Consider two players who bargain over a surplus initially
equal to a whole-number amount V, using alternating offers. That is, Player 1
makes an offer in round 1; if Player 2 rejects this offer, she makes an offer
in round 2; if Player 1 rejects this offer, she makes an offer in round 3; and
so on. Suppose that the available surplus decays by a constant value of c 5 1
each period. For example, if the players reach agreement in round 2, they
divide a surplus of V 2 1; if they reach agreement in round 5, they divide a
surplus of V 2 4. This means that the game will be over after V rounds, because
at that point there will be nothing left to bargain over. (For comparison,
remember the football-ticket example, in which the value of the ticket to the
fan started at $100 and declined by $25 per quarter over the four quarters of
the game.) In this problem, we will first solve for the rollback equilibrium to
this game, and then solve for the equilibrium to a generalized version of this
game in which the two players can have BATNAs.
(a) Let’s start with a simple version. What is the
rollback equilibrium when V 5 4? In which period will they reach agreement?
What payoff x will Player 1 receive, and what payoff y will Player 2 receive?
(b) What is the rollback equilibrium when V = 5?
(c) What is the rollback equilibrium when V 5 10?
(d) What is the rollback equilibrium when V = 11?
(e) Now we’re ready to generalize. What is the rollback
equilibrium for any whole-number value of V?
Now consider BATNAs. Suppose that if no agreement is
reached by the end of round V, Player A gets a payoff of a and Player B gets a
payoff of b. Assume that a and b are whole numbers satisfying the inequality a
1 b , V, so that the players can get higher payoffs from reaching agreement
than they can by not reaching agreement.
(f) Suppose that V 5 4. What is the rollback equilibrium
for any possible values of a and b? (e). If you get stuck, try assuming
specific values for a and b, and then change those values to see what happens.
In order to roll back, you’ll need to figure out the turn at which the value of
V has declined to the point where a negotiated agreement would no longer be
profitable for the two bargainers.)
(g) Suppose that V 5 5. What is the rollback equilibrium
for any possible values of a and b?
(h) For any whole-number values of a, b, and V, what is
the rollback equilibrium? (i) Relax the assumption that a, b, and V are whole
numbers: let them be any nonnegative numbers such that a 1 b , V. Also relax
the assumption that the value of V decays by exactly 1 each period: let the
value decay each period by some constant amount c . 0. What is the rollback
equilibrium to this general problem?