a**********n
发帖数: 5 | 1 Currently, you have $100. You go to play a game.
Each time, flip a coin, which appears head with probability 0.75, appears
tail with probability 0.25.
Each time, you can bet x dollars, if it's head, you get x , if it's tail,
you lose x.
You can play 10 times. What's the maximum expected payoff after 10 times? | a**r
发帖数: 19 | 2 这个问题有答案了吗？是不是应该用stochastic dynamic programing来解？
【在 a**********n 的大作中提到】
: Currently, you have $100. You go to play a game.
: Each time, flip a coin, which appears head with probability 0.75, appears
: tail with probability 0.25.
: Each time, you can bet x dollars, if it's head, you get x , if it's tail,
: you lose x.
: You can play 10 times. What's the maximum expected payoff after 10 times?
| p*****k
发帖数: 318 | 3 assume one can only bet at most all the money in her hand (so once ruined in the middle of the 10 rounds, you cannot play the remaining rounds), and the money you won from last round can be used to bet in the current round.
if i induct backwards, i.e., first say one can only play for 1 round which is trivial, then apply the result from the case that one can play for n round to the case that one is allowed to play for n+1 round. seems to me in order to achieve the maximum expected payoff, one wo | D**u
发帖数: 204 | 4 Notice that, if you play optimally, your expected payoff is a multiple of
your initial asset, where the multiple is independent of your initial asset.
So your optimal strategy is on the boundary of your strategy space. In this
problem, you should either bet everything or 0 on each step. Then betting
everything each step is the optimal strategy.
So the expected payoff of 10 rounds is:
100 * (2 * 0.75)^10 = 100 * (3/2)^10.
【在 a**********n 的大作中提到】
: Currently, you have $100. You go to play a game.
: Each time, flip a coin, which appears head with probability 0.75, appears
: tail with probability 0.25.
: Each time, you can bet x dollars, if it's head, you get x , if it's tail,
: you lose x.
: You can play 10 times. What's the maximum expected payoff after 10 times?
| c**********e
发帖数: 2007 | 5
> however, the risk is extremely huge as well.
No. You have 5.63% chance of winning $102,300,
and 94.47% chance of losing $100. The expected gain is
$5666.5. The risk can not be considered huge.
【在 p*****k 的大作中提到】
: assume one can only bet at most all the money in her hand (so once ruined in the middle of the 10 rounds, you cannot play the remaining rounds), and the money you won from last round can be used to bet in the current round.
: if i induct backwards, i.e., first say one can only play for 1 round which is trivial, then apply the result from the case that one can play for n round to the case that one is allowed to play for n+1 round. seems to me in order to achieve the maximum expected payoff, one wo
| a**r
发帖数: 19 | 6
asset.
~~~~~~~~~~~~~~~~~~~~why? 是因为这个game是可以线性叠加或者拆分的吗？
this
～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～
我用stochastic dynamic programming也是这个结论
～～～～～～～～～～～～～～～～～～～～
我同意这个结论。
【在 D**u 的大作中提到】
: Notice that, if you play optimally, your expected payoff is a multiple of
: your initial asset, where the multiple is independent of your initial asset.
: So your optimal strategy is on the boundary of your strategy space. In this
: problem, you should either bet everything or 0 on each step. Then betting
: everything each step is the optimal strategy.
: So the expected payoff of 10 rounds is:
: 100 * (2 * 0.75)^10 = 100 * (3/2)^10.
| D**u
发帖数: 204 | 7 Suppose player A starts with X dollars and always bets optimally at each
step. If you start with Y, you can just "mimic" A's optimal play by always
betting Y/X times what A's betting at each step. Due to the symmetry between
X and Y, you are playing optimally as well.
This easily leads to the conclusion: In order to maximize your expected
final payoff, it is equivalent to maximize your expected payoff for EACH
step. So you should bet everything or non on each step, and easy to check
that betting
【在 a**r 的大作中提到】
:
: asset.
: ~~~~~~~~~~~~~~~~~~~~why? 是因为这个game是可以线性叠加或者拆分的吗？
: this
: ～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～
: 我用stochastic dynamic programming也是这个结论
: ～～～～～～～～～～～～～～～～～～～～
: 我同意这个结论。
|
|
