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I found this question from Ian Hacking's book on probability and induction.

Diogenes is a cynic. He thinks the Maple Leafs will come in last in their league next year. His betting rate that they will come in last (proposition B) is 0.9. His betting rate that they not come in last (proposition ~B) is 0.2. Make a sure-loss contract against Diogenes.

It seems that a sure-loss contract is one wherein Diogenes loses every time. I don't understand how to make one from the given information though. Are you supposed to simply state prices that would provide Diogenes with a net loss every time?

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  • $\begingroup$ How exactly do "betting rates" work here? Are they odds or probabilities? $\endgroup$ – hmakholm left over Monica Mar 17 '17 at 11:59
  • $\begingroup$ I copied the exact question, but I'm assuming it's going to be probabilities. $\endgroup$ – Andrew Raleigh Mar 17 '17 at 12:15
  • $\begingroup$ Still confused. If these numbers mean that Diogenes offers a bet that will cost me \$9 and he pays out \$10 if $B$ happens (i.e. for the purpose of that bet he believes the probability of $B$ is $0.9$), and that he also offers a bet that will cost me \$2 and he pays out \$10 if $\neg B$ happens (for that he believes the probability of $\neg B$ is $0.2$) -- then it appears the implied house spread favors him and not the player. $\endgroup$ – hmakholm left over Monica Mar 17 '17 at 12:35
  • $\begingroup$ I'm also unsure. The betting rate is his personal probability, would giving him an unfair rate make him lose when combining the chances of the leafs coming in last and when they don't come in last? $\endgroup$ – Andrew Raleigh Mar 17 '17 at 12:44
  • $\begingroup$ Wait, is Diogenes a bookmaker (and our choice is to take the bets he offers or go elsewhere), or a player and we're the bookmaker (so we can propose a bet that he will either accept or decline)? $\endgroup$ – hmakholm left over Monica Mar 17 '17 at 12:47
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If we're the bookmaker and Diogenes is a customer (as clarified in comments), then offer him two bets

  • In the first bet he pays in \$89 and we promise to pay him \$100 if $B$ happens. Since he thinks the probability of $B$ is $0.9$ he should think the chance of winning \$100 is worth \$90 to him, so he would take the bet.

  • In the second bet he pays in \$19 and we promise to pay him \$100 if $\neg B$ happens. Since he thinks the probability of $\neg B$ is $0.2$ he should think the chance of winning \$100 is worth \$20 to him, so he would take the bet.

If he takes both bets, he will be paying us \$108 in total, and we must pay him \$100 no matter what happens -- a sure profit for us.

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  • $\begingroup$ That makes a lot of sense, thank you! Does the same apply to this question? math.stackexchange.com/questions/2190251/… $\endgroup$ – Andrew Raleigh Mar 17 '17 at 13:41
  • $\begingroup$ More or less -- except in order to exploit Epicurus' inconsistent probabilities you would probably need to offer him "opposite" bets where you pay him a fixed amount to start with and then he pays you back something if such-and-such happens. With his probabilities you should be able to work out a hedge where you're sure to win \$100 but you paid less than \$100 to make that happen. $\endgroup$ – hmakholm left over Monica Mar 17 '17 at 13:49

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