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“How Many Chances Do I Get”?

One stereotypical example quoted in behavioural finance is about our irrational risk aversion. Look at the Nobel Prize winning Kahneman

“In my classes, I say: ‘I’m going to toss a coin, and if it’s tails, you lose $10. How much would you have to gain on winning in order for this gamble to be acceptable to you?’

“People want more than $20 before it is acceptable. And now I’ve been doing the same thing with executives or very rich people, asking about tossing a coin and losing $10,000 if it’s tails. And they want $20,000 before they’ll take the gamble.”

In other words, we’re willing to leave a lot of money on the table to avoid the possibility of losing.

That is precisely what it is not.

The correct question to ask, if you were asked for this gamble is:

“How many chances do I get?”

If the chances are infinite, my answer would be $11. (Assuming it’s a fair coin). I have to have some capital – say $1,000 – to counter a really bad run, but in the end I’ll end up winning and winning big money.

If the chances are 10, then I have to be willing to lose $100 (max 10 losses), and to be paid for the risk I might demand $50 more, so I want $15.

If you’re going to tell me I have to be on it exactly once, then I’ll still want to get reasonably well paid, and if I’m going to lose $10 then I better gain $20 at least – I would actually probe to find out what’s the maximum you would pay and if it’s not attractive I won’t bother.

The concept of loss aversion is one thing, but you can’t look at it as a silo. Typically our choices are one-shots. How many careers can you have? So you choose your profession based on what you think will make sense, not on probabilities. In fact, given that most startups fail, no one should ever choose entrepreneurship at all – but we do, because we have the ability to influence what we create and therefore it makes complete rational sense to try and to not be discouraged by probabilities.

Probabilities only make sense if you take enough chances. When you don’t have enough chances, the probability of success and the odds you get are about as useful, in your decision making, as wood chips.

Here’s another one often quoted:

In an oft-cited experiment, the psychologists asked a group of subjects to imagine the outbreak of an unusual disease, expected to kill 600 people, and to choose between two public health programs to combat it.

Program A, the subjects were told, had a 100 percent chance of saving 200 lives. Program B had a one-third chance of saving 600 lives and a two-thirds probability of saving no lives.

Offered this choice, most of the subjects preferred certainty, selecting Program A.

But when the identical outcomes were framed in terms of lives lost, the subjects behaved differently. Informed that if Program A were adopted, 400 people would die, while Program B carried a one-third probability that no one would die and a two-thirds probability that 600 people would die, most subjects chose the less-certain alternative.

This is mentioned as “Oh If I framed the argument differently, people make different decisions”.

But it’s not that at all. That’s not why people made decisions differently.

It’s the fact that you had one chance. Just one.

If I were asked the same thing, I’d still chose Program A in the first instance and take the chance that I could potentially save the rest. Because I have just one chance, I bet on the fact that the probability would simply not work, and that I can change the outcome.

And I’d choose Program B in the second instance because, again, I would rather not see 400 people surely die, because there’s a chance I can save them and screw up the probability metric.

People don’t see probabilities as applying to them if they can somehow influence the outcome.

If I were given 100,000 such bets, then I would use the probabilities to make the decision. Otherwise the probability is useless, other than to add all sorts of eye candy and long articles and big books on behavioural finance.


But in reality, if you ask “How many chances do I get?” your decisions will completely change.

If you’re going with probabilities, then you better make sure you have enough bets to make, and that you can make them. Meaning, diversify, put your eggs into different baskets, and make sure that a loss in any one doesn’t put you out of business.

If you have the ability to influence the outcome of your investing – like Buffett does by getting on the board of the companies he buys – don’t bother with generic probabilities, though you must quote them when you want people to adore you.

I’ve been trying to build my life so I get more chances, more doors to open. Because luck plays a huge role in any success. But in many cases, you find sitting ducks – a fire-sale that you know you can turn around, a great business idea where others don’t seem to be looking, or some such. These shouldn’t be evaluated using probabilities alone – otherwise I will always find myself denying there is money to be made, using the simple argument that if there was, someone else would have made it.

  • Nice post, Deepak.
    Also, it needs to be pointed out that loss aversion is often thought of as a bias, some sort of flaw in human rationality, which it is NOT. It’s an offshoot of your point “How many chances I get?”
    There will always be more opportunities in the future, as long as I preserve my capital. So it makes perfect rational sense to avoid losses even if it means foregoing profits in the short-term. It takes money to make money. If my capital goes to zero, I cannot get back in the game again.
    Loss aversion would be a bias only if there was only once chance. Given that there are many chances for those who have capital, loss aversion is perfectly rational.

  • Rajesh says:

    Nice Article. Thanks Deepak.
    IMHO , in investing and trading, betting on probabilities is a recipe for disaster.
    Generally trading is considered as a probability game. A traders edge is compared to a Casinos edge and the basic assumption is that this probabilistic edge will create a surplus if repeated long enough.The more we trade more money we will make.This approach is not going to work in markets where probabilities themselves shift.
    Winning probability of a trading system is something we can arrive in hindsight for analytical purpose only.It cannot be used to trade the markets forward. Focus should be on the factors that can make a trade a high probability one rather than betting on the basic probability on back testing the system.
    I had written an article in my blog long back about this. Have a look

  • Mehul says:

    I see the point about having number of chances.
    However, loss aversion IS a bias and a pretty strong at that, in a way that it tricks the mind in thinking that this is the only chance one is getting at the time of investment decision making. Probability of loss in that ONLY (perceived) transaction becomes restrictive in taking the decision.
    I think it is a strong bias, and requires training of mind to overcome it. A post like this (highlighting importance of creating opportunities for allowing luck to play its part), to be kept in ‘reminder / alert’ slot of mind at the decision making time could be of help. Probably.

  • DJ says:

    “Otherwise the probability is useless, other than to add all sorts of eye candy and long articles and big books on behavioural finance.”
    I’ll take the bait as usual. Thank you. 🙂
    Kahneman in his book: Thinking, Fast and Slow covers the multiple bets case under a section called Samuelson’s Problem. He writes:
    “The great Paul Samuelson – a giant among the economists of the twentieth century – 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 won’t bet because I would feel the $100 loss more than the $200 gain. But I’ll take you on if you promise to let me make 100 bets.” Unless you are a decision theorist, you probably share the intuition of Samuleson’s friend, that playing a very favorable but risky gamble multiple times reduced the subjective risk. Samuelson found his friend’s answer interesting and went on to analyze it. He proved that under some very specific conditions, a utility maximizer who rejects a single gamble should also reject the offer of many.”
    He goes on to discuss that loss aversion could persist with multiple bets due to another behavior pattern he calls narrow framing and argues that it can be countered by broad framing, and that traders try to do that. He writes “A rational agent will of course engage in broad framing but humans are by nature narrow framers”.
    Anyway, my point here is not to say that I disagree with this or that, because I don’t. But, all I wanted to say is that loss aversion under the multiple bets scenario is mentioned by Kahneman in his book. And, any discussion/criticism of that scenario (which is most welcome), should include whatever he has written about it, especially when referencing him on the same topic.
    And, here is Samuelson’s paper on this:

    • I can tell you a real world reason why you should reject a bet that’s so obviously in your favour if the other party is expected to be smart. It’s because the game is rigged 🙂
      I haven’t read the book – and I’m sure Kahnemann has done a lot more work. All I’m saying is that the appeal of the probability should be limited to when the probability makes sense (i.e. you have enough bets to make). In a way it’s probably a justification of what Kahneman says about narrow framing, like you say above – that we prefer the narrowly framed argument. Which is why I say we must ask that question.
      Which one of the choices you decide must require you to ask if the chances you have are one, or many; if it’s one, the probability doesn’t matter quite as much, because in the real world, statistics are compiled in horribly bad fashion. (Especially in medicine)
      I think discussing the exact behaviour might be the question for another post.
      Key to Samuelson’s paper, by the way, is that you choose to divide the bet into smaller numbers since hte loss of $100 hurts you. My point in this is exactly the same – you don’t bet if you don’t have the gumption to take on (number of bets) x (loss per bet) – or even a fraction of that – say 34 continuous losses out of 100 possible bets in a 2:1 odds situation.

      • DJ says:

        I quoted only the initial part of the section that starts by saying that under some conditions, one can still be loss averse with multiple bets. However, the much more relevant part (that I mentioned briefly) is about how broad framing can help alleviate loss aversion, which is the major point of that section. A part of it can be found here:
        But, the point of my quoting wasn’t to take on any of the main points in this post. My point was far more insignificant and devious:
        It was about your tongue in cheek about “big books on behavioral finance” and your initial presentation of the argument, which implicitly suggests that Kahneman only looked at loss aversion in the context of a single bet. So, I wanted to point out that he has written about the multiple bets scenario in his “big book on behavioral finance”. Regardless of agreement or differences, it seems unfair to reference him on the single bet case, but not the multiple bet case.
        Additionally, your point seems to be that the single bet case shouldn’t exist at all, but I’m not sure about that. If he goes on to talk about multiple bets and broad framing, then it doesn’t seem like a big deal to me. Except perhaps for the fact that the single bet case gets most of the air-time in media. That doesn’t bother me as much. More often than not, ideas are dealt with in an incomplete fashion in media. Or things are dumbed down for easy and quick consumption.
        Oddly, I haven’t finished reading his book even though I’ve had it for a year now (so I suppose there might be something about it being a “big book”). Well, thanks to you, I’ve read one more chapter.

  • Dinesh says:

    To put in another way its not really the number of chance I get but even if its only one chance it depends how much of your net-worth you put in.
    People buy lottery tickets because its minuscule amount of net-worth where probability is negligible and potential reward is huge.
    If I get 1 chance to bet $10 then I would do so even if I get only $11 if I win, won’t look for $20, but if you ask me to bet $100,000 then I wouldn’t take for $200,000 if I win since I cannot afford to loose even if I get lot more if I win.
    Number of chances just depend upon how much percentage of net worth you put in one bet. If not in one coin toss you will get many more in your life time.
    But for insignificant amount compared to your net-worth if you still demand twice amount of bet ($20 if I win for $10 bet), than it shows you loss aversion.