I personally chose b and d, but when i asked my younger sister she actually said a and d. She said that she chose a because the chance of a potential win was higher, where as she chose d because the chance of a loss was negligible.

Thanks for Allais-ing my boredom with these amazing videos!

Why do you assume we can apply the same expected utilities to both examples? Surely people would be a lot more upset ending up with $0 in D than getting the more expected $0 in B.

choose B and D just by counting expectation. Bumped into that sort of problems before (interviewed for data analytic, solved the problem but didn't get the job).

I think the problem is flip-flopping between expected gain and probability of wininng a life changing amount, where life changing is a personal, non-linear utility function for money. In particular, with A we have a low probability (11%) of a life changing amount ($1m) AND an expected gain ($110k) that is not a life changing amount for many people.

I'm here to write an essay on Taking Risks, from game theory perspective. I chose such a topic, mainly because I myself sitr clear of any risk, so I surely chose C. I try to minimise risk, so turning some risk (0.01) into no risk (0.00) seems to be the best choice. However, when I'm presented with large risks, I'm more likely to try and get as much as I can. I inheretly don't belive that I could win neither lottery A nor B, therefore I decided that I may as well swing for the fences, and try catching 5M. Moreover, I would feel worse if I lost when I could have got 1M no questions asked, but would not belive that i could've done any better if lost within A and B lotteries, cause 1% is such a small diffrence that it wouldn't matter to me.

To be honest I approached this similarly to what was expressed in #29 Knife-Edge Equilibria. No matter what is the absolute probability of earning 1M or 5M, both their probabilities are so similar (1% difference) that I really doubt that a random number would jump right in between both probabilities. This means it will rarely be a situation where I could've earned 1M if I hadn't picked 5M. So I picked 5M, option B. Same story with lottery 2, so option D, did not change opinion after explanation

Behavioural Economics has the answer. You can see Starmer, C., (2000), Developments in Non-Expected Utility Theory: The hunt for a Descriptive Theory of Choice under Risk, Journal of Economic Literature, Vol. XXXVIII, pp 332-382.

excellent lecture

while I chose the "right" answer I may vary well have chosen the "wrong" one if the values(both the odds and payouts) since my choice was based on the payouts multiplied by the odds of receiving said payout

@Markd315

7 жыл бұрын

Nope, if you use average payouts with the dollar amounts then you'll always have an independent result!

B and D seem like the lotteries I would choose

I knew this paradox, and I know my answers are inconsistent. But somehow I do prefer B and C. Maybe with "regret aversion theory" this makes sense?

@jvgama

8 жыл бұрын

As for the estimation of utilities, I don't need to do any calculations. If I violate the independence axiom, expected utility cannot justify my answers. So, whatever were the values I would assign (maybe log(w) would be a good idea..), my answers would change to become consistent.

@jvgama

8 жыл бұрын

I don't know what is behind my inconsistency, but I will try to use the framework of expected utility to explain it. However, the utility will not be f(w) regardless of the lottery (we know that would not violate independence over lotteries). Getting $0 is neutral, but getting $0 when you had the opportunity of getting $5M is awful. And if it was by choice, it is even worst. And the disutility of that outcome depends on the perception of how responsible you were. Therefore, the disutility of 0$ is not the same in the first and second set of lotteries. The second one gives a much higher disutility because you could have avoided the risk and you didn't. The second gives a smaller amount because if you loose, most likely (10/11) it was not because you choose to incur in that risk. If I am not mistaken, this is a good explanation. Once they did an experiment with lotteries A and B where you would know the value from 1 to 100, and be able to identify if you were responsible to loose the lottery (1% of the time). In this scenario, people would conform to the independence axiom much more. I think I would to. Knowing your greed made you loose a lot of money creates extra disutility on getting $0.

I go cowardly with A and C

A & D. Because: Between A and B, the chance of not winning anything (i.e. $0) was 0.89 and 0.90. A promises _any_ profit more than B. Between C and D, the chance of not winning anything (i.e. $0) was 0.00 and 0.01. D's risk to not win is negligible (0.01) and I pick that. If between A and B there had been a more negligible probability of not winning anything (similar to D's 0.01), I'd have probably picked that one. The idea is: Avoid risk, but _some_ risk is okay if the outcome is distinctly different (i.e. $5'000'000 as opposed to $1'000'000). I stand by my decision, even after your solving of the problem. The total output appears (without having discreetly calculated it) to be more profitable in D.

@user-ll7iz1hr8t

5 ай бұрын

It doesn’t make any sense. The difference in probability of winning nothing in both lotteries is the same - 0.01 (0.89vs0.9 and 0vs0.01). Why would this difference be negligible in lottery 2 but not in lottery 1?

I think that the problem lays within lotteries. Based on personal risk taking wilingness, each and every individual can choose to award seemingly equal outcomes of mathematical sense, diffrent utility values just based on certainty. One can prefer sure A over risky B, while he would have no problem to go for B if it was as certain as A. One way to invision it, is to think of someone's prefrenes graph to be a Hyperballic, while other's to be linear. Plausibly propatuated by the idea of overcoming greed. Important note is that humans don't think mathematically, and they shouldn't be treated as computers.

Great interactive video!

@Gametheory101

4 жыл бұрын

#h2p

@williamkibler592

4 жыл бұрын

@@Gametheory101 OMG H2P! I had no idea. I love your content, thank you

I chose A and C because they are essentially the same with regard to B and D.

Why can't you use expected value to select your preference of lottery? The expected value of lottery A is $110,000, while the expected value of B is $500,000. So wouldn't you rationally choose B over A? Likewise, the expected value of C is $1M versus $1,390,000 for D, so you should select D.

@Gametheory101

8 жыл бұрын

+starrychloe You can. There's nothing wrong with that, and it will *always* produce independent preferences---for this lottery or any other setup.

By expected value, I would think B and D have the best outcomes. However the marginal utility of the $5m relative to $1m isn’t worth the risk, so I would go with lottery A and C, which have the highest chance to payout something.

Great videos by the way, thanks William, B, C - why C - well we are talking real world and over time, nothing is independent and in my experience having a simple transaction - you always get paid. I'd be happy with a million bucks - that would see me through to death. The more complex the deal or agreement the more the person paying you can grift. that's real world. Didn't Kahnaman and Tversky get a Noble prize for that - sheeesh, I missed out again :( But it does illustrate the massive limitations of this 'lab' thinking and transferring it to the real world. Induction and hyperbole ( metaphor and hope) are used all the time when applying a simple maths puzzle to life.

In life .01% could be the difference between life and death! And a lot of us only need so much to survive before it’s extra cushioning or excess! I love the way as a bonus I get to think of how my own biases and reasoning fits into game theory! It makes the difference between the technique of it acknowledging reasons very blatant in life when people refuse to do such a thing, it’s the difference between discussion or something being outside of I guess

I chose Lottery B and C, because I wanted to avoid the 0.01(0) outcome in D. My personal utility of winning 1 million would for example be 90 and the personal utility for winning 5 millions would be 100, (due to decreasing marginal utility). It is important for you to understand that my personal utility for winning 0 would be -200, because of the disappointment that will last for a lifetime that I didn't go for the sure thing. Therefore, C would give me a utility of 90 for sure, while D would provide me an average utility of 88.1 (0.01x(-)200+0.1x100+0.89x90). Hence I choose C(90) over D(88.1).

@IsamBitar

2 жыл бұрын

Exactly what I was thinking. This is really well written. Thank you.

@user-ll7iz1hr8t

5 ай бұрын

If you still remember writing this comment, I’m interested - did you originally assign some other utility to winning 0 in 2nd lottery and changed it to -200 later so that choosing C would make sense? BTW, if your respective utilities for winning 5M, winning 1M and winning 0 are 100, 90 and -200 then optimal decisions would be choosing A and C (-168.1>-170 and 90>88.1). Choosing B and C would only make sense if 1% probability of winning 0$ in second lottery somehow had lower EV than 1% probability of winning 0$ in first lottery, and it is self-contradictory regardless of utilities you assign to results

Choice: A & D With the choice of choosing either A & B, I just thought about how upset I'd be by not taking that 1% added extra chance. I assigned values to these outcomes and I rate them as such, $1 M = 9 | $5 M = 10 | No money = -2 My happiness doesn't align with the amount of money because I'm expecting to lose the lottery. Winning the lottery is rare and I'd be happy winning with either of the two prizes. This is reflected back on the values I used to represent those prizes as there's only a slight difference between them. For C & D, the situation has changed. The situation shows that I'm almost guaranteed to win money and it only shows a 1% chance of losing everything; due to this, I'm more inclined to optimize the money I make. My values for the outcomes changes to reflect this, $1 M = 1 | $5 M = 5 | No money = -2 Now, my happiness does align with the money and it likely influenced my decision to pick D. I guess, if you were to look into what the 1% did in both pairs of choices, it would sort of make my decision make sense? For A, you're using that 1% to get *any* money. It's either you win life-changing-money or you go back to the status quo; why wouldn't anyone try to maximize their chances of winning that sort of money? For D, the 1% is used as a sort of threat as you'd stand to gain nothing but aside from that outcome, you're promised to win that life-changing-money or possibly even more. 1% is so negligible that it just makes you think about optimizing the money you earn from either of the choices. You can gain up to $5 M and just default back to $1 M if you don't; all of this with the slight caveat of losing the money on a 1% chance. A 1% chance is insignificant to me but I understand that each added percentage counts in games of chance.

@user-ll7iz1hr8t

5 ай бұрын

I don’t think you understand what utilities are. They represent how desirable each of the outcomes is for you. Outcomes, not games. Let’s say there are 2 alternate realities - in one you had to decide between lottery A and lottery B and in another you had to decide between lottery C and D. Let’s say you already decided and won 1M$ dollars in both these worlds. If you already won, would conditions of the game you played have any impact on how happy you would be that you won that amount? If not, your assigned utilities for winning 1M should be the same in both situations. And difference in chance of winning any money is 1% in both situations, so you can’t really neglect this 1% in one of the games and not neglect it in another. Even if your intuition says that difference between 0 and 0.01 is less important than difference between 0.89 and 0.9

I would go for A and C, A because the 11% of getting money is 1% more than get money option in lottery B. And C of course because there's always a little probability you wouldn't get no money in D versus complete certainty of getting money in C. I guess I'm risk averse!

B > A, D > C, C vs D > A vs B

B and C Strange to call this "wrong". You always go for the option where you have a sure thing, hence C over D, whereas a 0.01 difference in chance to either win 5 million or 1 million is worth the risk, hence B over A.

@adityasharma2

3 жыл бұрын

Exactly my approach too! 😃

@user-ll7iz1hr8t

5 ай бұрын

But difference between chance of winning and not winning is 0.01 in both cases. The difference between 100% and 99% is the same as difference between 10% and 11%. If we follow “always go for an option with a sure thing” strategy” then we would also have to choose 100% chance of getting 1$ over 99.999999% chance of getting a billion and 0.000001% chance of getting nothing. So yeah, B and C is definitely wrong

A and D. I was rushing to do the math in my head and I slipped a decimal point.

AD A > B: Even though expected value of B is higher I chose A to have a lower chance of walking away with nothing which I deemed to be the most likely outcome for both cases. D > C: Obviously if I chose D and actually rolled a 1/100 to get nothing I would feel really silly but that's a risk I am willing to take when 99% of the time I am expected to do no worse than C.

Ok, I think I can explain this. Our perception of probability is not linear if we are presented this in abstract form. For abstract probabilities lower than 0.5, we anticipate a probability X as x^4* 8. So if you say we should expect something 0.1 of the time, we in fact anticipate it as probability 0.0008. This is why people are so angry when their video game avatar misses 10% of the time with a 90% accuracy.

I'd choose B because if i win, I'd get more money. I'd choose D because there's a big chance of winning 1M, and a little chance of winning even more. I'd take the tiny risk of not winning anything.

1) Lottery A. I am indifferent to winning 5M over 1M as both are life changing amounts of money. Since Lottery A has a higher probability of winning, it is the better choice 2) Ohh you are evil :D It is (almost) certain that I get the life changing amount of money, regardless of what strategy I go with, meaning I can change my evaluation. I am no longer indifferent to the fact that D has a higher expected value than C, so I choose D.

BC B because at a glance, the extra risk is expected to pay off, but I'm still expecting to walk away with 0 as that's the most likely outcome. C because I'm more than happy with 1 million dollars. Here I don't feel the need to take the extra risk, because I'm now already expecting to walk away with at least 1 million. If I end up being unlucky, I would feel like I lost out on 1 million dollars. Walking away with 0 because I lost out on 1 million dollars does not feel the same as walking away with 0 because I didn't hit the 10% chance of winning 5 million dollars or 11% chance for 1 million. Translated to game theory: My utility score for the 0 outcome in A-B is not the same as for the 0 outcome in C-D.

@alexanderreusens7633

8 ай бұрын

Other explanation, I'm risk averse. But if I'm forced to take risks, I calculate them out. In AB I'm forced to take risks, so I might as well see what has the best expected outcome. In CD I have a risk free option, so I'll take those without calculating. Even after seeing that to be consistent in my risk profile, I should have picked A, or I should have sticked with my calculated risk taking and switch to D, I'm still happy with my choice. Especially because I can (pseudo-)rationally justify it with "My utility score for the 0 outcome in A-B is not the same as for the 0 outcome in C-D" after I've already made my decision haha

B (500K value vs. 110K expected value) D (500K+890K vs. $1M)

What's better: 50% of prize now, or 100% of prize monthly over 20 years? A. Are you guaranteed to live for 20(+) years? B. Can you find an investment avenue with 2^(1/20) ≈ 3.5¼ APR? C. Relates to both A and B: Will you draw from the 50% or will you draw money = invest less, monthly?

Choices made: B expectation value is higher, then D for same reason.

Valuations: 5 000 000 -> 500 1 000 000 -> 400 0 000 000 -> 350 (and thanks God, that I am still alive) by the way, I do not protect my privacy too much, so it is easy to check consistency of this declaration with life history.

is'nt it the opposite?? If the money was actually at stake, i think independence would be violated considerably more often rather than less often. a lot more people including me would pick C over D while also picking B over A. The risk to reward factor is a lot less in B and A, where we are expected to not get any money regardless. however, the the risk to reward factor is significant in C and D, where we go from guaranteed money to possibly no money. Most people will be set for years and years with 1 million guaranteed dollars . however, even a 1percent chance that they could get 0 dollars instead of a million would be a major major disappointment. i guess it also depends on how old is the person, his/her financial situation etc. so basically a lot of real world factors come into play

if it is a one-time thing, then B, C, if it is a repeated event then B, D is the more rational one.

B>A,D>C

B and D

B and C

I'd choose lottery B and D

This whole discussion of independence, treating the B+C choice as irrational or dumb like people just didn't think about it hard enough, ignores the pricing of risk inherent in the two different equations (A and C). They really are different. The increase in risk from 89% to 90% is marginal. The increase in risk from 0% to 1% is literally infinite. People have different risk tolerances. People who choose the higher risk-higher reward lottery will *ON AVERAGE* be rewarded for that choice. But as an individual, it is not irrational to take a discount in exchange for guaranteed money. We also have to contend with the law of large numbers here - individuals making decisions can maximize their probability all they want, and in most cases, that makes sense and it's all you can do. That's the nature of the choice between A and B. But the choice between C and D is fundamentally different. Because no one can predict with certainty the outcome of any one coin toss, it's a matter of preference how much any one person is willing to pay to eliminate risk from the outcome. Surgeons talk about this in terms of bad outcomes - patients hear a risk of 1% for losing their leg after knee surgery and think it's a minor risk they can ignore; but for that 1 person out of 100 who gets that result, it is 100% of their outcome. It's reasonable to think that through when making a decision - are you really prepared for that worst outcome to be 100% of your outcome? If not, then how much are you willing to pay to eliminate the risk? For someone who doesn't expect a huge difference in happiness between 1 and 5 million dollars - maybe they're already pretty happy and don't need a ton of money to live well, yet a million dollars would nevertheless be life-changing as compared to $0 - the difference in risk becomes more relevant than the difference in money. People pay to avoid risk all the time in real life, and that is a legitimate strategy. It's why the bond market exists. There's another element here - in earlier games, you talked about dominant strategies needing to guarantee and equal or better result than an alternative play. But D only weakly dominates C in that respect. You are most likely to do equal or better, but not always. D is still a better play, *on average* than C, but it's a significant difference from the choice between A and B, which doesn't have that problem. I think this paradox is really fascinating but the fact that some people choice B and C is not an indication that they're irrational or can't figure out the equation. One can calculate the "correct" mathematical answer and still conclude the other choice is better based on one's actual complete preferences. The paradox is more interesting to me as an indicator of what the model captures and what it doesn't.

@rhedhaering

3 жыл бұрын

Lol, I just saw that you cover most of this in video #53. Hahahahaha

TL;DR - B and C On the first lottery I chose B. I did so because that 1% does not make all that big of a difference and I value more the additional payoff of 4 million. More importantly, should I choose lottery B and loose there would be no way of knowing whether it was due to that 1% or not. It'd be easy to convince myself that I would have lost anyway. On the second lottery I chose C: 100% chance for 1M. Is it stupid and illogical? You bet. Same 1%, same 4 million. But should I loose after picking lottery D I will KNOW that it was because I was greedy. On top of that, every million that comes after the fist brings me less additional value than the previous one. Could I live the next 60 years with that million, without lifting a finger? The math says 'yes'. As far as I know myself, were I to be given this chance in REAL life, I would have picked C. Now, that said, the less obstacles there are in choosing (say you just press a button and the payoff is instant), the less time spent in anticipation, the more likely I am to risk it. I hope that made sense. I used a whole bunch of tenses and words, so I could have screwed up somewhere. EDIT: With the lotteries highlighted in red, I feel as though there is a hidden multiplier that changes the value of the outcome. Maybe call it 'transparency of impact of decision' or 'subjective estimation of impact'? The more visible the impact of your decision is, the more risk averse you become. (if you *know* that you *will* know that you fucked up you *won't want* to fuck up) I'll have to think on this some more.

Lotteries B and D would have been my choices

B and C would be the choice for me

if you change 'million' to 'thousand', their choices would much more logical

Lottery B and D

Average payouts below, but b and c still make the most sense, take 100pct anytime you can get it. Independence doesnt apply if playing one time Unless you play the game many times, then b and d $110,000 $500,000 $1,000,000 $1,390,000

I literally have no kdea

I picked B and D.

lottery b, lottery d

Choice 1: B Choice 2: D

Lottery A . Lottery C.

b and d

1,39 M > 1M so D, but bihevioral GT maybe have a different answer:-)

A and C

i would choose b, c

Ppl. Chose the bigger math problem not thinking saw no risk and greed pursued

Choice 1: lottery A Choice 2: lottery C

@nickchow3788

4 жыл бұрын

I prefer dying a painful death

lottery B, D

B then D

Q1) B Q2) D

B &D

Lottery B Lottery D

B & C

I'd say B and D and yolooooo!!

BD

B, D

Lottery a

B and C B: has same risk of loss as A but has higher outcome chance than A, if I risk losing at the same rate, might as well maximize winnings. C: guaranteed winnings; D's chance of losing is infinitely bigger than C's chance of losing.

B, C

A, C

Lottery A, Lottery D e: Pretty much, $1m vs $5m is far less of a hindrance in decision making than NO money vs $1m. However, an 11% chance vs 10% chance of winning a large some of money is proportionally much different than the 99% chance of money vs 100% chance of money. Eg, if you gave me a 1% chance vs 2% chance for less money, then i could DOUBLE my chances rather than increasing from 99% to 100% which is negligible

B,C.

@AA1969

7 жыл бұрын

After assigning utilities to the outcomes my preferenced changed to A, C. But a close race between A and B. The utilities I assigned to outcomes (5, 1, 0) was (40, 20, -2). The expected utilities assigned to lotteries (A,B,C,D) is then (0.42; 2.2; 20; 3.98). I don't know if it violates the model to assign negative utility but honestly if I had the opportunity to play this lottery for free and then lost, I would have the - highly irrational? - feeling that I was worse of than had I never had the chance... Cheers,

B - D

A,D

bd

0:25 no offence but you don't even pronounce it right ahah (i'm french)

B, d

B C

A C

b,c

bc

A D

B D

b; c

Bd.

BC

B and C

B and D

b and d

A and C

B & D

B & C

BD

A, C

B,C

B, D

bd