## Which Phone Would You Buy?

Galaxy S3 recently became the best-selling smart phone in America, trumping iPhone. So what might be the reason why more people choose Galaxy over iPhone?

Let’s say we are in another world where there are two competing mobile phone companies that each sells one certain model of mobile phone. Let’s also assume all people care about is getting the better phone; they don’t care how much they have to pay. Each person has their own private information to base their decisions upon, and they can observe other people’s decisions. Let’s call the two kinds of phones Phone I and Phone G.

Now let A= “Phone G is the better phone” and let the probability Pr[A]=p. Let B= “An observation that a people have purchased Phone G and b people have purchased Phone I based on their private signals”. We should introduce the idea of a private signal. Let signal H= “Phone G seems like a good phone and I should buy it” and Pr[H|A]=q > ½. Let signal L= “Phone G doesn’t seem like a good phone and I shouldn’t buy it” and Pr[L|not A]=q > ½. q is always greater than zero. If Phone G is the better phone, your payoff would be Vg>0 and if you choose the worse phone your payoff would be Vb<0. Initially we assume you are indifferent about buying Phone G, which leads to the equation Vg*Pr[A]+Vb*(1-Pr[A])=0. Note that “Pr[X|Y]” stands for “the probability of X given Y”.

Now after observing B (“a people have Phone G and b people have Phone I based on their private signals”), you want to figure out whether buying Phone G is the right decision. Your payoff now is going to be Vg*Pr[A|B]+Vb*(1-Pr[A|B]), and you care if Pr[A|B] is greater than p, because if it is greater your payoff {Vg*Pr[A|B]+Vb*(1-Pr[A|B])} is going to be a positive number. Applying Bayes’ Rule:

Pr[A|B]= Pr[A]*Pr[B|A]/Pr[B]

= p*qa(1-q)b/[pqa(1-q)b+(1-p)(1-q)aqb]

> p*qa(1-q)b/[pqa(1-q)b+(1-p)qa(q-1)­b]

= p

Note that Pr[B]= Pr[A]*Pr[H|A]a*Pr[L|A]b + Pr[not A]*Pr[H|not A]a*Pr[L|not A]b. The inequality is based on the assumption that a>b. Now if you’ve seen as many Phone G users as Phone I users (a=b), you should follow your own signal and buy the phone that you believe to be better because Pr[A|B]=p. If the number of Phone G users and the number of Phone I users you’ve seen differ by 1, you should still follow your own signal because you might be the one that equalizes the numbers of users of both phones. However, if a and b differ by 2, you should follow the majority of the people, assuming that the last person that bought a phone followed their own signal.

So there are two major factors in determining which phone you would choose: the private signal you receive and the numbers of users of both phones that you’ve seen. This is the reason why advertising is so important: it increases the possibility of a positive private signal, which in turn increases the number of people that would buy a certain phone. The more users a phone has, the more likely people are going to buy this phone, which makes sense.

JohnFannister