Which payoff do you want to go for?

Consider a game where you have to choose between 1 of 2 payoffs:

Payoff 1. You toss 100 coins.
You get $1 for each Head that appears.

Payoff 2. You toss 10 coins.
You get $10 for each Head that appears.

Which payoff structure would you choose? And why?

#QuantitativeFinance

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I assume each coin toss is independent from one another (so the covariance between each coin toss is zero), and used only fair coins.

Yes, the expected Payoff is the same for ONE and TWO. But their variance differ.

$V_1 = \text { Probability } \times \text { Payoff ONE }^2 \times \text { Number of trials } = \frac {1}{2} \times 1^2 \times 100 = 50$

$V_2 = \text { Probability } \times \text { Payoff TWO }^2 \times\text { Number of trials } = \frac {1}{2} \times 10^2 \times 10 = 500$

Because Payoff TWO has a higher variance which is not desired, I will go with Payoff ONE.

$\text{ I learned from the best tutor }$

Pi Han Goh - 6 years, 3 months ago

What term do we use to describe a person who choses to "minimize variance when expected value is equal"?

Your answer greatly depends on what it is that we wish to maximize / minimize. There could be people who may want to maximize variance, because of the higher likelihood of a huge payoff that they care about.

Calvin Lin Staff - 6 years, 3 months ago

What term do we use to describe a person who choses to "minimize variance when expected value is equal"?

Risk-averse investor

Pi Han Goh - 6 years, 3 months ago

Let me try.

Payoff 1. Chance to get head for all coins = 1 out of 10^{200} = Chance to get $100 in terms of $1.

Payoff 2. Chance to get head for all coins = 1 out of 10^{10} = Chance to get $100 in terms of $10.

Thus, I think Payoff 2 is better.

Shein Phyo - 6 years, 3 months ago

This is optimistic way I guess

vijay katta - 6 years, 3 months ago

That's a valid argument if your goal is to "maximize the probability of the highest payoff".

Since getting $99, $98, ... $91 is somewhat similar to getting $100, would this affect your answer? Why, or why not?

Note that the payoff 1 should be "10 ^ {100}"

Calvin Lin Staff - 6 years, 3 months ago

The probability of getting payoff between 91$ to 100$ increases in the first case. Hence, if I consider myself satiated by $99, $98, ... $91, I'd prefer case 1.

Amit Roy - 6 years, 3 months ago

It's less of a hassle to toss 10 coins.

If we compare this to stocks, it's easier to keep a portfolio of10 stocks than 100 stocks. And lower commissions. The risk of loss is higher, but the chance of higher returns is better too.

Steven Perkins - 6 years, 3 months ago

That's a good point! We might need to factor in convenience / the cost of making a flip!

Can you clarify what you mean by "chance of higher returns is better too"? How do you quantify that?

Calvin Lin Staff - 6 years, 3 months ago

Earlier posters have quantified the variance for the smaller vs. the larger set of coins. For the set of 10 coins, the chance of getting more than 9 heads is 1 in 1024. For the set of 100 coins, the chance of getting more than 90 coins is much lower than that. I'd have to get my combinatorics (or statistics) hat on to quantify it, but it's much less. On the order of more than 5-sigma from the mean.

For stocks, a stock club (or newsletter writer) would be silly to buy 1 share each of the 500 largest companies. A number of funds exists to do that with much less expense. If they want to try to beat the performance of the S&P 500, they pick a small number of stocks (using some rational criteria) and often do better than the "market".

Of course, the risk of loss (or underperformance) is higher too.

Steven Perkins - 6 years, 3 months ago

Expected outcome is same in both cases ,but in case 1 chances are that outcome is more closer to 50$ than in case2, So for more standard outcome I will choose case1.

For suppose if my need demands 60$ I would choose case 2.

CASE1

Chances to get 50$ is 100C50 *.5^{100}=.0786

49$ is 100C49*.5^{100 }=.0780 (same for 51$)

For 60$ it is .0108

By similar calculations(and adding), chances to get outcome from 40$ to 60$ is .946.

where as in case 2 chances reduce to .656 this proves that there are more chances to get outcome closer to 50$ in case 1.

probabilities to get 60$ are 0.108 for case 1

.205 for case 2.

So, if 60$ is my need I will choose case 2

vijay katta - 6 years, 3 months ago

You bring up a valid point, which is "What is the utility function that I care about". If the utility function is just "I only care if I have more than $60 or not", then your approach shows that we should go for payoff 2.

What kind of utility function makes sense? Do people value all amounts of money equally?

Calvin Lin Staff - 6 years, 3 months ago

They average out to the same payoff (which is $50.00), so I would flip a coin to decide which to use.

Python:

import random
coin = (True, False)
def sim_1():
    payoff = 0
    for flip in xrange(100):
        if random.choice(coin):
            payoff += 1
    return payoff
def sim_2():
    payoff = 0
    for flip in xrange(10):
        if random.choice(coin):
            payoff += 10
    return payoff
trials = 100000
payoff_1 = 0.0
payoff_2 = 0.0
for trial in xrange(trials):
    payoff_1 += sim_1()
    payoff_2 += sim_2()
print "Payoff 1:", payoff_1 / trials
print "Payoff 2:", payoff_2 / trials

Brock Brown - 6 years, 3 months ago

what good

juan mozo - 6 years, 3 months ago

If i understand the problem correctly, would be indifferent to the payoffs structure, cause both options give the same expected value which would be 50.

Jesus Ulises Avelar - 6 years, 3 months ago

playoff 1

Ganesh Deepak - 6 years, 3 months ago

I think payoff 1 is better because chances increses with more no of toss..

Febin Jose - 6 years, 3 months ago

In my opinion, it's a matter of risk: payoff 1 has more consistency but less probability of scoring extremely large amounts of money, whereas payoff 2 is a sort of high risk high reward model where you either get a lot or get a bit. Personally, I would choose payoff 1.

Andrew Yang - 6 years, 3 months ago

Pay off 2, because it will likely to get heads more often and earn $10 for each head than 100 coin tosses. Who knows that in a toss of 100 you will only get less than 50% heads.

Akasha Diyunuge - 6 years, 3 months ago

Payoff 2 because I'll make decent money and i Don't have time to flip 100 coins. Stop over complicating things with math people.

Chase Parker - 6 years, 3 months ago

I will chose payoff 2 because while tossing coins 10 time we may get HEAD more than 1 or 2 times and the pay is more. But in payoff 1 tossing coins 100 times may give a HEAD 20 times and u get 20$. But in payoff 2 you can get more than 20$.

Praveen Kunka - 6 years, 2 months ago

Payoff 2 as it takes less time to toss 10 coins and the value is same for both

Augustine J - 6 years, 1 month ago

What are the statistics given that heads side of a coin is slightly heavier than tails? Giving tails more of a chance of facing up? Is that true? I heard that somewhere back maybe in 4th grade haha. Or has that been proven incorrect? if it is true, I would rather chose b since I would want less of a chance of getting tails side up? Can you explain to me how I am wrong (I probably am) like you would explain it to a 3rd grader?

This is why I'm more of an artist than math person.

Moriya Ferris - 5 years, 11 months ago

That is a very good point. How would your answer change if we didn't have a fair coin? For example, if the coin was totally biased and only gave tails, then both payoffs are the same. Similarly, if the coin was totally biased and only gave heads, then both payoffs are the same. But in between, the payoffs are different. So, does that mean that the answer could change depending on the probability of head/tail? If so, how and why?

How is payoff 2 "less chance of getting tails side up"? Just because of the absolute number? But, the first mostly likely has many more heads appearing, just a smaller payoff each time.

Calvin Lin Staff - 5 years, 11 months ago

Less probability of a heads since the gravitational pull of the heavier side. Sorry I missed stated. On the other hand, my fiance makes a good point. The more you flip a coin your results are more likely to reflect the chance of a 50/50 so it is better and more reliable to flip more times than less so option 1 of 100 flips is more reliable then just 10 in case you get a lucky streak of a ten or an unlucky streak with only 2 out of 10.

Moriya Ferris - 5 years, 11 months ago

@Moriya Ferris – As it turns out, the Expected Value of both payoffs would be the same, regardless of the probability of landing heads / tails. For example, if the coin is fair, and it will land heads 50% of the time, then the expected payoff in 1 is $50, and the expected payoff in 2 is $50. More generally, if it land heads $p\%$ of the time, then the expected payoff will be $\$p$ .

So, the question then becomes, pick your favorite probability $p$ of landing heads, which payoff would you want? Is there anything else to consider?

Calvin Lin Staff - 5 years, 11 months ago

Payoff 1 so that I can get more number of chances

Bhawana Bamal - 6 years, 3 months ago

I'd go for #2. I'm a risktaker.

David Tipoe - 6 years, 2 months ago

You will still need to define what you mean by "risktaker". There are possible definitions of "risktaker" which would make option 1 more valuable to them.

Calvin Lin Staff - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$