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A union and a firm argue over how to split the surplus from a contract. Every week they fail to agree, the pie shrinks: lost output, lost wages. That shrinking is the discount factor, and it decides who gets what long before anyone walks out.

Bargaining and negotiation

Real negotiation, a labour contract, a house sale, a merger, is alternating offers over a pie that decays with delay. Rubinstein’s model turns that into a sharp prediction: the split depends on patience and on each side’s outside option (BATNA), not on who talks loudest.

From earlier lessons

Two quick ones before we start.

In alternating offers, delay is costly because the pie...

Your BATNA is your best option if the talks...

Predict, then play

Negotiate against a bot over a pie that shrinks with every round of delay. Tune both players’ discount factors and the responder’s BATNA, propose a split, and see what the bot will accept.

Why did that happen?

Take a guess, then reveal the reason

Each side can reject and counter, but rejecting costs a round of shrinkage. A patient player (delta near 1) barely minds waiting, so they can hold out for more; an impatient player (low delta) settles quickly and cheaply. The subgame-perfect split balances these exactly. Outside options set a floor only when they exceed the continuation value from bargaining. A strike or lockout is a costly period of disagreement, not itself an outside option. When information is complete in this model, the parties settle immediately. Delays require an added friction, such as private information.

The real strategy

Definition, Rubinstein alternating offers, and Nash bargaining

In Rubinstein’s model, player 1 discounts by $\delta_1$ and player 2 by $\delta_2$ . The unique subgame-perfect equilibrium gives the first proposer $(1-\delta_2)/(1-\delta_1\delta_2)$ and the responder $\delta_2(1-\delta_1)/(1-\delta_1\delta_2)$ , agreed in the first round. With equal patience $\delta_1=\delta_2=\delta$ , these reduce to $1/(1+\delta)$ and $\delta/(1+\delta)$ . An outside option changes the solution only when it exceeds the continuation payoff it replaces.

At $\delta = 0.9$ the proposer gets $1/1.9 \approx 0.526$ and the responder $0.9/1.9 \approx 0.474$ . The first-mover advantage is real but small, and it shrinks to nothing as both sides grow patient. Change a BATNA and the whole split moves: improve your outside option and you capture more; worsen it and you concede. That is why negotiators spend more effort improving their alternatives than polishing their arguments. Strikes are not proof that bargaining is irrational; with full information the parties would settle at once. They arise when each side is unsure of the other’s true reservation value and testing that costs a round of the shrinking pie.

Work it out

Patience, BATNA, and the split

Start with no outside options and $\delta = 0.9$ . The proposer offers themselves $1/(1+\delta) = 0.526$ and the responder $0.474$ , and the responder accepts, because rejecting and countering next round would give them only $0.526$ of a pie already shrunk by $0.9$ , which is worse. As $\delta \to 1$ , both approach $0.5$ : patience erases the first-mover edge.

At equal $\delta=0.9$ , the responder already receives about $0.474$ , so a BATNA of $0.4$ does not bind and changes nothing. Give the responder a BATNA of $0.6$ instead: rejecting for that outside deal is better than continuation, so any acceptable agreement must give at least $0.6$ . A strike is a period of disagreement with lost output and wages, not itself a BATNA. Another job offer or an alternative supplier is a BATNA; strike costs affect delay payoffs and may reveal private information about willingness to wait.

Key idea

In alternating-offers bargaining, patience, move order, and any binding outside option affect the split. The first-mover edge fades as both grow patient. Delay does not arise in the complete-information benchmark, so strikes require an added friction such as private information.

Check yourself

Check yourselfapply

Two negotiators alternate offers with delta = 0.9. Roughly how does the pie split, and why?

Check yourselfanalyse

Why does improving your BATNA get you a better deal even if talks succeed?

Check yourselfbrings back: bargaining

If rational bargainers settle immediately, why do real strikes and lockouts happen?

Sources

Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica, 50(1), 97-109.
Nash, J. F. (1950). The bargaining problem. Econometrica, 18(2), 155-162.