Let’s Make a Bet

In my career, I’ve learned that there can be vastly different strategies for something you do only once versus something you get to do repeatedly.

Suppose I have a unfair coin. It comes up heads 60% of the time and tails 40% of the time. You have $25. I’ll let you bet as often as you want, and you want to maximize your money. What do you do?

I first encountered this problem on Paul Butler’s website, which has a simulator which lets you place bets over and over and tracks what happens to your bankroll. I did awfully at this. According to this reference from the website, a lot of people do terribly at this problem, so I don’t feel too badly.

Here was my reasoning: If I only got to make a single bet, I figured the best strategy was to go all-in. If I won (60% chance), I’d have $50, but if I lost (40% chance), I’d have $0 — the expected value of my “all in” strategy was $30. You can check different values of what you could bet, and you can see that “bet it all” gives you the highest expected value if you know you’ve got a better than 50% chance of winning.

I knew the wrinkle in this game is I could place multiple bets, so getting wiped out was bad. If I lost, I’d want to have some money left to bet again and try to earn more money. I figured there was a 40% chance of losing, so I’d ratchet down my “bet it all” strategy by 40% and only bet 60% of my money.

As I said, this strategy doesn’t work. Try it in the simulator. When you bet 60% of what you’ve got, the losses hit too hard. You have a 16% chance of two losses in a row, and that event reduces your bankroll by almost two thirds, making it hard to make up lost ground. (A small bankroll means you can only make small bets, so even when you start winning again you’re not winning as much.)

I’ll spare you the math, but it turns out the right amount to bet in this game is 20% of your bankroll. I was shocked that there was such a divergence between the optimum strategy if you could only make one bet (bet it all!) and the optimum strategy if you could make multiple bets (bet just 20%).

I think about this sometimes when it comes to my job. Every now and then, I find myself having influence over task allocation — deciding who works on what. When I’m in this situation, I think of advice I heard from one of my managers at Microsoft. He told me not to just assign tasks to the person who is best able to do the task. Instead, I should focus on who is going to learn the most by working on the task.

My former manager’s insight here comes back to different optimum strategies for things you do just once and things you are going to do repeatedly. Assigning tasks to the people who are most capable of finishing those tasks is the optimum “play the game once” strategy. With the most capable person working on any given task, the project will finish in the least time and with the least risk — but also, there will be the least growth in the capabilities of the team. Nobody will be stretched by the work they do. Assigning tasks based upon who’s going to learn the most from the task is a “play the game over and over” strategy. In exchange for a little bit of risk / time on the current project, you’re going to have increased capabilities on the team for the next project. Over time, the crew be able to take on more and more ambitious projects.

Recognizing the difference between single and repeated play strategies — and taking the time to figure out what kind of game you’re playing! — can profoundly impact outcomes.