23506 Chap.10 Sequential Bargaining: Time Discounting, Outside Options and Bargaining Power
Sequential Bargaining: Time Discounting, Outside Options and Bargaining Power
Sequential bargaining adds a protocol to a division problem: who proposes, who responds, what rejection leads to and how delay changes value. With discount factor δ in (0,1), a payoff z received k periods after the evaluation date is worth δ^kz. The exponent counts delays, not the calendar-period label. An acceptance threshold compares a current share with the responder’s discounted continuation value.
In the unit-pie three-period game, Player 1 claims x1 in period 1. After rejection, Player 2 offers x2, where x2 still denotes Player 1’s share. A second rejection produces default (y,1−y) in period 3. At period 2, Player 1 accepts when x2≥δy. Player 2 therefore offers x2*=δy and keeps 1−δy. Back at period 1, Player 2’s rejection value is δ(1−δy), so accepts x1≤1−δ+δ²y. Player 1 chooses equality and agreement occurs immediately.
Equilibrium payoffs are (1−δ+δ²y, δ−δ²y). A higher y strengthens Player 1’s fallback; with y=1/2, greater patience reduces the initial proposer’s share because Player 2 is more willing to wait. These effects belong to the stated protocol. In the pie-of-eight problem with terminal default (2,2), the re-derived offers are x2*=2δ and x1*=8−8δ+2δ². The terminal default sums to four because that is what the problem states; no missing complement should be invented.
What this chapter covers
- 01Bargaining protocols and reservation values
- 02Exponential discounting and delay count
- 03Period-2 acceptance x2≥δy
- 04Initial claim 1−δ+δ²y
- 05Default and patience as bargaining power
- 06Pie of eight with default (2,2)
Unit-pie SPE
- 2Calculate Player 1’s period-2 rejection value and acceptance rule.
- 2Find Player 2’s smallest accepted offer and verify agreement beats rejection.
- 2Discount Player 2’s period-2 share to period 1.
- 2Find Player 1’s maximum accepted first claim.
- 2Compare with rejection and state equilibrium payoffs and complete thresholds.
Key terms
- Bargaining protocol
- The order of proposals, responses, delays and terminal default.
- Reservation value
- The value a responder can obtain by rejecting the current offer.
- Discount factor
- The multiplier δ applied for each one-period delay.
- Acceptance threshold
- The boundary current offer at which a responder is indifferent to rejection.
- Outside option
- A fallback payoff available after rejection or exit.
- First-proposer advantage
- The proposer’s ability here to keep surplus above the responder’s discounted continuation value.
Sequential Bargaining: Time Discounting, Outside Options and Bargaining Power FAQ
What does δ measure?
Patience: a payoff one period later is worth δ times its amount at the current date.
Why is a period-8 payoff discounted by δ⁷ from period 1?
Seven one-period delays separate delivery from the evaluation date.
Whose share is x2?
Player 1’s share in Player 2’s period-2 offer under the course convention.
What is Player 1’s period-2 threshold?
Accept x2≥δy and reject below it, using acceptance at equality.
Why does Player 2 offer exactly δy?
It is the smallest accepted share for Player 1, so it maximises Player 2’s complement.
What is the equilibrium first claim?
x1*=1−δ+δ²y for the unit pie.
Why is agreement immediate?
The accepted first claim exceeds Player 1’s delayed receipt by 1−δ, while Player 2 is held to its rejection value.
How does y affect bargaining power?
A higher y raises Player 1’s terminal fallback and increases x1* by δ² for each unit increase.
What changes for the pie of eight?
With default (2,2), x2*=2δ and x1*=8−8δ+2δ².
Must the terminal default exhaust eight?
No. The problem explicitly gives (2,2), so its total is four and should not be replaced.
How can I avoid share and discount errors in bargaining?
Draw the proposal timeline and write beside every variable whose share it denotes. At the last response, compare acceptance now with the responder’s own payoff after rejection, discounted only for delays from that node. State the weak threshold under the maintained equality-acceptance rule. The proposer offers the smallest accepted responder share only after verifying that immediate agreement beats deliberate rejection. Replace the solved subgame with its equilibrium shares. Move one period backward, apply exactly one additional discount to the earlier responder’s continuation, and derive the maximum first claim it will accept. Compare that accepted claim with the first proposer’s own payoff after rejection. Report threshold response functions for every possible offer, not only the proposals observed on path. Check that agreement shares sum to the current pie, while respecting any terminal default that does not exhaust that pie. Endpoint tests as δ approaches zero or one provide an independent economic check of the symbolic expression.
Exam move
Draw the protocol as a three-box timeline and label each offer with whose share it denotes. At the final response, write current acceptance payoff, discounted rejection payoff and the inequality; make the proposer constraint bind and verify agreement beats deliberate rejection. Replace that subgame by its shares and move back exactly one date. Keep δ symbolic until the complete formula is obtained, then test δ near zero and one for economic plausibility. Re-derive the pie-of-eight case from (2,2) so no complement is invented. Finish by writing both threshold response functions, not merely the on-path proposals.
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