(Non) Equilibrium Selection, Similarity
Judgments and the “Nothing to Gain / Nothing
to Lose” Effect
Jonathan W. Leland
The National Science Foundation*
June 2007
*The research discussed here was funded by an Italian Ministry of Education
“Rientro dei Cervelli” fellowship. Views expressed do not necessarily
represent the views the National Science Foundation nor the United States
government. Not for quote without permission
Motivation
“Many interesting games have more than
one Nash equilibrium. Predicting which of
these equilibria will be selected is perhaps
the most important problem in behavioral
game theory.” (Camerer, 2003)
Games with Multiple Equilibria
The Matching Game
The Stag Hunt Game
Player 2
Player 2
L
Player
1
U $80, $80



$10, $10
D
$10, $10

R
$50,$50
L
Player
1
R
U $80, $80
$10, $50
D $50, $10
$50,$50
Two pure strategy equilibria
One pareto superior, one pareto inferior
Incentives are compatible – problem is coordination
Stag-hunt is harder – achieving pareto outcome is riskier
Equilibrium Selection Criteria

Payoff Dominance
choose the equilibrium
offering all players their
highest payoff – predicts UL
The Stag Hunt Game


Security-mindedness
Player 2
Player
1
L
R
U
$8, $8
$1, $5
D
$5, $1
$5,$5
choose the strategy that
minimizes the worst possible payoff – predicts DR


Risk Dominance

choose the strategy that minimizes loses incurred by players
as a consequence of unilaterally deviating from their eq.
strategy – predicts DR
An Alternative Approach
Similarity Judgments in Choice
R:{$X
S:{$M
X similar /
dissimilar M
,
,
p
q
p similar /
dissimilar q
Favor R,
Favor S,
Inconclusive,
Inconsequential
;
;
$Y
$N
,
,
Y similar /
dissimilar N
1-p
1-q
}
}
1-p similar /
dissimilar 1-q
Favor R,
Favor S,
Inconclusive,
Inconsequential
Choose R (S) if it is favored in some comparisons and
not disfavored in any, otherwise choose at random.
Similarity Judgments
The “Nothing to Gain/Nothing to Lose” Effect
R:{$10
S:{$9
,
,
10 ~x 9
.90
.90
;
;
.90 ~ p .90
$0
$9
9 >x 0
Inconsequential
R’:{$10 ,
S’:{$1
,
10 >x 1
Favors R’
,
,
.10
.10
}
} antg
.10 ~ p .10
Favors S
.10
.10
.10 ~ p .10
;
;
$0
$1
1 ~x 0
,
,
.90
.90
.90 ~p .90
Inconsequential
} antl
}
Similarity Judgments and the Allais
Paradox
S:{$3000 ,
R:{$6000,
.90
.45
;
;
6000 >x 3000 .9 >p .45
$0
$0
0 ~x 0
Inconclusive
S’:{$3000,
R’:{$6000,
.10
.55
}a
}
.55 >p .10
Favors S
.02
.01
6000 >x 3000 .02 ~p .01
Favors R’
,
,
;
;
$0
$0
,
,
0 ~x 0
.98
.99
.99~p .98
Inconsequential
}
} antl
Similarity Judgments and
Intertemporal Choice
T1:{$20,
T2:{$25 ,
S or D?
$25 >x $20
T11:{$20,
T12:{$25 ,
S or D?
$25 >x $20
1 month
2 months
}
}
S or D?
2 >t 1
11 month
12 months
S or D?
12 ~t 11
Inconclusive
Choose either
}
}
Nothing to lose
Choose T12
Sources of prediction in Similarity
based models


Intransitivity of the similarity relation

E.g., 20 ~x 17, 17 ~x 15 but 20 >x 15

Theoretically inconsequential manipulation of prizes,
probabilities, dates of receipt may have consequences if they
influence perceived similarity or dissimilarity.
Framing of the choice

Framing determines what is compared with what –
theoretically inconsequential changes in the description of
the choice may influence what is compared with what.
Application to Games - Preliminaries

Assume


Players all have the same Bernoullian utility function
Let:

>x mean “is dissimilar and greater than”


A strict partial order (asymmetric and transitive)
~x mean “is similar to”

Symmetric but not necessarily transitive (e.g., 20 ~x 15, 15
~x 10 but 20 >x10.
Similarity Judgments in Games
Player 2
L
R
Player
1



U
h, t
l, c
D
m, b
m, c
Player 2
L
R
Player
1
U
8, 8
2, 5
D
5, 2
5, 5
Payoffs to Player 1: h(igh) > m(edium) > l(ow)
Payoffs to Player 2: t(op) > c(enter) > b(ottom)
Decision process





Do I have a dominating strategy, if so, choose it.
Do I have dominating strategy in similarity – if so, choose it.
Does Other have dominating strategy, if so, best respond
Does Other have dominating strategy in similarity, if so, best respond
?
Similarity Judgments in the StagHunt

Player 1 (2):
Checks for dominance
U
Player
 Checks for dominance in
1
similarity
D
 Checks for dominance for
P2 (1), and best responds
 Checks for dominance in
similarity for P2(1), best responds
 Chooses at random

L
R
8, 8
2, 5
5, 2
5, 5
An Example


If Other L and You U Y=$8,
D
$5
Favors U,D,I


If Other R and You U Y=$2
D
$5
O=$8
$2
Favors L,R,I
O=$2
$5
Favors U,D,I Favors L,R,I
If U(D) favored in some and not disfavored in any, Choose
U(D), otherwise
If L(R ) favored in some and not disfavored in any, best
response to L(R ), otherwise random.
The “Nothing to Lose” Effect and
the Payoff Dominant Eq.




Decrease m and c.
For Player 1:
h >x m ~x l,
Choose U – ntl
For Player 2:
t >x c ~x b,
Choose L – ntl
Outcome is payoff
dominant UL
Player
1
L
R
8, 8
2, 2.1
2.1, 2
2.1, 2.1
U
D
The “Nothing to Gain” Effect and
the Security-minded Eq.




Increase m and c
For Player 1:
h ~x m >x l,
Choose D – ntg
For Player 2:
t ~x c >x b,
Choose R – ntg
Outcome is securityminded DR
Player
1
L
R
8, 8
2, 7.9
7.9, 2
7.9, 7.9
U
D
The “Nothing to Gain/Nothing to
Lose” Effect and Non-eq. Outcomes




Increase m, decrease c.
For Player 1:
Player
x
x
h ~ m > l,
1
Choose D – ntg
For Player 2:
t >x c ~x b,
Choose L – ntl
Outcome is non-equilibrium
DL
L
R
8, 8
2, 2.1
7.9, 2
7.9, 2.1
U
D
Predictions in the Stag Hunt
x
x
h > m, m ~ l
x
t >x c, c ~x b
Increasing c
t >x c, c >x b,
t ~x c, c ~x b
t ~x c, c >x b
UL
ntl, ntl
UL
ntl, br
UR
ntl, ntg
UL
br, ntl
??
DR
br, ntg
DL
ntg, ntl
DR
ntg, br
DR
ntg, ntg
x
Increasing h > m, m > l ,
m
h ~x m, m ~x l
h ~x m, m >x l
Testing the Ntg/Ntl Effect –
Experiment Details





76 students at the University of Trento
Experiment consisted of 3 parts, 1st of which
involved games.
9 games – 5 stag hunts, 3 matching pennies
games, 1 additional stag hunt (always last)
Order otherwise randomized
Subjects played 1 of games at end of session –
payouts between 1.20 and 8.00 euro.
Games and Individual Results
Increase c
I
n
c
r
e
a
s
e
m
L
R
U 8.00, 8.00 2.00, 2.10 37 (97%)
D 2.10, 2.00 2.10, 2.10 1 (3%)
37 (97%) 1 (3%)
ntl
ntl
U
D
L
R
U 8.00, 8.00 2.00, 2.10 17 (45%)
D 7.90, 2.00 7.90, 2.10 21 (55%)
35 (92%) 3 (8%)
U
D
L
8.00, 8.00
5.00, 2.00
26 (68%)
?
L
8.00, 8.00
7,90, 2.00
20(53%)
R
2.00, 5.00 18 (47%)
5.00, 5.00 20 (53%)
12 (32%)
?
R
2.00, 5,00 11 (29%)
7,90, 5.00 27 (71%)
18 (47%)
L
R
U 8.00, 8.00 2.00, 7.90 14 (37%)
D 7.90, 2.00 7.90, 7.90 24 (63%)
13 (34%) 25 (66%)
Results Regarding Game Outcomes
I
n
c
r
e
a
s
i
n
g
P1 U
95%
3%
37 (97%)
D
3%
0%
1 (3%)
37 (97%) 1 (3%)
L
R
P1 U
32%
15%
18 (47%)
D
36%
17%
20 (53%)
26 (68%) 12 (32%)
L
R
m P1 U
41%
4%
17 (45%)
D
51%
4%
21 (55%)
35 (92%) 3 (8%)
L
R
P1 U
15%
14%
11 (29%)
D
37%
34%
27 (71%)
20(53%) 18 (47%)
L
R
P1 U
13%
24%
14 (37%)
D
22%
42%
24 (63%)
13 (34%) 25 (66%)
Performance Relative to Proposed
Selection Criteria
Game
I-I
Predominant Eq. Predicted Based on
Player' Responses UL (95%)
Equilibrium Selection Criterion
Mixed Strategies
DR
Payoff Dominance
UL
Security-mindedness
DR
Risk Dominance
UL
ntg/ntl
UL
III-I
III-III
DL (51%)
DR (42%)
25% each
UL
DR
UL or DR
DL
UL
UL
DR
DR
DR
Games of Pure Conflict
Player’s interests
are diametrically opposed
 No equilibrium in pure
strategies, only a mixed
strategy

Player
1
L
R
h, b
m, t
l, c
h, b
U
D
Games of Pure Conflict and Ntg/Ntl
Effects







Player 1 compares:
L
R
high and low and
U
Player
h, b
m, t
high and middle
1
Increasing m produces
D
l, c
h, b
“Nothing to Gain” effect
- choose U
Player 2 compares top and bottom and bottom and
center. Decreasing c produces “Nothing to Lose”
effect – choose R
Predictions in Conflict Games
Decreasing c
t >x b, c ~x b
t >x b , c >x b
h >x l,
h >x m
Increasing
m
h >x l,
h ~x m
DR
br, ntl
??
UR
ntg, ntl
UR
ntg, br
Results
IV-III
P2
P1
U
D
m=
c=
P1
U
D
P1
U
D
L
7.5, 3.5
3.5, 7.4
VI-I
P2
R
3.6, 7.5
7.5, 3.5
L
R
U
7.5, 3.5
7.49, 7.5
P1
D
3.5, 3.51
7.5, 3.5
3.6
7.49
7.4
3.51
Results and Implied Outcome Frequencies
P2
P2
L
R
L
R
13%
50%
U
90%
24 (63%)
5%
P1
8%
29%
D
0%
5%
14 (37%)
8 (21%)
30 (79%)
2 (5%)
36 (95%)
Mixed Strategy Probablilities and Implied Outcome Frequencies
P2
P2
L
R
L
R
24%
25%
U
0%
0%
49%
P1
25%
26%
D
0%
100%
51%
49%
51%
0%
100%
36 (95%)
2 (5%)
0%
100%
Additional Results in Conflict Games
C11
Player 2
# P1/P2
L
Player 1
U
D
10
1
23%
7%
R
1
1
0
1
30%
55%
15%
10
0
78%
U(ntl)
34/34
22%
70%
R(ntl)
m=c=1
C13
Player 2
L
Player 1
U
D
10
1
82%
15%
R
9
9
0
1
97%
L(ntg)
3%
0%
10
0
85%
U(ntl)
31/31
15%
3%
m=1, c=9
C33
Player 2
L
Player 1
U
D
10
9
10%
90%
100%
L(ntg)
R
9
9
0
9
0%
0%
0%
10
0
10%
90%
29/29
D(ntg)
m=c=9
Across Game Results
P2
1
P1
L
R
U
8.00, 8.00
2.00, 2.10
D
2.10, 2.00
2.10, 2.10
P2
3
P1
P2
2
L
8.00,
8.00
7.90,
2.00
R
2.00, 2.10
7.90, 2.10
P2
4
L
R
L
R
U
8.00, 8.00
2.00, 7.90
7.5, 3.5
7.49, 7.5
D
7.90, 2.00
7.90, 7.90
3.5, 3.51
7.5, 3.5
P1 Predicted Pattern U D D U
P2 Predicted Pattern L L R R
Predicted
1 Off
2 Off
3 Off
16
11
9
2
Player 1
42%
29%
24%
5%
Player 2
22
58%
14
37%
1
3%
1
3%
Total
38
25
10
3
50%
33%
13%
4%
Across Game Results cont.
P1 Choice Patterns
Predicted
1 off
2 off
3 off
1
U
U
U
U
U
U
D
2
D
U
D
U
U
U
U
3
D
D
U
U
D
U
D
4
U
U
U
U
D
D
U
N
16
6
5
8
1
1
1
%
42%
16%
13%
21%
3%
3%
3%
100%
1
L
L
L
R
2
L
L
L
R
3
R
L
L
R
4
R
R
L
L
N
22
14
1
1
%
58%
37%
3%
3%
100%
P2 Choice Patterns
Predicted
1 off
2 off
3 off
Other Implications - The Relativity
of Similarity Judgments
P2
P1
U
D
L
80, 40
40, 80
48%
P2
R
40, 80 48%
80, 40 52%
52%
P1
U
D
P2
P1
U
D
L
R
320, 40 40, 80 96%
40, 80 80, 40 4%
16%
84%
L
44, 40
40, 80
80%
R
40, 80 8%
80, 40 92%
20%
Similarity Judgments and Framing
Effects In Choice Under Uncertainty
A
B
1-20 (20%) 21-40 (20%) 41-80 (40%) 81-100 (20%)
$5
$5
$0
$13
$0
$12
$5
$0
A'
B'
1-20 (20%) 21-40 (20%) 41-80 (40%) 81-100 (20%)
$0
$13
$5
$0
$5
$5
$0
$12
A
B
A'
31 (53%)
5 (8%)
36 (61%)
B'
18 (31%)
5 (8%)
23 (39%
49 (89%)
10 (17%)
Framing Effects in Games Own First vs Other First and non-Equilibrium
Outcomes
III-I
Player 2
L
8.00, 8.00
7.90, 2.00
U
D
Player 1
III-I*
Player 2
R
2.00, 2.10
7.90, 2.10
L
7,20, 7,20
7,10, 1,20
U
D
Player 1
Increasing c
x
h >x m, m ~x l
x
Increasing
m
x
x
x
Increasing c
x
x
t > c, c ~ b
t > c, c > b,
t ~x c, c ~x b
t ~ c, c > b
UL
ntl, ntl
UL
ntl, br
UR
ntl, ntg
x
h >x m, m ~x l
x
h > m, m > l ,
x
x
h ~ m, m ~ l
h ~x m, m >x l
R
1,20, 1,20
7,10, 1,30
x
UL
br, ntl
??
DR
br, ntg
DL
ntg, ntl
DR
ntg, br
DR
ntg, ntg
Increasing
m
x
x
x
x
x
t > c, c ~ b
t > c, c > b,
t ~x c, c ~x b
t ~ c, c > b
UL
br, br
UL
ntl, br
DL
br, br
UL
br, ntl
??
DR
br, ntg
UR
br, br
DR
ntg, br
DR
br, br
x
h > m, m > l ,
x
x
h ~ m, m ~ l
h ~x m, m >x l
Framing and Question Format in
Games
If Other chooses
If Other chooses
L
R
and you choose
and you choose
U
You receive
7,20
and Other receives
7,20
D
You receive
7,10
and Other receives
1,20
U
You receive
1,20
and Other receives
1,30
D
You receive
7,10
and Other receives
1,30
Given the payoffs Other faces, what choice do you predict he or she will make? L)____ or R______
Please indicate which choice you would like to make. U_____ or D_____
Results and Implications for Quantal
Response Models
Responses and Implied Outcome Frequencies
III-I
Player 2
Player 1
U
D
L
41%
51%
35 (92%)
R
4%
4%
3 (8%)
III-I*
Player 2
17 (45%) Player 1
21 (55%)
L
U
54%
D
14%
26 (68%)
R
25%
7%
12 (32%)
30 (79%)
8 (21%)
What Would You Choose?
For Player 1
If other chooses
W
and you choose
U
D
You receive
6
12
and Other receives
13
12
If other chooses
X
and you choose
U
D
You receive
5
8
and Other receives
5
8
If other chooses
Y
and you choose
U
D
You receive
2
6
and Other receives
7
6
If other chooses
Z
and you choose
U
D
You receive
13
2
and Other receives
2
2
Level-1 Bounded Rationality vs.
Similarity
EQ and Similarity and Level-1 predict DW
Player 2
W
Player 1
U
D
2
12
X
13
12
Y
13
5
8
8
12.5
5
6
Z
7
6
6.5
7
2
2
2
6.5
6.75
7
2
Similarity and EQ predicts U, Level-1 predicts D
Player 2
W
Player 1
U
D
13
2
X
13
12
5
12
Y
5
8
12.5
7
8
Z
7
6
6.5
2
6
2
2
6.5
6.75
7
2
EQ and L1 predict D, Similarity allows U
Player 2
W
Player 1
U
D
7
12
X
13
12
12.5
Y
5
5
8
8
6.5
2
6
Z
7
6
6.5
13
2
2
2
2
6.75
7
A Speculation - the social benefit of
individual irrationality?
I-I
P2
P1
U
D
P1
U
D
I-II
I-III
II-I
II-II
II-III
III-I
P2
III-II
III-III
P2
L
8.00, 8.00
16
2.10, 2.00
4.10
L
8.00, 8.00
16
7.90, 2.00
9.90
R
2.00, 2.10
4.10
2.10, 2.10
4.2
R
2.00, 2.10
4.10
7.90, 2.10
10
P1
U
D
L
8.00, 8.00
16
7.90, 2.00
9.90
R
2.00, 7.90
9.90
7.90, 7.90
15.80
A Speculation - the social benefit of
individual irrationality? (cont.)
Player 2
L
Player 1
U
D
10
1
11
2
R
1
1
0
1
10
1
10
0
C13
Player 2
L
Player 1
U
D
10
1
19
10
R
9
9
0
1
10
1
10
0
C33
Player 2
L
Player 1
U
D
10
9
19
18
R
9
9
0
9
10
9
10
0
A Speculation - the social benefit of
non-strategic thinking and limits to
learning
III-I
Player 2
Player
1
U
D
L
8.00, 8.00
7.90, 2.00
R
2.00, 2.10
7.90, 2.10
III-I*
Player 2
Player
1
U
D
L
7.20, 7,20
7.10, 1.20
R
1.20, 1.20
7.10, 1.30
Some Other Speculations and
Conjectures

Things will matter that shouldn’t
Time, recalibration and regret
 Differences in similarity perceptions and acrimony in
negotiations

Conclusions


Many choice anomalies can be explained if people
employ “nothing to gain/nothing to lose reasoning”
The same reasoning process applied to games predicts:

play in coordination and conflict games and


the successes and failures of equilibrium selection criteria and mixed
strategy choice
systematic differences in play as a consequence of
theoretically inconsequential changes in the way strategy
choices are elicited.
References
















Camerer, C. Behavioral Game Theory: Experiments on Strategic Interaction,
Princeton, 2003.
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Testing the Ntg/Ntl Effect –
Question Format
If Other chooses L and you choose U
D
You receive
You receive
8,00 and Other receives 8,00
5,00 and Other receives 2,00
If Other chooses R and you choose U
D
You receive
You receive
2,00 and Other receives 5,00
5,00 and Other receives 5,00
The Problem


“existing deductive selection rules have been shown to
do poorly in experiments” (Haruvy & Stahl, 2004)
Should we be surprised?


“…Game theory is the study strategic interactions among rational
players..”
We know people behave irrationally in risky and
intertemporal choice situations – why would we expect
them to do better in complex strategic settings?
A Speculation the problem with being strategic in a
non-strategic world
III-I
Player 2
L
8.00, 8.00
7.90, 2.00
U
D
Player 1
III-I*
Player 2
R
2.00, 2.10
7.90, 2.10
L
7,20, 7,20
7,10, 1,20
U
D
Player 1
Increasing c
x
x
x
x
x
h > m, m ~ l
Increasing
m
h > m, m > l ,
h ~x m, m ~x l
h ~x m, m >x l
x
x
x
R
1,20, 1,20
7,10, 1,30
Increasing c
x
x
t > c, c ~ b
t > c, c > b,
x
x
t ~ c, c ~ b
t ~ c, c > b
UL
ntl, ntl
UL
ntl, br
UR
ntl, ntg
UL
br, ntl
??
DR
br, ntg
DL
ntg, ntl
DR
ntg, br
DR
ntg, ntg
x
x
x
x
x
h > m, m ~ l
Increasing
m
h > m, m > l ,
h ~x m, m ~x l
h ~x m, m >x l
x
t > c, c ~ b
t >x c, c >x b,
x
x
t ~ c, c ~ b
t ~x c, c >x b
UL
br, br
UL
ntl, br
DL
br, br
UL
br, ntl
??
DR
br, ntg
UR
br, br
DR
ntg, br
DR
br, br