Rubinstein Bargaining

A non-cooperative alternating-offers model of bargaining with a unique subgame-perfect outcome.

Ask the Game theory assistant1 min read · Updated September 9, 2026

Definition

Two players alternate offers to split a pie of size $1$ ; discount factors $\delta_1,\delta_2\in(0,1)$ reflect impatience.

The unique SPE has player 1 offering $x_1^*=(1-\delta_2)/(1-\delta_1\delta_2)$ and it being accepted immediately.

Impatient players concede faster; first-mover advantage vanishes as frictions disappear.

The ability to delay is an implicit threat that drives the equilibrium split.

With $\delta_1=\delta_2=\delta$ , the proposer gets $1/(1+\delta)$ and the responder $\delta/(1+\delta)$ .

As $\delta\to 1$ , the outcome converges to the symmetric Nash bargaining solution $(1/2,1/2)$ .

Proved by stationary backward-induction: equate each player's offer to what they could obtain by waiting one period and proposing themselves.

Provides the "Nash program" microfoundation for the Nash bargaining solution.

Microfoundations for search-matching, wage determination, and bilateral trade.

Extensions incorporate outside options, risk of breakdown, and incomplete information.

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