Strategic Effects and Incentives in Multi-issue Bargaining Games

time τ passes between the end of the first bargaining stage (an acceptance) and the
beginning of a new bargaining stage (a new proposal). Moreover, the first proposer
at the second stage is the last responder at the initial stage. Player i’s rate of time
preference is indicated by ri > 0 (and i =1, 2). To take into account that there are
intervals of time of different lengths, we define players discount factors as follows:
player i’s within-cake discount factor δ_i =exp(-r_i∆) applies after a rejection and his
between-cake discount factor αi =exp(-riτ) applies after an acceptance. Moreover,
players are allowed to have different valuations of the issues. A positive parameter
λ_i indicates the relative importance of cake i to player i (see payoff functions below),
with i =1, 2.

The implementation of the agreement is assumed to be sequential, that is, a
division is implemented as soon as it is agreed. If an agreement is not reached on the
partition of a cake, players get zero payoffs (disagreement) at that stage. Then, if
disagreement takes place at the first stage, the second stage cannot take place and
players’ overall payoffs are zero. In our framework, we consider two agendas, agenda
i states that cake i is negotiated first, with i =1, 2. In this section we focus on
agenda 1. If, after t rounds, an agreement is reached on the division of the first cake,
(x, 1 - x), where x is the share player 1 obtains, and after n +1 periods (a period of
length τ and n periods of length ∆) another agreement is reached (1 - y, y), then the

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