Conditional Credences and Conditional Betting Odds Come Apart
It has been pointed out that credences and betting odds sometimes come apart: one’s credence in A cannot always be measured by one’s fair betting odds for A. (See “What are Degrees of Belief?” by Hajek and Erikkson.) So it should not be surprising that one’s conditional credence, P(B|A), cannot always be measured by one’s fair betting odds for a conditional bet on B given A. Suppose my conditional credence, P(There isn’t and won’t be any indication that my friend has betrayed me | My friend has betrayed me) is high. Nonetheless, it won’t be wise for me to bet any (positive) amount of money that there isn’t and won’t be any indication that my friend has betrayed me, given that she has. For suppose we find out that my friend has not betrayed me. In such a case, the bet is called off, and I don’t gain or lose any money. But suppose we find out that my friend has betrayed me. Then it is false that there isn’t any indication that my friend has betrayed me, in which case I lose the bet. So the conditional bet is one that I will not win, and might well lose. Conditional credences and conditional betting odds come apart.