EXPERIMENT
-- Pay careful attention to the instructions. I will collect the
money from you if you bid $1000.
--I need a volunteer to be the seller (my representative)
--I will roll a six-sided die which will determine the reservation
price of the good to the seller: 1 = $1, 2= $2, …, 6 = $6.
--The seller will only sell you the good if your bid exceeds his
reservation value.
--If you own the good, you will get this money times 1.5 minus
what you paid for the good. For example, if the die falls on 4
and you own the good, you get 4 x 1.5 = 6 minus what you paid
the seller.
-- Only the seller (the volunteer) sees the outcome of the die.
-- The seller will only sell you the good if your bid exceeds his
reservation value.
-- I will pay you your winnings and collect from you your bid
-- Go ahead and submit a bid. Your chances of winning the item
are independent of others’ bids. This is not an auction but a twoplayer bargaining.
1
Bargaining
Two or more parties attempt to reach an agreement on dividing a
good.
Examples:
 Organizational buyer and organization seller – how to split
revenues or costs
 Buyer and Seller – i.e. car dealer
 Employer and employee – Labor negotiations
 Court settlements
2
The Nash bargaining game
 A two-player game where two players attempt to divide a
good
 Each player requests an amount of the cake.
 If their requests are compatible, each player receives the
amount requested
 If not, each player receives nothing.
 Assume the utility function for each player to be a linear
function in what they get
 An infinite number of Nash equilibria exist for this game.
Given any request, the corresponding strategy of the
equilibrium pair simply requests the remainder of the cake.
 If the first person did not request the entire cake for herself,
we have a strict Nash equilibrium.
3
The ultimatum game
 Two players attempt to divide a good.
 One player (the proposer) has sole possession of the
cake and offers a certain amount of the cake to the
second player (the receiver), keeping the rest for
himself.
 The second player has only two choices: take the offer
or leave it.
 If player 2 takes the offer, each player receives the
amount of cake due.
 If player 2 chooses to leave it, each player receives
nothing.
 An infinite number of Nash equilibria exist.
 “Fair” outcomes
 We say an equilibrium is subgame perfect if the
strategies present in that equilibrium are also in
equilibrium when restricted to any subgame.
 Subgmae perfect equilibrium: proposers demand the
entire cake minus epsilon (if the cake is infinitely
divisible) or N-1 pieces (if the cake has N pieces).
Receivers accept any nonzero offer.
4
The Rubinstein-Stahl Bargaining Model
Rubinstein (1982) / Stahl (1972):
Two players must agree on how to share a pie of size 1.
In periods 0, 2, 4, …, player 1 proposes a division (x, 1-x) that
player 2 accepts or rejects
If player 2 accepts any offer that game ends.
If player 2 rejects, he gets to propose a division in the next period
This is an infinite horizon game of perfect information
“Stages” are not “periods” – Each period has two stages
δ1, δ2 are discount rates
Therefore, utilities at time t are: 1t x,  2t (1  x)
Many Nash equilibria in this game.
Example: Player 1 always demands x = 1 and refuses all smaller x;
player 2 always offers x = 1 and accepts any offer
However, it is not subgame perfect: If player 2 rejects player 1’s
offer and offers a share x > δ1, player 1 should accept.
5
The subgame perfect equilibrium:
(1   i )
when it is his turn to
(1   1  2 )
 (1   i )
make an offer. He accepts any share greater than i
and
(1   1  2 )
refuses any smaller share.
Player i always demands a share
Proof:
(1   i )
 (1   i )
=1   i
is the highest
(1   1  2 )
(1   1  2 )
share that will be accepted by the opponent.
Player i’s demand of
Making a lower demand is bad— it will be accepted and result in a
lower payoff.
Making a higher demand is bad – it will be rejected and accepting
the next period’s opponent’s offer is worse:
 i (1 
(1   i )
(1    i )
(1    i )
)   i2

(1   1  2 )
(1   1  2 ) (1   1  2 )
Similarly, it is optimal for player i to accept any offer of at least
(1   i )
i
and to reject lower shares because if he rejects he
(1   1  2 )
(1   i )
can only hope to get
next period.
(1   1  2 )
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Fairness and reciprocity
Typically in ultimatum games, offers are around 60:40 and are
accepted.
Fairness explanation?
Reciprocity explanation?
7
Deadlines in bargaining and strategic delays
 Last minute agreements are believed to be common in
negotiation settings.
 For example, labor agreements tend to be reached just before
contracts expire and court settlements tend to be reached just
before deadlines.
 The predominant view in negotiations is that it is undesirable
and disadvantageous for a negotiating party to be under time
pressure (Stuhlmacher, Gillespie, & Champagne, 1998), and
that a negotiating party should not disclose its time pressure
(Brodt, 1994; Cohen, 1980).
 In contrast, others have argued that by disclosing its own
time pressure, a party can induce greater symmetry in
negotiation, thereby improving the outcome for itself (Moore,
2000).
 In experimental settings, subjects tend to drag on negotiations
until just before the deadline (Roth et al., 1988).
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THE GAME
In its simplest form, the RUG involves two players. Player 1 proposes a division
of a fixed amount (in our case 25 tokens) to player 2 in the form of an integer offer of x
tokens. If player 2 accepts the x tokens, then the game ends with this division as the
outcome. If player 2 rejects the offer, player 1 is then allowed to make another offer, as
long as that offer is strictly higher by a minimum increment (1 token), and as long as both
players’ shares remain strictly positive. In addition, player 1 may end the bargaining at
any point, in which case both players receive 0 tokens. That is, the game ends either
when player 2 accepts a proposal, or when, following a rejection, player 1 declines to
make a better offer. The game would also end if player 2 rejects the highest feasible offer
player 1 is able to make.
We call the game a “reverse” ultimatum game because player 2’s rejection of an
offer is a form of “reverse” ultimatum, that may be interpreted as meaning “give me
more, or we will each get nothing,” and because the subgame perfect equilibrium division
between proposer and responder is the reverse of that in the ultimatum game.
The proof that the unique subgame perfect equilibrium gives one token to the
proposer and the remainder (N-1 tokens) to the responder is as follows. At a subgame
following a proposal that gives the proposer only a single token, the responder will
accept, because the requirement that both players’ shares must be positive implies that the
proposer cannot make another offer if this proposal is rejected, so a rejection will lead to
both players receiving zero. At a subgame following a rejection of any other proposal, the
proposer is therefore faced with a choice of stopping the bargaining and receiving zero,
or making a new proposal and eventually receiving a payoff of (at least) one token. At
any subgame following a proposal that gives the proposer k>1 tokens, the responder is
therefore faced with a choice of accepting it and receiving N-k tokens, or rejecting it and
eventually receiving N-1 tokens.
That is, in the reverse ultimatum game, the proposer is potentially faced with a
series of small ultimata, in contrast to the ultimatum game in which it is the responder
who is faced with one ultimatum. In both kinds of ultimatum games there are many
imperfect equilibria supported by “non-credible” threats, but in the reverse ultimatum
game it is the proposer whose threat (to end the negotiations) is not credible, when
players are taken to be concerned solely with their own monetary payoffs.
Adding a deadline to the RUG reverses the subgame perfect equilibrium
prediction. The proposer is able to wait to the last second to make an offer, thereby
resulting in a conventional “take it or leave it” ultimatum.
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Average accepted offer in bargaining agreements
Standard deviation is shown in parentheses and N is the number of agreements.
One
One
Average One Responder Responder Responder
over
No
3 Minute
1 Minute
games Deadline(1RND) Deadline
Deadline
(1R3minD) (1R1minD)
13.34
11.48
10.52
Games
(1.43)
(3.30)
(2.57)
1-25
N = 262
N = 359
N = 351
12.92
11.44
10.74
Games
(1.00)
(3.08)
(1.76)
21-25
N = 59
N = 79
N = 70
10
Bargaining Agreements
1-minute deadline
350
300
250
200
150
100
50
0
0-10
11-20
21-30
31-40
41-50
51-60
Time in seconds
Bargaining Agreements
3-minute deadline
300
250
200
150
100
50
0
0-20
21-40
41-60
61-80 81-100
101120
121140
141160
161180
Time in seconds
9
8
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Round
No deadline
Bargaining Agreements
responder
Share accepted by
Share of responder
80
70
60
50
40
30
20
10
0
0-20 21- 41- 61- 81- 101- 121- 141- 161- 181- 201- 221- 241- >260
40 60 80 100 120 140 160 180 200 220 240 260
Time in seconds
11
The Winner’s Curse in negotiation
 Participants engage in a bilateral negotiation over the transfer
of a company with a value v uniformly distributed between 0
and 100.
 The value to the seller (reservation value) is v
 The value to the buyer is 1.5 v
 Only the seller knows the true value of the company.
 The seller will sell you the company if your bid is greater
than his reservation value.
 Participants consistently bid amounts that are greater than 0.
 Most bid between the expected value of the company (50)
and the ex-ante value of the company to the buyer (75).
12