Thanks for posting this. I’ve always been skeptical of the idea that you should offer two sided bets, but I never broke it down in detail. Honestly, that is such an obvious counter-example in retrospect.
That said, “must either accept the bet or update their beliefs so the bet becomes unprofitable” does not work. The offering agent has an incentive to only ever offer bets that benefit them since only one side of the bet is available for betting.
I’m not certain (without much more consideration), but it seems that Oscar_Cunningham’s solution of always taking one half of a two sided bet sounds more plausible.
Suppose David is willing to stake 100:1 odds against Trump winning the presidency (before the election). Assume that David is considered to be a perfectly rational agent who can utilise their available information to calculate odds optimally or at least as well as Cameron, so this suggests David has some quite significant information.
Now, Cameron might have his own information that he suspects that David does not and Cameron know that David has no way of knowing that he has this information. Taking this info into account, and the fact that Cameron offered to stake 100:1 odds, he might calculate 80:1 when his information is incorporated. So this would suggest that David should take the bet as the odds are better than Cameron thinks. Except, perhaps David suspected that Cameron had some inside info and actually thinks the true odds are 200:1 - he only offered 100:1 to fool David into thinking it was better that it was—meaning that the bet is actually bad for Cameron despite his inside info.
Hmm… I still can’t get my head around this problem.
The offering agent has an incentive to only ever offer bets that benefit them
Right, and with two-sided bets there’s no incentive to offer them at all. One-sided bets do get offered sometimes, so you get a chance for free information (if the other agent is more informed than you) or free money (if you think they might be less informed).
Thanks for posting this. I’ve always been skeptical of the idea that you should offer two sided bets, but I never broke it down in detail. Honestly, that is such an obvious counter-example in retrospect.
That said, “must either accept the bet or update their beliefs so the bet becomes unprofitable” does not work. The offering agent has an incentive to only ever offer bets that benefit them since only one side of the bet is available for betting.
I’m not certain (without much more consideration), but it seems that Oscar_Cunningham’s solution of always taking one half of a two sided bet sounds more plausible.
Partial analysis:
Suppose David is willing to stake 100:1 odds against Trump winning the presidency (before the election). Assume that David is considered to be a perfectly rational agent who can utilise their available information to calculate odds optimally or at least as well as Cameron, so this suggests David has some quite significant information.
Now, Cameron might have his own information that he suspects that David does not and Cameron know that David has no way of knowing that he has this information. Taking this info into account, and the fact that Cameron offered to stake 100:1 odds, he might calculate 80:1 when his information is incorporated. So this would suggest that David should take the bet as the odds are better than Cameron thinks. Except, perhaps David suspected that Cameron had some inside info and actually thinks the true odds are 200:1 - he only offered 100:1 to fool David into thinking it was better that it was—meaning that the bet is actually bad for Cameron despite his inside info.
Hmm… I still can’t get my head around this problem.
Right, and with two-sided bets there’s no incentive to offer them at all. One-sided bets do get offered sometimes, so you get a chance for free information (if the other agent is more informed than you) or free money (if you think they might be less informed).