# Measure of value of resources in a competitive game

Let we have a competitive survival game in which a player has choice between different resources to earn. The question here is which resource should he prefer to maximize the chance of survival. I tried to find out if there already established measures of such sort, but so far without success.

Some thoughts led me to the following measure of the resource' value:

$$V=\int_{t_0}^{t_1} e^{\int_0^t \log (p_1(u))du}\log \frac{p_1(t)}{p_0(t)} dt$$

where $p_0(t)$ is the probability density of survival without the resource, $p_1(t)$ is the probability density of survival with the resource and $(t_0,t_1)$ is the period of time through which the resource affects the probability of survival (the time for which the value is evaluated considered t=0)

I would like to know if there any similar measures already proposed and about the possible drawbacks of this proposed value. I also wonder to which extent this can be applied to real economics.

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Why has this got a close vote? –  Aryabhata Nov 12 '10 at 17:26
I'm not sure where the above formula came from, but (assuming all resources are equally difficult to earn) why not simply choose the resource that maximally increases the expected time of survival? –  RandomGuy Feb 22 '11 at 1:00
Yes, but how would you compare resources in this case? –  Anixx Sep 7 '11 at 23:57
How about formal values from cooperative game theory? i.e. the Shapley value or the Banzhaf power index? They seem to be pretty appropriate for this setting as far as I understand it. –  Yair Zick Dec 11 '12 at 16:52

## 1 Answer

I'm not sure about your formula. Nor about your goal. For example, assume your char survives 80 years by default. But if your strategy is to search a dangerous jungle for the Fountain of YouthTM, you die at the age of 78 from exhaustion with probability 0.99, but you live 1000000 years if you succeed in finding the fountain (with probability 0.01).

• Searching the fountain maximizes the expected survival time
• Searching maximizes the probability to survive 81 years
• Not searching maximizes the probability to survive 79 years
• in a game with one searching and one non-searching player, the non-searcher wins with probability 0.99
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