Horse racing bet Problems

I have always wanted to write a series about discrete Yang. The purpose is to demonstrate the abstract concept of Yang, which is actually applied in a variety of ways. I remember that when I was a beginner in the concept of Yang, I had to spend a day not getting used to it. It's hard to use a theorem to finish teaching materials. After a while, I forgot about it. This is due to the lack of examples. Only specific examples can make abstract theories lively and vivid. This series contains the following questions. They are all wonderful examples and the knowledge used is not complex.

 

• Horse racing bet Problems
• Spacecraft Space Jump
• Reverse intuition probability in the pattern
• Blind date problems
• Rocky sheep Problems

 

Of course, the discrete Yang theory can solve more than this problem, and the subsequent examples will be supplemented with time. The continuous Yang theory will be included in the series of random analysis.

 

This article will introduce the question of Horse Racing bet:

 

Assume that in a bet, the probability of a gambler winning is $ P> 1/2 $. If the bet is to be won, the bet will be lost if the bet is equal to the bet. Now a gambler performs $ N $ game with initial funds $ y_0 $. Here $ N $ is a given positive integer. Set the money for gamblers after the game ends to $ y_ I $, gamblers want to maximize the expected growth rate by $ \ frac {1} {n} e \ log \ frac {Y_n} {y_0} $ (equivalent to maximizing $ e \ log Y_n $ ), so every time you place a bet, how much money should the gambler make for his current bet? Note that the gambler's strategy can be changed according to the changes in the bet, and the bet ratio of each bet does not have to be the same.

 

The controversial question here is why we need to maximize the log expectation, rather than directly asking $ N $ to maximize the expected funds after the Bureau $ ey_n $. In fact, if $ ey_n $ is required to maximize, the bet strategy is to place all the funds on each bet, but this will bring a great risk: as the number of Bureaus increases, gamblers will almost lose a game, however, a single failure will result in no loss. Such a bet strategy is a bad one. We will see later that the optimal strategy for maximizing the corresponding growth rate will avoid this problem: the bet on each bet is strictly less than the principal.

 

Use a mathematical language to describe the problem:

 

$ N $ gambling results can be described using $ N $ independent Bernoulli random variables $ x_1, x_2, \ ldots, X_n $: \ [P (x_ I = 1) = P, \ quad P (x_ I =-1) = 1-P = Q. \]

The information given by the previous $ I $ bet is $ \ mathcal {f} _ n = \ sigma (x_1, \ ldots, X_ I) $. Set the bet of a gambler in the $ N $ bet to $ C_n $. $0 <C_n \ Leq Y _ {n-1} $ is required here. Then $ C_n $ about $ \ mathcal {f }_{ n-1} $ is measurable, and $ Y_n = Y _ {n-1} + c_nx_n $.

 

The key step is to prove the following conclusion:

 

Theorem:No matter what strategy $ \ {R_N \} $, $ Z_N = \ log Y_n-n \ Alpha $ is always an upper ring, here $ z_0 = Log y_0 $, $ \ alpha = p \ log P + Q \ log q + log2 $.

 

Horse racing bet Problems