interesting coin game

wilsonnl · February 23rd, 2013, 7:53 pm

Given $10 capital, you are to play a fair coin game where you are asked to first choose the number of flips, say M times. For every flip, head you win 2M, tail you loose M. During the game, if you run out of money then the game stops. How do you choose M to maximize your return? then for $ n capital?If just consider the expected return, it seems the higher the M the better (in the end we win more with head than loose). Shall we consider here something like a utility function?

KnutSchnute · February 23rd, 2013, 9:15 pm

Hi wilsonnl,I think we should consider a utility function for reflection of risk attitude. If I am not mistaken, this setup is called St. Petersburg paradoxon. If you play it forever, your expected gain is infinite.

wilsonnl · February 24th, 2013, 5:21 am

thanks for the reply. I guess the similarity between the two is that both have infinite expected returns. However, this coin game gives the payoff after each flip (in contrast to St. Petersbug where player get paid only at the end), so your wealth will change during the game, so does the stopping time. Any suggestion on how to express the expected return or expected utility function with stopping criteria included?

isometry · February 26th, 2013, 12:58 pm

Assuming you invest all your winnings back to the game , you should calculate the utility at the stopping times T1, T2, ...Then select the stopping time which gives maximum expected utility.