
From the graphs, notice that there is a high correlation between the PageRank score and the views associated with the video i.e the higher the PageRank score the higher the views for the particular video. On the other hand, if we take a look at the graph that compares the PageRank of a video to its video length, we notice that there is a relatively low correlation. This tells us that most of the videos that have a high-influence and high PageRank scores are usually a shorter length video compared to the other videos.

So, next time your uploading your youtube video and you find yourself asking “How can I get more views?”, be sure to remember this algorithm and focus on increasing the PageRank of your video!

References:

• http://snap.stanford.edu/class/cs224w-2014/projects2014/cs224w-37-final.pdf