Mathematical Games and Random Walks
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1 Mathematical Games and Random Walks Alexander Engau, Ph.D. Department of Mathematical and Statistical Sciences College of Liberal Arts and Sciences / Intl. College at Beijing University of Colorado Denver UNDERGRADUATE RESEARCH PROGRAM LECTURE, APRIL 25, 2013 Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
2 A Number Game (Unique Bid Auction) 1. Everybody get a pen and piece of paper. 2. Every player write down a positive integer {1, 2, 3,...}. 3. Lowest Unique Bid Auction: The player with the smallest number that has not been picked by somebody else wins the game (and a prize). 4. Highest Unique Bid Auction: The highest unique bid buys RMB 100. Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
3 The Factor Game (Taxman Problem) 1. Playing the Taxman: A player successively chooses numbers from {1, 2, 3,..., n}. After each move, the Taxman collects all the divisors of this number. 2. Paying the Taxman: The Taxman must always be paid: If there are no more numbers with divisors, the Taxman gets all remaining numbers. Can you beat the Taxman? n = 10 n = 15 Num Div Num Div Num Div Your Score 0 The Taxman 0 Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
4 Many People Can Beat The Taxman: Best Scores for n = 10 to n = 50 Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
5 Can You Meet Alex (Germany) and David (USA)? There are 6 slots left! Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
6 A New Casino Game (Factor-Game Roulette) 1. Imagine a roulette wheel with numbers {1, 2, 3,..., 36}. 2. You can bet any amount on any number. 3. If your bet divides the winning number, you get back your amount times the bet. What is your optimal playing strategy? Are there any fair numbers? Are there any losing numbers? Are there any winning numbers? Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
7 A Winning Strategy in Factor-Game Roulette 1. Bet RMB 1 on 2; if the winning number is even, you get 2 RMB 1: Win = Get Bet = 2 1 = 1 2. Every time the winning number is odd, double your bet on 2: 1 2nd bet 2 3rd bet 4 4th bet 8 kth bet 2 k 1 3. Eventually, a winning number will be even; let this happen in round k: Win = Get Bet = 2 k ( k 1) = 2 k ( 2 k 1 ) = 1 Warning: This strategy may not work for everybody! Do you have enough money to double your bet k times? What is the probability that you have to double your bet k times? Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
8 The Gambler s Ruin Starting with an initial capital of RMB 2, you play a game and in every round either win or lose RMB 1 with a known probability of p or 1 p. The game ends either if you reach RMB 4, or if you are out of money. This can be called Stochastic Process, Random Walk, or Markov Chain. What is the probability to win RMB 4, or to run out of money? What is the probability that this game will last forever? What is the expected duration of this game? Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
9 The Gambler s Ruin Scenario Tree 2 1 p p 1 1 p p 3 1 p p p p p p 3 1 p p p p Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
10 TicTacToe Decision Tree Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
11 Chess Endgame Decision Tree Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
12 Gambler s Ruin Scenario Tree Analysis 1. Probability to reach 0 or 4? p 0 = (1 p) 2 + 2p(1 p)p 0 (1 p)2 = (1 p) 2 + p 2 p 4 = p 2 + 2p(1 p)p 4 p 2 = (1 p) 2 + p 2 1 p p p p 1 2. Probability to play forever? lim [2p(1 n p)]n = 0 (also note that p 0 + p 4 = 1) 3. Expected duration of game? [ 2k (2p(1 p)) ( k 1 (1 p) 2 + p 2)] 2 = (1 p) 2 + p 2 k=1 Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
13 Probabilities and Expected Duration for Different Values of p Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
14 Gambler s Ruin Markov Chain Analysis Let S = {0, 1, 2, 3, 4} and P ij be the probability to go from state i to j: p 0 p 0 0 P = 0 1 p 0 p p 0 p We can compute matrix powers P n to find n-step transition probabilities: p p(1 p) 0 p 2 0 P 2 = P P = (1 p) 2 0 2p(1 p) 0 p 2 0 (1 p) 2 0 (1 p)p p This may seem complicated but only uses a little bit of linear algebra Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
15 Gambler s Random Walk (Monte Carlo Simulation) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
16 Social Random Walk (Divorce Rates in England and Wales, ) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
17 Financial Random Walk (Stock Market Averages Charts, ) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
18 Economic Random Walk (China Steel Inventory, , TaintedAlpha) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
19 Environmental Random Walk (Temperature Anomalies, ) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
20 The Mathematics of Policy and Decision-Making Math 3301/3302 Operations Research: develops methods to help or support policy and decision-makers in solving real-world problems Math 4390 Game Theory: develops mathematical models of conflict and cooperation for intelligent rational policy and decision-makers A Simple Decision Model ( one-person game against nature ) A = {a 1, a 2, a 3,...} be a set of available actions or behaviors S = {s 1, s 2, s 3,...} be the set of possible states of the world 1. A decision-maker chooses an action a A. 2. The world (or nature ) adopts a certain state s S. 3. The decision-maker earns a reward r(a, s) or incurs a cost c(a, s). The goal is to choose a A so to maximize reward and minimize cost. Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
21 The News Vendor Problem Every day, Bobby Bàotóng buys newspapers from a store for RMB 2 and sells them on the street for RMB 2.5. If daily demand is between 6 and 10, how many newspapers should Bobby supply to maximize his profit? Rewards A c t i o n s S t a t e s min max s S s S Best Worst Case: Best Best Case: max a A max a A min r(a, s) s S max r(a, s) s S (max-min criterion) (max-max criterion) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
22 The News Vendor Regret Criterion For each s S, compute max possible reward r (s) = max r(a, s) For each s S and a A, compute opportunity costs (or regrets): c(a, s) = r (s) r(a, s) The regret criterion minimizes disappointment of lost opportunities: min max a A s S c(a, s) = min a A max s S (r (s) r(a, s)) a A R e w a r d s r(a, s) R e g r e t s c(a, s) max s S Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
23 Expected Values and Demand Profiles Suppose the news vendor can estimate probabilities π(s) for all s S: max a A E r(a) = 10 s=6 π(s)r(a, s) min E c(a) = a A 10 s=6 π(s)c(a, s) A c t i o n s Expected Rewards E r(a) high uniform normal low Expected Regrets E c(a) high uniform normal low Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
24 A Mathematical Model for Two-Person Games Two players Rowley and Coleen choose from a finite set of moves: R = {1,..., m} C = {1,..., n} If Rowley and Coleen play i R and j C they receive r ij and c ij : (r 11,c 11 ) (r 12,c 12 )... (r 1n,c 1n ) (r 21,c 21 ) (r 22,c 22 )... (r 2n,c 2n ) (r m1,c m1 ) (r m2,c m2 )... (r mn,c mn) is called the payoff matrix. Rowley is called row player and Coleen is called column player. Game 1 [ ] (1, 0) (2, 2) (3, 1) (0, 3) Equilibrium Game 2 [ ] (1, 0) (2, 2) (0, 3) (3, 1) No equilibrium Game 3 [ ] (2, 3) (0, 0) (1, 1) (3, 2) Two equilibria Coop Games [ ] (0, 0) (0, 5) (5, 0) (1, 1) [ ] (5, 0) (0, 1) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
25 Battle of the Sexes Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
26 The Chicken Game Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
27 Dove and Hawk (Classic Example of Evolutionary Game Theory) Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
28 Boxed Pigs A litte pig and a big pig are boxed in a room. Pressing a button on left wall gives 10 units of food from right wall and costs 2 units of energy. If little pig presses, big pig eats 9, little pig gets 1. If big pig presses, little pig eats 5, big pig gets 5. If both pigs press, little pig eats 3, big pig eats 7. Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
29 The Prisoners Dilemma Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
30 The Three Prisoners Problem Three prisoners A, B, and C are facing 5-year sentences in prison. The judge has randomly chosen one of the prisoners to be released. Prisoner A begs the judge to tell him which of the two other prisoners will be kept in prison (if both will be kept, he shall say at least one). The judge tells prisoner A that prisoner C will be kept in prison. What is the probability that prisoner A will be released? The Two Goats Problem You play in a game show and can choose one of three closed doors. Behind one door is the main prize, behind the other doors are goats. You choose to open Door 1 and the game host opens Door 3: a goat. The host offers a deal: you can stay with Door 1 or switch to Door 2. Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
31 Which Door Do You Choose? Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
32 Think Critically, Study Hard, and Be Good! Alexander Engau, Ph.D. Mathematical and Statistical Sciences, UC Denver URP Lecture, International College at Beijing, April 25, of 32
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