Rubinstein Alternating-Offers Bargaining Equilibrium Formula

Patience is bargaining power: solve infinite alternating offers in one closed-form ratio

In Ariel Rubinstein’s (1982) infinite-horizon bargaining game over surplus pie S, Player 1 proposes a split in Round 1; if Player 2 rejects, one period elapses (shrinking Player 1’s valuation by discount factor delta_1 in [0, 1) and Player 2’s by delta_2 in [0, 1)) before Player 2 counter-proposes. Backward stationarity pins down a UNIQUE immediate Round-1 agreement: Player 1 receives x_1* = S * (1 - delta_2) / (1 - delta_1 * delta_2) and Player 2 receives x_2* = S * delta_2 * (1 - delta_1) / (1 - delta_1 * delta_2)!

Test three strategic regimes over pie S = 100 (B2): (1) when Proposer 1 is more patient (delta_1 = 0.90 in B3, delta_2 = 0.80 in B4), 1 - 0.72 = 0.28 so Player 1 captures 100 * (0.20 / 0.28) = 71.4286 and Player 2 accepts 28.5714 immediately; (2) under equal patience (delta_1 = delta_2 = 0.80), Player 1 still wins 100 / (1 + 0.80) = 55.5556 purely from moving first; (3) when Responder 2 has zero patience (delta_2 = 0), Rubinstein collapses to the Ultimatum Game (x_1* = 100.0000)!

Rubinstein Subgame-Perfect Proposer Payoff x_1* Formula

Require surplus S > 0 and discount factors strictly below 1 (0 <= delta_1, delta_2 < 1; if both equal 1, bargaining is costless and every split in [0, S] can be supported as an equilibrium!).

=LET(s_pie, B2, d_1, B3, d_2, B4, IF(OR(s_pie <= 0, d_1 < 0, d_1 >= 1, d_2 < 0, d_2 >= 1), "Require S > 0 and delta in [0, 1)", ROUND(s_pie * (1 - d_2) / (1 - d_1 * d_2), 4)))

Responder Equilibrium Payoff x_2* = S * delta_2 * (1 - delta_1) / (1 - delta_1 * delta_2) Formula

Notice that x_2* equals delta_2 times what Player 2 would secure as Proposer next round, making Player 2 exact indifferent between accepting today and waiting one period.

=LET(s_pie, B2, d_1, B3, d_2, B4, IF(OR(s_pie <= 0, d_1 < 0, d_1 >= 1, d_2 < 0, d_2 >= 1), "Require S > 0 and delta in [0, 1)", ROUND(s_pie * d_2 * (1 - d_1) / (1 - d_1 * d_2), 4)))

Further reading

• Arte Sheets Lesson — Shapley Value Three-Player Coalition Marginal Contribution Formula
• Microsoft Support — ROUND function