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Two companies compete for customers. Customers arrive one at a time, and each buys either Company $1$’s product (denoted by “iPhone”) or Company $2$’s product (denoted by “Android phone”), but not both. Initially, each company has had $1$ sale. Each customer chooses which product to buy with probability proportional to the square of the number of sales, e.g., (if $4$ iPhones and $3$ Android phones have been sold, then the next customer buys an iPhone with probability $\frac{16}{25}$ and an Android phone with probability $\frac{9}{25}$).

I'd like to show with probability $1$, either there is a customer after whom all future customers buy iPhones, or there is a customer after whom all future customers buy Android phones.

My approach is to let $X_1,X_2,\ldots$ denote the interarrival times between purchases of iPhones. We assume that they have distribution modeled by $X_j \sim \frac{1}{j^2}Expo$. In a similar way we define the Android buying interarrival times to be $Y_1,Y_2,\ldots$ which are distributed $Y_i \sim \frac{1}{i^2}Expo$. At this point, I'm not sure how to go further, does anyone have any ideas?

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The interarrival times are completely irrelevant to your question. All that matters is the history of outcomes; i.e., you should define a process $X_{n+1} \mid X_1, X_2, \ldots, X_n \sim \operatorname{Bernoulli}(p_{n+1})$, where $p_{n+1}$ is a suitably chosen function of some sufficient statistic (hint, hint) of the previous outcomes $X_1, X_2, \ldots, X_n$.

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