Conference rusure::math

>> Hence the value of this strategy is infinite.  But the game is
>> symmetric, so the value of the corresponding strategy for your
>> opponent is also infinite.  Although it's zero-sum, in the long
>> run both you and your opponent come out WAY ahead!
        
        I was going to post this as a reply in the humor topic,
        but eventually decided to give it its own topic.
        
        Dan

    I didn't catch how you determine who the winner is.

    Re .2:
    
    The winner is determined by the sign of i-j.  If the first player
    presents a larger i than the second player presents a j, then the first
    player gets i-j dollars from the second.  If the second player presents
    a larger j, then the second player gets j-i dollars.
    
    
    				-- edp

>Here's a nice game-theoretical paradox from G. Owen, _Game Theory_:
> 
>Consider the 2-person zero-sum game with payoff
>     a[i,j]=i-j, where i,j natural numbers.
>That is, you pick a natural number and your opponent picks one, 
>and the loser pays the winner the difference.  Note: the game is 
>symmetric, and there are an infinite # of pure strategies.

As stated, this is not a very interesting game. It's a variant of a game 
common to small children (who tire of it very quickly) called "name the 
biggest number" or "tell the biggest lie". This game, with the added 
stipulation that play is asynchronous, has been discussed elsewhere in this 
conference: the last player always wins!

One way of describing the given game is that it has a saddle point at 
infinity. As one might expect, many infinite games have this property, so 
it's not really paradoxical at all.

The game becomes more interesting if the payoff is made something like
1/(i-j), which encourages each player to try to name a number much closer
to his opponent's. I think the solution is the same, but the payoffs get
ridiculously small. 

	The fallacy in the paradox is the same one as in the Cravat Incident:

	Bill and Ben each receive a cravat for Christmas.   Each is very proud
of his cravat, and they get into a discussion about which was the more costly.
Eventually they agree that they will find out the prices, and whoever has
the more expensive cravat will give it to the other as a consolation.

	Bill reasons as follows.   He may have the more expensive tie or the
cheaper one.   If he has the more expensive cravat, he'll lose it, but if
his is cheaper, he will *gain* *a* *more* *expensive* *cravat*.   Ergo, he
can't lose.   But Ben can reason exactly the same way...


	There are many paradoxes of the infinite in Game Theory.   Some others
involve Measure Theory to resolve, other concerned with non-terminating games
are much deeper.   Conway's "On Numbers and Games" is at one level a
generalization of the concept of transfinite number to that of a game!

Regards,
Andrew.

The cravat "game" is, I think, neither a fallacy or (in the strict sense) a
paradox - it's just a non-zero-sum game, with "nature" providing the
difference between the costs of the two cravats. 

This situation is commonplace in economic models: I buy jelly from a grocer 
at a price that is cheaper than my cost for making it myself, while the
jelly maker (as well as the grocer) makes a profit, so everyone's payoff 
from playing the game is positive.

    The cravat game *is* a fallacy, I think.

    The error in the reasoning is that there exists a well defined
    distribution of the probability of cravats, whether the player knows
    what that distribution is or not.  If the player's cravat is low in
    price, relative to that distribution, then it is probable that the
    price of the opponent's tie is higher, and vice versa.  To see this,
    imagine that there are only three "models" of cravat: Cheap ($3), Nice
    ($18) and Fancy ($300), and assume that they are bought for gifts in
    equal numbers.  The payoff matrix is then:

	      C    N    F     total
	    +----+----+----+
    	  C |  0 | -1 | -1 |  -2
    	    +----+----+----+
    	  N |  1 |  0 | -9 |  -8
    	    +----+----+----+
    	  F |  1 |  9 |  0 |  10
    	    +----+----+----+ ----
			       0

    In this case if the player *knows* he has a C cravat his expectation is
    that he will lose $2 worth of value, because 2 times out of 3 he will
    lose its $3 value.  If he knows he has an N cravat his expectation is
    that he will lose $8 worth of value, because 1 time out of 3 he will
    lose the $18 cravat and 1 time out of 3 he will win -- but only a $3
    cravat.  If he knows that he has an F cravat than his expectation is
    $10.  If he does not know which category of cravat he has -- more
    specifically, if his evaluation of the probability of each kind of
    cravat is the same for him and his opponent, then his expectation is
    0.

    This differs from the "infinite" problem, in that it is not clear
    that there *is* a well defined distribution in that case.  If there
    is (which depends on interpretation, is there a well defined
    distribution of numbers which *might be named* in practice) then
    it is a fallacy, if there is not then it is a paradox of the infinite
    -- which, depending on your interpretation of foundations, may be
    taken as another fallacy (improper extension of finite methods to
    infinite sets) or as a weakness in the theory.

				    Topher