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Unformatted text preview: negative) Kullback-Leibler divergence between the
given distribution π and the dual variable ν .
13.7 Arbitrage and theorems of alternatives. Consider an event (for example, a sports game, political
elections, the evolution of the stock market over a certain period) with m possible outcomes.
Suppose that n wagers on the outcome are possible. If we bet an amount xj on wager j , and the
outcome of the event is i (i = 1, . . . , m), then our return will be equal to rij xj . The return rij xj is
the net gain: we pay xj initially, and receive (1 + rij )xj if the outcome of the event is i. We allow
the bets xj to be positive, negative, or zero. The interpretation of a negative bet is as follows. If
xj < 0, then initially we receive an amount of money |xj |, with an obligation to pay (1 + rij )|xj | if
outcome i occurs. In that case, we lose rij |xj |, i.e., our net is gain rij xj (a negative number).
We call the matrix R ∈ Rm×n with elements rij the return matrix. A betting strategy is a ve...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.
- Fall '13
- The Aeneid