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Sunday 5 July 2020

Statistics, Gambling, and Games of Chance

This is a proposal for a survey course on statistics that uses gambling extensively in examples. The target audience is senior undergraduates with a non-statistical background, but quantitative students will also find enough novelty to be interested. The goal of the course is not to encourage gambling, but to use it as a vehicle for a broad range of otherwise difficult statistical topics.

For example, roulette is used to deliver a lesson on expected value, but many of the bets you can make in roulette have the same expected value despite the widely different payouts. Likewise, the probability of poker hands can be calculated exactly with combinatorics, and the odds offered on sport events can be used to explain log-odds, the basics logistic regression, and confidence intervals.
There is opportunity for some larger projects, like predicting sports, but a lot of the smaller assignment questions could be 're-skinned' from probability courses.

Because of the wide range in depth, and the diversity in prospective students, as well as the stigma of gambling and of teaching people to gamble, this would probably be best as a zero-credit course or an independent lecture series. However, it could also be used as an outreach tool to bring talent into statistics and data science that might not otherwise consider it.

The syllabus is given in parts because the timing hasn't been worked out yet. This could be anywhere from 20 to 80 hours of lectures depending on pacing and topic selection.

Draft Syllabus:

Part 1: Preliminaries
- What is gambling?
- PDF, CDF, continuous and discrete distributions
- Means, medians, variance, and skew

Part 2: Classic games of chance
- Monty Hall problem: Sample spaces
- Random packs of cards, loot boxes: Geometric and negative binomial distributions

- Craps, Yahtzee, dice games: Independent events, binomial distribution, normal approximation 

- Simulating dice events with Borel.
- Bingo and discrete distributions.

- Roulette, expected value, and house advantage (See also Vig/Juice)

- Blackjack and Transition Matrices (See also Snakes and Ladders, Monopoly)

Part 3: Combinatorics                              
- Counting principles, factorials, and Stirling's approximation.
- Combinations and Permutations.
- Playing Cards: conditional probability, information, and 'blockers'.
- Blackjack again: Counting cards, shoes, and asymptotics.
- The hypergeometric distribution. 
- Poker: Straights, flushes, pairs, and triples. 
- Poker: Accounting for community cards, and for drawing rounds.

Part 4: Utility
- Marginal utility, utility curves.
- Lotteries and Insurance: (different utility for you and the company).
- Future value, present value, discounting rates.

Part 5: Prediction Markets and Sports Betting
- Decimal vs American Odds: log-odds, logistic regression, ordinal logistic regression
- Odds movements, stock price movements: time-series models
- Parlays and correlated bets
- Over/Unders, Vig/Juice, Exotic or Prop bets
- Pre-game and in-game betting: Probabilities changing over time
- Fantasy sports: expected and variance of sums, opportunity cost
- Hedging strategies and arbitrage

Part 6: Common sports models
- Soccer and Hockey: Poisson, Zero-Inflated Poisson, and Diagonally-Inflated Poisson
- Basketball: Bivariate normal
- Head-to-head matchups (team sports): Elo, Glicko, and other ratings systems
- Many-player matchups (races): Microsoft TrueSkill

Part 7: Game theory for player-vs-player
- Zero-Sum games vs win/win (Nash equilibria)
- Rock-paper-scissors
- Prisoner's dilemma
- Jeopardy: Strategy and Decision flowcharts

Part 8: Texas Hold'em  
- Pot odds, implied odds, the fundamental theorem of poker
- Game Theory Optimal, Ranges, Bluffing
- Independent Chip Model

Part 9: Bankroll management
- Kelly Stakes Criterion, Bayesian adjustments, Simultaneous Kelly
- Drawdown, Value-At-Risk, Extreme Value Functions

Part 10: Simulations and Machine Learning
- Monte Carlo simulations, Monte Carlo Markov Chain
- How AIs for games like backgammon are trained
- Minimax algorithms

Historical / Cultural hooks
- The birth of probability comes from an unresolved game of chance
- Chaturanga, a very old version of chess which used dice to determine pieces to move.
- Parcheesi is an older, dice-based version of Sorry!
- The MIT Blackjack team
- The Royal Game of Ur
- Hubbub and Chekutnak: Indigenous Games of Chance
- Mahjong and trick-taking games.

Draft Introduction:

What is Gambling
What legally constitutes gambling changes with locations and time. By some definitions, anything that involves a degree of skill (e.g. choosing players in a fantasy sports league) is a competition and therefore not gambling. For the purposes of this class, we'll use the opposite extreme: Everything with financial uncertainty is gambling, including things aren't commonly considered to be games at all.

Buying a home is the stereotypical 'responsible purchase', and yet it's an *enormous* gamble. The long-term value of a house has a lot of variance. If there is variance, there is risk, and for the purposes of this course this gambling.

Most investments carry financial risk, but unlike most games of chance they carry relatively low risk and a positive expected value. Even bonds, which are just loans to corporations or governments, even if re-payment is guaranteed, carry risk in their resale value related to exchange rates.

Actuarial science is a whole field, closely related to statistics, that deals with financial uncertainty. Traditionally it deals with insurance and the premium payments that support it, but it also deals with the investment of funds until they are needed for payments of claims. If you enjoy this course and wish to pursue further theoretical depth, it's recommended that you look further into statistics and especially actuarial science.

Reading Sources:

 "Weighing the Odds in Sports Betting" by King Yao
"The Theory of Poker" by David Sklansky
The Game of Borel.

If this course proposal gives you ideas as an educator, see also my previous course proposal on "Statistics in Politics and Polling"  for a similar outreach / cross-disciplinary approach.

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