So now, we've shown that there's only one internally consistent strategy for Bea.

She should reject any offer below a third and

she should accept any offer above a third.

And armed with this understanding,

Abe will offer exactly one-third and she will say yes.

What this really means is the person making the offer gets two-thirds of

the pie and the person receiving the offer gets one-third.

On Monday that's Abe, but if things go wrong on Monday and

it's Bea's turn to make an offer on Tuesday,

then she'll be able to get two-thirds of the remaining 50, which is 33.3.

The same as what she would've received on Monday.

How much more the person who makes the offer gets compared to the person who

receives the offer depends on how fast the pie shrinks?

With our pie shrinking at the Indy 500 speed of 50% per day,

the imbalance is two-thirds for the proposer, one-third for the receiver.

If the pie only shrinks at 1% per day, then the imbalance is much smaller.

The proper gets 100 over 199 or 50.25% and the receiver gets 99 over 199 or 29.75%.

The power in the game is limited to one's ability to cause a day's worth of delay.

If a day's delay doesn't cost that much,

then the two sides aren't pretty equal footing.

In our example where the pie shrinks by half each day,

Abe has the power to get all that is lost on Monday.

Namely, 50%.

But on Tuesday, he is in a weak position as now Bea can cause a delay.

That leaves Abe with one-third of the remaining half.

Therefore, Abe as the person going first can get all of the first half

plus a third of the second half, which is two-thirds in total.

As the shrinkage rates get lower and lower,

the ability to move first becomes less important.

And thus, the solution to the bargaining game converges to 50, 50.

If the pie shrinks from one to delta each day, then the solution to the game is 1

over 1 plus delta for the proposer and delta over 1 plus delta for the receiver.

Thus, when delta is a half,

you get 1 over three-halves or two-thirds to the proposer.

When delta is 99%, then you get 1 over 1.99,

which is 50.25% to the proposer.

And if delta is 99.9%, then you get 1 over 1.999,

which is 50.025% to the proposer.

This makes perfect sense.

As the pie shrinks less between rounds,

there's less an advantage from going first and so the division gets closer to 50, 50.

We've been changing the amount by which the pie shrinks from day to day,

but it probably makes more sense to think in terms of offers

being made more frequently.

If offers and counteroffers can be made an hour apart, then the pie

will shrink a lot less between offers than if it takes a month to go back and forth.

What matters is the shrinkage, not the time.

If the offers can be made with little loss between them,

we expect the pie will be divided very close to 50, 50.