Tuesday, June 14, 2011

Is utility unbounded?

 A thought-experiment.

If utility is unlimited, then an infinite amount of money, offered at any finite odds, is infinitely more desirable than any finite amount of money.  We would notice this effect in that infinite prizes in lotteries would be surprisingly tempting.  Are you currently stealing from your friends in order to play high-stakes lotteries?  Would you gamble everything to play the lottery if the jackpot were infinite?  If you answer "no" and "yes," then maybe your utility curve is unlimited.  People do play lotteries more when the jackpot rises.  This change suggests that people are able to manage their probabilities, make side bets, divide potential lottery winnings and otherwise make real sense out of quantities of money far greater than they have ever earned or consumed.  Apparently, when faced with a lottery which pays hundreds of millions of coin at thousand-million-to-one odds, a 50% increase in the jackpot increases the players' interest.  What would happen if the odds remained fixed at thousand-million-to-one but the jackpot were infinite?  We might prepare for the shock to the global economy and government as the world receives its monarch.  Would you, in addition, try to win?  Would you give up every comfort in your life to win?  Would you sell, steal, borrow?  Would you attempt grand crimes to obtain vast sums with which to win the lottery?  Would you abuse the trust of your family and friends?  If not, then you have put a value c on your own comfort and the good you can do as a good citizen, and you have put a value v on economic omnipotence such that

  v < a thousand million times c. 

v is the utility bound -- the net sum of all the good you could do and all the fun you could have with unlimited resources.  On the other hand, if you are reading this and if your conscience whispers "Yes, I would beg borrow and steal for a chance at v" then you have valued v > a thousand million times c.  That wouldn't prove that utility is unbounded.  To prove that utility is unbounded, we should give you worse odds, such as a trillion-trillion to one, and better comforts, say c' = all the good you can do and all the fun you can have when you step into the role of someone -- anyone -- who you think seems to have a lot of fun and/or do a lot of good.  If your heart knows that would happily give up all that person's pleasures and good work in order to ruin his life with your gambling addiction, then you believe that

  v > a trillion trillion times c'. 

We could check that v exceeds any finite bound by setting the odds arbitrarily low.  A person who believes in unbounded utility offers to behave badly if he or she believes in the existence of an infinite lottery.  Such a person would happily suffer any finite setback and cause any finite amount of damage in exchange for a chance at winning v.  Perhaps your neighbors are such people; your neighbors act normally because they do not believe in an infinite lottery and they are unwilling to behave badly in order to win the kinds of lotteries which they are offered.  If the neighbor is not stealing from you and buying lottery tickets with the stolen coins, it is because the neighbor values

a lottery jackpot * lottery odds < the cost of stealing a coin . 


It's a strange value in cost-benefit analysis because it continues to have its value ay any odds, and it outweighs any finite cost.  We avoid this by believing in no infinitely-valuable properties.  If a thought experiment asks us to accept that something (the prize in a lottery) might have a value infinitely greater than anything else in the universe, we can reject that notion with the Archimedian principle that everything is comparable to everything else -- all pains and pleasures, goods and and evils are comparable to each other.  Or, if some things are infinitely better than others, we can ask whether there is a world of top goods which are all comparable. -- perhaps my own good citizenship has far-reaching and "infinite" consequences.


Someone who says "utility is unbounded" can depress the utility(coins) function to the point where we cannot tell the difference between his bets and those of someone who believes utility to be bounded by examining data-points with realistic expected values.  You can fit to a scatter-plot of observations functions which are bounded or unbounded.  Maybe we can find some data about willingness to play the lottery for very high values, and this would show the upper end of the utility curve.  The surprising result seems to be that people will play a lottery for low expected value if the prize is high, suggesting that they value the marginal coin more than a coin in the hand; that makes sense to me only in terms of investing that coin, not consuming it.  I wrote about consumption, investment, and gambling when discussing the utility of gambling.

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