Quote:
Originally Posted by Vantek
Beat me to it, although you should
start with part 1 Way too much unnecessary swearing in all Angry Video Game xxxx type videos, including this imitator of the original, but I think it's a fairly accurate demonstration of the game's entertainment value.
I don't think the Kelly criterion applies to this game at all... Kelly's formula applies to a bet and has to do with optimizing the size of the bet for a specific bankroll for an outcome that will either lose the bet or win at X odds. There is no bankroll in DoND and only speculative money is risked during the course of the show, so the player is freerolling.
When a player turns down an offer he is required to open a set number of cases which represents anywhere from a fifth to half of the current remaining cases. What he is doing each time he rejects an offer is playing a random game revealing more information about cases which aren't his. A formula could be made for it, but the basic idea is that every case he opens, he is randomly altering his odds. The higher the player runs above standard expectation in terms of his luck opening cases (reveals low values instead of high values), the higher the chance that opening the next case could greatly alter his EV for the game as determined by (sum of all case values remaining / cases, including his, remaining).
As has been said, the point of the game is that for every player, there comes a point where it is +lifeEV to take an offer where the value of the potential negative variance of opening more cases represents such a significant amount that it is desirable to settle for a percentage of the Mean of the remaining values than eliminate points from the set. That is what makes it interesting, because the offers are always based on the Mean of the remaining money, but a situation will inevitably arise where none of the data points are anywhere near the mean.
To help people understand the EV of it... if a $1, $5, $10, $50k, and a $100k case remain (one of them is yours)... your EV over infinite trials is the mean of these values, $30k, and the offer will likely be around 70-90% of the mean. Now, if you have to open one more case it puts the system into one of 5 possible states. the average new EV of all these possible outcomes is still around $30k, but the actuality of the situation is that the new EV will be ~$37k 60% of the time (if any of the 3 small values are removed), $25k 20% of the time, and $12.5k 20% of the time. So, oddly, opening a case here is an almost 0EV situation (even with an offer of 100% of the mean), but because the cases contain discrete values the EV is necessarily changing by at least $5k in some direction or another once one is eliminated.