BARGAINING
• Cooperative bargaining (a la Nash)
• Noncooperative bargaining (a la Rubinstein)
What is bargaining?
Ex: Seller values his car at 10 billion TL.
Payoff if
sold:
payoff if not sold:
price - 10
0
Buyer values the car at 12 billion TL.
Payoff if
bought: 12 - price
payoff if not bought: 0
They bargain on a price
between 10 and 12 billion TL. ???
What will be the outcome?
In general
Agents (people, countries, firms, etc.)
can benefit from cooperation (reaching an agreement)
total payoff they create together > sum of their individual payoffs
i.e. agreement creates a surplus
Bargaining: how to divide the surplus
Agreement => surplus is shared
Disagreement => get individual payoffs
(surplus lost)
How is the surplus divided?
Hard to answer
Depends on many things
Edgeworth (Mathematical Psychics, 1881)
Two approaches to analyzing bargaining situations:
Cooperative: player jointly agree (maybe with an arbitrator)
Noncooperative: players choose strategies: equilibrium
Nash’s cooperative bargaining solution
(Nothing to do with the Nash equilibrium)
Ex:
Andy: microchip (worth $900)
Bill : software (worth $100)
Together worth
$3000
Surplus
$2000 (how to divide?)
Have a look at the problem:
to simplify the analysis, assume that
each player’s payoff is the amount of money he gets
3000
Slope: in what ratio
they share the surplus
Bill
2000
2000
100
900
3000
Andy
General case:
Bargaining problem: A and B
trying to split a value v
If disagreement: A gets a (A’s disagreement payoff)
B gets b (B’s disagreement payoff)
NOTE: disagreement payoff = BATNA = backstop payoff
Agreement creates surplus: a + b < v
Surplus: v - a - b
Solution: A and B shares the surplus at ratios h : k
A gets
x = a + h (v-a-b)
B gets
y = b + k (v-a-b)
(h + k = 1)
Interpretation: h & k are A and B’s relative bargaining powers
y
v
y–b k
––––– = –
x–a h
Q
b
P
x+y=v
a
FIGURE 16.1 The Nash Bargaining Solution in the Simplest Case
v
x
Copyright © 2000 by W.W. Norton & Company
How did Nash come up with this?
The general problem:
y
y=f(x)
b
a
The general rule:
maximize
(x-a)h (y-b)k
subject to y = f(x)
x
y
(x – a)h (y –b )k =
Q
c3
b
c2
c1
P
y = f (x)
a
FIGURE 16.2 The General Form of the Nash Bargaining Solution
x
Copyright © 2000 by W.W. Norton & Company
Nash’s principles for bargaining:
1. If the scale in which the payoffs are measured changes linearly,
the outcome changes the same way
i.e.
multiply agent 1’s payoffs by an s (s>0)
multiply agent 2’s payoffs by a t
(t>0)
Under such changes of payoffs, the mixed (and pure)
strategy equilibria of a noncooperative game are unchanged
1
s = 2 and t = 1
a and b
1
2
2. The outcome should be efficient
Agreement can’t be here
*
Efficient frontier: agreement
must be on this line
Ex: What is the efficient frontier when v = x2 + y2
y=sqrt(x2 - v)
3. If some unchosen alternatives are omitted, the outcome should
not change
NOTE: Related to Arrow’s independence of irrelevant alternatives
property
Still the same point chosen
a and b
Nash’s bargaining theorem:
If a bargaining rule satisfies these three properties,
(i) it assigns player 1 a bargaining power of h,
(ii) it assigns player 2 a bargaining power of k,
and then
(iii) for every bargaining problem, it chooses the solution
(x*,y*) to
maximize
(x-a)h (y-b)k
subject to
y = f(x)
NOTE: y=f(x) is the equation for the problem’s efficient frontier
How to calculate the Nash bargaining solution (with h and k) to:
(x-a)h (y-b)k
b
a
Max (x-a)h (y-b)k
?
x+y=v
subject to
x+y=v
=>
Max (x-a)h (v-b-x)k
Take derivative w.r.t x and equate to 0:
h (x-a)h-1 (v-b-x)k - (x-a)h k (v-b-x)k-1 = 0
h (v-b-x) - k (x-a) = 0 =>
h / k = (x-a) / (y-b)
h (y-b) - k (x-a) = 0 =>
y
v
If you could affect the disagreement payoffs
y–b k
––––– = –
x–a h
What could you do?
Q
Q'
P
b
P1
x+y=v
P2
P3
a
FIGURE 16.3 Bargaining Game of Manipulating BATNAs
v
x
Copyright © 2000 by W.W. Norton & Company
The noncooperative approach
Alternating offers game
A and B trying to share a surplus v
A makes an offer
B accepts or makes a counter-offer
(if B makes a counter-offer) A accepts or makes a counter-offer
etc. etc. etc.
In real life, delay is costly
How to capture that?
Version 1: v decays
(the pie shrinks with each offer)
Ex: Basketball fan vs. scalpter (i.e. karaborsaci)
Fan is willing to pay $25 for each quarter of the game
Quart 1: Scalpter makes an offer
(for the whole game)
If fan rejects, he watches the first quarter on TV, then
Quart 2: Fan makes an offer
(for the three quarters)
Quart 3: Scalpter makes offer
(for the two quarters)
Quart 4: Fan makes offer
(for the last quarter)
What is the subgame perfect Nash equilibrium?
ROLLBACK
In general: allocate v
v = x1 + x2 +…+ x8 + x9 + x10
Period 1:
must allocate v
A makes offer
if offer rejected, v becomes v2 = v - x1
Period 2:
B makes offer
must allocate v2
if offer rejected, v2 becomes v3 = v2 - x2 = v - x1 - x2
…
…
…
if offer rejected, v8 becomes v9 = v - x1 -…- x8 = x9 + x10
Period 9:
A makes offer
must allocate v9
if offer rejected v9 becomes v10 = x10
Period 10:
B makes offer
must allocate v10
if offer rejected, v10 becomes v10 - x10 = 0
GAME ENDS
Rollback:
Period 10: B gets
x10
out of x10
Period 9: A gets
x9
out of x9 + x10
(must give x10 to B for acceptance)
Period 8: B gets
x8 + x10
out of x8 + x9 + x10
…
Period 1: A gets (nearly) all of x1 + x3 + x5 + x7 + x9
NOTES:
1. Zero disagreement payoffs in this game. With positive
disagreement payoffs: last period, B gets x10-a, etc.
2. Gradual decay vs. sudden decay (the ultimatum game)
Version 2: future v’s are discounted (time is valuable)
Ex: sharing $1 with common discount rate of d=0.95
How long will the game last?
The players are in identical situations when making an offer
(once the time comes, it doesn’t matter whether you are at
period 5 or 100, the amount to split is still $1)
At equilibrium, both players will make the same offer, x,
when it is their turn to make an offer
Suppose A is making offer:
A must give B at least 0.95x (what B will get next period)
A must get x
x = 1- 0.95x
=>
x = 0.512
B gets 1-x = 0.488 when he accepts.
If he rejects, he will receive x=0.512 next period.
Discount it to compare with the present gain of 0.488:
0.95 0.512 = 0.488 (they are the same).
The subgame-perfect Nash equilibrium:
Both players offer x=0.512 (for themselves)
Both players accept offers which give them at least 1-x = 0.488
Note that:
1. In equilibrium, the first offer A makes gets accepted.
2. A gets more than half of the pie. Compare to
ultimatum game
and
other values of discount rate
What happens when players have different discount rates?
A discounts future with 0.90
B discounts future with 0.95
The players are not in symmetric positions now:
In equilibrium A offers x
and
B offers y
What is A’s offer?
Must give B at least 0.95y => A gets 1-0.95y
x = 1- 0.95y
What is B’s offer?
Must give A at least 0.90x =>
y=1-0.90x
B gets 1-0.90x
Solving together:
x = 0.345
y = 0.690
The subgame-perfect Nash equilibrium:
A offers x=0.345 (for himself)
A accepts an offer which gives him at least 1-0.69=0.31
and
B offer y=0.690 (for himself)
B accepts an offer which gives him at least 1-0.345=0.655
Intuition: being patient is always good in bargaining! Or is it not?
General treatment:
A can get interest r
[discounts with a = 1/(1+r)]
offers x
B can get interest s
[discounts with b= 1/(1+s)]
offers y
A’s offer satisfies:
x=1-by
B’s offer satisfies:
y=1-ax
Solving:
x = 1 - b (1 - a x) => x =
y = 1 - a (1 - b y) => y =
1-b
1-ab
1-a
1-ab
s+rs
=
=
r+s+rs
r+rs
r+s+rs
Notes:
1. At equilibrium, the first offer is accepted
2. Each player gets more when he is the one to make the first offer
3. At equilibrium
x+y>1
What happens when the time between periods get shorter?
(ex: from a week to a day or to a minute or to a second …)
In this case, both r and s will converge to 0
rs converges much faster and can be ignored
Approximately, x = s / (r + s)
and
y = r / (r + s)
Approxim. add up to 1 ( compare to Nash bargaining rule)
4. Since y / x = r / s, more patience (smaller r) increases A’s share
Additional Items:
1. A symmetric Nash bargaining rule
symmetry: treat both agents identically
: both agents have the same bargaining power
i.e. if the bargaining problem is symmetric, the
agents get the same payoffs
2. Sharing a cake
implications of the Nash principles on the solution
3. Sharing a house
PIERCE’S PIZZA PIES
DONNA’S
DEEP DISH
EXERCISE 16.3
High
Medium
High
156, 60
132, 70
Medium
150, 36
130, 50
Copyright © 2000 by W.W. Norton & Company