Because I, seeing both things side by side, still prefer 29⁄36 over 7⁄36 by the easy reasoning “better win $2 then nothing”, and, if I’m only playing once, I claim it is a good strategy
You are not only playing once. If you follow a consistent decision making strategy, your choice should hold every time you face similar odds in the future. Or at least, for as long as you plan to play similar games.
Mathematically, $2 is the same as 50% chance to win $4, is the same as 25% chance to win $8. So every time I am offered $2 or 25% chance to win $9, I should choose the latter. (Most people however are extremely risk adversive—I think the ratio is something like 2:1, which leads to missed opportunities).
This holds for me, until someone offers me a million dollars or 50% chance to win whatever. I can retire on a million, everything above that is imaginary. Or until I am stranded in the mall with no wallet and desperately need $1.50 for a bus ticket. Which is why every investor needs 6 months worth of expenses—you don’t want to close positions at a bad time, just so you can pay rent.
With that in mind—and if $9 still seems low—at what price would you choose the second bet?
One should factor in the odds of similar games occurring multiple times throughout one’s life (unless one is a frequent visitor of casinos). I claim that *these* are too low for the situation to “add up to normality”.
Answering the question asked… I could start *considering* the second choice at 25% chance of 15 (probably properly 16 but my gut feeling has, of course, rounded it) and preferring it at… well, maybe never?
One should factor in the odds of similar games occurring multiple times throughout one’s life (unless one is a frequent visitor of casinos). I claim that these are too low for the situation to “add up to normality”.
No, one shouldn’t. Playing a game of chance once or a thousand times does not influence the outcome of the next round (aka the gambler’s fallacy). If a bet is a good idea one time, it’s a good idea a thousand times. And if a bet is a good idea a thousand times, it is a good idea the first time.
How could you consider betting a thousand times to be a good idea, if you think each individual bet is a bad idea?
Besides that, even if you don’t encounter exactly the same odds, you will encounter some odds. The numbers change, the way we make decisions remains the same.
The point of probabilities is not that the expected outcome occurs every time. The point is that it IS the expected outcome.
You might never be asked to participate in a betting game at the Mall. But at the end of your life, the sum of all your bets, big and small, will (probably) add up to normality.
Answering the question asked… I could start considering the second choice at 25% chance of 15 (probably properly 16 but my gut feeling has, of course, rounded it) and preferring it at… well, maybe never?
So, to summarize (and simplify a bit) - you would start considering letting go of (almost) certain $2 only after the expected utility is (around) $4.
Now, just because behaviour is common doesn’t mean it’s wrong. The loss aversion bias is called a bias, because it does lead to missed opportunities.
For example, a combination of loss aversion, risk aversion, status quo bias and lazyness leads to otherwise conscientious people keeping their savings in the bank. For them, any form of investment is deemed too risky, the topic too stress inducing to research or consider. Of course, inaction is also a decision and there is a thing called inflation—but you cannot put a price on your piece of mind./s
I realize the irony of writing about investments in 2020 - but when advising my best friend to buy gold (aka going long on fear) the response was “You know, I do not really believe in such things.” I’ve raised the topic once with each of my friends, the lack of interest is almost universal. Paper just seems so much… safer… if you do not think about it.
and preferring it at… well, maybe never?
Never?!!!
You do realize never includes, but is not limited to, $1000, $100,000, $1,000,000 or, indeed, $3^^^3?
Never!!!!
I hope you won’t take offence, but I don’t know how else to say it—you have expressed a preference for the (let’s say) certainty of winning $2 so extreme, that I find it hard to believe you would stick to it in practice.
No, one shouldn’t. Playing a game of chance once or a thousand times does not influence the outcome of the next round (aka the gambler’s fallacy). If a bet is a good idea one time, it’s a good idea a thousand times. And if a bet is a good idea a thousand times, it is a good idea the first time. How could you consider betting a thousand times to be a good idea, if you think each individual bet is a bad idea?
Gambler’s ruin. The bets are the same, but you are not. If bets are rare enough or large enough, they do not justify they simplifying assumption of ergodicity (‘adding up to normality’ in this case) and greedy expected-value maximization is not optimal.
Correct. Which is why risk management and diversification are crucial and why you should never bet more than you can afford to lose. I have this as an implicit rule, but I should have mentioned it. Thank you for pointing this out.
Edit: And I should probably read up more on the Gambler’s ruin. I can see how expected value maximization doesn’t hold up in the extreme cases, but it had to be pointed to me first.
The bets are the same, but you are not. If bets are rare enough or large enough, they do not justify they simplifying assumption of ergodicity
Can you expand on this a little? I don’t understand why the number of sequential bets you’re offered makes it easier to assume ergodicity, or indeed changes optimal bet sizing at all. (I strongly agree with the rest of your post fwiw)
You are not only playing once. If you follow a consistent decision making strategy, your choice should hold every time you face similar odds in the future. Or at least, for as long as you plan to play similar games.
Mathematically, $2 is the same as 50% chance to win $4, is the same as 25% chance to win $8. So every time I am offered $2 or 25% chance to win $9, I should choose the latter. (Most people however are extremely risk adversive—I think the ratio is something like 2:1, which leads to missed opportunities).
This holds for me, until someone offers me a million dollars or 50% chance to win whatever. I can retire on a million, everything above that is imaginary. Or until I am stranded in the mall with no wallet and desperately need $1.50 for a bus ticket. Which is why every investor needs 6 months worth of expenses—you don’t want to close positions at a bad time, just so you can pay rent.
With that in mind—and if $9 still seems low—at what price would you choose the second bet?
One should factor in the odds of similar games occurring multiple times throughout one’s life (unless one is a frequent visitor of casinos). I claim that *these* are too low for the situation to “add up to normality”.
Answering the question asked… I could start *considering* the second choice at 25% chance of 15 (probably properly 16 but my gut feeling has, of course, rounded it) and preferring it at… well, maybe never?
No, one shouldn’t. Playing a game of chance once or a thousand times does not influence the outcome of the next round (aka the gambler’s fallacy). If a bet is a good idea one time, it’s a good idea a thousand times. And if a bet is a good idea a thousand times, it is a good idea the first time. How could you consider betting a thousand times to be a good idea, if you think each individual bet is a bad idea?
Besides that, even if you don’t encounter exactly the same odds, you will encounter some odds. The numbers change, the way we make decisions remains the same.
The point of probabilities is not that the expected outcome occurs every time. The point is that it IS the expected outcome. You might never be asked to participate in a betting game at the Mall. But at the end of your life, the sum of all your bets, big and small, will (probably) add up to normality.
So, to summarize (and simplify a bit) - you would start considering letting go of (almost) certain $2 only after the expected utility is (around) $4.
I am sure you will not be surprised to learn you are not atypical in your preference: “Some studies have suggested that losses are twice as powerful, psychologically, as gains.[1]” (https://en.m.wikipedia.org/wiki/Loss_aversion). You might also be interested in the following articles: https://en.m.wikipedia.org/wiki/Risk_aversion and https://theintactone.com/2018/05/04/bf-u3-topic-5-loss-aversion-gamblers-fallacy/
Now, just because behaviour is common doesn’t mean it’s wrong. The loss aversion bias is called a bias, because it does lead to missed opportunities.
For example, a combination of loss aversion, risk aversion, status quo bias and lazyness leads to otherwise conscientious people keeping their savings in the bank. For them, any form of investment is deemed too risky, the topic too stress inducing to research or consider. Of course, inaction is also a decision and there is a thing called inflation—but you cannot put a price on your piece of mind./s
I realize the irony of writing about investments in 2020 - but when advising my best friend to buy gold (aka going long on fear) the response was “You know, I do not really believe in such things.” I’ve raised the topic once with each of my friends, the lack of interest is almost universal. Paper just seems so much… safer… if you do not think about it.
Never?!!!
You do realize never includes, but is not limited to, $1000, $100,000, $1,000,000 or, indeed, $3^^^3?
Never!!!!
I hope you won’t take offence, but I don’t know how else to say it—you have expressed a preference for the (let’s say) certainty of winning $2 so extreme, that I find it hard to believe you would stick to it in practice.
Edit: formatting.
Gambler’s ruin. The bets are the same, but you are not. If bets are rare enough or large enough, they do not justify they simplifying assumption of ergodicity (‘adding up to normality’ in this case) and greedy expected-value maximization is not optimal.
Correct. Which is why risk management and diversification are crucial and why you should never bet more than you can afford to lose. I have this as an implicit rule, but I should have mentioned it. Thank you for pointing this out.
Edit: And I should probably read up more on the Gambler’s ruin. I can see how expected value maximization doesn’t hold up in the extreme cases, but it had to be pointed to me first.
Can you expand on this a little? I don’t understand why the number of sequential bets you’re offered makes it easier to assume ergodicity, or indeed changes optimal bet sizing at all. (I strongly agree with the rest of your post fwiw)