# 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).

• This is an in­ter­est­ing idea. Essen­tially you pe­nal­ise dishon­esty by mak­ing the pot smaller.

This works pro­vided one player doesn’t pre­dictably have a lower max­i­mum bet and can then in­crease their max­i­mum bet (and there­fore the over­all pot) while si­mul­ta­neously mis­rep­re­sent­ing their be­lieved odds.

Did you con­sider us­ing the Kelly Cri­te­rion for the mini-bets in­stead of us­ing a flat rate? I’m not sure how this would af­fect the re­sult but I sus­pect it might have some nice prop­er­ties.

• I like your sum­mary of the method. Good point as well. Per­haps you would want a norm that play­ers don’t dis­cuss their max­i­mum bets be­fore en­ter­ing them.

I’m not fa­mil­iar with the Kelly Cri­te­rion so I’ll check that out.

• How does this ac­tu­ally work me­chan­i­cally? You don’t ac­tu­ally make an even-money bet about a prob­a­bil­ity, you make a weighted bet about an out­come, right? There’s no such thing as a \$2 bet that 66% is cor­rect. There’s a \$3-against-\$2 bet that the thing hap­pens. (that is, the per­son who says “more likely than 66%” wins \$2 if it hap­pens, and loses \$3 if it doesn’t hap­pen. The per­son say­ing “less likely than 66%” is the ex­act op­po­site.)

I’m guess­ing the “max wa­ger” is what each wants their max­i­mum loss to be, and if you dis­tribute that evenly across the range, it ends up ex­actly equiv­a­lent of a sin­gle bet at the mid­way-point be­tween the stated be­liefs.

And that means, as­sum­ing you know the di­rec­tion of dis­agree­ment (if you’re A and give some­thing a 60% prob­a­bil­ity, and you know B thinks it’s more prob­a­ble), you’re in­cented to OVERSTATE your differ­ence from your op­po­nent (A is go­ing to win if the event doesn’t hap­pen, so wants to get bet­ter odds on more bets, so claims a 0% prob­a­bil­ity, and gets a way bet­ter dis­tri­bu­tion of ac­tual wa­gers).

• By \$2 bet at 66% odds, I mean that the Yes po­si­tion costs \$2*66% and the No po­si­tion costs \$2*34%.

You’re right that “max wa­ger” is meant to be max­i­mum loss. I think you’re pick­ing up on the fact that I made a mis­take in calcu­lat­ing loss for each player. I was calcu­lat­ing the po­ten­tial loss for “\$2 bet at 66%” as 2 dol­lars for both play­ers, but that’s ob­vi­ously wrong, and no rea­son afaik that the play­ers should have the same max­i­mum loss. Thanks for the ob­ser­va­tion.

Let’s say A gives event E 60% odds and B gives E 90% odds. For a bet at even odds:

EV_A(YES) = .4 * -.5 + .6 * .5 = .1

EV_A(NO) = .6 * -.5 + .4 * .5 = -.1

From A’s per­spec­tive the no po­si­tion on the 5050 bet (or any bet where the no po­si­tion costs more than 40 cents on the dol­lar) is nega­tive EV. So if A sub­mit­ted 0% odds, they’d be forc­ing them­selves to take a lot of nega­tive EV bets.

• In fact, I mi­s­un­der­stood your pro­posal—I am in­cor­rect in say­ing that ly­ing helps. You’d get more bets on the same side as your “good” bets, but at un­fa­vor­able pay­outs so you’d lose too much when you lose.

• Lovely idea.

Minor point: it feels to me the av­er­age bet isn’t the usual av­er­age but in­stead the har­monic mean of all bets taken. The differ­ence might be small and more im­por­tantly there’s no rea­son why the ar­ith­metic av­er­age is fairer than the har­monic av­er­age, but it was just a small thing I no­ticed ὡC

• Good point.