Bargaining Theory Introduction
Bargaining Theory is a part of Game Theory. It helps us understand how people make deals when:
- Each person wants the best result for themselves
- They must find an answer that both sides accept
The final deal depends on what each side wants, how long they can wait, and what happens if they walk away.
The Nash Bargaining Solution
The Nash Bargaining Solution is a rule that finds a fair deal.
It looks for a choice that:
- Gives both sides more than if they just gave up
- Balances things out fairly when both people have the same power
To find this deal, we multiply the extra gains of both sides. The best deal is where that number is the highest.
Example: Selling a Car
Sarah wants to sell her car. Tom wants to buy it.
- Sarah won't take less than $2,000
- Tom won't pay more than $3,000
There's $1,000 of extra value between them. We call this the surplus.
- If Tom pays $2,500, both gain $500
- If Tom pays $2,000, Sarah gains $0 extra
- If Sarah gets $3,000, Tom gains $0 extra
Let's look at the extra value each person gets at different prices:
| Price | Sarah's Gain (Price − $2,000) |
Tom's Gain ($3,000 − Price) |
Nash Product (Sarah × Tom) |
|---|---|---|---|
| $2,000 | $0 | $1,000 | 0 |
| $2,250 | $250 | $750 | 187,500 |
| $2,500 (Fair Deal) | $500 | $500 | 250,000 (MAX) |
| $2,750 | $750 | $250 | 187,500 |
| $3,000 | $1,000 | $0 | 0 |
If they can't agree, they both get $0 extra.
The Nash formula picks the $2,500 price because 250,000 is the biggest number. Both sides get an equal share of the prize!
Example (cont): Sarah gets another buyer who is willing to pay her $2,200
- Now, Sarah's fallback is $200 higher. She won't accept anything less!
- The remaining surplus to bargain over is now only $800 (the gap between $2,200 and Tom's $3,000 limit)
- Splitting this new surplus 50/50 gives them each $400 of extra value
The new fair price becomes $2,600 (Sarah gets $600 extra, Tom gets $400 extra). Having a better alternative gave Sarah more bargaining power!
Working Together (Cooperation)
Sometimes bargaining isn't just about how to split a pie. It is about making the pie bigger first!
In Cooperative Bargaining, players find ways to make more total value together than they could alone.
Example: A Business Team
Two companies think about teaming up.
WizardEd makes great math apps but has no sales team.
School Gear sells tools to schools but has no tech team.
- Alone, WizardEd makes $300,000
- Alone, School Gear makes $200,000
Apart, they make $500,000 total.
But if they join forces, School Gear can sell WizardEd's apps to all their schools. Together, they earn $700,000!
By working together, they created $200,000 of new value. Now they just have to agree on how to split that extra cash.
When Deals Fall Apart
Not every talk ends in a deal. If someone won't budge, the talk fails.
When this happens, both sides get nothing extra and must fall back on what they had before.
Example: Worker Strike
A business and its workers try to agree on pay.
- If they agree, workers get paid and the shop stays open
- If they fail, workers strike and the shop closes
During a strike, the boss loses sales and workers lose pay. Nobody wins.
Even a simple deal can be better than no deal at all.
Summary
- Fairness helps people agree
- Making more value together is better than fighting over less
- Be ready to change your offer so everyone wins