# Consider two players who bargain over a surplus initially equal to a whole-number amount V, using…

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?