The authors (Cason, Friedman and Hopkins, Reviews of Economics Studies, 2014)
claimed a result that the treatments (using simultaneous matching in discrete
time) replicate previous results that exhibit weak or no cycles. After correct
two mathematical mistakes in their cycles tripwire algorithm, we research the
cycles by scanning the tripwire in the full strategy space of the games and we
find significant cycles missed by the authors. So we suggest that, all of the
treatments (using simultaneous matching in discrete time) exhibit significant
cycles.Comment: 2 pages, Keywords: experiments, cycles, mixed equilibrium, discrete
  time. JEL numbers: C72, C73, C92, D8

Wang, Zhijian

Xu, Bin

Zhu, Siqian

English

arXiv

The authors (Cason, Friedman and Hopkins, Review of Economic Studies, 2014) claimed that control treatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak or no cycles. After correcting two mathematical mistakes in their cycles tripwire algorithm, we study the cycles by scanning the tripwire in the full strategy space of the games and we find significant cycles that were omitted by the authors. So we suggest that all of the treatments exhibit significant cycles

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
A Comment on ”Cycles and Instability in
a Rock-Paper-Scissors Population Game:
A Continuous Time Experiment”
Zhijian Wang and Siqian Zhu and Bin Xu
Experimental Social Science Laboratory, Zhejiang University, Chu
Kochen Honors College, Zhejiang University, Public Administration
College, Zhejiang Gongshang University, KITPC, Institute of
Theoretical Physics, Chinese Academy of Sciences
26. November 2013
Online at http://mpra.ub.uni-muenchen.de/51691/
MPRA Paper No. 51691, posted 28. November 2013 07:43 UTC
1A Comment on “Cycles and Instability in a Rock-Paper-Scissors
Population Game: A Continuous Time Experiment”
Zhijian Wang1,∗ Siqian Zhu1,2, Bin Xu3,1,4,
1 Experimental Social Science Laboratory, Zhejiang University.
2 Chu Kochen Honors College, Zhejiang University.
3 Public Administration College, Zhejiang Gongshang University.
4 SKLTP, ITP, Chinese Academy of Sciences.
*correspondence: wangzj@zju.edu.cn
Abstract
The authors (Cason, Friedman and Hopkins, Review of Economic Studies, 2014) claimed that control
treatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak or
no cycles. After correcting two mathematical mistakes in their cycles tripwire algorithm, we study the
cycles by scanning the tripwire in the full strategy space of the games and we find significant cycles that
were omitted by the authors. So we suggest that all of the treatments exhibit significant cycles.
JEL numbers: C72, C73, C92, D83;
Keywords: experiments, learning, cycles, mixed equilibrium, discrete time
The existence of cycles in mixed equilibrium games has been a cutting edge question in the field crossing
game theory [1] and evolutionary game theory [2] for decades. It is a milestone contribution of Cason,
Friedman and Hopkins [3] that they confirmed the existence of cycles by quantitative measurement in
control experiments, but we can hardly agree with one of their conclusions in their latest article.
In their article [3], they claimed that control treatments (using simultaneous matching in discrete
time) replicate previous results that exhibit “weak or no cycles”, which depended on the result ——
the stable discrete time treatments (SD) do not exhibit clear cyclical behavior indicated by CRI’s not
significantly different from 0 (see their Table 3).
Two mathematical mistakes in the measurement algorithm were found after we checked their result
above.1 Then we corrected the mistakes (see Appendix). Using the refined measurement, if the start
point (α, β) of the tripwire for counting cycles (see Fig. 1) was set at ( 14 ,
1
4 ) —— Nash equilibrim (NE)
of the games as [3], CRI of SD will still not be significantly different from 0 (see up panel in Table 1 and
compare it with their Table 3). So, for SD, we agree that there are only weak or no cycles around NE.
However, if we set (α, β) at (0.23, 0.26) for SD-Mixed and at (0.22, 0.40) for SD-Pure and use the CRI
measurement as criterion still,2 we can find that CRI of both SD-Mixed and SD-Pure are significantly
different from 0 , indicating that there are indeed cyclical behaviors (see low panel in Table 1).
Meanwhile, if we use the accumulated counting number of cycles C as the index (see the Eq.(2) in [4]
for detailed explanation) instead of CRI, we also find that all C¯ (of the experimental blocks) for the
treatments are significantly different from 0 (see Table 2). This, again, indicates the existence of cycles.
So, for SD, we doubt the validity of their conclusion that there are “weak or no cycle”. We suggest
that the treatments (using simultaneous matching in discrete time) do exhibit significant cycles rather
than “weak or no cycles”.3
1We thank the authors’ [3] confirmation on this point during the ESA-NA (2014) Conference.
2 Our pilot results suggest that using angular motion of the transits (e.g. the θ in Eq.(10) in [4]) is an efficient way of
observing cycles too. We hope to return to this issue in a future study.
3We would like to point out the followings. (1) The start point (α, β) for the two SD games is close to the actual mean
observed value of the aggregated social strategy, which seems to tell us that cycles are actually around the mean observed
value instead of NE. (2) Their treatments of RPS games are economical since all they need are 6 groups of 8 subjects and a
period of time of 15 minutes. If cycle could be obtained, such treatments could be great exemplified classroom experiments
for teaching evolutionary game theory.
2Number of Number of Cycle
Game Condition (α, β) Counter-Clock Clockwise Rotation p-value
-wise Transits Transits Index (CRI)
S Continuous-Instant ( 1
4
, 1
4
) 25.2 5.5 0.65* 0.028
S Continuous-Slow ( 1
4
, 1
4
) 9.4 0.8 0.87* 0.027
S Discrete-Mixed (SD-Mixed) ( 1
4
, 1
4
) 2.3 1.2 0.38 0.116
S Discrete-Pure (SD-Pure) ( 1
4
, 1
4
) 1.0 1.0 0.13 0.753
Ua Continuous-Instant (
1
4
, 1
4
) 31.9 0.9 0.94* 0.027
Ua Continuous-Slow (
1
4
, 1
4
) 8.2 0.0 1.00* 0.014
Ua Discrete-Mixed (
1
4
, 1
4
) 2.3 0.2 0.79* 0.027
Ua Discrete-Pure (
1
4
, 1
4
) 1.9 0.3 0.79* 0.027
Ub Continuous-Slow (
1
4
, 1
4
) 0.3 8.5 -0.93* 0.028
S Discrete-Mixed (SD-Mixed) (0.23, 0.26) 2.3 1.1 0.41* 0.035
S Discrete-Pure (SD-Pure) (0.22, 0.40) 4.1 3.3 0.14* 0.028
Table 1 Mean Transits and CRI (Update Table 3 in [3] with Refined Measurement).
*Denotes CRI Index is significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test.
Game Condition (α, β) B1
† B2 B3 B4 B5 B6 C¯ ‡ p-value
S Continuous-Instant ( 1
4
, 1
4
) 128 53 120 74 58 157 98.3* 0.028
S Continuous-Slow ( 1
4
, 1
4
) 49 47 45 55 24 38 43.0* 0.028
S Discrete-Mixed (SD-Mixed) (0.23, 0.26) 3 0 3 2 9 18 5.8* 0.035
S Discrete-Pure (SD-Pure) (0.22, 0.40) 6 4 2 4 8 1 4.2* 0.027
Ua Continuous-Instant (
1
4
, 1
4
) 158 114 218 195 175 70 155.0* 0.028
Ua Continuous-Slow (
1
4
, 1
4
) 45 53 48 39 28 34 41.2* 0.028
Ua Discrete-Mixed (
1
4
, 1
4
) 9 16 10 6 6 17 10.7* 0.027
Ua Discrete-Pure (
1
4
, 1
4
) 5 11.5 1.5 13 5.5 11 7.9* 0.028
Ua Discrete-Pure (
1
4
, 1
4
) -52 -29 -43 -49 -40 -33 -41.0* 0.028
Table 2: The accumulated counting number of cycles (C) in each block. † B indicates block, number
1 ∼ 6 indicates the block number in related treatment. ‡ C¯ indicates the mean accumulated counting number of
cycles of an experimental block. *Denotes C index is significantly (p-value < 5%) different from 0 according to
2-tailed Wilcoxon test.
3Appendix
CRI was defined as [3] CRI=CCT−CTCCT+CT , and CCT and CT can be interpreted as follows. Supposing the
Poincare section (“tripwire”) is the segment between Pc:≡(α, β) as shown in Fig. A1 (referring to the
Fig. 1 in [3]) and Pe=(α, 0), and a transit is a directional segment from state (x1, y1) observed at t to
state (x2, y2) at (t+ 1). These two segments could cross at X as
X :≡ (Xx, Xy) =
(
α, y1 +
(y2 − y1)(α− x1)
x2 − x1
)
. (1)
Accordingly, CCT =
∑
Ct>0
Ct and CT =
∑
Ct<0
|Ct| in which Ct value of the transit (at time t) should
be4
Condition 1 Ct = 0 if X 6∈ (Pc, Pe] ∪ x2 = x1
Condition 2 Ct = 1 if X ∈ (Pc, Pe] ∩ x2 > α > x1
Ct = −1 if X ∈ (Pc, Pe] ∩ x2 < α < x1
Condition 3 Ct =
1
2 if X ∈ (Pc, Pe] ∩ x2 > x1 ∩ (x1 = α ∪ x2 = α)
Ct = − 12 if X ∈ (Pc, Pe] ∩ x2 < x1 ∩ (x1 = α ∪ x2 = α)
At the same time, the accumulated counting number of cycles is C:=
∑
Ct (exactly the same as Eq.(2)
in [4]). According to [4], when C serves as an index, the criterion for testing the existence of cycles is
that: if C is significantly different from 0, then there are cycles; otherwise, no cycles.
There are two mathematical mistakes in the measurement algorithm (see Result 3.do in their paper’s
supplement). Mistake-1: For Condition 2, a necessary condition for Ct = ±1 was wrongly set as y1+y22 <
β. Mistake-2: For Condition 3, Ct was wrongly set as 0.
Algorithm Index Mean
Game Condition (α, β) refined chosed Index p-value
(before/after) CRI/C value
S Discrete-Pure (SD-Pure) (0.22, 0.40) before CRI 0.07 0.173
(0.22, 0.40) before C 2.17 0.113
(0.22, 0.40) after CRI 0.14* 0.028
(0.22, 0.40) after C 4.20* 0.027
Table A1: Explanation for Necessity of Correction for the Measurement Algorithm.
*Denotes CRI Index is significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test.
These mistakes need to be corrected. Table A1 exhibits both results, algorithm not refined and refined.
Obviously, without refined algorithm we cannot find cycles even if we set (α, β) at (0.22, 0.40). On the
contrary, with the refined algorithm, the existence of cycles can be verified by both CRI and C. This
comparison implies that the algorithm correction is necessary.
Programs for the replication of the results in this paper are provided in supplements.
Acknowledgments
We thank John Ledyard, Charles Plott, Hai-Jun Zhou, Daniel Friedman and Qiqi Cheng for helpful
discussion. This work was supported by Grants from 985 at Zhejiang University, SKLTP of ITP CAS
(No.Y3KF261CJ1) and Philosophy & Social Sciences Planning Project of Zhejiang Province (13ND-
JC095YB).
4For a careful description of this measurement, see the Eq.(2) for the accumulated counting number C for cycles in [4].
We thank the authors’ [3] confirmation on this point.
40.25 0.5 0.75 1
0.25
0.5
0.75
1
(
1
4
,
1
4
)
(α,β)
(0.82,0.10)
(x1, y1)
(x2, y2)
C
0
Scissors Rock
Paper
Figure A1: Illustration of the tripwire for counting cycles (Poincare section) [3]. A tripwire for
counting cycles is a segment constructed between the reference point (α, β) and the simplex edge, as the vertical
dashed line extending below the point (α, β). The reference point of the right tripwire should be at (α, β)=(0.82,
0.10), while the authors [3] set it at (α, β) = ( 1
4
, 1
4
) (the Nash equilibrium), the reference point of the tripwire in
the middle, and thus discovering weak or no cycles for stable discrete time treatments (SD). The arrow indicates
a transit from (x1, y1) to (x2, y2), crossing the tripwire from the left side.
References
1. Kuhn, Harold W, John C Harsanyi, Reinhard Selten, Jorgen W Weibull, and Eric van Damme.
1997. “The work of John Nash in game theory.” Economic Sciences, 1991-1995 :160.
2. Smith, J.M. 1982. Evolution and the Theory of Games. Cambridge university press.
3. Cason, T. N., D. Friedman, and E. Hopkins. 2014. “Cycles and Instability in a Rock-Paper-
Scissors Population Game: a Continuous Time Experiment.” Review of Economic Studies 81,
doi:10.1093/restud/rdt023.
4. Xu, B., H.-J. Zhou, and Z. Wang. 2013. “Cycle frequency in standard Rock-Paper-Scissors games:
Evidence from experimental economics.” Physica A: Statistical Mechanics and its Applications
392 (20):4997 – 5005, doi:10.1016/j.physa.2013.06.039.


A Comment on "Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment"

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveA Comment on ”Cycles and Instability ina Rock-Paper-Scissors Population Game:A Continuous Time Experiment”Zhijian Wang and Siqian Zhu and Bin XuExperimental Social Science Laboratory, Zhejiang University, ChuKochen Honors College, Zhejiang University, Public AdministrationCollege, Zhejiang Gongshang University, KITPC, Institute ofTheoretical Physics, Chinese Academy of Sciences26. November 2013Online at http://mpra.ub.uni-muenchen.de/51691/MPRA Paper No. 51691, posted 28. November 2013 07:43 UTC1A Comment on “Cycles and Instability in a Rock-Paper-ScissorsPopulation Game: A Continuous Time Experiment”Zhijian Wang1,∗ Siqian Zhu1,2, Bin Xu3,1,4,1 Experimental Social Science Laboratory, Zhejiang University.2 Chu Kochen Honors College, Zhejiang University.3 Public Administration College, Zhejiang Gongshang University.4 SKLTP, ITP, Chinese Academy of Sciences.*correspondence: wangzj@zju.edu.cnAbstractThe authors (Cason, Friedman and Hopkins, Review of Economic Studies, 2014) claimed that controltreatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak orno cycles. After correcting two mathematical mistakes in their cycles tripwire algorithm, we study thecycles by scanning the tripwire in the full strategy space of the games and we find significant cycles thatwere omitted by the authors. So we suggest that all of the treatments exhibit significant cycles.JEL numbers: C72, C73, C92, D83;Keywords: experiments, learning, cycles, mixed equilibrium, discrete timeThe existence of cycles in mixed equilibrium games has been a cutting edge question in the field crossinggame theory [1] and evolutionary game theory [2] for decades. It is a milestone contribution of Cason,Friedman and Hopkins [3] that they confirmed the existence of cycles by quantitative measurement incontrol experiments, but we can hardly agree with one of their conclusions in their latest article.In their article [3], they claimed that control treatments (using simultaneous matching in discretetime) replicate previous results that exhibit “weak or no cycles”, which depended on the result ——the stable discrete time treatments (SD) do not exhibit clear cyclical behavior indicated by CRI’s notsignificantly different from 0 (see their Table 3).Two mathematical mistakes in the measurement algorithm were found after we checked their resultabove.1 Then we corrected the mistakes (see Appendix). Using the refined measurement, if the startpoint (α, β) of the tripwire for counting cycles (see Fig. 1) was set at ( 14 ,14 ) —— Nash equilibrim (NE)of the games as [3], CRI of SD will still not be significantly different from 0 (see up panel in Table 1 andcompare it with their Table 3). So, for SD, we agree that there are only weak or no cycles around NE.However, if we set (α, β) at (0.23, 0.26) for SD-Mixed and at (0.22, 0.40) for SD-Pure and use the CRImeasurement as criterion still,2 we can find that CRI of both SD-Mixed and SD-Pure are significantlydifferent from 0 , indicating that there are indeed cyclical behaviors (see low panel in Table 1).Meanwhile, if we use the accumulated counting number of cycles C as the index (see the Eq.(2) in [4]for detailed explanation) instead of CRI, we also find that all C¯ (of the experimental blocks) for thetreatments are significantly different from 0 (see Table 2). This, again, indicates the existence of cycles.So, for SD, we doubt the validity of their conclusion that there are “weak or no cycle”. We suggestthat the treatments (using simultaneous matching in discrete time) do exhibit significant cycles ratherthan “weak or no cycles”.31We thank the authors’ [3] confirmation on this point during the ESA-NA (2014) Conference.2 Our pilot results suggest that using angular motion of the transits (e.g. the θ in Eq.(10) in [4]) is an efficient way ofobserving cycles too. We hope to return to this issue in a future study.3We would like to point out the followings. (1) The start point (α, β) for the two SD games is close to the actual meanobserved value of the aggregated social strategy, which seems to tell us that cycles are actually around the mean observedvalue instead of NE. (2) Their treatments of RPS games are economical since all they need are 6 groups of 8 subjects and aperiod of time of 15 minutes. If cycle could be obtained, such treatments could be great exemplified classroom experimentsfor teaching evolutionary game theory.2Number of Number of CycleGame Condition (α, β) Counter-Clock Clockwise Rotation p-value-wise Transits Transits Index (CRI)S Continuous-Instant ( 14, 14) 25.2 5.5 0.65* 0.028S Continuous-Slow ( 14, 14) 9.4 0.8 0.87* 0.027S Discrete-Mixed (SD-Mixed) ( 14, 14) 2.3 1.2 0.38 0.116S Discrete-Pure (SD-Pure) ( 14, 14) 1.0 1.0 0.13 0.753Ua Continuous-Instant (14, 14) 31.9 0.9 0.94* 0.027Ua Continuous-Slow (14, 14) 8.2 0.0 1.00* 0.014Ua Discrete-Mixed (14, 14) 2.3 0.2 0.79* 0.027Ua Discrete-Pure (14, 14) 1.9 0.3 0.79* 0.027Ub Continuous-Slow (14, 14) 0.3 8.5 -0.93* 0.028S Discrete-Mixed (SD-Mixed) (0.23, 0.26) 2.3 1.1 0.41* 0.035S Discrete-Pure (SD-Pure) (0.22, 0.40) 4.1 3.3 0.14* 0.028Table 1 Mean Transits and CRI (Update Table 3 in [3] with Refined Measurement).*Denotes CRI Index is significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test.Game Condition (α, β) B1† B2 B3 B4 B5 B6 C¯ ‡ p-valueS Continuous-Instant ( 14, 14) 128 53 120 74 58 157 98.3* 0.028S Continuous-Slow ( 14, 14) 49 47 45 55 24 38 43.0* 0.028S Discrete-Mixed (SD-Mixed) (0.23, 0.26) 3 0 3 2 9 18 5.8* 0.035S Discrete-Pure (SD-Pure) (0.22, 0.40) 6 4 2 4 8 1 4.2* 0.027Ua Continuous-Instant (14, 14) 158 114 218 195 175 70 155.0* 0.028Ua Continuous-Slow (14, 14) 45 53 48 39 28 34 41.2* 0.028Ua Discrete-Mixed (14, 14) 9 16 10 6 6 17 10.7* 0.027Ua Discrete-Pure (14, 14) 5 11.5 1.5 13 5.5 11 7.9* 0.028Ua Discrete-Pure (14, 14) -52 -29 -43 -49 -40 -33 -41.0* 0.028Table 2: The accumulated counting number of cycles (C) in each block. † B indicates block, number1 ∼ 6 indicates the block number in related treatment. ‡ C¯ indicates the mean accumulated counting number ofcycles of an experimental block. *Denotes C index is significantly (p-value < 5%) different from 0 according to2-tailed Wilcoxon test.3AppendixCRI was defined as [3] CRI=CCT−CTCCT+CT , and CCT and CT can be interpreted as follows. Supposing thePoincare section (“tripwire”) is the segment between Pc:≡(α, β) as shown in Fig. A1 (referring to theFig. 1 in [3]) and Pe=(α, 0), and a transit is a directional segment from state (x1, y1) observed at t tostate (x2, y2) at (t+ 1). These two segments could cross at X asX :≡ (Xx, Xy) =(α, y1 +(y2 − y1)(α− x1)x2 − x1). (1)Accordingly, CCT =∑Ct>0Ct and CT =∑Ct<0|Ct| in which Ct value of the transit (at time t) shouldbe4Condition 1 Ct = 0 if X 6∈ (Pc, Pe] ∪ x2 = x1Condition 2 Ct = 1 if X ∈ (Pc, Pe] ∩ x2 > α > x1Ct = −1 if X ∈ (Pc, Pe] ∩ x2 < α < x1Condition 3 Ct =12 if X ∈ (Pc, Pe] ∩ x2 > x1 ∩ (x1 = α ∪ x2 = α)Ct = − 12 if X ∈ (Pc, Pe] ∩ x2 < x1 ∩ (x1 = α ∪ x2 = α)At the same time, the accumulated counting number of cycles is C:=∑Ct (exactly the same as Eq.(2)in [4]). According to [4], when C serves as an index, the criterion for testing the existence of cycles isthat: if C is significantly different from 0, then there are cycles; otherwise, no cycles.There are two mathematical mistakes in the measurement algorithm (see Result 3.do in their paper’ssupplement). Mistake-1: For Condition 2, a necessary condition for Ct = ±1 was wrongly set as y1+y22 <β. Mistake-2: For Condition 3, Ct was wrongly set as 0.Algorithm Index MeanGame Condition (α, β) refined chosed Index p-value(before/after) CRI/C valueS Discrete-Pure (SD-Pure) (0.22, 0.40) before CRI 0.07 0.173(0.22, 0.40) before C 2.17 0.113(0.22, 0.40) after CRI 0.14* 0.028(0.22, 0.40) after C 4.20* 0.027Table A1: Explanation for Necessity of Correction for the Measurement Algorithm.*Denotes CRI Index is significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test.These mistakes need to be corrected. Table A1 exhibits both results, algorithm not refined and refined.Obviously, without refined algorithm we cannot find cycles even if we set (α, β) at (0.22, 0.40). On thecontrary, with the refined algorithm, the existence of cycles can be verified by both CRI and C. Thiscomparison implies that the algorithm correction is necessary.Programs for the replication of the results in this paper are provided in supplements.AcknowledgmentsWe thank John Ledyard, Charles Plott, Hai-Jun Zhou, Daniel Friedman and Qiqi Cheng for helpfuldiscussion. This work was supported by Grants from 985 at Zhejiang University, SKLTP of ITP CAS(No.Y3KF261CJ1) and Philosophy & Social Sciences Planning Project of Zhejiang Province (13ND-JC095YB).4For a careful description of this measurement, see the Eq.(2) for the accumulated counting number C for cycles in [4].We thank the authors’ [3] confirmation on this point.40.25 0.5 0.75 10.250.50.751(14,14)(α,β)(0.82,0.10)(x1, y1)(x2, y2)C0Scissors RockPaperFigure A1: Illustration of the tripwire for counting cycles (Poincare section) [3]. A tripwire forcounting cycles is a segment constructed between the reference point (α, β) and the simplex edge, as the verticaldashed line extending below the point (α, β). The reference point of the right tripwire should be at (α, β)=(0.82,0.10), while the authors [3] set it at (α, β) = ( 14, 14) (the Nash equilibrium), the reference point of the tripwire inthe middle, and thus discovering weak or no cycles for stable discrete time treatments (SD). The arrow indicatesa transit from (x1, y1) to (x2, y2), crossing the tripwire from the left side.References1. Kuhn, Harold W, John C Harsanyi, Reinhard Selten, Jorgen W Weibull, and Eric van Damme.1997. “The work of John Nash in game theory.” Economic Sciences, 1991-1995 :160.2. Smith, J.M. 1982. Evolution and the Theory of Games. Cambridge university press.3. Cason, T. N., D. Friedman, and E. Hopkins. 2014. “Cycles and Instability in a Rock-Paper-Scissors Population Game: a Continuous Time Experiment.” Review of Economic Studies 81,doi:10.1093/restud/rdt023.4. Xu, B., H.-J. Zhou, and Z. Wang. 2013. “Cycle frequency in standard Rock-Paper-Scissors games:Evidence from experimental economics.” Physica A: Statistical Mechanics and its Applications392 (20):4997 – 5005, doi:10.1016/j.physa.2013.06.039.

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A Comment on "Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment"

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