January 28, 2001. March 3, 2002. October 30, 2003. Eric Rasmusen notes,
erasmuse@indiana.edu Http://Php.Indiana.edu/~erasmuse



 54. Ariel Rubinstein,  ``Perfect Equilibrium in a Bargaining Model, ''
Econometrica, 50: 97-109 (January 1982).






John Sutton has an easier proof of the main theorem  in

John Sutton, "Non-Cooperative Bargaining Theory: An Introduction," The
Review of Economic Studies, Vol. 53, No. 5. (October 1986), pp. 709-724,

and, originally I think (but I haven't checked),  in

Avner Shaked and  John Sutton, "Involuntary Unemployment as a Perfect
Equilibrium in a Bargaining Model,"   Econometrica, Vol. 52, No. 6.
(November 1984), pp. 1351-1364.

 I think I used that proof in Games and Information, but I don't
remember for sure. The proof in Rubinstein's paper is difficult because
Rubinstein is making it   general-- for any bargaining costs satisfying
A1 to A5. But in the end what is most commonly used in   models   are
the two special cases I and II, and for those, simpler proofs work  (but
see my other note, at the end of this file).

The Sutton (1986) and, I think, Shaked and Sutton (1984) discuss exit
options-- the possibility that a player can exit the game when it is his
turn to make an offer  and take an alternative payoff instead. This has
a peculiarly different effect than the conventional  threat point.

p. 299. N is 1, 2,3, etc., representing the period at which the
bargaining ends. A player prefers an earlier ending ordinarily.

A1. A bigger share of the pie  is better.

A2. Agreement sooner is better than agreement later, if the player gets
a positive share of the pie.

A3. If r in period t_1 is preferred by i to s a period later,    then
the same is true in period t_2, for any two periods t_1 and t_2.
(t_1<t_2 is not required.) This makes the problem stationary.

A4. If we have a sequence of shares for player 1 that approaches a limit
r, and each member of the sequence is preferred to share s, then the
limit is preferred to s also.

A5.  The extra share at time 1 that makes you indifferent between
getting your basic  share at time 0 or time 1 is  not any smaller if the
basic share  is bigger. Delay doesn't hurt less if what you eventually
get is bigger.

In I. Fixed Bargaining Costs, the original article has a typo. It said
(s_i -c_i dot t_i) instead of (s_i -c_i dot t_1). The typo is corrected
in the reader.

 REMARK. The remark points out that the assumptions impose stationarity
on the model-- a player's preferences between getting something at time
t and at t+1  do not change as t changes    and the pie shrinks,
bargaining costs mount up, some especially costly period of shrinkage is
reached. All periods look alike except that the pie may have shrunk.

p. 300. "The above equilibrium..." This paragraph points out that the
Nash equilibrium would require player 2 to reject player 1's demand for
epsilon more share, even though player 2 would lose more  by waiting a
period.

The f|s^1...s^T notation is confusing. Here is what it means. The old
notation said that viewed at period T+1,   1's strategy sequence
function at time t was f^{T+1}(s^1...s^T, r^1...r^{t-1}); that is, it
was a function  of all the behavior up to T and all the behavior from 1
period after T to t-1 periods after T.

 The new notation says that we will  use the notation f^t for that. f^t
will take as given the s^1...s^T, because that already has happened
before T+1. But it will be a function of r^1...r^{t-1}, because those
events haven't happened yet. Thus, the  equation could be abbreviated as

f^t(r) = f^{T+1}(s, r)

DEFINITION.

P1. A deviation by 2 from future-looking strategy of offering  fhat to f
would not be preferred by player 2, conditional on s^1 to s^T having
already been offered and rejected.

P2.    A deviation by 2 from saying YES to 1's period T offer, if
ghat^T=YES,    would not be preferred by player 2 to what would hapen if
he said NO.

P3.  A deviation by 2 from saying NO to 1's period T offer if ghat^T=
NO, would not be preferred by player 2 to what would hapen if he said
YES.

P4,P5, P6 are the analogous requirements for player 1.

p. 301. Lemma 2 is easy to prove because it is the analog of Lemma 1,
but for player 1 instead of player 2.

p. 302. Lemmas 1 and 2 say that the players won't deviate by  asking for
the wrong share.  Lemmas 3 and 4  say that the players won't deviate by
saying YES or NO  in the wrong circumstance.

  Proof of Lemma 3, Case B.  This is difficult, though the REMARK is
helpful in explaining it. I'll try again here. Suppose 1 offers a  and 2
accepts it in equilibrium, but we are thinking about  player 2 saying NO
and offering b instead. It can't be that player 2 could expect player 1
to accept b, because then player 1 would have deviated to offer b in the
first place, to prevent delay. So it must be that player 1 would say NO
to b, and to do that, he must expect to get a better share later.

p. 303. In the first line of the definition of d_2(y), the original
article has an x missing. It should read "for all x (x,0)",  as it does
in the reader.

 d_1(x) is the smallest number such that d_1 now is preferred to x in
one period.

d_1(x) <x and d_2(y) <y, because  smaller amounts now are preferred to
bigger amounts later. These two facts are worth mentioning, though not
stated till Proposition 3's proof,  because in Figure 1 they ensure that
the two dotted lines are both in the bottom right triangle.

 The significance of Figure 1 is that the equilibrium set Delta is where
the two d() functions intersect.

p. 304. To understand Proposition 3, it is helpful to look at the middle
panel of Figure 3 on page 305. Also, keep in mind that the "closed line
segment" can be a single point (and that is the ordinary case). Figure 2
is mean to illustrate Propositions 3 and 4, though it is never
referenced.

Proposition 4. Proposition 1 established that every point in Delta was
an equilibrium  partition.  Proposition 4 establishes, roughly,  that
every equilibrium partition is a point in Delta.Figure 2 is mean to
illustrate Propositions 3 and 4, though it is never referenced.

 Proof of Proposition 4. "Assume x_2<s" means to assume Proposition 4 is
false, since x_2 is defined here (tho not in so many words) as the
biggest element of Delta_1 and s is defined as  the  limit of the
biggest element of A.

p. 305. Figure 3. The reason these figures look so odd is that what
happens along the  sides of the box matters. Thus, in the first panel,
the dotted line runs along the horizontal axis at first and then turns
up diagonally. The original article seems to have had this messed up,
with some of the dotted lines missing. In panels 1 and 3, equilibrium
shares are indicated with black dots for the cases where the "weak"
player makes the first offer. (In each, if the "strong player makes the
first offer, the equilibrium shares are 0 and 1.)




          OTHER WAYS OF DISCOUNTING BESIDES SPECIAL CASES I AND II

     The   idea in approach I is that the game is one in which a pie
starts off at size 1, and if divided between the two players, their
shares  A and B will add up to A+B=1 and their utilities will be A and
B. If they wait a period, and have discount factors .9 and .7 (because
they suffer from the waiting), then their shares still add up to 1, but
now their utilities are .9A and .7B. If they wait two periods, their
utilities are .81A and .49B.

    If the pie is a pot of money, we need  the assumption that each
player's utility is linear in money-- that is, that they are risk
neutral. Otherwise,   the effect of the threat of one more period  delay
depends on  the current size of the pie. It may be that a player
starting with an offer of a share of .9 suffers  less from delay than a
player starting with an offer of a share of .1, simply because  a
player's utility drops precipitously if his wealth drops below .1--
perhaps that is the subsistence diet. Thus, when the payoffs from,   a
two-period delay are not .81A but .8U(A) for concave function U, be
wary of applying the 50-50 split result.

For a quite technical article exploring the relationship between time
preference and utility functions, see:

Peter C. Fishburn and  Ariel Rubinstein (1982)   "Time Preference,"
International Economic Review, Vol. 23, No. 3. (October  1982), pp. 677-
694.

    A   different situation is when the players are risk-neutral but
their discount factors change over time.  Thus, it might be that  Player
A's payoff from one period of waiting is .9A, but his payoff from two
periods of waiting is not .9*.9*A = .81A  but .9*.6*A = .54A.  There are
two ways this might come about.

First, it could be that his discount factor is independent of his
current position, but does depend on absolute time. His discount factor
for going from 2000 to 2001 is .9, but his discount factor for going
from 2001 to 2002 is .6 because everyone knows  market interest rates
will rise  sharply on January 1, 2001. This is     a   realistic
relaxation of the model, but it   takes us away from Rubinstein's Model
I, since we cannot  expect to describe the results in terms of just two
numbers for the player's discount factors. And we will need an infinite
record of the    discount rates for each period in the future.

Second,  it could be that Player A's discount factor is independent of
time, but does depend on his current position. His discount factor for
payments a year from the present might be .9, and for two years from now
might be .6.  Starting in 2000, this means that his discount factor for
going from 2000 to 2001 is .9  but his discount factor for going from
2001 to 2002 is .6.  But starting in 2001, this means his  discount
factor for going from 2001 to 2002 is .9  but his discount factor for
going from 2002 to 2003 is .6.  There is inconsistency across time. This
is the subject of the literature on "hyperbolic discounting". See

Ariel Rubinstein (2003)  "'Economics and Psychology'? The Case of
Hyperbolic Discounting," forthcoming,  International Economic Review
(2003), http://www.princeton.edu/~ariel/papers/delta3.pdf.

 Michael   Rabin  and Richard Thaler (2001), "Anomalies: Risk Aversion."
Journal of Economic Perspectives, 15: 219-232.

  Adding these complications means that the idea in Rubinstein (1982) of
using the stationarity of the two players' problem across time may well
fail, and so it may be difficult or impossible to find the equilibrium
from the infinitely repeated bargaining game. Remember, however, that it
is much easier to find the equilibrium of the T-period game, however
large T may be, and it may well happen that the equilibrium is unique
and converges to some value as T goes to infinity. See Chapter 12 (in
the 3rd or 4th forthcoming edition)  of my book, Games and Information,
to see how such a model works.


Another article that you may find useful in thinking about discounting
and bargaining is:

 Ken Binmore, Ariel Rubinstein and  Asher Wolinsky  (1986) "The Nash
Bargaining Solution in Economic Modelling,"The RAND Journal of
Economics, Vol. 17, No. 2. (Summer  1986),pp. 176-188.

  Finally, Ariel Rubinstein is unusually thorough in providing material
via the Web. See

 http://www.princeton.edu/~ariel/vitae.html

or back up to his homepage,

 http://www.princeton.edu/~ariel/