Deriving the unique Subgame Perfect Equilibrium of alternating-offer bargaining under discounting.
Two players must divide a surplus of size 1 between them. Neither can seize it unilaterally — they need agreement. The protocol is alternating offers:
P1 offers a split (s₁ , 1 − s₁). P2 either accepts (game ends) or rejects (clock advances).
P2 offers (1 − s₂ , s₂). P1 either accepts or rejects.
Delay is costly. A dollar received at round t is worth less than a dollar now. Each player i has a discount factor δᵢ ∈ (0,1). A payoff of u received at period t is worth δᵢᵗ⁻¹ · u in present-value terms. The closer δ is to 1, the more patient the player.
Why can we actually solve this infinite game? The key insight is stationarity. Consider what happens after a rejection at t = 1:
The sub-game starting at t = 3 (P1 proposes again) is structurally identical to the game at t = 1.
Both sub-games begin with Player 1 proposing, followed by an infinite sequence of alternating offers with the same discount factors. The only difference is that payoffs at t = 3 are discounted by an extra factor of δ₁ · δ₂ relative to t = 1. In an SPE, strategies depend only on the game's structure, not its calendar date. So the equilibrium behaviour at t = 3 must replicate that at t = 1.
This gives us the crucial modelling shortcut: instead of solving an infinite recursion, we can characterise the equilibrium by a pair of fixed-point equations.
Define two unknowns describing SPE shares:
Now the logic of indifference. When Player 1 proposes, Player 2 will accept if and only if the offered share is at least as good as waiting one round and getting y (discounted). At the margin, P1 offers P2 exactly this indifference value to minimise the concession:
Symmetrically, when Player 2 proposes, Player 1 accepts if the offered amount matches the discounted value of proposing next round:
Expand each step below to walk through the substitution method:
Use the sliders to explore how patience (δ) shapes bargaining power. The formulas update in real time.
P1 captures 81.1% — more than half. Being the proposer confers leverage because the responder must discount the future.
P1 is more patient (δ₁ > δ₂), so they extract a larger share. Patient players can credibly threaten to wait.
As both players become perfectly patient, the cost of delay vanishes. The first-mover advantage shrinks to zero and the equilibrium converges to an equal 50 – 50 split — delay is costless, so neither side can leverage impatience. This limit corresponds to the Nash Bargaining Solution with equal bargaining power.
Setting both discount factors equal simplifies the formula to x* = 1/(1 + δ). The proposer always gets strictly more than ½ (since δ < 1), but this advantage vanishes smoothly as δ → 1. The entire strategic power structure reduces to one parameter: how costly it is to wait.