4Kolmogorov's Law of Large Numbers
Borel's law of large numbers no longer holds if we modify Protocol 1.1 by allowing Reality to choose her moves from the whole range of real numbers. In this case Reality can turn any bet by Skeptic, no matter how small, into an arbitrarily large loss for Skeptic: as soon as Skeptic makes his first nonzero move, say 
, Reality can make 
so far from zero in the direction opposite the sign of 
that the negative change in Skeptic's capital, 
, will bankrupt him.
To put Skeptic back into the game, we must offer him additional bets that hedge against such extreme moves by Reality. The most natural hedge is one that sets a price 
for the squared difference between 
and 
. When this hedge is available, Skeptic can choose as his gain function the sum of a multiple of ...
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