Samuelson’s problem

Writing is difficult. Writing well, even more so. Writing well requires a combination of a solid idea explained beautifully through simple words that hit the mark. 

Writing helps assimilate the thoughts. Part of the reason I think is because writing requires usage of a different part of the brain. And as this new part does the heavy lifting, new neural connections are formed leading to newer ideas and/or maybe a deeper and better understanding of the original idea.

We must write with an audience in mind. Warren Buffett says his letters are written such that even his sisters would understand and if you wanted to write but didn’t have sisters, you could borrow his. I think my intended audience would be my 10 years younger self- the perfect example of Charlie’s one legged man in an ass kicking contest. (Charlie said that if you don’t use big and powerful ideas from all disciplines you would be the one legged man in an ass kicking content). The difference between my younger self and me is that now I am aware that I am the one legged man.

Talking of which, here is a mental model – Samuelson’s Problem that was extremely simple to understand yet extremely profound.


In Daniel Kahneman’s book- Thinking Fast and Thinking Slow, he narrates the Samuelson’s problem.

“…Samuelson famously asked a friend whether he would accept a gamble on the toss of a coin in which he could lose $100 or win $200. His friend responded, “I wont bet because I would feel the $100 loss more than the $200 gain. I’ll take you on if you promise to allow me to make 100 such bets.”

As I thought about this, I found this grabbing because as we all know losses and gains are part of every venture and every life. Its not enough to think just in terms of gains and losses but also in terms of probabilities of winning/ positive outcomes and losing/ negative outcomes. Once you do that, you will realize that some ventures offer positive expected outcome (like the one above) and we should simply grab those. In the Samuelson’s problem, the loss on a single flip would be $100 and the gain could be $200 and both are equally likely to occur. Because of the asymmetric nature of the bet (positive expected outcome), it makes sense to play this kind of a game many many times over. If you could play this game 10 times, you are likely to end up with $1,000 and if you could play it 100 times, you would end up with $10,000. Also, the chance of a total loss (that is you go home with lesser money in your pocket that you set out with) is 50% for a single flip, 18% for ten flips and 0.04% for hundred flips. 

Daniel Kahneman then goes to say

…the aggregated gamble of hundred 50-50 lose $100/ gain $200 has an expected return of $5,000 with only a 1/2300 chance of losing any money and a 1/62,000 chance of losing more than $1000”

To me the big lesson was to think in terms of expected outcomes and not just gains/ losses and to seek out bets that offer positive expected outcomes and avoid ones with negative expected outcomes. 

Our investment portfolios should pretty much be like this. Each investment idea should be a bet offering a big upside and small downside with probabilities in our favor. A collection of such investment ideas will, on the whole, offer a great outcome and the chance of a total loss on such a portfolio drops significantly. 

Warren Buffett understands this. He built a portfolio of such high class companies such that even mistakes like Dexter Shoes didn’t wipe out Berkshire. Likewise, in the Reinsurance business, they underwrite policies such that if a catastrophe were to occur, they would no doubt face a loss on that policy but across several such policies and several years, they would have a great outcome.

Jeff Bezos understands this. At Amazon, they make several bets on new ideas; an idea that clicks like AWS could add billions in revenue and profit and yet an idea like the Fire phone that failed, hardly causes a blip. And he intends to continue making such bets with significant upsides and minimal downsides.

Likewise, in sports like Cricket, a long drawn series is usually a better indicator of the relative strengths of the two teams than a single match alone. Because a single match like the single coin toss could throw up an ‘unexpected’ result; but over many such flips/ games, the outcome would get closer and closer to the mathematically expected outcome.

You could use this idea to avoid trouble as well. For example, if you are going to take up an activity that has a negative expected outcome such as rash driving on Indian roads, you should completely avoid or minimize your exposure to them. Say by driving rashly your gain is time saved and your losses would be minor/ major accident, damage to property or life, traffic violation ticket or worse. Lets put a $ value- gains = $100 and losses = $200. And say, the chances of such a venture would be 50-50. If you drive rashly once, the chance of your total loss is 50%. And if you drive rashly 10 times, the chances of your total loss increases to 82%!! 

So you see, once this idea hit me, I looked at the world around me and realized that the very successful people like Buffett, Munger, Bezos were into ‘games’ that had extremely positive expected outcomes. And for my own well being, I need to arrange my affairs such that it gives me positive expected outcomes. Here are a few that have clicked for me:

  • Investing with a long term horizon
  • Practicing law of karma
  • Regular exercise
  • Company of smart/super smart people
  • Driving with margin of safety and with more than adequate protection to minimize negative outcomes

Coming back to Charlie’s one-legged man. I would like to claim that I was/ am the one legged man, because despite a Masters in Mathematics I didn’t get this simple yet profound idea.


Note: In all the examples above, I have deliberately left out the effect of feedback loops. When you toss a fair coin, there is no positive or negative feedback from the first coin toss to the next. But in investing, your second bet can be better than the first and third better than the second. Feedback loops can make the increase the probability of an outcome of the successive bet. This makes investing even more of an irresistible game.


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