## An army of gamblers and a martingale

Start flipping a coin. How long should you wait (on average) before observing the pattern ${HHTHHTT}$ !? Easy you might say, that’s simply the hitting time of the state ${H}$ starting from state ${A}$ of the following Markov chain.

Indeed, one can solve this kind of problem using a first step analysis. In other words, if ${a}$ is the expected number of steps before reaching ${H}$ starting from ${A}$, ${b}$ is the expected number of steps before reaching ${H}$ starting from ${B}$, ${c}$ is the expected number of steps before reaching ${H}$ starting from ${C}$, etc… one can readily see that ${a,b, \ldots, h}$ verify the system of equations

$\displaystyle \begin{array}{rcl} a &=& 1 + \frac12(a+b), \quad b = 1 + \frac12(a+c), \quad c = 1 + \frac12(c+d) \\ d &=& 1 + \frac12(a+e), \quad e = 1 + \frac12(a+f), \quad f = 1 + \frac12(c+g) \\ g &=& 1 + \frac12(e+h), \quad h = 0. \end{array}$

After a finite amount of time and coffee one then come up with the answer ${a=128}$. That was a little bit tedious and I don’t really want to know how long one should wait on average before observing the sequence ${HHTHTTTHTHHHHTTHTHTTH}$.

It turns out that there is a very neat way to approach this kind of problems without solving any linear system. Apparently, this trick was discovered in the 80s by Shuo-Yen Robert Li. As often in probability, this boils down to introducing a clever martingale. Suppose that one wants to know how long it takes on average before observing the magic coin pattern ${x_1x_2 \ldots x_n}$ of length ${n \geq 1}$ with ${x_i \in \{H,T\}}$ for all ${1 \leq i \leq n}$. To this end, one will introduce an army of gamblers, a casino, a game of chance and a martingale. The martingale ${M}$ describes the amount of money earned by the casino. At each round (starting at round ${t=1}$) the casino tosses a coin and the gamblers can make bets. If a gambler starts betting just before round ${k}$, he wins if the coin shows ${x_1}$. If he decides to bet again at round ${k+1}$, he wins if the coin shows ${x_2}$, etc… Suppose that there are infinitely many gamblers ${G_0, G_1, G_2, \ldots}$ and that gambler ${G_k}$ starts betting just before round ${k+1}$. The game is fair in the sense that if a gambler bets an amount ${A}$, with probability ${1/2}$ he gets ${2A}$ and with probability ${1/2}$ he losses his bet. One supposes that gambler ${G_k}$ starts betting ${1}$ dollar just before round ${k+1}$ and keeps reinvesting his total capital until he either losses everything (total loss equals ${1}$ dollar) or observes the magic coin pattern ${x_1x_2 \ldots x_n}$ (total earning equals ${2^n-1}$). Indeed, the amount of money earned by the casino is a martingale and ${M_0=0}$. For example, ${M_1=1}$ if the first gambler ${G_0}$ loses his bet and ${M_1=-1}$ if the wins. Consider the time ${\tau}$ when the magic coin pattern first appears. This is a stopping time and the optional stopping theorem for martingales shows that ${\mathbb{E}[M_{\tau}] = 0}$. Indeed, one can also compute this quantity another way. By time ${\tau}$, all the losers have given ${1}$ dollar to the casino. Also, if the gambler ${G_k}$ is still in the game he has won a total amount ${2^{\tau-k}-1}$. in other words,

$\displaystyle M_{\tau} = \tau - \sum_{j=0}^{\tau-1} 2^{\tau-k} \, \delta_j$

where ${\delta_j=1}$ if the gambler ${G_j}$ is still in the game after the round ${\tau}$ and ${\delta_j=0}$ otherwise. The fundamental remark is that the quantity ${\sum_{j=1}^{\tau} 2^{\tau-k+1} \, \delta_j}$ is not random so that ${\mathbb{E}[\tau] = \sum_{j=0}^{\tau-1} 2^{\tau-k} \, \delta_j}$. Indeed, right after round ${\tau}$ one can know with certainty whether ${G_{k}}$ is still in the game or not. Gambler ${G_{\tau-k}}$ is in the game only if the ${k}$ last letters of the magic word ${x_1x_2 \ldots x_n}$ are the same as the first ${k}$ letters. For example, if the magic word is ${HHTHHTH}$ then the gamblers who have seen the pattern ${H}$,${HHTH}$ and ${HHTHHTH}$ at time ${\tau}$ are still in the game when the magic word first appear. In this case this shows that ${\mathbb{E}[\tau] = 2^1+2^4+2^7}$.

For example, this immediately shows that we have to wait on average ${2^{21}+2^1}$ coin tosses before observing the pattern ${HHTHTTTHTHHHHTTHTHTTH}$.

1. #### how i quit gambling said,

April 14, 2013 at 8:45 pm

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2. #### how i quit gambling said,

April 17, 2013 at 6:57 pm

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November 20, 2015 at 3:05 am

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