qr 38 bargaining, 4/24/07 i. the bargaining problem and nash solution ii. alternating offers models
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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