nivra
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Post by nivra on Jul 19, 2013 8:20:35 GMT
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tiav
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Post by tiav on Jul 23, 2013 14:27:50 GMT
whereas you are technically correct, you are optimizing in first order dominance only (you are optimizing in expectation, and not accounting for risk). As most players might tend to be slightly risk averse (especially noticeable at higher costs), you need to account for their risk considerations. For instance, if something has an expected cost of 200k with an average distribution interval in the 90% range of 160k-380k, vs. 220k, with an average range of 185k-280k, someone may be willing to spend slightly more to lower the downside risk. In effect, because you are just choosing a price and a risk level, this is basic finance.
This problem is nothing more than the sum of binomial distributions. It is a basic optimal control problem where you attach some penalty for risk and choose the piecewise control at each stage. As it stands, you are making fundamental assumptions that just aren't true...you are assuming people to be risk neutral, and they are not. Partially because, you do not enchant enough in this game to diversify away the risk, and moreso because you are increasing risk exposure with level due to increasing costs and enchanting levels, which fundamentally prevents diversification.
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nivra
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Post by nivra on Jul 23, 2013 15:40:54 GMT
This problem is nothing more than the sum of binomial distributions. It is a basic optimal control problem where you attach some penalty for risk and choose the piecewise control at each stage. Wow. Fascinating. So the part quoted above is a bit too dense for me. I understand what a sum of binomial's is. I understand that each enchant level is kind of a binomial, but instead of the overall # of attempts being fixed, you're trying to reach a set # of successes (1 per enchant level). I have no idea what you mean by optimal control program or piecewise control. I'd love some elucidation on that. As for the overall gist of it. It makes a ton of sense. Thank you. So for each luck stone/enchant level combination, you have a whole distribution of outcomes for cost. I've calculated expected value, but don't have any metric for variance. I can calculate binomial variance for each one and then convert that into cost variance, I think. That should be similar to riskiness, I think. Not sure what to do from there. I've never looked into any literature regarding riskiness of statistical outcomes. |
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tiav
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Post by tiav on Jul 23, 2013 15:55:56 GMT
oops, my apologies, the distribution is negative binomial (geometric), not binomial. The mean is 1/p where p is percentage you force the enchant to go through (with or without stones) and the variance is (1-p)/p^2. The maximum variance occurs with very low p. It occurs when p=10%, and the variance in that case is 90. For a proper analysis, one should consider the expected cost which is
(Cost of Enchant + Cost of Stone)*(1/p)
Your analysis is based on this. However, it should also include a penalty to the risk. One way to do this is to maximize the reward to risk ratio, but the reward and risk need to be calculated in some form of utility function. To even begin that, you have to have a valuation of the enchant, you can calculate it on the range of cost of enchant to cost of enchant + a lvl 9 stone. Therefore, it should be noted that an enchant is worth at most (300k+cost) as this is the cost that one could achieve the enchant every time (this price can be guaranteed with guild shop lvl. 8 as lvl 8 stones can be purchased for 150 contribution). Beyond that, things get hairy. Any form of averaging requires a valuation. Whereas the range of the valuation is easy to determine, narrowing that valuation is subjective. If you set the valuation too high or low, you bias the optimization problem. One possible mechanism is to choose the lowest cost average valuation, and then risk adjust it to choose the optimal stone use. My fear is that this method will also create a bias when using a Sharpe-Ratio-style analysis.
By piecewise control, I mean that the programming problem should be solved independently for each enchant, as each decision and the optimal control (in this case the level stone you use) is an independent decision with an independent cost structure.
I guess I should note that I have a MS in Financial Engineering...so this kind of stuff is kind of my bread and butter.
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