c******r
发帖数: 300 | 1 There is a game where you can bet any amount of money you want, the
probability of your winning is p. Suppose you have 0.5 money to start with
and you want to reach 1 eventually.
What is the best strategy then? Hint: p < 0.5 and p >= 0.5 should be dealt
with differently. | p*****k
发帖数: 318 | 2 my naive thought is that if p>0.5, one uses kelly criterion and bet every
time 2p-1 of what he has. but for p<0.5, it's puzzling - how could one
reach 1 eventually with the expected payoff to be negative for each round? | s******s
发帖数: 40 | 3 直觉就是，<0.5全压，>0.5慢慢压，呵呵
【在 c******r 的大作中提到】
: There is a game where you can bet any amount of money you want, the
: probability of your winning is p. Suppose you have 0.5 money to start with
: and you want to reach 1 eventually.
: What is the best strategy then? Hint: p < 0.5 and p >= 0.5 should be dealt
: with differently.
| c******r
发帖数: 300 | 4 Sorry, I didn't make it clear, you want to maximize your probability of
reaching 1, for p < 0.5, you are right, it's not always possible. For p >=0.
5, I am not sure your strategy works since you may have some chance of
losing all the money, however, intuitively speaking, you should be able to
reach 1 almost surely.
【在 p*****k 的大作中提到】
: my naive thought is that if p>0.5, one uses kelly criterion and bet every
: time 2p-1 of what he has. but for p<0.5, it's puzzling - how could one
: reach 1 eventually with the expected payoff to be negative for each round?
| a**m
发帖数: 102 | 5 If p>0.5, the strategy could be that you bet equally for n times, where n is
a very large number. By the law of large number, after n bets, you almost
surely get 2p-1 money, and keep going for the same process, you will finally
end up to or beyond 1.
If p<0.5, my intuition is bet all the money on.
I think the point is when p>0.5, you bet as many times as you can to
minimize the variance (risk to lose); when p<0.5, you bet only once to
maximize the variance (chance to win).
【在 c******r 的大作中提到】
: There is a game where you can bet any amount of money you want, the
: probability of your winning is p. Suppose you have 0.5 money to start with
: and you want to reach 1 eventually.
: What is the best strategy then? Hint: p < 0.5 and p >= 0.5 should be dealt
: with differently.
| j******n
发帖数: 271 | 6 p=0.5 differs from p>0.5 and needs to be considered separately as well. | a**m
发帖数: 102 | 7 As for p=0.5, I think any strategy will lead to the same probability 0.5 to
win.
【在 j******n 的大作中提到】
: p=0.5 differs from p>0.5 and needs to be considered separately as well.
| p*****k
发帖数: 318 | 8 chopinor, just to clarify, when p>0.5, i think there are many strategies
that you can reach 1 a.s., so what do you want to optimize?
(btw, this is probably a spoiler, but im guessing that you have red & black
in mind?) | c******r
发帖数: 300 | 9 hi pcasnik, you are right for p > 0.5, there are many strategies that you
can reach 1 a.s., which is not very interesting part of the problem. I think
expected waiting time can be something you minimize, but I am sure this
then becomes a difficult perhaps open problem.
I think p=0.5 can still be done in roughly the same way since symmetric
random walk is zero recurrent, though LLN won't be useful, CLT definitely
will help out.
p<0.5 is the most interesting part, I don't have a complete solution
【在 p*****k 的大作中提到】
: chopinor, just to clarify, when p>0.5, i think there are many strategies
: that you can reach 1 a.s., so what do you want to optimize?
: (btw, this is probably a spoiler, but im guessing that you have red & black
: in mind?)
| l**********t
发帖数: 5754 | 10 for p>0.5, is it Kelly criterion that has the shortest expected waiting time
(or highest expected captial growth given a fixed time) with zero chance of
ruin? (refer to Edward Thorp's papers). | p*****k
发帖数: 318 | 11 littleshirt, that was my first thought too (see my first post). but a tricky part of this problem is that there is a fixed boundary one wants to reach. it's true that kelly betting beats all other strategies in the long run, but for this specific goal 1, the average stopping time is small if p gets large, so i would think the argument does not work unless p is very close to 0.5.
chopinor, i did not know you posed this problem by yourself. i happened to know a long list of literature for the c |
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