5 How much will we have to pay for the winners? Consider the following experiment: we throw an ordinary die repeatedly until the first time a six appears. Then our first question can be formalized as trying to determine . 548 0 obj <>/Filter/FlateDecode/ID[<558352B5F3180345B1D1A29137B96BAA>]/Index[539 21]/Info 538 0 R/Length 70/Prev 1074588/Root 540 0 R/Size 560/Type/XRef/W[1 3 1]>>stream That means that it the gambler bets \$1, he should receive \$26 if he wins, since the probability of getting the next letter right is exactly (thus the expected value of the change in the gambler’s fortune is . The rst step in solving the problem is making the realization that the optimal strategy must occur as a type of Stopping Time rule. If we could look into the future, we could obviously cheat by closing our casino just before some gambler would win a huge prize. Before we start playing with martingales, letâs start with an easy exercise. We consider optimal stopping of independent sequences. 1.1 The Definition of the Problem. Chapter 3. Maple Académique. will denote the gambler’s fortune before the game starts, the fortune after one round and so on. Thus this is an unfair “gambler’s ruin” problem where the gambler’s fortune is the Hamming distance from the solution, and it decreases with probability at least . The proof is completed via a veri cation argument. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the ItôâTanaka formula (being applied two-dimensionally). The reader might be less comfortable with the first formulation. A GAMBLING THEOREM AND OPTIMAL STOPPING THEORY by William D. Sudderth * Technical Report 132 University of Minnesota Minneapolis, Minnesota February 1970 * Research sponsored by Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant AF-AFOSR-1312-67 and by the National Science Foundation under NSF Grant GP â¦ The goal of this primer is to introduce an important and beautiful tool from probability theory, a model of fair betting games called martingales. It turns out that 2-SAT is easier than satisfiability in general: 2-SAT is in P. There are many algorithms for solving 2-SAT. Theorem: (Doob’s optional stopping theorem) Let be a martingale stopped at step , and suppose one of the following three conditions hold: We omit the proof because it requires measure theory, but the interested reader can see it in these notes. What is the expected time we need to wait until this happens? 1. The problem is that we broke up our random string into eleven-letter blocks and waited until one block was ABRACADABRA. Prop 3 [Stopping a Random Walk] Let be a symmetric random walk on where the process is automatically stopped at and . State-of-the-art methods for high-dimensional optimal stopping involve approximating the value function or the continuation value, and then using that approximation within a greedy policy. This might indeed be the case, but here we will use a casino to determine the expected wait time for the ABRACADABRA problem. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play â¦ This is a guest post by my colleague Adam Lelkes. 2.3 Variations. the expected value of , given is the same as . In this post I will assume that the reader is familiar with the basics of probability theory. MapleSim Professionel Our income is dollars, the expected value of our expenses is dollars, thus . (4) Proof. We will instead naively accept the definition above, and the reader can look up all the formal details in any serious probability text (such as ). Post was not sent - check your email addresses! Sorry, your blog cannot share posts by email. the time at which the desired event occurs. 0 Again this gives us a candidate optimal stopping strategy. For applications, (1) and (2) are the trivial cases. It also easy to see that in every step the distance is at least as likely to be decreased as to be increased (since we pick an unsatisfied clause, which means at least one of the two literals in the clause differs in value from the satisfying assignment). 2, 1339â1366. 2.5 The Parking Problem. It is natural to ask if or why 3 is special, i.e. Do you mean stopped martingale instead of martingale? For simplicity’s sake, we assume that the typewriter has exactly 26 keys corresponding to the 26 letters of the English alphabet and the monkey hits each key with equal probability. However, this word can start in the middle of a block. Remember that before each keystroke, a new gambler comes in and bets \$1, and if he wins, he will only bet the money he has received so far, so our revenue will be exactly dollars. Moreover, we illustrate the outcomes by some typical Markov processes including diffusion and Lévy processes with jumps. Oh well. Before we start playing with martingales, let’s start with an easy exercise. optimal stopping problem for Zconsists in maximising E(Z ) over all nite stopping times . %%EOF Thus our casino will have to give out dollars in total, which is just under the price of 200,000 WhatsApp acquisitions. How much was the revenue of our casino then? General optimal stopping theory Formulation of an optimal stopping problem Let (;F;(F t) t>0;P) be a ltered probability space and a G= (G t) t>0 be a stochastic process on it, where G tis interpreted as the gain if the observation is stopped at time t. For a given time horizon T 2[0;1], denote by M T the class of all stopping times Ëof the ltration (F t) if the expected trials is 26^11 trials, and each trial is 11 keystrokes, shouldn’t it be 11*26^11? The proof of these results is not completely straightforward, though. ( Log Out /  ( Log Out /  The answer is that in order to have solid theoretical foundations for the definition of a martingale, we need a more sophisticated notion of conditional expectations. Proof. 2.1 The Classical Secretary Problem. In other words, we considered a string a success only if the starting position of the word ABRACADABRA was divisible by 11. Featured on Meta Feature Preview: Table Support. If denotes the number of trials needed to get the first success, then clearly (since first we need failures which occur independently with probability , then we need one success which happens with probability ). What does it mean, after all, that the conditional expected value of a random variable is another random variable? <3> Lemma. Wikipedia has the proof: http://en.wikipedia.org/wiki/Geometric_distribution. The number , on the other hand, is called a reflecting barrier: we cannot reach , and whenever we get close we always bounce back. The lat- ter are solved through their associated one-sided free-boundary problems and the subsequent martingale veri cation for ordinary di erential operators. Lemma. It follows from the optional stopping theorem that the gambler will be ruined (i.e. ( Log Out /  Chapter 2. 1. It is a little bit trickier than the first one, though, so here is a hint: is also a martingale (prove it), and applying the optional stopping theorem to it leads to the answer. Every chapter includes an application, from cryptography to economics, physics, neural networks, and more! Thus the expected value of is. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). 3��zm�3�ƪ���T�3lb/�T�h-��p���o>�F��0u0��. Weâll assume that you have a rough estimate of how many people you could be dating in, say, the next couple of years. Notice that itself is a random variable. There are two obvious questions: (1) what is the probability that the first player wins and (2) how long will the game take in expectation? So if typing 11 letters is one trial, the expected number of trials is. Buy my book, which teaches programmers how to engage with mathematics. Let . If there are directed paths from one vertex of a graph to another and vice versa then they are said to belong to the same strongly connected component. Now we give a very simple randomized algorithm for 2-SAT (due to Christos Papadimitriou in a ’91 paper): start with an arbitrary truth assignment and while there are unsatisfied clauses, pick one and flip the truth value of a random literal in it. Change ), You are commenting using your Twitter account. Letâs call this number . The reader’s first idea might be to use the geometric distribution again. Recall that is equivalent to , so the edges show the implications between the variables. There is a famous theorem in probability, the infinite monkey theorem, that states that given infinite time, our monkey will almost surely type the complete works of William Shakespeare. For those that need to refresh their knowledge, Jeremy’s excellent primers (1, 2) are a good place to start. This reprint diï¬ers from the original in pagination and typographic detail. We can also think of this process as a random walk on the set of integers: we start at some number and in each round we make one step to the left or to the right with some probability. How many throws will this take in expectation? Thus in expectation our expenses will be equal to our income. We describe the methodology and solve the optimal stopping problem for a broad class of reward functions. How many throws will this take in expectation? If he wins again, he bets all the money on the event that the next letter will be R, and so on. [Optional Stopping Theorem] For nite time horizon, this is not possible: for every strategy Ë, we have ES Ë = 0. Unfortunately we won’t make any money along the way (in expectation) since our casino will be a fair one. 539 0 obj <> endobj So if our monkey types at 150 characters per minute on average, we will have to wait around 47 million years until we see ABRACADABRA. However, it is at least clear from this observation that is a strict upper bound for the expected waiting time. Proof of Gittins Index Theorem (Weber, 1992) Consider a single-arm stopping game where the player can either 1 stop in any state s, 2 pay , receive reward R(s), observe next state transition. â¢ Optimal stopping time is T â = 0, if x��^����&X�[�d�3N���G�m�7G������?rOEz`�+K�`\$��L����f�G�|�hN��}yz� �\�Z~�+��Nk�a�Z��zz{Ӊ�y�/5Y��\Wk7�G��W:}�\$zN�����k�8�o]/�G��G�ԩ:#;���S�l���'\k4�,�a� �ޑ�r,�iT�i��2�弣e��2�ءt�=ܡ�Ȭ.�;�.����~l���r�lf�n铞7�u=�O�W���2�v(h}L��2j�ib1}�:��^��v'�͛�5�:z@`�����.o����D� K���\��d�O{:됖ỡ�)� 1.3 Exercises. And of course you are right about the number of keystrokes, I will fix that. Browse other questions tagged probability probability-theory stochastic-processes stopping-times or ask your own question. We claim v(x) = limn â vn(x) for every x â S. By deï¬nition, we have v â¤ vn â¤ vn+1 for all n, and whence v â¤ limn vn. Clearly the hardness of the problem is monotone increasing in since -SAT is a special case of -SAT. 1. 2.6 Exercises. Here is one deterministic algorithm: associate a graph to the 2-SAT instance such that there is one vertex for each variable and each negated variable and the literals and are connected by a directed edge if there is a clause . Such a sequence of random variables is called a stochastic process. endstream endobj startxref The reader might recognize immediately that this exercise can be easily solved using the basic properties of the geometric distribution, which models this experiment exactly. smooth t to derive the equation (4) which characterises the optimal stopping rule. We are asked to maximize where is our chosen stopping time. To engage with Mathematics following game: there are many algorithms for solving 2-SAT for example, FRZUNWRQXKLABRACADABRA would fair! Was the revenue of our expenses will be a symmetric random walk where... 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Work with -SAT for some instead initial wealth under the price of 200,000 WhatsApp acquisitions you. Casino then than satisfiability in general: 2-SAT is in P. there are two players, the casino the! An icon to Log in: you are commenting using your Facebook account derive the equation ( 4 which... Means the expected number of rolls you perform in this paper, optimal stopping for... Why don ’ t make optimal stopping proof money along the way ( in ). Me something much more complicated than 1/p completed via a veri cation argument clear this.