Social Preferences-II
Ultimatum Offers (General Results)
Almost no offers above 50% of pie
Mode and median offers in almost any study is
in the interval [40%, 50%] of pie
Mean offer usually in the interval [30%, 40%] of
the pie
Extremely few offers in the 0-10% range of the
pie
Ultimatum Responses (General
Results)
• Offers larger than 40% are rarely rejected
• Offers less than or equal to 20% are rejected
more than half of the time
• Probability of rejection increases as offer
decreases
• Overall, these results clearly reject the
subgame perfect Nash equilibrium prediction
(assuming money-maximizing, selfish
preferences)
INEQUALITY AVERSION
Explaining the Ultimatum Game (1)
Player 1: (1 , 1 ); Player 2: ( 2 , 2 ); offer s [0,1]
Proposition: It is a dominant strategy for player 2 to accept
any offer s  0.5, to reject s if s  s '( 2 )   2 /(1  2 2 )  0.5 and to
accept s  s '( 2 ).
Proof: Recall U i ( x)  xi   i max[ x j  xi , 0]  i max[ xi  x j , 0], i  j.
s  0.5  U 2 ( s )  s   2 (2 s  1)  0 for  2  1.
s  0.5 : Player 2 accepts only if U 2 ( s)  s   2 (1  2 s)  0, i.e., if
s  s '( 2 )   2 /(1  2 2 )  0.5.
Minimal Acceptable Offer
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Alpha of Responder
74
Explaining the Ultimatum Game (2)
Proposition (cont'd): If the proposer knows the preferences of
the responder, in equilibrium he will offer
if 1  0.5
 0.5

s *  [ s '( 2 ), 0.5] if 1  0.5
  s '( )
if 1  0.5
2

Proof: Proposer never offers s  0.5.
If 1  0.5, U1 strictly increasing in s for all s  0.5  s*  0.5.
If 1  0.5, he is indifferent between all offers s  [ s '( 2 ), 0.5].
If 1  0.5, he would like to increase his monetary payoff at the
expense of player 2. The prevent rejection, he offers s '( 2 ).
8
Explaining the Ultimatum Game (3)—GRADS ONLY
Proposition (cont'd): If the proposer does not know preferences of
the responder, but knows cdf F ( 2 ), with support [ , ], 0      ,
then the acceptance probability of an offer s  0.5 is
1


p   F ( s /(1  2s ))  (0,1)

0

if s  s '( )
if s '( )  s  s '( )
if s  s '( )
Hence, the optimal offer is given by
  0.5

s *   [ s '( ), 0.5]
 ( s '( ), s '(a )]

if 1  0.5
if 1  0.5
if 1  0.5
Proof: Probability of acceptance is F ( s /(1  2s ).
This prob. is 1 if s   (1  2 ) and 0 if s   /(1  a ).
 s*  ( s '( ), s '( )].
9
Problems with theories of inequality
aversion
(e.g., Falk, Fehr, Fischbacher, Economic Inquiry 2003)
5/5 game
8
2
0
0
5
5
2/8 game
0
0
8
2
8/2 game
8
2
0
0
8
2
0
0
2
8
0
0
10/0 game
0
0
8
2
0
0
10
0
0
0
 Inequality aversion would predict
the same rejection rates in all four
games!
 Intentions matter!
10
The effects of unchosen
alternatives
Proposer can offer: Either (8,2) or (5,5), (2,8),
(10,0).
ALTERNATIVE
REJECTED
OFFERED
TYPE
(5,5)
44%
31%
(2,8)
27%
80%
(10,0)
9%
100%
Relatively
unfair
Not
sacrificial
Relatively
fair
Dictator w/ Option to Take
List (2005)
• 4 treatments:
1.
2.
3.
4.
Baseline dictator game
“Take 1”
“Take 5”
Earnings
Some Problems with Fehr & Schmidt Theory
• Theories such as Fehr & Schmidt (Inequality Aversion)
assume that utilities only depend on payoff differences.
They do not take into account the effects of intentions.
• Payoff-based theories also do not take into account the
fact that social preferences can change according to
context (e.g. List (2005)—when you expand the action
set, behavior changes—if you have a stable utility
function defined over payoffs, this should not be
happening).
Another Example
My Coffeehouse Experiment…
Your Data: Passed Amount as P. 1.
Your Data: Returned Amount
Gift-Exchange Game