Saying that the expected value of this strategy is undefined seems like underselling it. The expected value is positive infinity since the cumulative reward is increasing strictly faster than the cumulative probability of getting nothing.
This might be a disagreement about whether or not it’s appropriate to use “infinity” as a number (i.e. a value). Mathematically, if a function approaches infinity as the input approaches infinity, I think typically you’re supposed say the limit is “undefined”, as opposed to saying the limit is “infinity”. So whether this is (a) underselling it or (b) just writing accurately depends on the audience.
I agree with you that the limit of the EV of “bet until you win n times” is infinite as n→∞. But I agree with Guy Raveh that we probably can’t just take this limit and call it the EV of “always bet.” Maybe it depends on what precise question we’re asking...
Saying that the expected value of this strategy is undefined seems like underselling it. The expected value is positive infinity since the cumulative reward is increasing strictly faster than the cumulative probability of getting nothing.
This might be a disagreement about whether or not it’s appropriate to use “infinity” as a number (i.e. a value). Mathematically, if a function approaches infinity as the input approaches infinity, I think typically you’re supposed say the limit is “undefined”, as opposed to saying the limit is “infinity”. So whether this is (a) underselling it or (b) just writing accurately depends on the audience.
I agree with you that the limit of the EV of “bet until you win n times” is infinite as n→∞. But I agree with Guy Raveh that we probably can’t just take this limit and call it the EV of “always bet.” Maybe it depends on what precise question we’re asking...
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