1. Cognitive Load and Mixed Strategies Sean Duffy David Owens John Smith Rutgers-Camden Haverford Rutgers-Camden Psychology Economics Economics

2. 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

3. 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

4. Cognitive resources and mixed strategies We seek to better understand mixing behavior  By examining the role of cognitive resources 4

5. 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 Rosa- García (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

6. 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

7. How to think about the manipulation? Discovered crayon in Homer Simpson’s brain  Was causing cognitive shortcomings 7 Homer without crayon in brain Homer with crayon in brain

8. 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

9. 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 (2015) 9

10. 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

11. 11 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

12. Allred, Duffy, and Smith (2015) Play several one-shot games  under differential load Within-subject design  Subjects in both load treatments 12

13. Allred, Duffy, and Smith (2015) 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

14. Allred, Duffy, and Smith (2015) 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

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.” 15

16. 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 16 Computer’s Actions (Pursuer) UpDown Your Actions (Evader) Up01 Down20

17. Screenshot 17

18. Experimental Design Low load  1-digit number High load  6-digit number Also scanned all 130 right hands  Different paper 18

19. 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

20. Experimental Design Timing within each period: Given new number to remember Play game Receive feedback about that outcome Asked for number Repeat 20

21. Details 130 Subjects  78 Rutgers-Camden  52 Haverford 13,000 game observations z-Tree  Fischbacher (2007) Earned average $33  From $5 to $54 21

22. Hypotheses High load earn less against  Exploitative computers  and exploitable computers High load farther from equilibrium proportions High load more serial correlation 22

23. Summary statistics Correct  High load 90.7%  Low load 96.2%  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 BR in Naïve Pattern  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  33% is “optimal”  High load 52.3%  Low load 56.1%  p=0.03 23

24. Proportions and serial correlation Binomial chi-square against exploitative opponents High load different  p<0.001 Low load different  p<0.001 Not different  Two-sample Kolmogorov-Smirnov Test of runs against exploitative opponents One-sample K-S test High load not indep.  p<0.001 Low load not indep.  p<0.002 Not different  Two-sample Kolmogorov-Smirnov 24

25. Earned Overall  High load 0.737  Low load 0.730  M-W not significantly different Naive Pattern  High load 0.855  Low load 0.753  M-W p<0.001 Exploitative Mixture  High load 0.664  Low load 0.602  M-W p=0.02 Exploitative WSLS Naive 50-50  Not significantly different High load either  earned more  or no difference 25

26. Earned across rounds Round: period under same treatment (1-50) Coefficient estimates and p-values 26 DV: Earned Round0.00190 (p=0.006) 0.00190 (p=0.006) 0.00190 (p=0.006) High Load0.0584 (p=0.04) 0.0584 (p=0.04) 0.120 (0.004) Round*High Load-0.00201 (p=0.04) -0.00201 (p=0.04) -0.00201 (p=0.04) Repeated meas?NoYes Treatment dums?No Yes AIC31300.931270.531216.8 Higher earnings across periods Higher earnings for high load No improvement for high load

27. Response time across rounds Time remaining when decision was made Coefficient estimates and p-values 27 DV: Time remaining Round0.0227 (p<0.001) 0.0227 (p<0.001) 0.0227 (p<0.001) High Load0.234 (p<0.001) 0.234 (p<0.001) 0.664 (p<0.001) Round*High Load-0.005 (0.004) -0.005 (0.001) -0.005 (0.002) Repeated meas?NoYes Treatment dums?No Yes AIC46501.944326.444299.4 Faster decisions across periods Faster decisions for high load Slower increase for high load

28. 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

29. Conclusions Available cognitive resources  not necessarily related to increased earnings either not significant or negative 29

30. Conclusions Available cognitive resources  is related to improvements in earnings over time Subjects with greater available cognitive resources  will faster converge to optimal behavior 30