Bargaining under incomplete information

1
Bargaining under incomplete information

• Strategic manipulation of information

2
Set up

• A seller has a good that is free marginal cost
  is nil.
• He wants to sell it to the first customer that
  walks in
• The seller lives in a town where there are two
  classes of customers fanatics buyers and normal
  buyers.
• Half of the population is fanatic
• Fanatics receive a unit of pleasure for buying
  the good (vF1) while normal customers receive
  only half a unit of pleasure for buying the good
  (vN 0.6).
• When a customer comes in the seller does not know
  whether he is a fanatic or a normal customer.
• The seller has two chances for selling the good.
• He first claims a price p1 and if the customer
  accepts the game is over
• if the customer rejects he can claim another
  price p2 and if the customer accepts the game is
  over.
• if the customer rejects this second offer both
  receive nothing.
• Both buyers discount the future with the same
  discount factor d 0.95 while the seller does
  not discount the future.

3
Possible wrong strategies

• Naïve seller strategy
• Set a price that is a weighted average of the
  population in the first period (p1 0.8 ),
• If he does not sell at the price lower the price
  at the normal buyer level (p2 0.6 ).
• Half smart seller
• Set a price to capture the fanatics in the first
  offer (p1 1 ),
• If he does not sell at that price lower the price
  at the normal level (p2 0.6 ).
• What is the outcome for these strategies?
• None of them ever sell in the first period and
  everybody buys in the second period.

4
Why he never sells in the first period

• Naïve seller strategy
• normal customers will never buy
• Fanatics will make a surplus of vF - p1 1- 0.8
  0.2 ,
• If they wait till the second period they get a
  surplus of vF p2 1- 0.6 0.4
• Discounted to today with the discount factor so
  he gets a total surplus of 0.95 0.4 0.38.
• Half smart seller
• The normal seller will never buy in the first
  period as before
• Again the fanatic can make more surplus by
  waiting till tomorrow.
• Both strategies yield expected surplus of 0.6

5
Smart Seller forward looking

• If nobody buys in the first period and that does
  not give me any information about my customer
• the price that I set in the second period must be
  equal to the price that I set in the first
  period,
• but I know that I will not be selling anything
  because I did not sell in the first period
• that means that it is not a good idea to set the
  same price
• But in order for me to set different prices I
  have to learn something from my customer
• Therefore if I do not sell anything in the first
  period I MUST be learning something about my
  customer
• The only way to learn something about my customer
  when my customer does not buy is to have two
  different prices
• One price that only fanatics are willing to pay
  and another price that is targeting normal
  customers

6
Smart Seller prices

• Because normal buyer are not as prone to buy the
  good,
• If the seller sets a price that the normal buyers
  accepts, this price must be low enough for the
  fanatic to be willing to pay.
• This implies that the price targeting the fanatic
  must be higher than the price targeting the
  normal customer
• Will the seller set the lower price at the
  beginning and then the higher price in the second
  period?
• No, because that means that all transactions take
  place in the first period and
• there are no remaining customers in the second
  period.
• Therefore the higher price must be charged in the
  first period and the lower price in the second
  period.
• The second period price (low) targets normal
  customers and the first period price (high)
  targets fanatics.

7
Smart Seller backward looking I

• If the second period prize targets normal
  customers
• It is a good idea to squeeze as much as the
  seller can
• and set a price equal to the normal customer
  value (vN 0.6) p2 0.6.
• Now we know that if the customer that the seller
  faces is a normal customer he will buy in the
  second period
• We know this because the price he sets in the
  second period is p2 0.6 squeezes all from the
  normal customer and
• The price in the first period is higher
• Which prize should the seller set in the first
  period?
• The fanatic customer has a way to mislead the
  seller
• He can refuse to buy in the first period and buy
  in the second period at the price set for the
  normal buyer
• This will give him tomorrow a total surplus of vF
  p2 1 0.6 0.4.
• We have to discount that value to today which
  gives a present value of 0.95 0.4 0.38
• If the seller wants to sell to the fanatics in
  the first period he must set a price that leaves
  the fanatic with a total surplus of 0.38.

8
Smart Seller backward looking II

• So it must be that vF p1 0.38 so the seller
  squeezes as much as possible from the fanatic
  customer considering
• That the second period price is 0.6
• And the fanatic can always wait till the next
  offer and buy the good at 0.6.
• Therefore, the seller will set a price in the
  first period p1 0.62 and have a sale in the
  second period with a price p2 0.6
• Is this a good idea for the seller?
• The total expected profit that the seller
  generates is just 0.5 0.62 0.5 0.6 0.61.

9
Smart Seller equilibrium I

• If p2 gt 0.6 the normal customer will never buy in
  the second period,
• Then p2 gt 0.6 is not a good idea because only
  normal customer survive to the second period
  and the seller can not sell to them.
• If p2 lt 0.6 the normal customer will buy for sure
  but the seller is not squeezing as much as
  possible
• If p1 gt 0.62 the fanatic customer will never buy
  in the first period since he can wait to the
  second period and pretend to be a normal
  customer,
• This implies that all customers will buy in the
  second period at p2 0.6
• Total expected profit for this strategy is just
  0.6
• This is less than what he makes by setting p1
  0.62
• If 0.62 gt p1 gt 0.6 the fanatic customer will buy
  in the first period but the seller is not
  squeezing as much as he can
• If p1 lt 0.6 everybody will buy at that price and
  the profits will be just p1 lt 0.6

10
Smart Seller equilibrium II

• The normal buyer can not buy at the first period
  price p1 0.62
• The normal buyer can only buy at the second
  period price p2 0.6
• The fanatic buyer
• can buy in the first period and make a surplus of
  vF p1 1- 0.62 0.38
• Can wait and buy in the second period and make vF
  p2 1- 0.6 0.4 but he has to discount that to
  today the actual value is 0.8 0.4 0.38
• Because he is indifferent he takes the first
  period deal
• The buyer makes a total surplus of 0.61
• The fanatic makes a total surplus of 0.38
• The normal buyer makes no surplus

11
Smart Seller what happens

• In the first period the seller asks for a price
  p1 0.62
• If that price is accepted he knows that it was a
  fanatic and the game ends.
• If that price is rejected he knows that the
  customer he has in front of him is a normal
  customer.
• The seller sets a price p2 0.6 if the price in
  the first period is rejected.
• The customer buys and the game ends.

12
Conclusions

• When information is incomplete we need to add
  forward looking behavior
• If I offer this and is declined, who remains in
  the market?
• Sometimes bargaining outcomes can be delayed
• If the customer that enters the store is a normal
  customer he receives a price that he can not pay.
• Tomorrow the same customer comes in a he receives
  a price that he can pay
• A profitable trade was delayed
• Pretending is a source of power
• If a customer (fanatic) can pretend to be another
  customer (normal), and that affects the profits
  that the seller can make, the pretender can make
  a surplus the fanatic makes a surplus of 0.38.

13
Questions to think

• How can we draw a game tree for this situation?
• What happens if the percentage of fanatics and
  normal changes?
• What happens when we change the discount factor?
• What happens when we add a discount factor for
  the seller?

14
Fanatic buyer
Accept
Accept
Reject
Seller
Fanatic buyer
Fanatic buyer
Reject
Seller
15
Accept
Accept
Reject
Seller
Normal buyer
Normal buyer
Reject
Seller
Normal buyer
16
Fanatic buyer
Accept
Accept
Reject
Seller
Fanatic buyer
Fanatic buyer
Reject
Seller
Accept
Accept
Reject
Seller
Normal buyer
Normal buyer
Reject
Seller
Normal buyer
17
Fanatic buyer
Accept
Accept
Reject
Seller
Fanatic buyer
Fanatic buyer
Reject
Seller
Accept
Accept
Reject
Seller
Normal buyer
Normal buyer
Reject
Seller
Normal buyer
18
Fanatic buyer
Accept
Accept
Reject
Seller
Fanatic buyer
Fanatic buyer
Reject
Seller
Probability of fanatic buyer after all possible
prices claimed in the first period
Accept
Accept
Reject
Seller
Normal buyer
Normal buyer
Reject
Seller
Normal buyer