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# Two Measures of Incoherence: How Not to Gamble If You Must

**Mark J. Schervish, Teddy Seidenfeld and Joseph B. Kadane**

### Abstract:

*The degree of incoherence, when previsions are not
made in accordance with a probability measure, is measured by either
of two rates at which an incoherent bookie can be made a sure loser. Each bet
is considered as an investment from the points of view of both the
bookie and a gambler who takes the bet. From each viewpoint, we
define an amount invested (or escrowed) for each bet, and the sure
loss for incoherent previsions is divided by the escrow to determine
the rate of incoherence. Potential applications include the treatment
of arbitrage opportunities in financial markets and the degree of
incoherence of classical statistical procedures. We illustrate the
latter with the example of hypothesis testing at a fixed size.
**
** Keywords:* Bookie, Coherence, Escrow, Gambler, Rate of
arbitrage.

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