Am I stupid? I can't fit these two concepts into any type of harmony.

[BillC]

The expected value of the bankroll is a decreasing function of time. It has a negative derivative, i.e., a negative growth rate.

Do not confuse the expected bankroll's rate of change with positive expectation on a particular bet, or series of bets.

[PairTheBoard]

[ QUOTE ]
The expected value of the bankroll is a decreasing function of time. It has a negative derivative, i.e., a negative growth rate.

Do not confuse the expected bankroll's rate of change with positive expectation on a particular bet, or series of bets.

[/ QUOTE ]

Just to be clear here. Playing with a fixed positive edge and proportional betting which exceeds twice the Kelly optimum proportion. You say,

" The expected value of the bankroll is a decreasing function of time. "

Calling David Sklansky. Do you still agree with Bill here David? Anybody else?

PairTheBoard

[PairTheBoard]

[ QUOTE ]
BillC --
<font color="white"> ,
</font> The expected value of the bankroll is a decreasing function of time. It has a negative derivative, i.e., a negative growth rate.



[/ QUOTE ]

FALSE

Let me give a simple example that proves you are wrong here Bill.

Consider a weighted coinflip where you have a 55% chance of winning. Regular 1-1 payoff. Let your initial bankroll be 1 unit and make one coinflip bet per unit of time. Your proportional betting will be 100% of your current bankroll.

What is the expected value of your bankroll at time t=n?

E[Bn] = (.55)^n * 2^n = 1.1^n

The "expected value of your bankroll as a function of time" not only is NOT decreasing, it is an Increasing Function of Time. In fact, the expected value of your bankroll as a function of time Grows Exponentially to Infinity.

This is true even though the "Growth Rate" for your Bankroll is Negative and your Bankroll at time t=n converges in probabilty to zero as n--&gt;infinity.

PairTheBoard

[PairTheBoard]

btw, If you insist on the proportional betting being strictly less than 100% of the Bankroll, then in the above example make it 99%. The expected value of your bankroll at time t=n flips is then greater than the RHS below:

E[Bn] &gt; .55^n * 1.99^n = 1.0945^n

Now you never go broke, the "Growth Rate" of your bankroll is negative, your bankroll still converges in probabilty asymptotically to zero, but the Expected Value of your Bankroll grows exponentially toward infinity.

PairTheBoard

[David Sklansky]

Please let your uncle use the computer again.