You, however, do not know if it is a fair coin, and are offering me a fair bet. I only have 100 dollars to my name, and I am can bet as much as I want (up to 100 dollars) in either direction at even odds.
If I bet 100 dollars on heads, heads-me gets 200 dollars, and tails-me gets nothing. If I bet 100 dollars on tails, tails-me gets 200 dollars, and heads me gets nothing. If I bet nothing, both versions of me get 100 dollars.
However, every dollar in the hands of heads-me is worth 1.5 times as much as a dollar in the hands of tails-me, since heads-me exists 1.5 times as much. (I am ignoring here any diminishing returns in my value of money.)
Thus, to maximize value I should bet 100 dollars on heads. However, maybe it is better to think of tails-me as the rightful owner of 40 percent of my resources. When I bet 100 dollars on heads, I am seizing money from tails-me for the greater good, since heads-me has the (proportionally greater) existence necessary to better take advantage of it.
Alternatively, I could say that since 60 percent of me is heads-me, heads me should only control 60 dollars, which can be bet on heads. Tails me should control 40 dollars, which can be bet on tails. These two bets partially cancel each other out, and the net result is that I bet 20 dollars on heads.
If you are especially fast at maximizing expected logarithms, you might see where this is going.
Wow I have been looking for an intuitive explanation of Kelly Betting for years, and this is the first one that really hit from an intuitive mathematical perspective.
Is there no way to salvage it via a Nash bargaining argument if the odds are different? Or at least, deal with scenarios where you have x:1 and 0:1 odds (i.e. you can only bet on heads)?
Yes, I got down to the Nash Bargaining part which is a bit harder and got confused again, but this helped as a very simple math intuition for why to Kelly Bet, if not how to calculate it in most real world betting situation.
Wow I have been looking for an intuitive explanation of Kelly Betting for years, and this is the first one that really hit from an intuitive mathematical perspective.
Thanks.
Be warned that this explanation only applies if the environment is offering both sides of every event at the same odds.
Is there no way to salvage it via a Nash bargaining argument if the odds are different? Or at least, deal with scenarios where you have x:1 and 0:1 odds (i.e. you can only bet on heads)?
Where by the “same odds,” I mean if you can take 3:2 for True, you can take 2:3 for False.
Yes, I got down to the Nash Bargaining part which is a bit harder and got confused again, but this helped as a very simple math intuition for why to Kelly Bet, if not how to calculate it in most real world betting situation.