QR 38

Bargaining, 4/24/07

I. The bargaining problem and Nash solution

II. Alternating offers models

I. The bargaining problem and Nash solution

Bargaining constant in IR, but haven’t said much directly about it.

What exactly is bargaining; how to represent it using game theory?

• Bargaining is in large part a coordination problem.

• Two parties need to agree on the distribution of a good.

• Many equilibria, but disagreement over which is preferred (distributional conflict).

Bargaining problems

But failure to reach an agreement leaves all parties worse off. So bargaining involves:

1. Potential for mutual gains

2. Conflict over how to divide these gains

• Bargaining is not zero-sum: a surplus exists, compared to the situation where no bargain is reached.

Solutions

A solution to a bargaining problem involves:

• Specification of situations in which a bargain will be reached

• How the surplus will be divided

• Most models, and Schelling’s discussion, focus on the second question

Types of bargaining models

Two types of bargaining models, drawing on cooperative and non-cooperative game theory.

• Nash devised a cooperative solution

• Later the Nash bargaining solution was shown to be the equilibrium of a non-cooperative game as well

Two-player bargaining model

Consider a two-person bargaining situation.

• If the parties reach an agreement, they get a total value v to split between themselves.

• If they don’t reach an agreement, A gets a and B gets b.

Payoffs• a and b are called backstop payoffs,

BATNA, or reservation points.

• Often set these equal to zero to simplify the problem.

• The surplus equals the total benefit from reaching an agreement: • surplus=v-a-b

Solutions

Assume that each player gets BATNA plus a fraction of the surplus:

• A gets the fraction h

• B gets k (=1-h).

• Let x be the total A gets:– x=a+h(v-a-b)– x-a=h(v-a-b)– This says that the additional benefit A gets

from an agreement (x-a) is some fraction h of the total surplus.

Solutions• Let y be the total B gets:

– y=b+k(v-a-b)– y-b=k(v-a-b) (the benefit B gets is fraction k of

the surplus)

• These are the Nash formulas. • Think of them as dividing the surplus in the

proportion h:k• Can write (y-b)/(x-a)=k/h

– Then think of k/h as the slope of the line specifying the solutions

Nash bargaining solution

A’s payoff (x)

B’spayoff(y)

a

b(a, b)

Line withslope k/h

v(=x+y)

v

Q

a’

(a’, b)

Q’

Nash bargaining solution

A nice way to think about the problem, but it doesn’t tell us where h and k come from.

• Can think of h and k as bargaining strengths.

• Need more context to use this solution.– Nash assumed h=k; then get determinate

solution for x and y.

Nash bargaining solutionNote that being able to move the reversion

point (a,b) in your direction provides you with a higher payoff.

• What would this mean in IR?– Usually making a threat that would hurt

yourself if you had to implement it, like a trade war.

II. Alternating offers model

To get more insight, need a model with more context.

• An important general model is an alternating offers model.

• A dynamic model, with some number of periods.

Alternating offers model

• In each period, one player has the opportunity to make an offer to the other.

• The other can accept or make a counteroffer.

• This process continues until an offer has been made and accepted.

Alternating offers model• With a finite number of periods, can use

rollback to find the equilibrium. • But in an infinitely-repeated game, why

would this process ever end? • Have to assume that the surplus becomes

less valuable over time; discounting.– This could result because the surplus itself is

shrinking (some probability it will disappear, e.g.), or because the players are impatient.

– The two are conceptually similar, although D&S present separately.

Solving alternating offers model

Assume that two players are bargaining over the division of a dollar.

• A dollar tomorrow is as good as having only 95 cents today. – Remember how we used discount factors to

address situations like this (repeated games).

• Assume BATNAs are zero.

Solving alternating offersThe player making the offer suggests that he gets

x. • We want to solve for x using backward induction • That is, x is the equilibrium outcome.

Let A start. • A knows that B will get x in the next round,

because x is the equilibrium payoff to the player making the offer.

• So A has to offer something today that is worth the same as getting x in the next round.

Solving alternating offers• So A has to offer B 0.95x now. • Leaves A with 1-0.95x.• But we called what A is offering x:• So x=1-0.95x x=1/1.95 x=0.512• So, the equilibrium is for the player who

gets to make the first offer to get 0.512• The player who goes second will get

1-.512=0.488

Solution• The equilibrium is reached immediately

even though an unlimited number of counteroffers are allowed

• That is, the outcome is efficient; the surplus does not decay.

• A first-mover advantage results: the player making the first offer gets more (x>1/2).

Solution with different discount rates

• This example assumed that the two players had the same discount factor (.95).

What if the two players have different discount rates?

• For A, let a dollar tomorrow may be worth only 0.90 today. B’s discount rate is .95.– So A is willing to accept a smaller

amount in order to be paid sooner. • In equilibrium, the more impatient player

gets less.

Different discount rates• Let x be the amount A gets when he goes

first

• Let y be the amount B gets when he starts

• A has to offer B 0.95y. – So x=1-0.95y

• B has to offer A 0.90x– So y=1-0.90x

Different discount rates

• We can solve these equations for x and y:

x=1-.95y y=1-.9x

x=1-.95(1-.9x)=1-.95+.855x=.05+.855x

.145x=.05

x=0.345

y=1-.9(.345)=0.690

Different discount rates

• So if A goes first, A gets .345 and B gets 1-.345=.655

• If B goes first, A gets .31 and B gets .69

• So A gets less than B because of impatience, even if A goes first

Generalized solution

• A sees $1 today as worth $(1+r) tomorrow

• B sees $1 today as worth $(1+s) tomorrow

• Means that A sees $1 tomorrow as worth $1/(1+r) today

• x=(s+rs)/(r+s+rs)

• y=(r+rs)/(r+s+rs)

Generalized solution• rs is usually very small• So it is approximately true that x=s/(r+s)• y=r/(r+s)• Then we can see x and y as the shares

that go to each player (verify that x+y=1).• Write as y/x=r/s• The shares that players get are inversely

proportional to their rates of impatience.

Solution

a=1/(1+r) b=1/(1+s)

When A makes an offer, has to give B the equivalent of getting y the next period; this is by. So:

x=1-by y=1-ax

x=1-b(1-ax)=1-b+abx

x-abx=1-b

x(1-ab)=1-b

Solution

x=(1-b)/(1-ab)=(1-(1/1+s))/(1-(1/1+r)(1/1+s))

=(1+s-1)/(1+s-(1/1+r))=s/(1+s-(1/1+r))

=s(1+r)/((1+r)+s(1+r)-1)

=(s+rs)/(1+r+s+rs-1)

=(s+rs)/(r+s+rs)s/(r+s)

y=(r+rs)/(r+s+rs)r/(r+s)