A. Shaked

J. Sutton

Ken Binmore

English

Research Papers in Economics

American Economic Association
Testing Noncooperative Bargaining Theory: A Preliminary Study
Author(s): K. Binmore, A. Shaked, J. Sutton
Source: The American Economic Review, Vol. 75, No. 5 (Dec., 1985), pp. 1178-1180
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1818658
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http://www.jstor.orgTesting Noncooperative Bargaining Theory: 
A Preliminary  Study 
By K. BINMORE, A.  SHAKED, AND J. SUTTON* 
Bargaining  theory has  received much  at- 
tention  of late. There has also been a grow- 
ing interest in experimental work on bargain- 
ing, notably by Reinhard Selten (1978), and 
by Alvin Roth, M. Malouf, and J. Murnighan 
(1981).  This  work  confirms a  view  that  is 
common among social psychologists: namely, 
that  subjects tend to seek a "fair" outcome 
to  bargaining  problems.  The  thrust of  the 
inquiry  is  then  to  determine what the sub- 
jects will regard as fair in a given situation. 
A tension exists between this work and the 
theoretical approach revitalized by Ariel Ru- 
binstein  (1982).  (See  also  Binmore,  1982, 
1983;  Shaked  and  Sutton,  1984.) This  new 
approach  involves  modeling  the  process  of 
offer  and  counteroffer by  means  of  which 
agreement can be reached, as a formal non- 
cooperative  game;  and studying agreements 
that  can  be  sustained  as  equilibria of  this 
game. 
The  tension  is  sharply  illustrated  by 
a recent experimental study of W. Guth, R. 
Schmittberger, and B. Schwarze (1982). (See 
also  Guth,  1983.)  Two  subjects  have  to 
divide  a  sum  of  money (the "cake"), using 
the  following  primitive procedure: Player 1 
makes  a  demand,  which Player 2  can  then 
accept or refuse. This concludes the game. If 
the demand  is refused, both players receive 
nothing.  A  strategic analysis assigns all (or 
nearly all) of the cake to Player 1, but experi- 
ments show that a much "fairer" division is 
usual. 
The work of Giuth  et al. seems to preclude 
a predictive role for game theory insofar as 
bargaining  behavior is concerned. Our pur- 
pose  in  this  note  is  to  report briefly on  an 
experiment that shows that this conclusion is 
unwarranted. (Only  the briefest account  of 
the  experiment  is  offered  here;  for  a  full 
account, see our 1984 paper.) 
This  does  not  mean  that our  results are 
inconsistent  with those of Giith et al. Under 
similar conditions, we obtain similar results.' 
Moreover,  our  full  results  would  seem  to 
refute  the  more  obvious  rationalizations of 
the  behavior  observed  by  Gulth et  al.  as 
"optimising  with complex motivations." In- 
stead, our results indicate that this behavior 
is not stable in the sense that it can be easily 
displaced  by  simple  optimizing  behavior, 
once  small changes are made in the playing 
conditions. 
I. The Experiment 
In the present work, we went beyond the 
one-stage  " ultimatum" game of  Guth et al. 
and examined a two-stage game, as follows: 
Stage I: The cake is of  size 100 pence. 
Player  1  makes  a  proposal  (X);  Player  2 
accepts (1 receives X, 2 receives 100 -  X)  or 
rejects (game continues). 
Stage  II: The cake is of  size 25 pence. 
Player  2  makes  a  proposal  (X');  Player  1 
accepts (1 receives X', 2 receives 25 -  X') or 
rejects (1 receives 0, 2 receives 0). 
A  game-theoretic  analysis  requires  that 
Player  1  makes  an  opening demand in  the 
range 74-76  pence, and Player 2 accepts any 
opening  demand of 74 pence or less (for he 
cannot  do  better  by  refusing,  even  if  he 
obtains  the entire cake in the second stage). 
*London  School  of  Economics, Houghton St., Lon- 
don  WC2A  2AE,  UK.  We gratefully acknowledge the 
financial  support  of  the  International Centre for Eco- 
nomics  and  Related  Disciplines  (Suntory-Toyota 
Foundation),  and the hospitality of the Psychology De- 
partment  at  LSE,  where  our  experimental work  was 
conducted,  under the immediate supervision of Yasmin 
Batliwala, Mimi Bell, and Maria Herrero. We also thank 
Werner  Guth,  Alvin  Roth,  and  particularly Reinhard 
Selten for comments on an early draft.  'See  fn. 2 below. 
1178 VOL. 75  NO. 5  BINMORE ETAL.: NONCOOPERATIVE BARGAINING THEORY  1179 
50  50  50 
GAMEA  GAME  B 
40-  40- 
30 
30-  .-  30- 
.....,  ,  .....:.'  .  ..  .  .  .  .  ..  ..  .:: 
@ 20-  16  20- 
E 
~~~12 13  1  4 
... 
; 
.~~~~~~~~~~~~.......  ,.,......-,,,,.,.. 
? 10-  10-  :  8  : 
X  ~~~~~~~~~~~~~~...  ...... 
E  ~  ~3  2  :2  :  : 
Z  0-  0-  .  :: 
5op  7%5p  loop  50p  TiSp  loop 
Opening  demands  Opening  demands 
FIGURE 1.  OPENING DEMANDS  FOR  MAIN RESULTS 
We  studied  the game, using subjects who 
were isolated from each other, and who com- 
municated  their decisions via linked micro- 
computers.  Following  lengthy pilot  studies, 
in which we solicited players' comments after 
they  had  played  the  game,  we  decided  to 
extend the design, as follows. We invited the 
subject who had filled the role of Player 2 to 
play the game again, but this time he would 
fill  the  role  of  Player 1.  We recorded only 
his  opening  demand  in  this  second  game 
(Game B). 
II. The  Results 
We  focus  here  on  the  main  features of 
interest. The opening demands made in Game 
A  and  Game  B,  respectively, are shown in 
the  histograms  in  Figure 1. They  exhibit a 
marked change of behavior between Game A 
and  Game  B. A  tendency to "play  fair" in 
Game A becomes  a strong tendency to play 
"like a game theorist" in Game B. 
This  marked  change  in  behavior  con- 
stitutes the first of the two main findings of 
the present study. The null hypothesis is that 
the opening demands in each game are drawn 
from the same population and is rejected at 
the 0.1  percent level (Kolmogoroff-Smirnoff 
two-tailed test). 
Focusing  on  those subjects who filled the 
role of Player 2 in Game A, we looked at the 
TABLE  1 
Opening  Response to High Opening Demands 
Demand  (63 ?  a:!? 77) in Game A 
in Game B  No  Yes 
b<62  1(F)  2 
b ?63  2  17(G) 
subsample  who  faced  a  "high" demand in 
that game. A  fair player would reject a high 
demand, and would not himself make a high 
demand  (when  offered the chance to act as 
Player 1, in Game B). The results (shown in 
Table  1)  indicate little support for  the view 
that a substantial proportion  of the population 
are  "fairmen"  as  opposed to  "gamesmen." 
The  table shows  the relationship between a 
subject's  response  to  the  opening  demand 
made  of  him  in  Game A,  and the opening 
demand which he later makes when acting as 
Player 1 in Game B. Cell G denotes Games- 
men,  cell  F  denotes  Fairmen.  We  chose 
the midpoint  between 50 and 75 as our di- 
viding  line between low and high demands. 
The  table  refers only  to  the  subsample of 
our population  who  faced high demands in 
Game A. 
What,  then,  of  the players who filled the 
role  of  Player  1  in  Game A,  and who  ex- 
hibited  a marked tendency to make fair de- 1180  THE AMERICAN ECONOMIC REVIEW  DECEMBER 1985 
mands?  While  we  have  considered  various 
possible  explanations, the interpretation that 
we  favor is  this: subjects, faced with a new 
problem,  simply  choose  "equal division" as 
an "  obvious" and "acceptable" compromise 
-an  idea familiar from the seminal work of 
Thomas Schelling (1960). We suspect, on the 
basis  of  the  present experiments, that such 
considerations  are easily displaced by calcu- 
lations  of  strategic advantage, once  players 
fully appreciate the structure of the game. 
Finally,  it is important to note that Guith 
et  al. did in  fact study subjects playing the 
one-stage ultimatum game for a second time, 
without observing any marked change in be- 
havior.2  Thus,  it  is  not  only  this  feature 
which distinguishes our results from theirs. 
The key feature to note, in this respect, is 
that responses to opening demands in Game  A 
were  strongly  biased  in  favor of  "rational- 
ity."  (Of  22  opening  demands in the range 
63 < a < 77,  only  3  were rejected.) On  the 
other hand, at the second stage of  Game A 
-following  a refusal at the first stage-sub- 
jects  showed a strong tendency to reject high 
demands (as in the study of Guth et al). 
Our suspicion is that the one-stage ultima- 
tum game is a rather special case, from which 
it is dangerous to draw general conclusions. 
In the ultimatum game, the first player might 
be  dissuaded  from making an  opening  de- 
mand  at,  or  close  to,  the "optimum" level, 
because  his  opponent  would  then  incur  a 
negligible cost in making an "irrational" re- 
jection.  In the two-stage game, these consid- 
erations are postponed  to  the second stage, 
and so their impact is attenuated.3 
2Opening  demands were slightly higher, and refusals 
of  these demands  more  frequent. 
3There  remains  the  possibility  that  the  difference 
between  our results and those of  Giuth et al. might be 
traced  to  differences  in  the experimental environment 
rather than to differences in the game played. Giuth  et al. 
operated in an open environment within which subjects 
could  see  each  other  (although  the  identity  of  their 
current opponent was, of course, a secret). Our assistant, 
Yasmin  Batliwala, has run a controlled experiment to 
check  for  this  possibility  (which will be  reported sep- 
arately).  Replicating  our  experimental conditions,  she 
compared the behavior of subjects playing our two-stage 
game with that of a control group playing the one-stage 
ultimatum  game.  Broadly,  the  results  confirmed  our 
present  interpretation. Behavior in  the two-stage game 
was  similar to  that reported in this paper. Behavior in 
the  one-stage  ultimatum game was consistent with the 
observations  of  Giuth et  al. in that game theory was a 
poor predictor of outcomes. 
REFERENCES 
Binmore, K. G., "Perfect  Equilibria in  Bar- 
gaining  Models,"  ICERD  Discussion  Pa- 
per, London School of Economics, 1982. 
,  ("  Bargaining  and  Coalitions  I," 
ICERD  Discussion  Paper, London School 
of Economics,  1983. 
, Shaked,  A. and  Sutton,  J., "  Fairness or 
Gamesmanship in Bargaining: An Experi- 
mental Study," ICERD Discussion Paper, 
London School of Economics, 1984. 
Guth,  W.,  " Payoff  Distributions  in  Games 
and the Behavioral Theory of Distributive 
Justice," mimeo., Koln, 1983. 
, Schmittberger,  R. and  Schwarze,  B., "An 
Experimental Analysis of Ultimatum Bar- 
gaining," Journal of Economic  Behavior  and 
Organization, 1982, 3, 367-88. 
Roth,  Alvin, Malouf, M.  and Murnighan,  J., 
"Sociological  versus  Strategic Factors  in 
Bargaining," Journal of Economic Behavior 
and Organization, 1981, 2, 153-77. 
Rubinstein, Ariel, "Perfect  Equilibrium in  a 
Bargaining Model," Econometrica,  January 
1982,  50, 97-109. 
Shaked, A. and Sutton, J., "Involuntary Un- 
employment  as a Perfect Equilibrium in a 
Bargaining  Model,"  Econometrica,  No- 
vember 1984,  52, 1351-64. 
Selten,  Reinhard, "The  Equity  Principle  in 
Economic  Behavior,"  in  H.  Gottinger 
and  W.  Leinfellner, eds.,  Decision  Theo- 
ry  and  Social  Ethics,  Issues  in  Social 
Choice,  Dordrecht:  Reifel  Publishing, 
1978, 289-301. 

An Experimental Analysis of Ultimatum Bargaining,&quot;

Bargaining and Coalitions I,&quot; ICERD Discussion Paper, London School of Economics,

Fairness or Gamesmanship in Bargaining: An Experimental Study,&quot; ICERD Discussion Paper, London School of Economics,

Involuntary Unemployment as a Perfect Equilibrium in a Bargaining Model,&quot; Econometrica,

Payoff Distributions in Games and the Behavioral Theory of Distributive Justice,&quot; mimeo.,

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Testing Non-Cooperative Bargaining Theory: A Preliminary Study

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