Strategic Effects and Incentives in Multi-issue Bargaining Games

1). Under these conditions also the other effects of δj on v_i (with i, j =1, 2) is as
in the standard one-cake bargaining theory, that is, negative for i 6= j, and positive
otherwise⁴ . When these conditions are relaxed, the effects of the δ_i on equilibrium
payoffs may be ambiguous (see points 4 and 5 of corollary 1). For the case of corner
solutions, we can obtain similar effects as in the standard single-cake bargaining game
when player 1 values the second bargaining stage more than his opponent does, that
is, α₁∕λ₁ ≥ α₂λ₂ (see points 9 of corollary 1) either he is more impatient than his
opponent (expression (19) is negative if δ₁ <δ₂) or the interval of time between an
acceptance and a new proposal goes to zero (again (19) is negative for ∆ → 0).

A common assumption in the literature is to assume that parties have the same
discount factors δ_i = δ, with i =1, 2, in this case the interplay of the forces in the
bargaining process with SPE defined by part 2 of proposition 1 is greatly simplified.
As a result player i’s payoff does not depend on αi with i =1, 2. Moreover, player 2’s
payoff is also independent of his relative valuation of cake 2 (λ₂), while the relative
importance of the first cake between players λ1 still plays a role (as indicated in point
3 of Corollary 1). For the equilibrium outcome defined by the demand xe₁ (defined in
(10)) the assumption of a common discount factor does not have a great impact on
the interplay of the forces, since this is already simplified by the fact that a player
⁴The effect of δ₂ on v₂ is negative, since α₁ + λ₁ >α₁ - α₂λ₁λ₂ and 1 - δ₁δ₂ >δ₁ - δ₂ ≥
δ1 - δ2 - δ2 (1 - δ₁²).

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