1. Cognitive Load and Mixed Strategies: On Brains and Minimax
Sean Duffy J.J. Naddeo David Owens John Smith
Rutgers-Camden Rutgers-Camden Haverford Rutgers-Camden
Psychology Economics Economics Economics

2. Beauty contest-Laboratory outcomes

3. Strategic behavior and cognitive ability
Examine relationship between
measures of cognitive ability and
strategic behavior
Al-Ubaydli, Jones, and Weel (2016), Ballinger et al. (2011), Baghestanian and Frey (2016), Bayer and Renou (2016a,2016b), Benito-Ostolaza, Hernández, and Sanchis-Llopis (2016), Brañas-Garza, Espinosa, and Rey-Biel (2011), Brañas-Garza, Garcia-Muñoz, and Hernan Gonzalez (2012), Brañas-Garza and Smith (2016), Burks et al. (2009), Burnham et al. (2009), Carpenter, Graham, and Wolf (2013), Chen, Huang, and Wang (2013), Corgnet et al. (2016), Coricelli and Nagel (2009), Devetag and Warglien (2003), Fehr and Huck (2016), Georganas, Healy, and Weber (2015), Gill and Prowse (2016), Grimm and Mengel (2012), Jones (2014), Jones (2008), Kiss, Rodriguez-Lara, and Rosa-García (2016), Lohse (2016), Palacios-Huerta (2003b), Proto, Rustichini, and Sofianos (2014), Putterman, Tyran, and Kamei (2011), Rydval (2011), Rydval and Ortmann (2004), and Schnusenberg and Gallo (2011).

4. Experimental Differences between Spock and Homer
Rather than measure cognitive ability
We manipulate it
Advantage to manipulating cognitive ability
Cognitive ability related to lots of other things
Maybe X determines strategic sophistication
And X merely related to cognitive ability
Differences between Spock and Homer
Are not confined to cognitive ability

5. How to think about the manipulation?
Discovered crayon in
Homer Simpson’s brain
Was causing cognitive shortcomings
Homer without crayon in brain
Homer with crayon in brain

6. 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?

7. 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)
Samson and Kostyszyn (2015)
Allred, Duffy, and Smith (2016)

8. Duffy and Smith (2014) JoBEE
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

9. Duffy and Smith (2014) JoBEE
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

10. Allred, Duffy, and Smith (2016) JEBO
Play several one-shot games
under differential load
Within-subject design
Subjects in both load treatments

11. Allred, Duffy, and Smith (2016) JEBO
What are the beliefs about the
distribution of the cognitive load?
effect of the cognitive load on opponent?

12. 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).

13. Cognitive resources and mixed strategies
We seek to better understand mixing behavior
By examining the role of cognitive resources

14. Experimental Design 100 repetitions of Hide-and-Seek Game
Block of 50 under high load
Block of 50 under low load
Block of 50 playing naive computer
Either Up-Down-Down or 50-50
Block of 50 playing exploitative computer
Either BR to mixture or BR to WSLS
Computer’s Actions
(Pursuer) Up
Down
Your Actions
(Evader) 1 2

15. 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.”

16. Screenshot

17. Experimental Design Low load High load
1-digit number
High load
6-digit number
Also scanned all 130 right hands
Different paper

18. Experimental Design Strongly incentivized memorization task
Performance in memorization task
unrelated to payment for game outcome in that period
Paid for 30 randomly selected game outcomes
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 BR in Naïve Pattern Correct Down in Exp. WSLS
High load 62.8%
Low load 55.1%
p<0.001
Down in Exp. WSLS
33% is “optimal”
High load 55.9%
Low load 56.8%
p=0.60
Down in Exp. Mix
High load 52.3%
Low load 56.1%
p=0.03
Correct
High load 88.0%
Low load 97.9%
p<0.001
Down in Naïve 50-50
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

23. Proportions and serial correlation
Test of runs against exploitative opponents
One-sample K-S test
High load not indep.
p<0.001
Low load not indep.
p=0.07
Not different
Two-sample Kolmogorov-Smirnov
p=0.42
Binomial chi-square against exploitative opponents
High load different
p<0.001
Low load different
Not different
Two-sample Kolmogorov-Smirnov
p=0.37

24. Earned by treatment Coefficient estimates and p-values DV: Earned
High Load
0.0626
(p=0.03)
0.0692
(p=0.04)
0.0791
(p=0.02)
Strategy dums?
Yes
Repeated meas? No
Treatment dums?
AIC
Higher earnings
for high load

25. Earned across rounds Round: period under same treatment (1-50)
Coefficient estimates and p-values
DV: Earned
Round
(p=0.006)
High Load
0.114
(p=0.003)
0.120
(p=0.004)
0.130
(p=0.002)
Round*High Load
(p=0.04)
Strategy dums?
Yes
Repeated meas? No
Treatment dums?
AIC
Higher earnings
across periods
Higher earnings
for high load
No improvement
for high load

26. Response time across rounds
Time remaining when decision was made
Coefficient estimates and p-values
DV: Time remaining
Round
0.0227
(p<0.001)
High Load
0.519
0.664
0.593
Round*High Load
-0.005
(p=0.004)
(p=0.001)
(p=0.002)
Strategy dums?
Yes
Repeated meas? No
Treatment dums?
AIC
Faster decisions
across periods
Faster decisions
for high load
Slower increase
for high load

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
related to improvements in earnings over time
Subjects with greater available cognitive resources
exhibit more learning

30. Cognitive load: What next?
Settings where learning is important
Bandit problems
Search
Coordination games
Monotonic?
1, 3, 6, 9, 12, 15 digits

31. Cognitive load Cognitive load manipulation Can help give clues
about implications of limited cognitive resources

32. Definitions: target, start, response

33. Central Tendency Bias (CTB)
Stylized summary of hundreds of
visual judgment papers
Slope = 1
Short lines
overestimated
Response
Long lines
underestimated
Target

34. Why economists should care about visual judgments?
Real decision
payoffs and probabilities are not given
Unlike what we do in the lab
First there is a judgment
then there is a decision
Present these quantities as a visual judgment
Do estimates of probability or payoffs have same biases?
Other applications that will occur to you?

35. Cognitive load affects visual judgments
Allred et al. (2016)
Judgments of line length
While under cognitive load
Some judgments under high load
Remembering 6 digit number
Some judgments under low load
Remembering 2 digit number
Randomized start line

36. Allred, Crawford, Duffy, Smith (2016)
Worse visual judgments
under high load
Slope = 1
Low load
High load
Response
Cognitive load
increases the
CTB
Target