Introduction
The empirical content of dominance solvability N. Hanaki, N. Jacquemet, S. Luchini, Adam Zylbersztejn) U. Nice, PSE (U. Paris 1), U. Lyon 2 (GATE), AMSE
Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Introduction
Dominance solvability is a fundamental solution concepts in game theory. It is based on two principles: dominance: players disregard dominated strategies; iterated dominance: players always act as if others used dominance; In dominance-solvable games, the iterated elimination of dominated strategies leads to the unique Nash equilibrium. Crawford (2004): human players often display less strategic sophistication than is needed to justify many application of iterated dominance (and related concepts) to model human decision making.
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Rosenthal’s game
Testing dominance solvability in the lab Rosenthal (1981): one of the earliest and most classic examples of the inconsistency between standard theory and human behavior. B A
L R
l (8.50 ; 3.00) (6.50 ; 4.75)
B r ( 8.50 ; 3.00) (10.00 ; 5.00)
A
L R
Game 1
l (9.75 ; 8.50) (3.00 ; 8.50)
r ( 9.75 ; 8.50) (10.00 ; 10.00)
Game 2
This game clearly captures both key facets of dominance solvability: using dominant strategies: for player B, r (weakly) dominates l; best responding to dominant strategies: for player A, best response to r is R (and best response to l is L); The game is one-step dominance-solvable: (R, r ) . Numerous lab implementations: large and systematic deviations from the theoretical solution ) failures to use both dominance and iterated dominance.
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Previous experiments
Table: Overview of existing experimental evidence Experiment
Form (L)
Payo↵ (R, r )
(R, l)
L
Outcomes (%) R, r R, l
Beil–Tr.1 Beil–Tr.2 Beil–Tr.3 Beil–Tr.4 Beil–Tr.5 Beil–Tr.7
Seq Seq Seq Seq Seq Seq
(9.75; (9.00; (7.00; (9.75; (9.75; (58.50;
3.0) 3.0) 3.0) 3.0) 6.0) 18.0)
(10; 5.0) (10; 5.0) (10; 5.0) (10; 5.0) (10; 5.0) (18.0; 28.50)
(3; 4.75) (3; 4.75) (3; 4.75) (3; 3.00) (3; 3.00) (60; 30.0)
66 65 20 47 86 67
29 35 80 53 14 33
6 0 0 0 0 0
Beard et al.–Tr.1 Beard et al.–Tr.2
Seq Seq
(1450; 450) (1050; 450)
(1500; 750) (1500; 750)
(450; 700) (450; 700)
79 50
18 32
Goeree, Holt–Tr.1 Goeree, Holt–Tr.2 Goeree, Holt–Tr.3
Ext Ext Ext
(80; 50) (80; 50) (400; 250)
(90; 70) (90; 70) (450; 350)
(20; 10) (20; 68) (100; 348)
16 52 80
Cooper, Van Huyck–Tr.9 Cooper, Van Huyck–Tr.9
Str Ext
(4; 1) (4; 1)
(6; 5) (6; 5)
(2; 4) (2; 4)
JZ, JZ, JZ, JZ, JZ, JZ,
Str Str Str Str Str Str
(10; 5.0) (10; 10.0) (10; 10.0) (10; 10.0) (10; 10.0) (10; 8.5)
Beard, Beard, Beard, Beard, Beard, Beard,
2014–BT1 2014–ET1 2014–ET3 2014–ET4 2014–ET2 2014–BT2
(9.75; (9.75; (9.75; (8.50; (8.50; (8.50;
3.0) 5.0) 5.5) 5.5) 8.5) 7.0)
(3.0; (5.0; (5.5; (5.5; (6.5; (6.5;
4.75) 9.75) 8.50) 8.50) 8.50) 7.00)
Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
r |R
r
83 100 100 100 100 100
— — — — — —
3 18
83 64
— —
84 36 16
0 12 4
100 75 80
— — —
27 21
— —
— —
— —
86 84
51 54 39 25 26 26
41 33 48 61 70 70
8 13 13 14 4 4
84 72 79 82 94 94
81 73 76 82 94 94
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Previous experiments
Lessons from existing experimental studies: the use of (iterated) dominance depends on personal incentives; in some cases, player As adapt their behavior to player Bs’ incentives (but not in our experiments); deviations from standard predictions do not fade away under repetition; deviations from standard predictions cannot be explained by inequality aversion;
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Strategic uncertainty
Strategic uncertainty at work? Failure to best respond to other players’ dominant strategy might be due to strategic uncertainty: player As may know how to use iterated dominance . . . . . . but fail to do so due to the uncertainty about player Bs’ actual use of dominance. In order to explore this dimension, we vary the degree of strategic uncertainty about player Bs’ behavior player As face: Hum(an) environment where player Bs are represented by human subjects; Rob(ot) enviroment in which player Bs are computerized and always play their dominant strategy, which is common knowledge.
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Cognitive skills
Di↵erences in cognitive skills at work? Recent experimental studies in both psychology and economics: high cognitive skills predict strategic sophistication. First, people with high cognitive skills make more accurate predictions about other people’s intentions (Burks et al., 2009; Ibanez et al., 2013; Carpenter et al., 2013). Second, they apply more sophisticated reasoning and are more apt in strategic adaptation. Evidence from another dominance-solvable game, the p-beauty contest game: Branas-Garza et al. (2012): subjects with higher cognitive skills make more steps of reasoning on the equilibrium path; Gill and Prowse (2015): they adapt their behavior to other players’ cognitive skills; Fehr and Huck (2015): they adapt their behavior to their beliefs about other players’ cognitive skills; Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Cognitive skills
We use two measures of cognitive skills (Branas-Garza et al., 2012; Corgnet et al., 2015). Cognitive Reflection Test (Frederick, 2005) ”measures cognitive reflectiveness or impulsiveness, respondents’ automatic response versus more elaborate and deliberative thought” (Branas-Garza et al., 2012): 1
A notebook and a pencil cost 1.10 Euros in total. The notebook costs 1 Euro more than the pencil. How much does the pencil cost?
2
If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
3
In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
Raven’s Progressive Matrices Test: a measure of fluid intelligence, or ”the capacity to think logically, analyse and solve novel problems, independent of background knowledge” (Mullainathan and Shafir, 2013); There are 16 items of an increasing difficulty to be solved within 10 minutes. Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Cognitive skills
Figure: Example of a question from Raven’s test
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Main results
Preview of the main findings Finding 1. We observe systematic and sizable deviations from standard predictions based on dominance solvability, across games and despite repetition. Finding 2. Cognitive skills are systematically associated with strategic behavior: the use of dominance; the ability to best respond to the use of dominance by others; the sensitivity to uncertainty about others’ behavior. Finding 3. Raven’s test score is a better predictor of strategic behavior than the CRT score.
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Experimental procedures
Experimental procedures
B A
L R
l (8.50 ; 3.00) (6.50 ; 4.75)
B r ( 8.50 ; 3.00) (10.00 ; 5.00)
A
L R
Game 1
l (9.75 ; 8.50) (3.00 ; 8.50)
r ( 9.75 ; 8.50) (10.00 ; 10.00)
Game 2
2 ⇥ 2 factorial design: Game 1 or Game 2, Hum or Rob; Hum: subjects’ roles held constant – either A or B for the entire game, one-shot interactions (common knowledge); Rob: player As playing with automated player Bs are told that ”the computer will choose r in every round, with no exception.” perfect stranger design, indefinite repetition (10 rounds in reality); one round compensation rule (random draw at the end); CRT and Raven’s test carried out after the experimental game (flat fee of 5 Eur); Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Experimental procedures
Experimental data for each payo↵ matrix
B A
L R
l (8.50 ; 3.00) (6.50 ; 4.75)
B r ( 8.50 ; 3.00) (10.00 ; 5.00)
A
L R
Game 1
l (9.75 ; 8.50) (3.00 ; 8.50)
r ( 9.75 ; 8.50) (10.00 ; 10.00)
Game 2
Hum: 3 sessions, each involves 10 player As and 10 player Bs; Rob: 2 sessions, each involves 20 player As (accept for 1 session with 18 subjects); between-subject design: each participant only took part in a single session; all sessions computerized, run in Paris; duration of 45-60 minutes, average payo↵: 23.83 Eur (including 5 Eur show-up fee and 5 Eur post-experimental tasks fee); Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Results
Table: Aggregate behavior in experimental games Round 1
2
3
Game 1 Game 2
0.333 0.200
0.600 0.333
0.667 0.400
Game 1 Game 2
0.767 0.833
0.800 0.933
0.867 0.900
Game 1 Game 2
0.500 0.300
0.733 0.333
0.700 0.400
Game 1 Game 2
0.700 0.500
0.750 0.575
0.750 0.725
4
5
Overall 6
7
Pr (R, r ) in Human condition 0.700 0.567 0.600 0.433 0.400 0.433 0.500 0.500 Pr (r ) by player B in Human condition 0.900 0.800 0.800 0.700 0.933 1.000 0.933 0.933 Pr (R) by player A in Human condition 0.767 0.767 0.800 0.700 0.400 0.433 0.533 0.533 Pr (R) by player A in Robot condition 0.725 0.800 0.800 0.800 0.575 0.800 0.700 0.700
Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
8
9
10
0.633 0.433
0.567 0.467
0.700 0.533
0.580 0.420
0.833 0.900
0.867 0.900
0.800 0.933
0.813 0.920
0.767 0.500
0.700 0.500
0.867 0.533
0.730 0.447
0.825 0.775
0.800 0.775
0.775 0.775
0.773 0.690
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Results
Cognitive skills in the experimental sample (standard findings) average Raven’s test score: 8.679 (SD 3.117), average CRT score: 0.479 (SD 0.852); same distributions across four experimental conditions: KW test for Raven’s (CRT) scores gives p = 0.275 (p = 0.502); a moderate (yet highly significant) correlation between both scores: ⇢ = 0.306 (p < 0.001); males score higher than females (Raven’s test: 9.382 with SD 0.341 vs 8.014 with SD 0.384, p = 0.009; CRT: 0.676 with SD 0.111 vs 0.291 with SD 0.087, p = 0.007; two-sided t-tests); CRT: 70% provide 0 correct answers; 16% – 1, 8% – 2, and 6% – 3 (in line with Branas-Garza et al., 2012; lower bound of the samples reported by Frederick, 2005) ) we split our sample into CRT = 0 and CRT > 0; Raven’s test: 1st tertile < 7 correct answers (out of 16), 3rd terile > 10 answers;
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Results
Cognitive skills and strategic behavior Figure: CRT score and aggregate behavior across rounds and treatments
Bootstrap proportion tests: p = 0.126, p = 0.235, p = 0.037, respectively.
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Results
Cognitive skills and strategic behavior Figure: Raven’s test score and aggregate behavior across rounds and treatments
Bootstrap proportion tests: Player As Hum: 1st tercile vs. 2nd terile: p = 0.255, 2nd vs. 3rd: p = 0.580, 1st vs 3rd: p = 0.565; Player As Rob: p = 0.001, p = 0.735, p < 0.001, respectively; Player Bs: p = 0.064, p = 0.057, p < 0.001, respectively. Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Results
Cognitive skills and strategic behavior
Table: Cognitive predictors of strategic behavior: regression analysis
Const. 1[CRT > 0] Raven 1[Game 2] 1[Male]
R2
Pr(R) by player A in Human condition in Robot condition Model 1 Model 2 Model 3 Model 1 Model 2 Model 3 0.423*** 0.552** 0.563** 0.573*** 0.242* 0.240* (0.080) (0.176) (0.197) (0.088) (0.135) (0.135) 0.131 0.152 0.062 (0.024) (0.095) (0.121) (0.102) (0.102) 0.013 0.018 0.0426*** 0.0434*** (0.025) (0.027) (0.013) (0.012) -0.270* 0.263 0.266 0.068 0.054 0.056 (0.129) (0.139) (0.136) (0.083) (0.076) (0.079) 0.132 0.187 0.158 0.096 0.072 0.077 (0.126) (0.100) (0.107) (0.090) (0.076) (0.089) + Round dummies 0.151 0.141 0.160 0.050 0.139 0.140
Pr(r ) by player B in Human condition Model 1 Model 2 Model 3 0.705*** 0.430*** 0.444*** (0.027) (0.103) (0.099) 0.109* 0.046 (0.047) (0.036) 0.0313** 0.0287** (0.009) (0.008) 0.100 0.132* 0.129* (0.052) (0.056) (0.056) 0.025 0.024 0.017 (0.046) (0.046) (0.047) 0.060
0.108
0.111
Note. Standard errors (in parantheses) are clustered at the session level in Human treatments (3 clusters per game matrix, 6 in total) and individual level in Robot condition (40 clusters per game matrix, 80 in total) and computed using the delete-one jackknife procedure. */**/*** indicate significance at the 10%/5%/1% level.
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Results
Cognitive skills and strategic behavior Altogether, cognitive skills predict certain components of strategic behavior: the use of dominant strategy (reflected in player Bs’ behavior); the ability to best respond to other player’s dominant strategy (reflected in player As’ behavior in Robot condition); Each time, Raven’s test score is a better predictor of behavior than CRT score. However, Raven’s test score loses its power to predict player As’ behavior once player Bs’ behavior becomes uncertain (i.e. when moving from Robot to Human condition). Is there an interplay between the degree of strategic uncertainty, behavior in the experimental games, and individual cognitive skills?
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Results
Cognitive skills and strategic behavior
Figure: Aggregate proportion of decisions R across rounds and treatments by Raven’s test score tertile
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Results
Cognitive skills and strategic behavior
Table: The e↵ect of strategic uncertainty and cognitive skills: evidence from player As’ behavior in Human and Robot conditions
Constant 1[Robot] 1[CRT > 0] 1[Male] 1[Game 2]
R2
Raven’s test score tertile 1st 2nd 3rd 0.277* 0.592*** 0.330** (0.147) (0.065) (0.135) 0.044 0.158* 0.428** (0.195) (0.090) (0.155) 0.002 0.038 0.016 (0.262) (0.066) (0.188) 0.144 0.033 0.212 (0.145) (0.063) (0.179) 0.034 -0.245** (0.176) (0.146) (0.092) (0.155) + Round dummies 0.048 0.173 0.298
Note. Standard errors (in parentheses) are clustered at the session level in Human treatments (3 clusters per game matrix, 6 in total) and individual level in Robot condition (40 clusters per game matrix, 80 in total) and computed using the delete-one jackknife procedure. */**/*** indicate significance at the 10%/5%/1% level. Hanaki, Jacquemet, Luchini, Zylbersztejn (U. Nice, PSE (U. Empirical Paris 1), U. content Lyon of 2 (GATE), dominance AMSE) solvability
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Conclusion
Conclusion In line with the previous literature, we find a clear relationship between strategic behavior and cognitive skills. In a classic 2 ⇥ 2 dominance-solvable game, subjects with higher cognitive skills: are more likely to play dominant strategy;
more likely to best respond to other’s dominant strategy; display a greater strategic sophistication (sensitivity to the presence of uncertainty about others’ behavior). However, notwithstanding the previous studies comparing the performance of Raven’s test and CRT in predicting strategic behavior (Branas-Garza et al., 2012; Corgnet et al., 2015), we show that Raven’s test outperforms CRT.
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Conclusion
Conclusion More broadly, these results contribute to the ongoing debate on the relationship between rationality and intelligence (Stanovich, 2009). Stanovich and West (2014) distinguish between two aspects of rational behavior: instrumental rationality, or the ”ability to take appropriate action given one’s goals and beliefs”, and epistemic rationality which enables agents to hold ”beliefs that are commensurate with available evidence”. In our strategic environment, instrumental rationality can be associated with the ability to solve the game, while epistemic rationality – with the ability to play it with others. Our data suggest an important relationship between fluid intelligence (rather than reflective thinking) and both of these facets of rationality in strategic settings.
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