The zone between the walk aways

The supplier needs at least its alternative; you pay at most yours. The zone between them has width $z=b-s$, where $z$ is the width and $b$ your next best price and $s$ the supplier's walk away.

The even split

Equal gains put the price at the zone's middle: $p^{*}=s+\frac{b-s}{2}=\frac{s+b}{2}$, where $p^{*}$ is the even split. Either end moves it half as far, since $\frac{\partial p^{*}}{\partial s}=\frac{1}{2}$.

The middle of the openings

Average the openings: $m=\frac{a+o}{2}$, where $m$ is their middle and $a$ the ask and $o$ the offer. Its gap from the even split is $g=m-p^{*}$, where $g$ is that gap. Both sides can accept that middle only while $s\le m\le b$.

When there is no zone

When $b<s$, the supplier's alternative exceeds yours. No price meets both limits; the openings still have a middle.

A worked example

1. The zone: $z=1100-900=200$.
2. Its even split: $p^{*}=900+\frac{200}{2}=1000$, also the average of the walk aways, $\frac{900+1100}{2}=1000$.
3. The openings meet at: $m=\frac{1200+950}{2}=1075$.
4. The gap between middles: $g=1075-1000=75$.
5. That middle is inside the zone: $900\le 1075\le 1100$.

Two changes, two different effects

An ask of 1,300 takes the middle outside the zone: $m=\frac{1300+950}{2}=1125$.

A supplier walk away of 1,000 moves the even split half as far: $p^{*}=\frac{1000+1100}{2}=1050$.

Each pair moves its own middle.

What this does not model

No discovery or verification of either alternative.

No unequal weights, patience, breakdown risk or gains during delay.

No tactics or sequence of offers; an ask below the offer remains valid.

The sweep

Across supplier alternatives from 700 to 1,200 in steps of 100, the split rises by 50 each step until the supplier's alternative exceeds 1,100 and no zone remains.

Allowed amounts

Whole amounts from 0 to 1,000,000,000, all in one currency; changing supplier belongs in your next best price.

Boundary

The page compares four stated amounts; it neither values alternatives nor recommends a position.

Sources

Abhinay Muthoo, Bargaining Theory with Applications, explains the split around an alternative. Avinash Dixit, Susan Skeath and David Reiley, Games of Strategy, treats weighted surplus sharing; here both weights are a half.

Scale

Amounts through 1,000,000,000; whole amounts give exact halves

Basis

Two sides, one price, equal gains above their alternatives

Assumptions

Whole amounts in one currency; the supplier's alternative is your estimate

Model version

2.0.0