A method for fair bargaining over odds in 2 player bets!

Alice and Bob are talk­ing about the odds of some event E. Alice’s odd’s of E are 55% and Bob’s are 90%. It be­comes clear to them that they have differ­ent odds, and be­ing good (and com­pet­i­tive) ra­tio­nal­ists they de­cide to make a bet.

Essen­tially, bet con­struc­tion can be seen as a bar­gain­ing prob­lem, with the gap in odds as sur­plus value. Alice has pos­i­tive EV on the “No” po­si­tion for bets at >55% odds. Bob has neu­tral or bet­ter EV on the “Yes” po­si­tion for bets at <90% odds.

Naive bet con­struc­tion strat­egy: bet with 5050 odds. Nega­tive EV for Alice, so this bet doesn’t work.

Less naive bet con­struc­tion strat­egy: Alice and Bob ne­go­ti­ate over odds. The prob­lem here, in my eyes, is that Alice and Bob have an in­cen­tive to strate­gi­cally mis­rep­re­sent their pri­vate odds of E in or­der to ne­go­ti­ate a bet­ter bet. If Alice is hon­est that her odds are 50%, and Bob lies that his odds are 70%, so they split the differ­ence at 60%, Bob takes most of the sur­plus value.

If both were hon­est and bar­gain­ing equitably, they’d have split the differ­ence at 72.5% in­stead. So I’ll call 72.5% the “fair” odds for this bet.

A nicer and more ra­tio­nal­ist al­igned bet con­struc­tion strat­egy wouldn’t re­ward dishon­esty! So, here it is.

1. Alice and Bob sub­mit their max­i­mum bets and their odds.

2. Take the min­i­mum of the two max­i­mum bets. Let’s say its $198.

3. Con­struct 99 mini bets*; one at 1% odds of E, 2% odds of E… 99% odds of E. Each player au­to­mat­i­cally places 2$ on each mini bet that is fa­vor­able ac­cord­ing to their odds ($198/​99 = $2).

*99 cho­sen for sim­plic­ity. You could choose a much higher num­ber for the sake of gran­u­lar­ity.

So, in this case, Alice ac­cepts the No po­si­tion on all bets at =>55% odds, and Bob ac­cepts the Yes po­si­tion on all bets at =<90% odds, so they make 35 $2 bets, the av­er­age odds of which are 72.5%, which is the fair odds.

Ob­serve that there is no in­cen­tive for ei­ther player to have mis­rep­re­sented their odds. If Alice over­rep­re­sented her odds as 60%, she would just deny her­self the abil­ity to bet on bets at 56% through 59%, which have pos­i­tive EV for her.

Note that Alice and Bob only bet $70 -- less than half of the max­i­mum bet. If Bob wanted Alice to bet more money than she was re­ally will­ing to risk, he might try to con­vince her that his odds were close to hers, such that a high max­i­mum bet would still lead to a low ac­tual bet. Does this seem like a prob­lem to you? I think this method is still an im­prove­ment.

*The mini bets can be ab­bre­vi­ated an­a­lyt­i­cally as one bet at av­er­age odds, I just like the mini bets con­cept for mak­ing the in­tu­ition clear.

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I am not sure how ex­cit­ing this method is to any­one. I like it be­cause mis­rep­re­sen­ta­tion of value is a core prob­lem in 2 player bar­gain­ing, and I re­al­ized bet­ting is a bit of a spe­cial case. Other spe­cial cases ex­ist too—any­thing where play­ers would be will­ing to ac­cept un­cer­tainty over the size of the trade (and are trad­ing a con­tin­u­ous good such that that’s even pos­si­ble).