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x ∈ Rn , with as components xj the amounts we bet on each wager. If we use a betting strategy
80.0 Table 1: Odds for the 2000 European soccer championships.
x, our total return in the event of outcome i is equal to
vector Rx. n
j =1 rij xj , i.e., the ith component of the (a) The arbitrage theorem. Suppose you are given a return matrix R. Prove the following theorem:
there is a betting strategy x ∈ Rn for which
Rx ≻ 0
if and only if there exists no vector p ∈ Rm that satisﬁes
RT p = 0, p 0, p = 0. We can interpret this theorem as follows. If Rx ≻ 0, then the betting strategy x guarantees a
positive return for all possible outcomes, i.e., it is a sure-win betting scheme. In economics,
we say there is an arbitrage opportunity.
If we normalize the...
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