Company size vs talent: which one is more important? (Math problem)

There are always tradeoffs when picking one company over another. Imagine for a second you are thinking about picking between a standard position at a straight-forward, large company versus a fun position at an exciting smaller business. Which one is more appealing to you?

There are many reasons people pick one or the other. For instance, larger companies often offer stronger compensation packages at first, and the expectations can be clearer when joining a well-established business.

But is it really safer to join a large company rather than going for a talented startup?

I came across a delightful math problem about this very topic that illustrates something very interesting.

Here is the problem:

Two businesses, A and B, are competing for accounts in the same industry. A is a large company with 50 accounts, and B is a small one with 20 accounts.

Competition is fierce and turnover is high: periodically a random account decides to move from one company to another. It just so happens that B has an edge in talent and tends to win more accounts: 52 percent of the time the account will shift from A to B (and 48 percent from B to A).

How do you think the competition will play out in the long-run? That is, what is your prediction about the future of each company?

Can you figure it out?

If you are struggling on how to start, check out solution method 1 in my puzzle about a drunkard and a cliff.

I will post an answer in the comments section by the end of the day on Thursday.

(If you prefer to read the answer now, check out the answer to problem 3 in section 4.15, which appears on page 188 of Finite Mathematics. I have slightly re-worded the problem for this post. Note the book provides an answer, but you’ll have to read the entire section to understand why that answer is correct)



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  • Greg

    Rather than solving this mathematically, I simulated it in Excel…

    After 20,000 “competitions” then it is clear that the trend is for both companies to stay at a healthy size, with Company B having 52% of the 70 accounts.

  • Matthias Daues

    Hmmm. Quite in opposite to Greg I slavishly applied the formula for unfair coin flip with finite resources (http://en.wikipedia.org/wiki/Gambler's_ruin#Unfair_coin_flipping) and ended up with a probability for company B (the rather talented but smaller one) to lose all accounts in the long run against company A (the rather less talented incumbent) with a probablity of 98,5%.

    My conclusion therefore would be to join the bigger company as it might eat up all of the market in the long run.

    The interesting thing is that this model is too constrained to be used as a basis for real life decisions. Time, for example, doesn’t play a role in it: How long is the long run? Can’t tell.

    There’s other obvious gaps in it, but I’d conclude that sticking to your personal preferences would guide you better than following the mathematical model.

    It helps to justify the safe (cowardly) bet of joining the larger one – assuming a non-changing market it will inevitably win at the end.

    Cheers, M.

  • Matthias Daues

    …and after a bit of thinking along the impulse: This must be false, it occurred to me that I flipped p and q. So after recalculating company A will with a probability of roughly 80% end up without customers.

    The conclusion about time being no part the model still is valid. The “safe-bet-conclusion” is moot.

    Sorry for being sloppy.

  • Dan

    I looked at the Finite Math example and this is not the same problem. Movement in the “drunkard cliff” or “unfair gambling” problems are independant of the current state. The way the example is written has a random account followed by a random movement, not just the random movement. This leads to the answer given by Greg.

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Thanks for pointing that out Dan, I have re-worded the problem so I hope it resembles the unfair gambling.

    The solution I had in mind was this: the game was meant to be like the gambler’s ruin with B winning 52 percent of the time. The future is that one company will end up winning all the accounts. Company B wins 80 percent of the time compared to A only 20 percent.

  • Peter

    I have a theorical question about the exercise:
    In such a scenario, a company will always breakdown. Imagine the winner company have to face a new company everytime its rival is defeated. Is the sum of all contestants the same that a big company with infinite size? If so, all companies are going to close, does not matter how much edge one could have.

    A extreme version of that question is what does happen if a company with 100% of edge face a company with infinite resources?

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