Expected value – Some “solutions”

(Follow-up to Expected value.)

This problem is known as the St. Petersburg paradox, and it was studied in the 18th century by Daniel Bernoulli and other mathematicians. In this post we will do a simple expected value analysis, but any kind of realistic analysis must consider the phenomenon of risk aversion.

Expected value

Let’s call Hi the event where a head appears at the i-th toss, and Hi the event where a tail appears at the same toss number. If we call Pn the probability that n tosses are required to get a head, we have:

\displaystyle P_1 = P(H_1) = \frac{1}{2}

\displaystyle P_2 = P(T_1) P(H_2) = \frac{1}{2}\frac{1}{2} = \frac{1}{4}

\displaystyle P_3 = P(T_1) P(T_2) P(H_3) = \frac{1}{2}\frac{1}{2} \frac{1}{2} = \frac{1}{8}

and, generalizing,

\displaystyle P_n = P(T_1) P(T_2) \hdots P(T_{n-1}) P(H_n) = \left(\frac{1}{2}\right)^n = \frac{1}{2^n}

As the bettor receives $ 2n from the house if n tosses are required, the expected balance is

\displaystyle E[H] = \sum_{n=1}^\infty P_n 2^n - 100 = \sum_{n=1}^\infty \frac{2^n}{2^n} - 100 = \sum_{n=1}^\infty 1 - 100 = \infty.

So it would seem a perfect bet to make: what can be better than a bet with “infinite expected value”? 😀

Simulation

Let’s make a simulation to check if this result is reasonable:

import random

N = 10000

def get_tosses_until_head():
    tosses = 0
    while True:
        coin = random.randint(0, 1)
        tosses += 1
        if coin == 0:
            break
    return tosses

def get_balance():
    balance = -100
    tosses = get_tosses_until_head()
    balance += 2 ** tosses
    return balance

def main():
    random.seed(0)
    min_balance = max_balance = accum_balance = get_balance()
    for i in xrange(1, N):
        balance = get_balance()
        if balance < min_balance:
            min_balance = balance
        if balance > max_balance:
            max_balance = balance
        accum_balance += balance
    print 'Statistics after %d tries:' % N
    print '  Minimum balance: %d' % min_balance
    print '  Average balance: %d' % (float(accum_balance) / N)
    print '  Maximum balance: %d' % max_balance

if __name__ == '__main__':
    main()

Running the simulation we get:

Statistics after 10000 tries:
  Minimum balance: -98
  Average balance: -75
  Maximum balance: 32668

The average seems quite negative, how can we reconcile this with an “infinite expected value”? The answer is the same as in the case of the martingale: low probability events with huge returns skew the expected balance.

Let’s see how the normal expected value estimator works with a random variable X, with finite expected value \bar{x} and finite variance \sigma^2_X:

\displaystyle \widehat{x} = \frac{1}{N}\sum_{i=0}^N X_i

\displaystyle E[\widehat{x}] = E[\frac{1}{N}\sum_{i=0}^N X_i]

\displaystyle E[\widehat{x}] = \frac{1}{N}\sum_{i=0}^N E[X_i] (linearity of expectation)

\displaystyle E[\widehat{x}] = \frac{1}{N}\sum_{i=0}^N \bar{x} (X_i are identically distributed)

\displaystyle E[\widehat{x}] = \bar{x} (the estimator is unbiased)

\displaystyle \sigma^2(\widehat{x}) = E[\widehat{x}^2 - E[\widehat{x}]^2]

\displaystyle \sigma^2(\widehat{x}) = E[\widehat{x}^2 - \bar{x}^2] (\bar{x} is an unbiased estimator)

\displaystyle \sigma^2(\widehat{x}) = E[\widehat{x}^2] - \bar{x}^2 (linearity of expectation; \bar{x} is a number)

Replacing \widehat{x} by its definition and focusing in E[\widehat{x}^2]:

\displaystyle E[\widehat{x}^2] = E\left[\left(\frac{1}{N}\sum_{i=0}^N X_i\right)^2\right]

\displaystyle E[\widehat{x}^2] = \frac{1}{N^2}E\left[\left(\sum_{i=0}^N X_i\right)^2\right] (linearity of expectation)

\displaystyle E[\widehat{x}^2] = \frac{1}{N^2}E\left[\sum_{i=0}^N\sum_{j=0}^N X_i X_j\right] (square of a multinomial)

\displaystyle E[\widehat{x}^2] = \frac{1}{N^2}\sum_{i=0}^N\sum_{j=0}^N E[X_i X_j] (linearity of expectation)

\displaystyle N^2 E[\widehat{x}^2] = \sum_i E[X_i^2] + \sum_{\substack{i, j \\ j \ne i}} E[X_i] E[X_j] (X_i is independent of X_j if i \ne j)

\displaystyle N^2 E[\widehat{x}^2] = N E[X_i^2] + (N^2 - N) \bar{x}^2 (X_i are identically distributed)

Integrating this result in the \sigma^2(\widehat{x}) expression:

\displaystyle \sigma^2(\widehat{x}) = \frac{N}{N^2} E[X_i^2] + \frac{N^2 - N}{N^2} \bar{x}^2 - \bar{x}^2

\displaystyle \sigma^2(\widehat{x}) = \frac{N}{N^2} E[X_i^2] - \frac{N}{N^2} \bar{x}^2

\displaystyle \sigma^2(\widehat{x}) = \frac{1}{N} \sigma^2_X

This guarantees the convergence of the estimator to the expected value. But in our case there is no expected value (“it’s infinite”) and, consequently, we cannot even define the variance. Then the simulation is not very useful to get the expected value, as the estimator doesn’t need to converge to the expected value.

House without unbounded wealth

One (very realistic) situation where even the expected balance is negative is when the house has reasonably bounded wealth. For example, if we are betting against an agent whose possessions are merely equal to the entire world (!), the amounts we can be paid are bounded by ~$ 200·1012, giving a very different expected value:

\displaystyle E[H] = \sum_{n=1}^\infty P_n \min(2^n, 200\cdot10^{12}) - 100

\displaystyle E[H] = \sum_{n=1}^{47} \frac{2^n}{2^n} + \sum_{n=48}^\infty \frac{200\cdot10^{12}}{2^n} - 100

\displaystyle E[H] = 47 + \frac{200\cdot10^{12}}{2^{47}}\sum_{n=1}^\infty \frac{1}{2^n} - 100

\displaystyle E[H] = 47 + \frac{200\cdot10^{12}}{2^{47}} - 100 < 47 + 2 - 100 = -51

Expected value

(In memory of Toxie 😀 )

Let’s suppose the following bet were proposed:

  • The bettor pays $ 100 to the house.
  • An honest coin is tossed until a head appears.
  • The bettor receives $ 2n from the house, being n the number of coin tosses.

Is this bet worth making? Why?

An analysis will be given in the next post.

Blue-eyed islanders: a solution

Follow up to Product & Sum.

As in the “three wise men” puzzle, it’s useful to begin by analyzing simplified variants of the problem. Lets start analyzing the variant where only one islander has blue eyes and the other 999 have brown eyes.

When the foreigner gives his message, as the blue-eyed islander can see that the other islanders are all brown-eyed, he now knows his own eye color and must commit suicide the following day at noon. But, when the blue-eyed islander kills himself, all the other islanders know that they are brown-eyed and they must commit suicide at noon the next day (we can assume that each islander knows that his eye color is either blue or brown).

Now we can examine the variant where two of the 1000 islanders are blue-eyed. In this case the foreigner’s message does not give any information immediately, as any inhabitant of the island knows by observation that there is at least one blue-eyed islander. But the absence of a suicide at noon the following day shows to each of the two blue-eyed islanders that the other one is not the only blue-eyed inhabitant of the island. Then, in the second noon after the message, the blue-eyed islanders commit suicide, followed by the rest of the tribe the next day.

This reasoning can be extended to the full case, where the 100 blue-eyed islanders will commit suicide at the 100th noon since the foreigner’s message and the 900 brown-eyed ones will do the same at the following noon (the 101st).

Solving the product & sum riddle

Followup to Product & Sum.

Contradicting what was said in the previous post, the blue eyed islanders puzzle will be solved in a future post to avoid making this post excessively long. 😀

Product & Sum

One way to visualize the structure of this problem is to take each assertion by P and S as a filter to be applied over the initial set of integer pairs. This solution will then represent each of these filters as a block of Python code, with additional text explaining how each filter is connected with the associated assertion.

Given are X and Y, two integers, greater than 1, are not equal, and their sum is less than 100. S and P are two mathematicians; S is given the sum X+Y, and P is given the product X*Y of these numbers.

It is clear that we can restrict the first number to be strictly less than the second number, as the order of the numbers cannot be determined from the given data. Then we can get all the allowed pairs with the following Python code:

all_pairs = [(x, y)
             for y in range(2, 100) for x in range(2, y)
             if x + y < 100]

– P says “I cannot find these numbers.”

This implies that there are multiple allowed pairs whose products match the value that was given to P. We can start counting the number of pairs with each possible product:

num_pairs_by_prod = {}
for x, y in all_pairs:
    if x * y not in num_pairs_by_prod:
        num_pairs_by_prod[x * y] = 0
    num_pairs_by_prod[x * y] += 1

The pairs allowed by P’s assertion are those whose product is shared with other pairs:

pairs_1 = [(x, y) for x, y in all_pairs if num_pairs_by_prod[x * y] > 1]

– S says “I was sure that you could not find them.”

Then we know that the value of the sum is enough to know that the product cannot identify the pair of integers. The set of sums of integer pairs that can be identified by their products is:

identif_by_prod_pairs_sums = set(x + y for x, y in all_pairs
                                 if num_pairs_by_prod[x * y] == 1)

As the sum is enough to know that the integer pair cannot be identified by its product, the sum of the pair cannot be in the above set:

pairs_2 = [(x, y) for x, y in pairs_1
           if x + y not in identif_by_prod_pairs_sums]

– P says “Then, I found these numbers.”

This indicates that in pairs_2, the list of pairs restricted by the first two assertions, the correct pair can be identified by its product. Then we can do essentially the same we did in the first step but now keeping the pairs identifiable by their products:

num_pairs_by_prod_2 = {}
for x, y in pairs_2:
    if x * y not in num_pairs_by_prod_2:
        num_pairs_by_prod_2[x * y] = 0
    num_pairs_by_prod_2[x * y] += 1
pairs_3 = [(x, y) for x, y in pairs_2 if num_pairs_by_prod_2[x * y] == 1]

– S says “If you could find them, then I also found them.”

This implies that the pair can be identified by its sum from the list restricted by the first three assertions. We can get the final pairs list repeating the previous step, but now searching for a unique sum:

num_pairs_by_sum_3 = {}
for x, y in pairs_3:
    if x + y not in num_pairs_by_sum_3:
        num_pairs_by_sum_3[x + y] = 0
    num_pairs_by_sum_3[x + y] += 1
pairs_4 = [(x, y) for x, y in pairs_3 if num_pairs_by_sum_3[x + y] == 1]

Putting the code together, adding a print statement to get the final list of pairs and running the code we get

[(4, 13)]

matching the results in the literature.

Product & Sum

Follow up to Three wise men.

Solution to “three wise men”

A certain king wishes to determine which of his three wise men is the wisest. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. In fact all three spots are white. He then offers his favor to the one who will first tell him the color of his spot. After a while, the wisest announces that his spot is white. How does he know?

One way of visualizing the solution is to think hypothetical scenarios where some of the men have black spots in their foreheads. Lets start supposing that two of the men have black spots: then it will be obvious to the third that he has a white spot, as at least one of the three must have one.

Now suppose that only one of the men has a black spot and take the point of view of the “wisest” (quicker at least) man with a white spot. He cannot directly infer the color of his spot, as he sees one spot of each color. But, as we have seen in the last paragraph, the man in front of him with a white spot would have known the color of his spot if he (the observer) would have had a black spot in his forehead. As he didn’t speak, then he knows that the color of his spot must be white.

Finally, lets attack the full problem taking the place of the wisest man. He knows that, if he would have had a black spot in his forehead, one of the other men would have spoken, by the previously mentioned reasons. As they haven’t done that, he can conclude that the spot in his forehead must be white.

Blue eyed islanders puzzle

A similar, but more difficult, problem is the blue eyed islanders puzzle (this version was stolen from Terry Tao):

There is an island upon which a tribe resides. The tribe consists of 1000 people, 100 of which are blue-eyed and 900 of which are brown-eyed. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, one resident can see the eye colors of all other residents but has no way of discovering his own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical, highly devout, and they all know that each other is also highly logical and highly devout. One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe. One evening, he addresses the entire tribe to thank them for their hospitality. However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, mentioning in his address “how unusual it is to see another blue-eyed person like myself in this region of the world”. What effect, if anything, does this faux pas have on the tribe?

Product & sum

This is another classical problem, first proposed (apparently) by Hans Freudenthal en 1959, and also called the “Impossible” Puzzle:

Given are X and Y, two integers, greater than 1, are not equal, and their sum is less than 100. S and P are two mathematicians; S is given the sum X+Y, and P is given the product X*Y of these numbers.
P says “I cannot find these numbers.”
S says “I was sure that you could not find them.”
P says “Then, I found these numbers.”
S says “If you could find them, then I also found them.”
What are these numbers?

Hint: a computer is very useful to solve this problem.

Solution for both puzzles in the next post.

Three wise men

An interesting traditional puzzle (this formulation was stolen from John McCarthy):

A certain king wishes to determine which of his three wise men is the wisest. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. In fact all three spots are white. He then offers his favor to the one who will first tell him the color of his spot. After a while, the wisest announces that his spot is white. How does he know?

400 teracycles, 200 gigabytes, 7 collisions

(Follow-up to Sumando subconjuntos – La respuesta.)

Introduction

In a previous post (and in a comment from Guille [ɡiˈʒe] 😀 ) we have seen how the pigeonhole principle implies that a set of 70 numbers in the range [1018, 1019) must have two subsets with equal sum. But this is a non-constructive proof, as it doesn’t give us the two subsets with the same sum. To rectify this omission, in this post we will see how this “sum collision” can be obtained.

Finding collisions: simple cases

We can start by defining the problem in a more general way: given a sequence of elements xi and a function f(), find two elements of the sequence, xj and xk, such that f(xj) = f(xk). In this way the sum collision problem can be reduced to finding a duplicate in the associated sequence of sums, f(xi).

Two common ways to get duplicates in a sequence are the following:

def find_duplicate_sort(g):
    sl = sorted(g)
    prev = None
    for e in sl:
        if e == prev:
            return e
        prev = e
    return None
def find_duplicate_set(g):
    es = set()
    for e in g:
        if e in es:
            return e
        es.add(e)
    return None

The first one has O(n log n) complexity and the second one has O(1) complexity if we use a hash-based set. As the set-based approach also has a lower constant, we will use this approach in the rest of this post.

This algorithm works well if the sequences to be inspected for duplicates can be fit entirely in RAM, but in this case we have seen that tens of billions of elements must be inspected to have a high probability of finding a collision. In the next section we will analyse how this restriction can be evaded.

Finding collisions in big sets

Each of the subsets to be evaluated in this problem can be encoded using 70 bits and, to allow a simple and fast sum calculation algorithm to be used, this was rounded up to 10 bytes. Then, if we want to inspect 20 billion subsets of 35 elements to get a very high probability of not wasting the calculation time, we will need 200 GB to store the whole sequence. 200 GB of data cannot be stored in the RAM of an usual machine, but it’s quite easy to store this amount of data in a hard disk nowadays.

To allow a fast hash-set based RAM collision search while keeping the bulk of the data in disk, we can take the following approach:

  1. Generate in RAM a big number of random subsets and sort them by their sums.
  2. Find a vector of numbers (to be called “pivots”) aplitting the sorted subsets vectors by their sums in approximately equal-sized segments. (The segment n will be composed by all the subsets whose sum is between pivot n-1 and pivot n).
  3. Generate in RAM a big number of random subsets and sort them by their sums.
  4. Split the sorted subset vector in segments using the previously generated pivots and append each segment to an associated segment file (for example, append segment 3 to 0003.bin).
  5. Repeat steps 3 and 4 until getting enough subsets.
  6. Check each segment file at a time for collisions.

If we choose enough pivots, the size of each segment file will be small enough to allow easily doing step 6 with a hash-based set (each segment file won’t have the same size, as the generated subsets are random; but the law of large numbers ensures that their sizes won’t be very different).

Source code & parameters

The (ugly) C code that checked for collisions can be found in the repository associated with this blog. The chosen parameters were:

  • Number of subsets: 2·1010.
  • Number of subsets in RAM: 107.
  • Number of elements in each subset: 35 (constant).
  • Number of segments: 1000.
  • Number of slots in the hash-set: 227.

Results

The first stage (segment file generation) elapsed time was approximately 41 hours, somewhat over my original estimation of 36 hours, and the segment file range ranged from 194827630 bytes to 206242980 bytes. The second stage (collision detection inside each segment file) lasted for 12-18 hours.

The output of the second stage (discarding files where no collisions were found) was:

Processing file 218...  Collision between identical elements.
 DONE in 40.754850s.
Processing file 363...  Collision between different elements!!!
  ed940f4f5710c6351a00
  a35377a5a74a03961c00
 DONE in 38.585990s.
Processing file 394...  Collision between different elements!!!
  9ab5dfa6312e090e2a00
  6558c9c8eab667233400
 DONE in 35.570039s.
Processing file 409...  Collision between different elements!!!
  2429d70f6ff500ab2c00
  423cf8c758740ac73b00
 DONE in 34.499926s.
Processing file 434...  Collision between different elements!!!
  5a8bf5a532915f213100
  496ebc281e9958713e00
 DONE in 32.610608s.
Processing file 475...  Collision between different elements!!!
  c35d3f427a7c488c0e00
  4e37132b717a287a1900
 DONE in 21.971667s.
Processing file 655...  Collision between different elements!!!
  4653023676ea8e5f1900
  6abb914e641ba45a3900
 DONE in 21.514123s.
Processing file 792...  Collision between different elements!!!
  8a4a63d85de377130600
  853d3d45b15e0c471e00
 DONE in 21.506716s.

Each set is represented as a byte string with bit number increasing when advancing through the byte string. For example, ed940f4f5710c6351a00 represents the bit string 10110111001010011111000011110010111010100000100001100011101011000101100000000000 and, consequently, the subset with indices 0, 2, 3, 5, 6, 7, 10, 12, 15, 16, 17, 18, 19, 24, 25, 26, 27, 30, 32, 33, 34, 36, 38, 44, 49, 50, 54, 55, 56, 58, 60, 61, 65, 67, 68. Its elements are

5213588008354709077 9115813602317594993
1796745334760975384 3579709154995762145
2312952310307873817 3627590721354606439
5763055694478716846 2730952213106576953
4868653895375087301 9737387704190733086
9262565481673300485 5968266991171521006
6752113930489992229 3772194655144630358
9029836747496103755 3318990262990089104
9205546877037990475 9849598364470384044
1376783969573792128 1108556560089425769
7820574110347009988 6951628222196921724
4776591302180789869 7999940522596325715
2290598705560799669 7835010686462271800
8998470433081591390 9131522681668251869
9096632376298092495 5295758362772474604
5953431042043343946 3151838989804308537
8643312627450063997 3624820335477016277
3782849199070331143

and its sum is 203743882076389458417.

In the same way, the set a35377a5a74a03961c00 has elements

5213588008354709077 9011219417469017946
3579709154995762145 3627590721354606439
5941472576423725122 4317696052329054505
2730952213106576953 5014371167157923471
9737387704190733086 9262565481673300485
5968266991171521006 5917882775011418152
5866436908055159779 9233099989955257688
3772194655144630358 3318990262990089104
9990105869964955299 2664344232060524242
1376783969573792128 1108556560089425769
7820574110347009988 9889945707884382295
7652184907611215542 8082643935949870308
4271233363607031147 6415171202616583365
6393762352694839041 2290598705560799669
7481066850797885510 5295758362772474604
5953431042043343946 9929081270845451034
7207546018039041794 3624820335477016277
3782849199070331143

and 203743882076389458417 as its sum, the same value as the previous different subset. 😀

JS puzzle

The previous post asked what happened when the JS code

(function(f){f(f)})(function(g){g(g)})

was executed. In first place we can observe that the two anonymous functions are really the same, as they only differ in the name of a dummy parameter. Let’s call this function u() and observe what happens when we apply u() over a function f():

u(f) -> f(f)

It’s clear that u() applies its argument over itself. As in this case we are applying u over itself, the result will be

u(u) -> u(u)

giving us an infinite recursion.