there is a game:
you can get 1 free bottle. or you bet this bottle, for a chance of 1/2 for 3 bottles or 0 bottle.
what will you do?
In logic, you should bet the bottle, coz the expectation is 1.5 bottle, while the other one is 1 bottle only.
However, don't for get what is "expectation", which is the mean gain from large number of trials.
if you only get one chance, what will you do?
may be we can introduce a utility for the bottle. say, you need a bottle so much. like a medicine, you need it for your life, so, the question will become weather you have a bottle for sure or chance for more or zero bottle. Thus, 3 bottles for you may be not so useful, attractive or necessary. therefore, if people have no bottle on their hands, they will prefer to have the guarantee bottle.
If people already has few bottle on their hands, thus, the guarantee bottle is not so attractive. so, why not gambling?
The utility function can be like this. it depends on the result, on how many bottles people already have.
For no bottle people, their utility function may be 1 for the guarantee bottle and also 1 for the 3-unsure-bottles. means both of them is equal attractive. thus, the expectation of utility is 1 to 1/2, ok, pick the guarantee one.
For had some bottles people, the 1 bottle may be only 1 for them, but 3 for the 3 bottles, which means, the 3 bottles is more attractive. the expected utility is 1 to 3/2. Gambling.
How about the 3 bottles changed to 100 bottles?
the utility changed. and you can see, different people has different utility.
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Let's talk about utility.
although utility is different from different people, it can be compare within same people.
what does it mean i like coke more than milk? if my utility function for coke is 3, milk is 1. thus. i like to trade 1 coke for 3 milk in order i got satisfy.
wait! how about i hate milk? at this case, the utility of milk will be negative, say -1. Thus, no matter how, i will not trade a coke for milk!
next time, if you want to compare somethings logically, try to think how many of these can trade for that.
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back to the game.
in order people will gambling even they got no bottle, the utility must be 1 for the 1-bottle and at least 2 for the 3-bottles. This means, you like to use 2 guarantee-bottles for 3 unsure-bottles. the uncertainty lower the attractive to 2/3 for each unsure-bottle.
How about in 1-sure-bottle vs 100-unsure-bottles with 1/2 chance?
the utility for choosing 100-unsure-bottles is still at least >2 . and the attractive for each unsure-bottle becomes at least 1/50 due to the uncertainly. we can see, this de-valuation is so huge. many people will agree the attractive of a unsure-bottle should be more than 1/50. say, 1/2. Thus, from the utility, we can see the total utility for 100-unsure-bottles is 50, very easy to larger than 2. Therefore, people may even choose to take the chance even though they did not have any bottle.
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Other issue, in the game holder view point.
people can buy a coke or buy a chance to win 3 coke with same price.
............um.... i have to think about this......
How about apply this "trading" for politics? to solve conflict? like terrorism?
How much they can trade for the "Muslim" to stable life?
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