Network Diffusion – Xbox or Playstation?
Source: https://www.businessinsider.com/playstation-ps4-xbox-one-sony-microsoft-crossplay-explained-2018-3
INTRODUCTION:
Since the release of the original Xbox in 2001, both Microsoft and Sony have dominated the console gaming scene with their roughly equivalent consoles – Microsoft with Xbox, and Sony with PlayStation. One thing keeping these two giants in opposition is the lack of technological support for playing games across platforms. In other words, if I were playing a game on Sony’s PlayStation, I would only be able to play online with other PlayStation players and would oftentimes not be able to connect and play that game with Xbox players, because Microsoft and Sony dont allow for it (in most games). This leads to an issue, as Gamers are forced to make a decision between buying either an Xbox or PlayStation based on what their friends choose so they can play together. Business Insider described this issue in detail in a recent article, discussing the competition between Sony and Microsoft, their two similar, but different “independent online ecosystems,” Sony’s being PlayStation Network and Microsoft’s being Xbox Live, and how simply “there’s no way to play games across these two platforms.” Furthermore, they bring up the problem of how oftentimes Gamers are left “playing alone” when playing on another platform than their friends. As a way to solve this, Gamers look to choose a platform that their friends also play. By being on the same console, Gamers benefit from not only playing all games together online, but they also have access to additional features like party chat (which allows them to talk online) with each other. In this blog post, we explore how this dynamic reflects a diffusion model, how that model works in action, as well as how this dynamic relates to my own life experiences and how it applies to today’s context.
EXPLANATION AND EXAMPLE: Coordination Game
The lack of cross platform play with most games leads to a coordination game. For example, let’s say Gamer A, represented by node A, has a fraction f of his friends already using Xbox and a fraction (1-f) of friends using PlayStation. The individual payoff each node receives for choosing Xbox is x, while the individual payoff each node receives for choosing PlayStation is p. These individual payoffs can be visualized in the game theory matrix below, where nodes only gain a payoff if they match (meaning they are using the same console) and receive 0 payoff if they don’t match (meaning they are using opposite consoles). As a result of these individual payoffs, node A will receive an overall payoff X = fdx for choosing to play on the Xbox platform or an overall payoff P = (1-f)dp for choosing to play on the PlayStation platform, where f = the fraction of friends using Xbox (as defined above), d = A’s number of friends, and x or p = the individual payoff for each node that chooses Xbox or Playstation respectively. The question here is, what should Gamer A do? From here, we can put the overall payoffs for A choosing either Xbox or PlayStation, X and P, against each other. By setting X ≥ P, we can determine the threshold of neighbors needed playing on Xbox for Gamer A to also choose to play on Xbox over PlayStation. More specifically, if the fraction of A’s neighbours exceeds this threshold value, A will choose to play Xbox. After some math, where the values for d cancel out on both sides, we are left with the inequality f ≥ (p)/(x + p), meaning that Gamer A will choose to play on the Xbox platform if the fraction of his friends using the platform is greater than or equal to (p)/(x + p). A key idea here is that this threshold value is only determined by the individual payoffs of x and p.
Now let’s look at a more specific, real-world example. Let’s assume that Gamer A has 3 friends that play on Xbox and 4 friends that play on PlayStation. The network structure is pictured below. The individual payoff both Xbox players and PlayStation players receive is 1, given both consoles are roughly similar with roughly the same games, hardware, and performance. Given the properties about payoffs mentioned above (payoff only gained if they match, payoff is 0 if they don’t match), the game theory payoff matrix is pictured below as well. The overall payoff for Gamer A choosing Xbox is X = fdx = (3/7)(7)(1) = 3, while the overall payoff for Gamer A choosing PlayStation is P = (1-f)dp = (4/7)(7)(1) = 4. When applied to the threshold formula established earlier, Gamer A will choose to play on the Xbox platform if the fraction of his friends using the platform is greater than or equal to (1)/(1+1) = ½. In this case, however, the fraction of his friends playing on Xbox is 3/7, which is less than 1/2. As a result, Gamer A will go with PlayStation. This makes sense, as it would be optimal to be able to play online with the majority of your friends, or in other words, more than ½ of them.
EXPLANATION AND EXAMPLE: Diffusion Cascade
With this threshold rule in mind, the use of one console can spread across a larger network. See the network below, Figure 0. There are 2 clusters of friends, or in this example “Gamers,” labeled A and B. All nodes are currently using PlayStation, except for nodes 5 and 6 who, for some alternative reason, choose to use Xbox. As we solved earlier, the threshold value is ½. This value works in this network too because the threshold value, as explained earlier, is determined only by the individual payoffs of x and p, and these payoffs are still the same: 1. This means that for all nodes, if ½ of their friends switch to Xbox, they will follow suit. Immediately, nodes 7 and 10 realize that ⅔ of their friends are using Xbox, and immediately switch to Xbox as well because ⅔ > ½. Soon after, Gamer 9 comes to the same exact realization and also switches to Xbox. Finally, Gamer 8 realizes 3 of his 5 friends are on Xbox, which is more than the threshold value of ½, so he switches to Xbox as well. At this point, the entirety of cluster B has switched from PlayStation to Xbox, as seen in figure 1.
Now let’s look at cluster A. At some point, Gamer 3 realizes that 2 of his 4 friends, exactly half, play Xbox. Because ½ ≥ ½, Gamer 3 will switch to Xbox. Gamers 2 and 4 will come to the exact realization and switch from PlayStation to Xbox as well. Finally, Gamer 1 will realize that 2 of his 2 friends have switched, more than ½, and will do the same. This growth can be seen in figure 2.
Finally, let’s look at cluster C. Unfortunately, this is where the cascade of switching from Xbox to PlayStation stops. Despite the multiple possible entry points at nodes 11, 16, and 15, there are no instances where a node in cluster C has more than ½ of his friends using Xbox. As a result, all nodes in this cluster will stick with using PlayStation. The reason that this cascade of switches to Xbox does not penetrate cluster C is because of cluster C’s high density p. This means that in this cluster, each node has a high amount of friends also in the cluster. A network of density p is defined as a set of nodes such that each node in the set has at least a fraction of its neighbors in the set. According to the cluster theorem, if a network contains a cluster of a density greater than the threshold f, then the cascade will not invade that cluster. In this case, the cluster C has a density ¾, which is greater than the threshold value of ½, which means the cascade of switches to Xbox will not spread into cluster C.
APPLICATION: My Own Life
Going beyond just an explanation of diffusion, though, this dynamic has applied to my own life throughout my childhood and into my teenage years. I was born more than a year after the PlayStation 2 was released and only a month before the original Xbox was released (when the Xbox vs. PlayStation rivalry really began), so these generations of consoles didn’t have a big presence in my lifetime. However, as I grew up, I got into playing video games like many other kids. The first decision I faced was to get the Xbox 360 or PS3. At this point in my life, all everyone talked about in the halls at school were what games they were playing on their Xbox 360s, so the decision to me was clear. Connecting back to the network model with the threshold ½, the number of friends I had that used the Xbox platform far exceeded the amount that played on PS3, far more than the value of ½ I needed to make my decision. For the next few years I played on my Xbox 360 with my brother and friends. In late 2013 the new Xbox and PlayStation consoles came out, the Xbox One and PS4. A couple years after I was faced with a similar decision to before: Xbox One or PS4? At this point though, there seemed to be a growing cascade of people deciding to switch to the PlayStation console. More and more people seemed to have switched to the PlayStation platform, and instead of friends only talking about Xbox in the halls at school, it was a more balanced ratio of Xbox to PS4. After asking around and talking with some friends, though, I realized that the majority of my friends still played on the Xbox platform. In other words, the fraction of my friends that played Xbox was still greater than the threshold value of ½, and as a result I decided to buy an Xbox One.
Even this past November, this question has been raised again with the release of the Xbox Series X and the PlayStation 5. In today’s world, with online multiplayer games having become more popular than they’ve ever been, the question of Xbox or PlayStation seems to carry even more weight. However, with popular games such as Call of Duty, Rocket League, and Fortnite recently beginning to support cross-platform play, Gamers don’t seem to have to worry as much about making the right console decision based on what their friends chose when it comes to games. Though, the debate still exists. External features such as party chat (allowing you to speak with your friends online) are still exclusive to each console, and it doesn’t look like they will change anytime soon. Overall, it will be interesting to see how the decision between Xbox and PlayStation will play out in the next few years for Gamers across the country. Will there be a cascade across the entire gaming social network to one console? Only time will tell.