21.13 Bargaining
Books:
Abhinay Muthoo - Bargaining Theory with Applications (1999) (check books folder)
Josh Nash - Nash Bargaining (1950)
Allocation of scare resources
| Buyers & Sellers | Type |
|---|---|
| Many buyers & many sellers | Traditional markets |
| Many buyers & one seller | Auctions |
| One buyer & one seller | Bargaining |
Allocations of
Determining the share before game-theoretic bargaining
Use a judge/arbitrator
Meet-in-the-middle
Forced Final: If an agreement is not reached, one party will use take it or leave it
Art: Negotiation
Science: Bargaining
Game theory’s contribution: to the rules for the encounter
Area that is still fertile for research
21.13.1 Non-cooperative
Outline for non-cooperative bargaining
- The rules
Take-it-or-leave-it Offers
Bargain over a cake
If you accept, we trade
If you reject, no one eats
Under perfect info, there is a simple rollback equilibrium
In general, bargaining takes on a “take-it-or-counteroffer” procedure
If time has value, both parties prefer to trade earlier to trade later
- E.g., labor negotiations - later agreements come at a price of strikes, work stoppages
Delays imply less surplus left to be shared among the parties
Two-stage bargaining
I offer a proportion, \(p\) , of the cake to you
If rejected, you may counteroffer (and \(\delta\) of the cake melts)
Payoffs:
In the first period: 1-p, p
In second period: \((1-\delta) (1-p),(1-\delta)p\)
Since period 2 is the final period, this is just like a take-it-or-leave-it offer
- You will offer me the smallest piece that I will accept, leaving you with all of \(1-\delta\) and leaving me with almost 0
Rollback: then in the first period: I am better off by giving player B more than what he would have in period 2 (i.e., give you at least as much surplus)
You surplus if you accept in the first period is \(p\)
Accept if: your surplus in first period greater than your surplus in second period \(p \ge 1 - \delta\)
IF there is a second stage, you get \(1 - \delta\) and I get 0
You will reject any offer in the first stage that does not offer you at least \(1 - \delta\)
In the first period, I offer you \(1 - \delta\)
Note: the more patient you are (the slower the cake melts) the more you receive now
Whether first or second mover has the advantage depends on \(\delta\).
If \(\delta\) is high (melting fast), then first mover is better.
If \(\delta\) is low (melting slower), then second mover is better.
Either way - if both players think, agreement would be reached in the first period
In any bargaining setting, strike a deal as early as possible.
Why doesn’t this happen in reality?
reputation building
lack of information
Why bargaining doesn’t happen quickly? Information asymmetry
- Likelihood of success (e.g., uncertainty in civil lawsuits)
Lessons:
Rules of the bargaining game uniquely determine the bargain outcome
which rules are better for you depends on patience, info
What is the smallest acceptable piece? Trust your intuition
delays are always less profitable: Someone must be wrong
Non-monetary Utility
each side has a reservation price
- LIke in civil suit: expectation of wining
The reservation price is unknown
One must:
probabilistically determine best offer
but - probability implies a chance that non bargain will take place
Example:
Company negotiates with a union
Two types of bargaining:
Union makes a take-it-or-leave-it offer
Union makes a n offer today. If it’s rejected, the Union strikes, then makes another offer
- A strike costs the company 10% of annual profits.
Probability that the company is “highly profitable”, ie., 200k is \(p\)
If offer wage of $150k
Definitely accepted
Expected wage = $150K
If offer wage of $200K
Accepted with probability \(p\)
Expected wage = $200k(p)
\(p = .9\) (90% chance company is highly profitable
best offer: ask for $200K wage
Expected value of offer: \(.9 *200= 180\)
\(p = .1\) (10% chance company is highly profitable
best offer: ask for $200K wage
Expected value of offer: \(.1 *200= 20\)
If ask for $10k, get $150k
not worth the risk to ask for more
If first-period offer is rejected: A strike costs the company 10% of annual profits
Strike costs a high-value company more than a low value company
Use this fact to screen
What if the union asks for $170k in the first period?
Low profit firms ($150k) rejects - as can’t afford to take
HIgh profit firm must guess what will happen if it rejects
Best case: union strikes and then asks for only $140k (willing to pay for some cost of strike), but not all)
In the mean time: strike cost the company $20K
High-profit firm accepts
Separating equilibrium
only high-profit firms accept the first period
If offer is rejected, Union knows that it is facing a low-profit firm
Ask for $140k
What’s happening
Union lowers price after a rejection
Looks like giving in
looks like bargaining
Actually, the union is screening its bargaining partner
Different “types” of firms have different values for the future
Use these different values to screen
Time is used as a screening device
21.13.2 Cooperative
two people diving cash
If they do not agree, they each get nothing
They cant divide up more than the whole thing
21.13.3 Nash (1950)
Bargaining, bilateral monopoly (nonzero-sum two -person game).
Non action taken by one individual (without the consent of the other) can affect the other’s gain.
Assumptions:
Rational individuals (maximize gain)
Full knowledge: tastes and preferences are known
Transitive Ordering: \(A>C\) when \(A>B\), \(B>C\). Also related to substitutability if two events are of equal probability
Continuity assumption
Properties:
\(u(A) > u(B)\) means A is more desirable than B where \(u\) is a utility function
Linearity property: If \(0 \le p \le 1\), then \(u(pA + (1-p)B) = pu(A) + (1-p)u(B)\)
- For two person: \(p[A,B] + (1-p)[C,D] = [pA - (1-p)C, pB + (1-p)D]\)
Anticipation = \(p A - (1-p) B\) where
\(p\) is the prob of getting A
A and B are two events.
\(u_1, u_2\) are utility function
\(c(s)\) is the solution point in a set S (compact, convex, with 0)
Assumptions:
If \(\alpha \in S\) s.t there is \(\beta \in S\) where \(u_1(\beta) > u_2(\alpha)\) and \(u_2(\beta) > u_2(\alpha)\) then \(\alpha \neq c(S)\)
- People try to maximize utility
If \(S \in T\) , c(T) is in S then \(c(T) = c(S)\)
If S is symmetric with respect to the line \(u_1 = u_2\), then \(c(S)\) is on the line \(u_1 = u_2\)
- Equality of bargaining
21.13.4 Iyer and Villas-Boas (2003)
- Presence of a powerful retailer (e.g., Walmart) might be beneficial to all channel members.
21.13.5 Desai and Purohit (2004)
2 customers segment: hagglers, nonhagglers.
When the proportion of nonhagglers is sufficient high, a haggling policy can be more profitable than a fixed-price policy