Optimal gambling systems for favorable games
Kelly Betting is a prescription for optimal resource allocation among a sequence of gambles which are typically repeated in an independent and identically distributed manner. a mathematical theory for committing resources in favorable gamesintroduction: coin tossingminimizing the probability of ruinthe kelly criterionthe advantages of maximizing e log xna stock market examplewarrant hedgingportfolio selection using e log xthe kelly criterion and deficiencies in the markowitz theory of portfolio selectionreferences. part i. favorable games. blackjack. baccarat. roulette. the wheel of fortune. the stock market. part ii. a mathematical theory for committing resources in favorable games. introduction: coin tossing. minimizing the probability of ruin. the kelly criterion. the advantages of maximizing e log x n. a stock market example. warrant hedging. portfolio selection using e log x.
Portfolio Performance Measurement
So we can see that this aggressive strategy is almost surely 5 going to result in ruin. There is a great book called Fortune's Formula 1 that details the stories and adventures surrounding these brilliant minds. This is kind of abstract, so let's take a look at an example from Wikipedia. The point is that you get to pick the winning side! Kelly's optimal betting criterion is an incredibly interesting mathematical result.
Portfolio Performance Measurement
Money management. For a speculative investor, there are two aspects to optimizing a trading strategy. The first and most important goal of a trader is to achieve a positive expected risk-adjusted return.
Once this has been achieved, the trader needs to know what percentage of his capital to risk on each trade. The underlying principals of money management apply to both gambling and trading, and were originally developed for the former. He noted that, as a consequence of this, when profits are reinvested, in order to measure the value of risky propositions one should calculate the geometric mean. The paper was later translated into English Bernoulli Now a classic book, this is the work upon which modern-day game theory is based.
The paper established the discipline of information theory and became a classic. In short, he developed the concepts of information entropy and redundancy. Shannon asserted that binary digits could be transmitted over a noisy channel with an arbitrary small probability of error if the binary digits were suitably encoded. He showed that in order to achieve maximum growth of wealth, at every bet a gambler should maximize the expected value of the logarithm of his capital, because it is the logarithm which is additive in repeated bets and to which the law of large numbers applies.
Money management systems which maximize the expected value of the capital are said to employ the Kelly criterion. Breiman proved that using the Kelly criterion is asymptotically optimal under two criteria: 1 minimal expected time to achieve a fixed level of resources and 2 maximal rate of increase of wealth.
My last post was about some common mistakes when betting or gambling, even with a basic understanding of probability. This post is going to talk about the other side: optimal betting strategies using some very interesting results from some very famous mathematicians in the 50s and 60s.
I'll spend a bit of time introducing some new concepts at least to me , setting up the problem and digging into some of the math. We'll be looking at it from the lens of our simplest probability problem: the coin flip.
A note: I will not be covering the part that shows you how to make a fortune -- that's an exercise best left to the reader. There is an incredibly fascinating history surrounding the mathematics of gambling and optimal betting strategies. The optimal betting strategy, more commonly known as the Kelly Criterion , was developed in the 50s by J. Kelly , a scientist working at Bell Labs on data compression schemes at the time. In , he made an ingenious connection between his colleague's Shannon work on information theory, gambling, and a television game show publishing his new findings in a paper titled A New Interpretation of Information Rate whose original title was Information Theory and Gambling.
The paper remained unnoticed until the s when an MIT student named Ed Thorp told Shannon about his card-counting scheme to beat blackjack. Kelly's paper was referred to him, and Thorp started using it to amass a small fortune using Kelly's optimal betting strategy along with his card-counting system.
Thorp and his colleagues later went on to use the Kelly Criterion in other varied gambling applications such as horse racing, sports betting, and even the stock market. Thorp's hedge fund outperformed many of his peers and it was this success that made Wall Street take notice of the Kelly Criterion. There is a great book called Fortune's Formula 1 that details the stories and adventures surrounding these brilliant minds.
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