aa r X i v : . [ s t a t . A P ] A ug Cyclic motions in Dekel-Scotchmer Game Experiments
Zhijian Wang
Experimental Social Science Laboratory, Zhejiang University, Hangzhou, 310058, ChinaState Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
Abstract
TASP (Time Average Shapley Polygon, Bena¯ım, Hofbauer and Hopkins,
Journal of Economic Theory , 2009), asa novel evolutionary dynamics model, predicts that a game could converge to cycles instead of fix points (Nashequilibria). To verify TASP theory, using the four strategy Dekel-Scotchmer games (Dekel and Scotchmer,
Journalof Economic Theory , 1992), four experiments were conducted (Cason, Friedman and Hopkins,
Journal of EconomicTheory , 2010), in which, however, reported no evidence of cycles (Cason, Friedman and Hopkins,
The Review ofEconomic Studies , 2013). We reanalysis the four experiment data by testing the stochastic averaging of angularmomentum in period-by-period transitions of the social state. We find, the existence of persistent cycles in Dekel-Scotchmer game can be confirmed. On the cycles, the predictions from evolutionary models had been supported bythe four experiments.JEL classification: C72; C73; C92; D83
Keywords:
Experiments; Dekel and Scotchmer game; period by period transition; angular momentum; stochasticaveraging
Contents1 Introduction 12 Methods 3
While facing a game, the first step is to look for Nashequilibra (fixed points) [3], but in some condition, in-stead of fix points, a game could converge to cycles. ∗ email: [email protected] PS Dekel−Scotchmer cycle D R PS TASP D R Figure 1: Cyclic trajectory and its angular momentum. (a) IdealDekel-Scotchmer cycle (refer to the Figure 1 in p396 in [1]) and (b)Time Average Shapley Polygon (TASP) (refer to the Figure 1 in p2313in [2]) for unstable RPSD game. The frequencies of strategies P and S are on the horizontal axes and of strategy D on the vertical axis.The red arrows indicate the accumulated angular momentum (vector)respects to the cycles. Preprint submitted to Elsevier May 20, 2018 s an example in evolutionary game theory catalog,recently, a dynamic theory — Time Average ShapleyPolygon (TASP) theory [4] — is built giving a preciseprediction about non-equilibrium play in games. Totest TASP theory, four exemplified experiments of theDekel-Scotchmer game [1], called also as Rock-Paper-Scissors-Dumb (RPSD) games [2], were conducted byCason et al . [2]. The four experiments game were multiround repeated, set as discrete time (instead of continu-ous time as [5]), at the same time the matching protocolare randomly pair-wised which called as evolutionaryprotocol [6, 7]. In the four experiments, some evidencessupporting TASP are found. But no cycle is reported in[2], which is emphasized by the authors in their recentliterature [5]. In fact, this is the second time for the cy-cles in the Dekel-Scotchmer game was declined in ex-periments. This game had firstly been tested in experi-ments, no cycles was found which was also emphasisedby the authors [6] (We go back this point in discussion,section 4). All these seems to suggest that there is nocycle in the four Dekel-Scotchmer game experiment.As illustrated in Fig. 1, cycle should along R , P , S , R , P ... in the RPS plane in the tetrahedronthe state spaceof the Dekel and Scotchmer game. Since the gamewas designed [1], not only TASP, variants evolutionarymodels has expected the cycle [4, 1, 8, 9, 10]. Empiri-cally, as in biology system [11, 12], in experimental eco-nomics, evolutionary models have been supported ex-tensively [13, 7, 14, 15, 16, 17]. Recently, in the discretetime protocol, the cycles have been constantly tested out[18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. So, it is enigmathat using the similar protocol in the four experiments,why does the RPS cycle not exist. The objective of thispaper was to study whether or not cycle exists the fourDekel-Scotchmer game experiments.The four experiments have a 2 × D ). The payo ff matrix can be presented as R P S DRPSD a b cb a c b a cd d d .For Unstable and Stable games, [ a b c d ] equals to [90120 20 90] and [60 150 20 90], respectively. Both gameshave the same unique Nash-Dumb (the probability tochoose Dumb is 1 /
2, shown as the redline in Fig. 2). The
Nash Dumb P D ( a v e r age p l a y o f D ) L x , L y , L z Figure 2: The relation between observed average angular momentum( ¯ L x , ¯ L y , ¯ L z ) and observed average play of D ( P D , see also the Table1 in [2]). Observed ( ¯ L x , ¯ L y , ¯ L z ) can be obtained from k = D -play). Color [yel-low, purple, blue, green] presents game-[0,1,2,3], respectively. Sym-bol ( (cid:3) , × , + ) represents ( ¯ L x , ¯ L y , ¯ L z ) respectively. Enlarged symbolsare the averages by games and smaller by sessions. second design is two conversion rates of Experimen-tal Francs (the entries in the game matrix) to US Dol-lars. In the High-pay (Low-pay) treatment, 100 EF = $5 ($2). In High-pay games, the monetary incentive foroptimal strategy is stronger and less noise is expected.Mainly, the settings of the four game are summarized asshown in Table 1. While these games are identical intheir equilibrium predictions, they di ff er quite substan-tially in terms of predicted learning behavior The stablegames (game-2,3) would converge to the Nash equilib-rium. At the same time, in the unstable game (game-0, 1), play will approach to a cycle (in RPS-plane, seegreen triangle in Fig. 3) in which there would be noweight placed on the strategy Dumb ( D ). So, the correc-tion of TASP can be evaluated [2] by the average play of D ( P D ). The main result [2], as illustrated in y -axis inFig 2, P D in game-1 leaves Nash Dumb the farthest andgame-2 the closest. These meet TASP theory well.Cycles are also expected by TASP theory (Fig.1) inthese four games. The theoretical arguments of the fic-titious learning model have been well analyzed [2] ——For the game treatment, as the basic argument (p.2312in [2]), in the stable game there would be convergenceto the Nash equilibrium. In the unstable game, however,there will be divergence from equilibrium and play willapproach a cycle in which no weight is placed on thestrategy Dumb (D). For the payo ff treatment, high-payhas the same e ff ect as an increase in the noise param-2 able 1: The four games (treatments) game i.d. Low pay High payUnstable ← ↓ ↓ Stable ← Each game has 3 repeated sessions with 12 subjects in each session. Gamein each session is about 80 times repeated. Matching protocol is randomlypairwise. On cycle expected by RPS-CH, a → b presents the strength ingame- a larger than that in game- b ( ¯ L a > ¯ L b ). The related empirical results areshown in Table 4. eter (p.2317 in [2]) —— Accordingly, the quantitativeexpectation is that the more deviation from equilibrium,the more cyclic motion will be, which forms the secondtestable point following presented by Table 1. For theobjective of this paper, the expectation on cycle couldbe decomposed into three testable hypothesis:(1) Cycles exist only along R , P , S , R ,... directionshown in Fig. 3 and Table 2. Explanation for thistestable points see section 2.4. The results, see section3.1, support this point.(2) Cycles’ strength depends on games, shown inTable 1, in which game-1 is the largest. Results, seesection 3.2, support this point.(3) Cycles’ persistence depends on games, in whichgame-1 performs the best. Results, see section 3.3,support this point.These three hypothesis are tested in this paper. Usingangular momentum (an observation of rotation in clas-sical physics), in the period-by-period transitions (PPT)of social state in the four experiments [2], we test thecycles and find that all the three theoretical argumentsare supported in significant. we hope our observationcan provides an exemplified evidence of the existenceof Dekel-Scotchmer cycles.
2. Methods
There are four pure strategies in the game, thereforewe use a four dimensional (4D) vector ( x , y , z , u ) to de-note a generic social state of the population, where x , y , z and u are the fractions of players using strategy R , P S and D , respectively [19, 16]. The fraction must be oneelement in (0 , / N , / N ...,
1) set in an N -players game.There are 4 pure social states which can be denoted as W i ( i ∈ { R , P , S , D } ). At the same time, the sum of thefractions is 1 ( x + y + z + u = Figure 3: Social state space of the RPSD game in 3D representation.Each social state is represented by a (blue) dot. Each k , k , k , k setting, see the x − y − z ; O -column in Table 2, is illustrate as a sub-figure. The green triangle is the RPS-plane. The red arrow indicating L -direction if the net notations are along R → P → S → R (expected byTASP theory see the ( L x , L y , L z )-columns in Table 2). For example, in k setting, the red arrow is (0,0,1) which means ( L x = , L y = , L z >
0) are expected.
3y the permuting D , R , P , S at O = (0,0,0) and otherthree at x , y , z -axis respectively, there could have fourways (denoted as k , k , k , k ) to realize the projections.See column-( x , y , z ; O ) in Table 2 for the assignments.The four 3D spaces can be presented graphically as atrirectangular tetrahedron lattice as illustrated in Fig. 3. In such lattice space, generically speaking, the ob-served social state depends on time. From one period ( t )to its next period ( t + x ( t ) to x ( t +
1) , called as one period-by-period transi-tion (PPT), can be observed. Each PPT is a 3D vector inthe lattice space. Successive PPT vectors form an evo-lutionary trajectory in 3D. For example, in a 80 roundsexperimental sessions, an evolutionary trajectory of 80nodes in which there are 79 PPT can be obtained.
For simplicity we consider first a particle (with mass m =
1) moving with respect to a specific reference point(denoted as o ). Consider one PPT, from x ( t ) to x ( t + L ( t ) can beexpressed as [21] L ( t ) = [ x ( t ) − o ] × [ x ( t + − o ] , (1)in which symbol × means cross product of the two vec-tors. So, L ( t ) has a magnitude equal to the area of theparallelogram with edges [ x ( t ) − o ] and [ x ( t + − o ],has the attitude of the plane spanned by [ x ( t ) − o ] and[ x ( t + − o ], and has orientation being the sense of therotation that would align [ x ( t ) − o ] with [ x ( t + − o ]. Itdoes not have a definite location or position [28]. Nolose of the generality, the random mixed (1,1,1,1) / This measurement can be interpreted as following.In equilibrium, in long run, the time average of L (de-noted as ¯ L ) is 0, because of the detailed balance in PPT[18, 29]. In non-equilibrium, ¯ L should be significantlydi ff erent from zero which provides combinative cycles’information (strength and direction of rotation) of the In the studied case of N =
12 and each subject can choose one infour pure strategy in one period, the total number of di ff erent observ-able social states is Q i = h i − ( N + i ) i = L vector is reference point ( o ) depended in one PPT. It is no dif-ficult to prove that, ¯ L of a closed loop is independent of referencepoint setting. To test the robustness of the results in Table 3, Table5 and Table 4, the reference point has been set for all the 455 states,respectively. The results are consistent. ” tumbling cycles ”. This way is to proceed from the mi-croscopic level motions to the macro level observationby stochastic averaging [30, 31, 32, 33] over PPT.In our studies case, there are three components( L x , L y , L z ) of an L . Each component describes the rota-tion along its own directions respectively. So, each PPTcan provide one sample in each of the three directionsrespectively. In su ffi cient samples, if a component ¯ L w ( w ∈ { x , y , z } ) deviates from 0 with the statistic signifi-cance, cycles exist in the direction.As mentioned above, to our study case, we have fourcoordination settings ( k , k , k , k ). For di ff erent set-tings, the observable ¯ L w ( w ∈ { x , y , z } ) of a experimentaltrajectory should be di ff erent. The testable points tables are Table 2 and Table 3.Lets see an example for constructing the testable pointstables.If the R , P and S are at (0,0,0), (1,0,0) and (0,1,0)respectively (setting k , see Table 2), in long run, theonly ¯ L components deviating from 0 with the statisticsignificance should be L z . Because, TASP predicts thatcycles exist only in the X-Y plane along R,P,S,R.... Ac-cording to L definition in Eq 1, observed L should haveonly the component on the Z direction upwards. Thiscan be shown in k setting in Fig. 3 and k = k , k , k , k )setting, expected observation of L can be represented asred arrows in Fig. 3. At the same time, in matrix form,testable expectations on these four settings are shown inTable 2 too.In summary, for the four games and four coordinationsetting, RPS-CH falls into 48 (3 L -components × × Respectively,present the evolutionary trajectory in four coordinationsettings, then we measure the L using Eq. 1 for the fourgames in in the experiments. Then, we can compare theactual motion with the three hypothesis above. Actually, disregarding the 4D →
3D projection, the game is 4D,cross production of two 4D vectors is an antisymmetric tensor having6 components. Each of the 6 components is an observable and inde-pendent. So, in four games, only 24 test points are independent. Forbrevity, the measurements and the results are presented without thiscompression. Regular 3-simplex (normal tetrahedron structure) rep-resentation is also suitable for a RPSD strategy game in general. Butdecomposing vector L in normal tetrahedron structure could lead toadditional complexity to visualize. able 2: Testable TASP hypothesis on ¯ L x , ¯ L y , ¯ L z Setting ( k ) x - y - z ; O ¯ L x ¯ L y ¯ L z R - P - S ; D + + + P - S - D ; R (cid:13) (cid:13) + S - D - R ; P (cid:13) – (cid:13) D - R - P ; S + (cid:13) (cid:13) Setting three of the four pure strategies along column x - y - z [ e x = (1 , , e y = (0 , , e z = (0 , , O state assigned at (0,0,0). Testablehypotheses (PRS-CH) are in last 3 columns in which ’ + ’ (’ – ’) or (cid:13) means the L w should along (oppose to) w -axis direction or not deviating from 0. Table 3: Experimental ( ¯ L x , ¯ L y , ¯ L z ) × − in four setting k game ¯ L x ¯ L y ¯ L z p x p y p z + ∗∗∗ + ∗∗∗ + ∗∗∗ + ∗∗∗ + ∗∗∗ + ∗∗∗ + ∗∗∗ + ∗∗∗ + ∗∗ + ∗∗∗ + ∗∗∗ + ∗∗∗ (cid:13) (cid:13) + ∗∗∗ (cid:13) (cid:13) + ∗∗∗ (cid:13) (cid:13) + ∗∗∗ (cid:13) (cid:13) + ∗∗∗ (cid:13) – ∗∗∗ (cid:13) (cid:13) – ∗∗∗ (cid:13) (cid:13) – ∗∗∗ (cid:13) (cid:13) – ∗∗∗ (cid:13) + ∗∗∗ (cid:13) (cid:13) + ∗∗∗ (cid:13) (cid:13) + ∗∗ (cid:13) (cid:13) + ∗∗∗ (cid:13) (cid:13) Superscript [ ( . ) , ( ∗ ) , ( ∗∗ ) , ( ∗∗∗ ) ] represents p less than [0.2, 0.1, 0.05, 0.01]. p w -values coming from one-sample ttest with null hypothesis that thepopulation mean ¯ L w is equal to 0. The sample sizes for each test point are thetotal number of PPT and are [237, 217, 237, 237] for game-[0,1,2,3],respectively. Table 4: | L | and cross game comparison for ( L x , L y , L z ) × − game | L | ◦ , – ∗∗ , – ∗ –, + ,– –,–, + + ◦ , + ∗∗ , + ∗∗ + , + , + ∗ Compare (Wilcoxon rank sum) cycle strength of the game in row to the gamein column. For example, the symbol (– ◦ ) in 3rd column means L x in game-0 issmaller than game-1 (at p < . Table 5: Time dependence of ¯ L (1 , , × − game ¯ L (1 , , , ¯ L (1 , , , ∆ ¯ L (1 , , Samples0 7.5 1.7 -5.8 ∗∗∗ (351, 351)1 7.1 6.9 -0.2 (351, 321)2 4.9 2.7 -2.2 (351, 351)3 5.1 4.3 -0.8 (351, 351) ( a , b ) in Samples column indicates the samples from (1st,2nd)-half periods inthe game sessions. Statistic uses ttest with ∆ ¯ L =
3. Results
Result : Cycles exist and only exist paralleling RPS-plane in all of the 4 game experiments. Cyclic evolu-tions are along
RPS R ... in all of the 4 game experi-ments.
Support material:
Statistics results of ( L x , L y , L z ),from 4 settings and games respectively, are shown inTable 3. In k setting, the full D strategy is settle at(0,0,0). All the three components ( L x , L y , L z ) > p < .
05) for all of the 4 games. In k set-ting, only L z > p < . exists andonly exist in RPS-plane too. This result is supported by k setting and k setting. Comparing the theoretical ex-pectations (Table 2) and empirical results (2), RPS-CHis supported at all of the 48 testable points.The direction of existed cycles can be distinguishedby taking the signal of ( L x , L y , L z ) into account. Empir-ical signal ( + or – in Table 3) of ( L x , L y , L z ), comparingwith RPS-CH signal ( + or – in Table 2) by the k -settingsand games respectively, one can find that RPS-CH issupported excellently at all of the 48 testable points too. Result : Strength of cyclic motion in game-1 (unsta-ble and High-pay) is the largest. Strength of cycles isnegatively dependent on P D (average play of Dumb). Support material:
The rotation strength of cycles canbe quantified by the vector mode | ¯ L |≡ ( ¯ L x + ¯ L y + ¯ L z ) / .The game-1 has the largest rotation strength | L | shownin the 2nd-column of Table 4. This result is alsosupported by the statistical test (Wilcoxon rank sum)by pair games comparison. In Table 4, over the 4game, the strength orders can be compared with thearrows in Table 1. All the results meet RPS-CH [2] well.Strength of cycles is negatively dependent on P D (av-erage play of Dumb). At the same time, the result inFig. 2 has to be explained —— Strength of cycles isnegatively dependent on P D (average play of Dumb).This finding is statistically significant. In session level, there is 12 samples ( n =
12) for each L w . OLE testresults is that the negative dependence significant with ( p x , p y , p z ) < (0 . , . , . .3. Cycle persistence Result : Persistence of cycle in game-1 performs best.Except game-0, persistence of cycle can not be rejectedby data.
Support material:
One way to test the persistence ofcycles is to compare L samples in early and latter peri-ods. In session level, the hypothesis ( L , = L , ) cannot be rejected in general. At game level to test the persistence, we can project( L x , L y , L z ) into (1,1,1)-vector in k setting to build acombinative scale L (1 , , . For L (1 , , , and L (1 , , , com-parison, both have 351 samples , and results are shownin Table 5. One unexpected result is: in two low-paytreatment, L (1 , , in game-0 declines significantly andmore strongly than that in game-2. Nevertheless, in thefour treatments, cycle persistence in game-1 has the bestperformance, which meets TASP theory again.
4. Discussion
In the four exemplified experiments [2], firstly in thispaper, the Dekel-Scotchmer cycles are reported. Welooked and we see behavior with many of the propertiesthe theorists [4, 1, 8, 9] told us that we would see. Tothe best of our knowledge, no only in Dekel-Scotchmergame, no cycle has been reported in any four strategygames before. These observed cycles, together with thecycles obtained in recent experiments [18, 19, 5], wewish, could provide a novel way to merger the expectedand the actual motions.In experiments, cycle, as the typical non-equilibriumphenomena, have been long sought but the necessarycondition for its existence is unclear [16, 5, 18]. In thisview, current paper could server as the third exempli-fied evidence between the two condition (the continu-ous time and full information environments [5] and thediscrete time and only local information environments[18]). In the experiment investigated here, the time isdiscrete but the information is full. Nevertheless, thenecessary conditions for cycle existence is still an openquestion. In total 36 samples (4 games × / game × L ), only one sample can be rejected ( ¯ L y in the second session ingame-0, p = < The 351 samples include the samples from 3 L -component × / session × / game. But game 1 is 30 samples less be-cause there are only 70 periods in its 3 sessions. See [2] for the details. Referring to TASP experiment designer’s expectation, in the un-stable treatments beliefs (on actions) should (more) continue to cycle.
In closed related literatures, as mentioned [5, 16], ex-perimental work is surprisingly sparse. But one result,which is straightly contrasty to our results reported here,has to be reminded. Before [2], a series of RPSD gameshad been tested experimentally in 1999 by Von Huyck et . al . [6]. One remarkable result is that: The subjectsdon’t exhibit the kind of correlated behavior predictedby the dynamic (p139 in [6]). Then, a conclusion wasemphasized:
A lesson from the experiment is that oneshould discount models that predict deterministic cycles (p148 in [6]). On the contrary, referring to the cyclesobserved from [5] data, we suggest that their results onthe cycle in their data [6] are worth of being rechecked. Using cycle in Dekel-Scotchmer game as the exempli-fied calibration, we suggest, there has many cycles hasbeen existed in existed experiment data.For further investigations on the cycles of social mo-tion, between laboratory experiments and evolutionarygame theory, one central question is still: Whether theactual motions coincide with the expected motions andvise versa? As illustrated in [18, 5] the evolutionarytrajectory can be calculated analytically or numericallybased on a evolutionary model, so cyclic behaviors canbe predicted theoretically. Together with stationary ob-servations of a game (e.g., mean observations [36] anddistribution in strategy space [19, 16]), observations ofcycles (e.g. L in this paper, frequency of cycle [18] andCRI [5]) can server as a set of calibrations to constraintgame models. Acknowledgements
Grant of experimental social science (985-project)for Zhejiang University and SKLTP of ITP-CAS (No.Y3KF261CJ1) support this research.
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