A very irrational problem
Formulate it in this way:
i) The probability of winning, p, is no greater than half, i.e. <=0.5
ii) Your utility function is increasing, but concave. Call it U(x), then U'>0, but U''<0
You start from x dollars. You are allowed to bet some money to maximize your expected utility.
Call it ax.
Then we want max(pU(x+ax)+(1-p)U(x-ax)) , subject to 0<=a<=1
Let f(a)=pU(x+ax)+(1-p)U(x-ax)
f'(a)=pxU'(x+ax)-(1-p)xU'(x-ax)
Note that p<=0.5 <=> p<=(1-p)
Also, U''<0 <=> U' is decreasing
Therefore, pxU'(x+ax)<(1-p)U'(x-ax)
f'(a)<0
The optimal betting strategy is not to gamble, even if p=0.5.
還有, 看不懂的是"每次输了都买从A元开始输了多少元的两倍。"
如果是這樣, 那你應該是 A, 2A(=(A*2)), 6A(=(A+2A)*2), 18A, etc.
而不是A,2A,3A,6A...
最後, 如果我沒搞錯的話, 你就是那些behaviorial finance 所說的人
i) gambler's fallacy: given you lose this game, you think you are less likely to lose the next one; but the two games are independent, so the losing probability keeps the same
-> given you lose your last game, the expected earning of your last game is still 0, if p=0.5, and negative if p<0.5
ii) focus on gains/losses, not final wealth
iii) convex utility in the negative region: people are risk-adverse in gains, but risk-seeking in losses. They think it is not worthwhile to gamble more (if you have 1 million, you are not quite likely to take out half of them for an outcome of either 0 or 2 million); but if you have lost money, you think you should gamble, because at most you lose all your remaining money, which may not be much, but you enjoy a big upward potential.
嘛...有錯請指教了
我認為最荒謬的其實是
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