Cognitive Load
and
Mixed Strategies:
On Brains and Minimax
Sean Duffy
J.J. Naddeo
David Owens
John Smith
Rutgers-Camden
Psychology
Rutgers-Camden
Economics
Haverford
Economics
Rutgers-Camden
Economics
Mixing is difficult for subjects

Often subjects have difficulty playing mixed
strategies in the laboratory


Individual mixing proportions
Actions with serial correlation

O'Neill (1987), Brown and Rosenthal (1990), Batzilis et
al. (2013), Binmore, Swierzbinski, and Proulx (2001),
Geng, Peng, Shachat, and Zhong (2014), Mookherjee
and Sopher (1994, 1997), O'Neill (1991), Ochs (1995),
Palacios-Huerta and Volij (2008), Rapoport and
Amaldoss (2000, 2004), Rapoport and Boebel (1992),
Rosenthal, Shachat, and Walker (2003), Shachat
(2002), Van Essen and Wooders (2013).
2
Does experience help?

Bring in subjects who have experience mixing
in other situations

Examine their behavior

Levitt, List, and Reiley (2010), Palacios-Huerta
and Volij (2008), Van Essen and Wooders (2013)
3
Cognitive resources and mixed strategies

We seek to better understand mixing
behavior

By examining the role of cognitive resources
4
Strategic behavior and cognitive ability

Examine relationship between
 measures of cognitive ability and
 strategic behavior

Ballinger et al. (2011), Baghestanian and Frey (2012), Bayer and
Renou (2012), Brañas-Garza, Garcia-Muñoz, and Hernan
Gonzalez (2012), Brañas-Garza, Paz Espinosa, and Rey-Biel
(2011), Burks et al. (2009), Burnham et al. (2009), Carpenter,
Graham, and Wolf (2013), Chen, Huang, and Wang (2013),
Devetag and Warglien (2003), Georganas, Healy, and Weber
(2013), Gill and Prowse (2015), Grimm and Mengel (2012),
Jones (2014), Jones (2008), Kiss, Rodriguez-Lara, and RosaGarcía (2014), Palacios-Huerta (2003), Proto, Rustichini, and
Sofianos (2014), Putterman, Tyran, and Kamei (2011), Rydval
(2011), Rydval and Ortmann (2004), and Schnusenberg and
Gallo (2011)
5
Manipulate cognitive resources

Rather than measure cognitive ability


We manipulate available cognitive resources
Advantage to manipulating available cognitive
resources

Cognitive ability related to lots of other things
6
How to think about the manipulation?


Discovered crayon in
Homer Simpson’s brain

Was causing cognitive shortcomings
Homer with crayon in brain
Homer without crayon in brain
7
How to manipulate cognitive resources?

Cognitive Load

Task that occupies cognitive resources



Unable to devote to deliberation
Observe behavior
Require subjects to memorize a number



Big number
Small number
Differences in behavior?
8
Cognitive load and games







Milinski and Wedekind (1998)
Roch et al. (2000)
Cappelletti, Güth, and Ploner (2011)
Carpenter, Graham, and Wolf (2013)
Duffy and Smith (2014)
Buckert, Oechssler, and Schwieren (2014)
Allred, Duffy, and Smith (2016)
9
Duffy and Smith (2014)

Repeated 4-player prisoner’s dilemma

Under differential cognitive load

Given number
Play game
Asked to recall number

Between-subject design



Subjects only in one treatment
10
Duffy and Smith (2014)

Choice of low load subjects



Differentially converged to SPNE prediction
Low load “closer” to equilibrium
Low load subjects better able to condition


on previous outcomes
Low load better able to sustain some periods of
cooperation
11
Allred, Duffy, and Smith (2016)

Play several one-shot games


under differential load
Within-subject design

Subjects in both load treatments
12
Allred, Duffy, and Smith (2016)

Two effects of cognitive load
1. Reduced ability to make computations
2. Subjects realized they were disadvantaged
in distribution of cognitive resources


Believed opponents more sophisticated
More likely to use available information



About load of opponent
Prompt to think harder
Work in opposite directions
13
Allred, Duffy, and Smith (2016)

What are the beliefs about the


distribution of the cognitive load?
What are the beliefs about the

effect of the cognitive load on opponent?
14
Experimental Design

Play against computer opponent

Subjects told
“How does the computer decide what to
play? A number of possible strategies have
been programmed. Some computer
strategies can be exploited by you. Some
computer strategies are designed to exploit
you.”

15
Experimental Design

100 repetitions of Hide-and-Seek Game
Computer’s Actions
(Pursuer)
Your Actions
(Evader)



Down
Up
0
1
Down
2
0
Block of 50 under high load
Block of 50 under low load
Block of 50 playing naive computer


Up
Either Up-Down-Down or 50-50
Block of 50 playing exploitative computer

Either BR to mixture or BR to WSLS
16
Screenshot
17
Experimental Design

Low load


High load


1-digit number
6-digit number
Also scanned all 130 right hands

Different paper
18
Experimental Design


Strongly incentivized memorization task
Performance in memorization task


Paid for 30 randomly selected game outcomes





unrelated to payment for game outcome in that
period
if 100 memorization tasks correct
Paid for 29 if 99 correct
…
Paid for 1 if 71 correct
Paid for none if 70 or fewer correct
19
Experimental Design

Timing within each period:

Given new number to remember
Play game
Receive feedback about that outcome
Asked for number
Repeat




20
Details

130 Subjects
 78 Rutgers-Camden
 52 Haverford

13,000 game observations

z-Tree


Fischbacher (2007)
Earned average $33

From $5 to $54
21
Hypotheses

High load earn less against


Exploitative computers
and exploitable computers

High load farther from equilibrium proportions

High load more serial correlation
22
Summary statistics









100% is “optimal”
High load 61.5%
Low load 58.5%
p=0.07
Down in Naïve Pattern




33% is “optimal”
High load 49.3%
Low load 52.4%
p=0.11




High load 62.8%
Low load 55.1%
p<0.001
Down in Exp. WSLS

Down in Naïve 50-50


High load 88.0%
Low load 97.9%
p<0.001
BR in Naïve Pattern

Correct



33% is “optimal”
High load 55.9%
Low load 56.8%
p=0.60
Down in Exp. Mix




33% is “optimal”
High load 52.3%
Low load 56.1%
p=0.03
23
Proportions and serial correlation


Binomial chi-square
against exploitative
opponents
High load different





Two-sample
Kolmogorov-Smirnov
p=0.37
Test of runs against
exploitative opponents
One-sample K-S test
High load not indep.



p<0.001
Low load not indep.

p<0.001
Not different


p<0.001
Low load different


p=0.07
Not different


Two-sample
Kolmogorov-Smirnov
p=0.42
24
Earned by treatment

Coefficient estimates and p-values
Higher earnings
for high load
DV: Earned
High Load
0.0626
(p=0.03)
0.0692
(p=0.04)
0.0791
(p=0.02)
Strategy dums?
Yes
Yes
Yes
Repeated meas?
No
Yes
Yes
Treatment dums?
No
No
Yes
AIC
31221.7
31199.0
31212.3
25
Earned across rounds


Round: period under same treatment (1-50)
Coefficient estimates and p-values
DV: Earned
Higher earnings
across periods
Higher earnings
for high load
No improvement
for high load
Round
0.00190
(p=0.006)
0.00190
(p=0.006)
0.00190
(p=0.006)
High Load
0.114
(p=0.003)
0.120
(p=0.004)
0.130
(p=0.002)
Round*High Load
-0.00201
(p=0.04)
-0.00201
(p=0.04)
-0.00201
(p=0.04)
Strategy dums?
Yes
Yes
Yes
Repeated meas?
No
Yes
Yes
Treatment dums?
No
No
Yes
AIC
31239.6
31216.8
31230.1
26
Response time across rounds


Time remaining when decision was made
Coefficient estimates and p-values
DV: Time remaining
Faster decisions
across periods
Faster decisions
for high load
Slower increase
for high load
Round
0.0227
0.0227
(p<0.001) (p<0.001)
0.0227
(p<0.001)
High Load
0.519
(p<0.001)
0.664
(p<0.001)
0.593
(p<0.001)
Round*High Load
-0.005
(p=0.004)
-0.005
(p=0.001)
-0.005
(p=0.002)
Strategy dums?
Yes
Yes
Yes
Repeated meas?
No
Yes
Yes
Treatment dums?
No
No
Yes
46386.3
44299.4
44298.9
AIC
27
Conclusions

Available cognitive resources



not related to standard measures of serial correlation
not related to standard measures of mixing
proportions
No evidence that available cognitive resources

related to standard results
28
Conclusions

Available cognitive resources

Not necessarily related to higher earnings
29
Conclusions

Available cognitive resources


Subjects with greater available cognitive
resources


related to improvements in earnings over time
exhibit more learning
Danke
30