The hot hand

The wager described in the previous post was motivated by an interesting recent paper (Miller and Sanjurjo, reference below) discussing the hot hand, or the perception in some sports, particularly basketball, that a player with a streak of successful shots (“He’s heating up!”) has an increased probability of making a subsequent shot and continuing the streak (“He’s on fire!”).

Whether being on a streak yourself, or trying to defend a player on a streak, on the court this certainly feels like a real phenomenon. But is the hot hand a real effect, or just another example of our human tendency to see patterns in randomness?

A famous 1985 paper (Gilovich, Vallone, and Tversky, reference below) argued the latter, analyzing the proportion of successful shots immediately following a streak of three made shots in various settings (NBA field goals, free throws, and a controlled experiment with college players). Not finding any significant increase in proportion of “streak-extending” shots made, the apparent conclusion would be that a past streak has no effect on current success.

But that’s where this puzzle comes in: even if basketball shots are truly iid with success probability p, we should expect a negative bias in the proportion of shots made following a streak, at least compared to the intuitively expected proportion p. Miller and Sanjurjo argue that the absence of this bias in the 1985 data suggests that the hot hand is not just “a cognitive illusion.”

Both papers are interesting reads. In presenting the problem here as a gambling wager, I simplified things somewhat down to a “win, lose, or push” outcome (i.e., were there more streak-extending successes than failures, or fewer), since the resulting exact expected return can be computed more efficiently than the expected proportion of successes following a streak:

Given n remaining trials (basketball shots, coin flips, whatever) with success probability p, noting outcomes following streaks of length s, and winning (losing) the overall wager if the number of streak-extending successes is greater (less) than the number of streak-ending failures, the expected return is v(n,0,0), computed recursively via

v(0,r,w) = sgn(w)

v(n,s,w) = p v(n-1,r,w+1)+(1-p)v(n-1,0,w-1)

v(n,r,w) = p v(n-1,r+1,w)+(1-p)v(n-1,0,w)

where, using the setup in the previous post where we flip n=100 fair coins with probability p=1/2 of heads, looking for heads extending streaks of length s=4, the expected return v(100,0,0) is about -0.150825.

References:

  1. Gilovich, T., Vallone, R., and Tversky, A., The Hot Hand in Basketball: On the Misperception of Random Sequences, Cognitive Psychology, 17(3) July 1985, p. 295-314 [PDF]
  2. Miller, Joshua B. and Sanjurjo, Adam, Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers, Econometrica, 86(6) November 2018, p. 2019-2047 [arXiv]
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Gambler’s fallacy fallacy?

The gambler’s fallacy is the belief that roulette wheels, dice, etc., have “memory,” so that, for example, having observed an unlikely streak of losses, the probability that the next outcome will be a win has increased, as a correction toward the expected long-term trend. The Wikipedia page provides a good example involving repeatedly flipping a fair coin:

“If after tossing four heads in a row, the next coin toss also came up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is [only] 1/32, a person might believe that the next flip would be more likely to come up tails rather than heads again. This is incorrect and is an example of the gambler’s fallacy.”

That is, having observed a streak of four heads in a row, we are actually just as likely to observe heads again on the subsequent fifth flip as we are to observe tails. Similarly, even after betting on red at the roulette wheel and losing four times in a row, we should still expect to win a fifth such bet on red the same stubborn 18/38 of the time (assuming a typical double-zero American wheel).

Right?

So, here is what I think is an interesting puzzle: let’s play a game. I will flip a fair coin n=100 times, and prior to each flip, if you have observed a current streak of s=4 or more consecutive heads, then make a note of the outcome of the subsequent flip. After all 100 coin flips, tally the noted “streak-following” flips: if there are more heads than tails, then I pay you one dollar. If there are more tails than heads, then you pay me one dollar. (If there are an equal number of heads and tails, then we push.)

If the gambler’s fallacy is indeed a fallacy, then shouldn’t this be a fair bet, i.e., with net zero expected return? But I claim that I have a significant advantage in this game, taking more than 15 cents from you on average every time we play! Following a streak of heads, we expect to observe a much larger proportion of “trend-correcting” tails than “streak-extending” heads.

And there is nothing special or tricky about this particular setup. Try this experiment with a different number n of coin flips, or a longer or shorter “target” streak length s, or even a roulette-like bias on the coin flip probability. Or instead of focusing only on streaks of consecutive heads (i.e., ignoring streaks of tails), look for streaks of either s or more heads or s or more tails, and note whether the subsequent flip is different. The effect persists: on average, we observe a larger-than-expected proportion of outcomes that tend to end the streak.

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Reproducing randomness

Introduction

We often need “random” numbers to simulate effects that are practically non-deterministic, such as measurement noise, reaction times of human operators, etc. However, we also need to be able to reproduce experimental results– for example, if we run hundreds of Monte Carlo iterations of a simulation, and something weird happens in iteration #137, whether a bug or an interesting new behavior, we need to be able to go back and reproduce that specific weirdness, exactly and repeatably.

Pseudo-random number generators are designed to meet these two seemingly conflicting requirements: generate one or more sequences of numbers having the statistically representative appearance of randomness, but with a mechanism for exactly reproducing any particular such sequence repeatably.

The motivation for this post is to describe the behavior of pseudo-random number generation in several common modeling and simulation environments: Python, MATLAB, and C++. In particular, given a sequence of random numbers generated in one of these environments, how difficult is it to reproduce that same sequence in another environment?

Mersenne Twister

In many programming languages, including Python, MATLAB, and C++, the default, available-out-of-the-box random number generator is the Mersenne Twister. Unfortunately, despite its ubiquity, this generator is not always implemented in exactly the same way, which can make reproducing simulated randomness across languages tricky. Let’s look at each of these three languages in turn.

(As an aside, it’s worth noting that the Mersenne Twister has an interesting reputation. Although it’s certainly not optimal, it’s also not fundamentally broken, but lately it seems to be fashionable to sniff at it like it is, in favor of more recently developed alternatives. Those alternatives are sometimes justified by qualitative arguments or even quantitative metrics that are, in my opinion, often of little practical consequence in many actual applications. But that is another much longer post.)

Python

Python is arguably the easiest environment in which to reproduce the behavior of the original reference C implementation of the Mersenne Twister. Python’s built-in random.random(), as well as Numpy’s legacy numpy.random.random(), both use the same reference genrand_res53 method for generating a double-precision random number uniformly distributed in the interval [0,1):

  1. Draw two 32-bit unsigned integer words (w_1, w_2).
  2. Concatenate the high 27 bits of w_1 with the high 26 bits of w_2.
  3. Divide the resulting 53-bit integer by 2^{53}, yielding a value in the range [0, 1-2^{-53}].

So given the same generator state, both the built-in and Numpy implementations yield the same sequence of double-precision values as the reference implementation.

Seeding is slightly messier, though. The reference C implementation provides two methods for seeding: init_genrand takes a single 32-bit word as a seed, while init_by_array takes an array of words of arbitrary length. Numpy’s numpy.random.seed(s) behaves like init_genrand when the provided seed is a single 32-bit word. However, the built-in random.seed(s) always uses the more general init_by_array, even when the provided seed is just a single word… which yields a different generator state than the corresponding call to init_genrand.

MATLAB

Although MATLAB provides several different generators, the Mersenne Twister has been the default since 2005. Its seeding method rng(s) is “almost” exactly like the reference init_genrand(s)— and thus like numpy.random.seed(s)— accepting a single 32-bit word as a seed… except that s=0 is special, being shorthand for the “default” seed value s=5489.

The rand() function is also “almost” identical to the reference genrand_res53 described above… except that it returns random values in the open interval (0,1) instead of the half-open interval [0,1). There is an explicit rejection-and-resample if zero is generated. It’s an interesting question whether this has ever been observed in the wild: it’s confirmed not to occur anywhere in the first 2^{21} random draws from any of the sequences indexed by the 2^{32} possible single-word seeds.

C++

Both Visual Studio and GCC implementations of the Mersenne Twister in std::mt19937 (which is also the std::default_random_engine in both cases) allow for the same single-word seeding as the reference init_genrand (and numpy.random.seed, and MATLAB’s rng excepting the special zero seed behavior mentioned above).

However, the C++ approach to generating random values from std::uniform_real_distribution<double> is different from the reference implementation used by Python and MATLAB:

  1. Draw two 32-bit unsigned integer words (w_1, w_2).
  2. Concatenate all 32 bits of w_2 with all 32 bits of w_1.
  3. Divide the resulting 64-bit integer by 2^{64}.

By using all 64 bits, this has the effect of allowing generation of more than just 2^{53} equally-separated possible values, retaining more bits of precision for values in the interval (0,1/2) where some leading bits are zero.

Another important difference here from the reference implementation is that the two random words are reversed: the first 32-bit word forms the least significant 32 bits of the fraction, and the second word forms the most significant bits. I am not sure if there is any theoretically justified reason for this difference. One possible practical justification, though, might be to prevent confusion when comparing with the reference implementation: even for the same seed, these random double-precision values “look” nothing at all like those from the reference implementation. Without the reversal, this sequence would be within the eyeball norm of the reference, matching in the most significant 27 or more bits, but not yielding exactly the same sequence.

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The distribution of “roll and keep” dice

Introduction

Some role-playing games involve a “roll and keep” dice mechanic, where you roll some number of dice, but only “keep” a specified number of them with the largest values, where the result of the roll is the sum of the kept dice. For example, rolling five dice and keeping the largest three (sometimes denoted 5d6k3) could yield a score between 3 and 18, inclusive. What is the probability of each possible score?

The motivation for this post is (1) to capture my notes on the general solution to this problem, and (2) to describe some potential optimizations in implementation to handle very large instances of the problem, such as this Hacker Rank version of the problem, where we may be rolling as many as 10,000 dice.

The solution

To start, let’s simplify the problem by just rolling and keeping all of the dice without discarding any, since the order statistics are what make this problem complicated. Given n dice each with d sides, the number r_{n,d}(s) of equally likely ways to roll a score s is given by

r_{0,0}(s) = 1

r_{n,d}(0) = 0

r_{n,d}(s) = \sum\limits_{k=0}^{\lfloor\frac{s-n}{d}\rfloor} (-1)^k {n \choose k} {{s-k d-1} \choose {n-1}}

so that the probability of rolling a score s is r_{n,d}(s)/d^n.

If we now consider keeping only m \leq n of the dice, then we can group the r_{n,d,m}(s) desired outcomes with score s by:

  • the smallest value a among the m largest dice that we keep,
  • the number b of kept dice that are strictly greater than a, and
  • the number c of discarded dice that are strictly less than a.

This yields the following summation:

r_{n,d,m}(s) = \sum\limits_{a=1}^d \sum\limits_{b=0}^{m-1} \sum\limits_{c=0}^{n-m} {n \choose {b,c}} (a-1)^c r_{b,d-a}(s-a m)

Implementation details

At this point, we’re done… except that if the number n of dice rolled is large, then a straightforward implementation of this nested summation will be pretty slow, involving calculation of a large number of very large multinomial coefficients.

My first thought was to speed up calculation of those very large coefficients via their prime factorization using Legendre’s formula. The paper by Goetgheluck linked at the end of this post describes a more optimized approach, but the following simple Python implementation is already significantly faster than the usual iterative algorithm for sufficiently large inputs (the implementation of primes(n) is left as an exercise for the reader, or a future post):

def binomial(n, k):
    """Large binomial coefficient n choose k."""
    if k < 0 or k > n:
        return 0
    c = 1
    for p in primes(n):
        c = c * p ** (v_fact(n, p) - v_fact(k, p) - v_fact(n - k, p))
    return c

def v_fact(n, p):
    """Largest power of prime p dividing n factorial."""
    v = 0
    while n > 0:
        n = n // p
        v = v + n
    return v

(It’s worth noting that the Hacker Rank problem actually only asks for the number of ways to roll a given total modulo a prime, suggesting that a similar but even more effective optimization using Lucas’s theorem is probably intended.)

But we can do much better than this, by observing that in the “usual” iterative algorithm, all of the intermediate products inside the loop are also binomial coefficients… and we need all of them for this problem. That is, we can rewrite the summation as

r_{n,d,m}(s) = \sum\limits_{a=1}^d \sum\limits_{b=0}^{m-1} {n \choose b} r_{b,d-a}(s-a m) \sum\limits_{c=0}^{n-m} {n-b \choose c} (a-1)^c

and then “embed” the usual iterative calculation of the binomial coefficients inside the nested summations. The following implementation eliminates all but a single explicit calculation of a binomial coefficient:

def rolls_keep(s, n, d, m):
    """Ways to roll sum s with n d-sided dice keeping m largest."""
    result = 0
    for a in range(1, d + 1):
        choose_b = 1
        for b in range(0, m):
            sum_c = 0
            choose_c = 1
            pow_c = 1
            for c in range(0, n - m + 1):
                sum_c = sum_c + choose_c * pow_c
                choose_c = choose_c * (n - b - c) // (c + 1)
                pow_c = pow_c * (a - 1)
            result = result + choose_b * rolls(s - a * m, b, d - a) * sum_c
            choose_b = choose_b * (n - b) // (b + 1)
    return result

def rolls(s, n, d):
    """Ways to roll sum s with n d-sided dice."""
    if s == 0 and n == 0:
        return 1
    if d == 0:
        return 0
    result = 0
    choose_k = 1
    for k in range(0, (s - n) // d + 1):
        result = result + (-1) ** k * choose_k * binomial(s - k * d - 1, n - 1)
        choose_k = choose_k * (n - k) // (k + 1)
    return result

Reference:

  1. Goetgheluck, P., Computing Binomial Coefficients, The American Mathematical Monthly, 94(4) April 1987, p. 360-365 [JSTOR]
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Snap card game probability

Introduction

In the children’s card game Snap, a deck of cards is shuffled and divided as evenly as possible among two or more players. In alternating turns, each player deals a card from her stack onto a face-up pile in front of her. If at any time, the top cards of any two players’ face-up piles have the same rank, the first player to shout “Snap!” takes both face-up piles and adds them to the bottom of her remaining stack. The objective is to accumulate all of the cards.

It is possible for the players to deal through their entire stacks without a single snap, i.e., at no time do the top cards of two piles have the same rank. Let us call such a game boring (a term that may be suggested by, say, your niece to whom you are introducing the game). What is the probability of a boring game of Snap?

One of the reasons I love combinatorics is that it is so easy to ask a hard question. That is, we can pose a series of very similar-sounding problems, all of which are easy to state and understand, but whose minor differences result in major changes in complexity of their solutions. The motivation for this post is to describe a simple children’s card game as an example of this phenomenon.

Frustration Solitaire

Before tackling the actual problem described above, let’s consider a slightly modified version of the two-player game that is easier to analyze:

  1. Instead of dividing a single shuffled deck in half, start with two separate complete shuffled decks, one for each player.
  2. Instead of alternating turns dealing a card from each player’s stack, at each turn both players simultaneously deal a card face-up from their remainder.

Dealing through the entirety of both decks without a snap is effectively equivalent to winning a game of Frustration Solitaire, where a single player deals cards from a single shuffled deck, saying ranks “Ace, two, three, …, king” repeatedly, losing the game if the dealt card ever matches the called rank.

Even within this already-modified context, there are three slight variations that range from very simple to– I think– intractable:

  1. If a snap requires that both cards match in rank and suit, then we are counting derangements, and the probability of a boring game is approximately 1/e, or about 0.367879.
  2. If a snap requires that both cards match in rank only, as in Frustration Solitaire, then we have discussed this problem before here, in the context of a Secret Santa drawing among 13 families each with 4 family members. In this case, the probability of a boring game is approximately 0.0162327.
  3. If a snap requires that both cards match in rank or in suit… well, although this is still effectively a problem of counting permutations with restricted positions, I think this problem is much harder in practice, since the board of restricted positions can’t be nicely decomposed into sub-boards with no rows or columns in common.

War

Let’s consider one more modified version of the game, this time actually rolling back one of the changes above: let’s return to playing with a single shuffled deck divided evenly among the two players, but retain the change where the players deal simultaneously from their stacks.

This version of the game has a lot of structure in common with another children’s card game, War. In this case, a boring game of Snap corresponds to War with no war– that is, dealing through both deck halves without any pair of cards matching in rank.

This is a relatively straightforward inclusion-exclusion problem; with a deck with r=13 ranks and s=4 suits, the probability of a boring game is

\frac{1}{(r s)!} \sum\limits_{j=0}^n (-1)^j {n \choose j} j!(r s-2j)! [x^j]g(x)^r

where

n = \lfloor\frac{r s}{2}\rfloor

g(x) = \sum\limits_{k=0}^{s/2} \frac{s!}{k!(s-2k)!} x^k

which for a standard 52-card deck yields a probability of a boring game of about 0.210214.

Counting “words” with prohibited subwords

Coming back finally to the original rules of Snap, counting boring games is complicated by the fact that the players alternate turning individual cards face-up, rather than simultaneously revealing a pair of cards at a time, so that a snap may “start” with either the first or the second player’s deal. Intuitively, consider skipping the initial separation of the deck into halves, and simply deal cards one at a time from the single shuffled deck; a boring game is one in which no two consecutively dealt cards are the same rank.

There is a wonderful paper by Jair Taylor (referenced below) describing a very general but more sophisticated technique for counting arrangements with these types of restrictions. Applying this technique to Snap, the probability of a boring game using a deck with r ranks and s suits is

\frac{(s!)^r}{(r s)!} \int_{t=0}^{\infty} e^{-t} g_s(t)^r dt

where

g_k(t) = (-1)^k \sum\limits_{i=0}^k (-1)^i {k-1 \choose k-i} \frac{t^i}{i!}

yielding a probability of approximately 0.0454763, or about once every 22 shuffles, that a game of Snap will be boring.

Once again, however, it’s very easy to make this problem much harder: if we allow a snap to involve adjacent cards matching in rank or in suit, what is the resulting probability of a boring game? What if there are more than two players?

Reference:

  1. Taylor, J., Counting words with Laguerre series [arXiv]
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Poker puzzle

A variant of this problem came up in discussion recently, that I think lends itself to attack via either mathematical “pencil and paper” or “write a program” analysis:

What is the median poker hand? More precisely, among all {52 \choose 5}=2598960 possible five-card poker hands, what hand ranks higher than exactly half of them?

Of course, the solution is possibly not quite unique, since the ranking of hands is a weak order: suits never decide the relative ranking of hands. For example, there are four possible royal flushes, and no one royal flush beats another. So, as a follow-up combinatorics problem: how many “distinct” hand ranks are there? That is, how many equivalence classes are there in the incomparability relation “hand x ties with hand y” on the set of possible hands?

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Calling Java from MATLAB with pass-by-reference arrays

MATLAB has a pretty intimate connection with Java, supporting the creation and manipulation of native Java objects directly in MATLAB source code. For example:

x = java.lang.Double(pi);
disp(x.isInfinite());

Contrast this with C++, where it is typically necessary to write an additional small amount of “wrapper” C code using the MEX API to bridge the gap between languages.

For the most part, conversions between MATLAB and Java data types are automatic, as expected, and as desired… with one exception: passing a primitive array by reference. For example, suppose that we want to call a static method in a third-party Java library that converts an input 6-state vector (as a primitive array of 6 doubles) from one coordinate frame to another, by populating a given “output” array of 6 doubles:

public static int convert(double[] x_in, double[] x_out) { ... }

As far as I can tell, it is impossible to use a method like this directly from MATLAB. The problem is that we can never construct the necessary output buffer x_out, and hold onto it as a reference that could be passed to convert. MATLAB insists on automatically converting any such reference to a Java array of primitive type into its “value” as a MATLAB array of the corresponding native type, leaving the reference to the actual “buffer” behind in the Java address space.

Interestingly, MATLAB’s built-in function javaArray seems like it might have been intended for situations like this. But the arrays created by this built-in function are not of the necessary primitive data types, but of their “boxed” counterparts, which seems almost entirely useless to me, since I don’t know of any Java libraries whose interface involves arrays of boxed primitive types.

So, it seems that we need at least some additional wrapper Java code to make methods like this usefully accessible from MATLAB. The objective of this post is to provide an implementation that I think fills this gap in a reasonably general way, by using Java’s reflection mechanism to allow creating primitive arrays and passing them by reference to any Java method directly from MATLAB.

(Aside: I suppose I get some masochistic enjoyment from repeatedly using the phrase “pass by reference” here, hoping for pedantic complaints arguing that “Java is pass by value, not pass by reference.”)

The code is available on GitHub, as well as the old location here. Following is a simple example showing how it works:

s = java.lang.String('foo');
dst = matlab.JavaArray('java.lang.Character', 3);
callJava('getChars', s, int32(0), int32(3), dst, int32(0));
dst.get()

The idea is similar to the libpointer and calllib interface to C shared libraries: a matlab.JavaArray acts like a libpointer, and callJava acts like calllib, automatically “unwrapping” any matlab.JavaArray arguments into their underlying primitive array references.

Finally, a disclaimer: the method dispatch is perhaps overly simple, keying only on the desired method name and provided number of arguments. If your Java class has multiple methods with the same name, and the same number of parameters– regardless of their type, or of the return type– the first method matching these two criteria will be called.

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