[Math] Proof that the Kelly Criterion is optimal

gamblinginformation theoryoptimization

Let's say you are betting on the outcome of some random variable. You can invest any amount $X$, and each outcome multiples that amount $X$ by a certain amount $ \ge 0$. You must choose $X$ less than how much capital you have. A Kelly Bet is one in which you bet an amount that maximizes the logarithm of your capital. The Kelly Bet will allow your capital to grow larger than any other betting scheme in the long term (i.e., as the number of bets approaches infinity).

What is the proof that previous statement?

Best Answer

The statement is false.

For example consider the bet where you double your money with probability $2/3$ and lose it otherwise. Now consider the strategy "bet \$1 more than the Kelly bet on the first turn, and then use the Kelly bet after that". If this wins on the first turn it will forever have more money than the Kelly strategy, and if it loses on the first turn it will forever have less money than the Kelly strategy. So no matter how large the number of bets becomes, it has a $2/3$ chance of having a larger amount than the Kelly strategy.

Related Solutions

Parimutuel betting: how are bettors incentivised to place their bets early

This is a well known problem of parimutual betting: It is foolish to place bets early. Not only can circumstances change before betting closes---which means that you might have made a different bet with the new information---but you also reveal you information by betting early and aggressively, which makes the odds shift. One paper in game theory and information economics therefore argues that rational and strategic bettors should only place their bets at the last possible moment:

Marco Ottaviani, Peter Norman Sørensen, Surprised by the parimutuel odds?, American Economic Review, Volume 99, Number 5, December 2009.

The solution is to use something called prediction markets. Here, one does not simply pay into a pool and the money gets divided among the winners after resolution of the outcome. Rather, you buy assets whose prices evolve over time. Here, you do have incentives to buy early if you have some good information, because early prices may be "off", and you can sell later at a profit when prices have moved to more reasonable levels.

Betting on game of the World Series

Step 0: Work backwards from the end

Step 1: If the score is $3-3$ you want the net position so far to be $0$, and you bet $1000$

Step 2: If the score is $3-2$ you want the net position so far to be $+500$, and you bet $500$; if the score is $2-3$ you want the net position so far to be $-500$, and you bet $500$

Step 3: If the score is $3-1$ you want the net position so far to be $+750$, and you bet $250$; if the score is $1-3$ you want the net position so far to be $-750$, and you bet $250$; if the score is $2-2$ you want the net position so far to be $0$, and you bet $500$ etc.

You want the position at a particular score to be the average of the two possible positions after the next game. You want to bet half the difference of the two possible positions after the next game. You should end up with this:

Bets:
=====
Score     0       1       2      3  
                        
0       312.5   312.5   250    125  
1       312.5   375     375    250  
2       250     375     500    500  
3       125     250     500   1000  

Positions:
==========                  
Score      0       1       2       3       4
                        
0          0     312.5   625     875    1000
1       -312.5     0     375     750    1000
2       -625    -375       0     500    1000
3       -875    -750    -500       0    1000
4      -1000   -1000   -1000   -1000

Best Answer

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Parimutuel betting: how are bettors incentivised to place their bets early

Betting on game of the World Series

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