ByDieter Balkenborgand Todd KaplanThe Watchman and the Thief -An Experiment on the Comparative Statics in Games

IntroductionMixed strategy: Randomizing between different optionsCentral role of mixed strategy in gamesMixed strategy as a tool for preventing to be outguessed by the opponentBart Simpson in episode 9F16 "Good ol' rock. Nuthin' beats that!. Lisa: Poor Bart. He is so predictable.

von Neumann on game theoryJacob Bronowski recalls in The Ascent of Man, BBC, 1973, a conversation with John von Neumann:

I (JB) naturally said to him, since I am an enthusiastic chess player You mean the theory of games like chess. No, no, he (vN) said. Chess is not a game. Chess is a well-defined form of computation. You may not be able to work out the answers, but in theory there must be a solution, a right procedure in every position. Now real games, he said, are not like that at all. Real life is not like that. Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do. And that is what games are about in my theory.

Mixed Strategies are a way of remaining unpredictablePenalty Kick:

92.91Kick RDive L58.394.9769.92Dive RKick L

R = strong side of kickerNash prediction for (Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R)Actual Data =(42.31L+57.69R, 39.98L+60.02R)Palacios-Huerta (2003), Volij & Palacios-Huerta (2006)

Q: How does increase in punishment affect equilibrium behaviour?WATCHMAN61564THIEF7WATCHMAN564564THIEF474High Punishment:Low Punishment:45The Watchman and the ThiefWatchWatchRestHomeStealRestHomeSteal????

A: Watchman gets more lazy, theft remains as before.WATCHMAN61564THIEFRestWatchSteal7WATCHMAN564564THIEFRestHomeSteal474High Punishment:Low Punishment:45WatchHome

Game Structures: The Watchman and the Thief00-P10-FR-1WATCHMANTHIEFNot watchingWatchingNot stealingStealingP >0, punishment for the thief.F >0, fine for the watchman.R >0, reward for the watchman.

Mixed Strategy Nash EquilibriaMixed Nash equilibria determined by the condition to make the opponent indifferentHow a player randomizes depends only on payoff of his opponent Counterintuitive ImplicationsRelevance for economics: Relevant economic implications (Dasgupta, Stiglitz 1980, Kaplan, Luski, Wettstein 2003) (More firms implies less innovation)

Main design featurePopulation design (Nagel, Zamir 2001):10 subjects are row players (thieves)10 are column players (watchmen)In each period every thief plays against every watchman and vice versa (10 plays)Placement random

QuestionsQualitative predictions of theory correct?Own Payoff effect to be expected. Will they be wielded out in market-like matching environment?Relevance of security strategies?

Literature:Experiments on normal form games with a unique mixed equilibrium

Atkinson and Suppes (1963)2x2, 0-sum O'Neill (1987)4x4, 0-sum Malawski (1989)2x2, 3x3 game Brown & Rosenthal (1990)4x4, 0-sum Rapoport & Boebel (1992)5x5 Mookerherjee & Sopher (1994)matching pennies Bloomfield (1995)2x2, r. dev, Ochs (1995) 2x2, compet., r.d., r.m. Mookerherjee & Sopher (1997)4x4 games, const sum Erev & Roth (1998)many data sets McKelvey, Palfrey & Weber (1999)2x2 Fang-Fang Tang (1999, 2001, 2003) 3x3 Rapoport & Almadoss (2000)investment game

Literature (cont): Goree, Holt, Palfrey (2000) 2x2 , 0-sum Bracht (2000)2x2 , 0-sum Bracht & Ichimura (2002)2x2, 0-sum Binmore, Swierpinski & Proulx2x2, 3x3, 4x4, 0-sum Shachat (2002)4x4, 5x5 games, 0-sum Shachat & Walker 2x2 games Rosenthal, Shachat & Walker (2001)2x2 games, 0-sum Nagel & Zamir (2000)2x2 games Selten & Chmura (2005)2x2 gamesEmpirical: Walker and Wooders (2001)tennis Ciappori, Levitt, Groseclose (2003)soccer Palacios-Huerta (2003)soccer Volij & Palacios-Huerta (2006)soccer

OutlineExperimental DesignAggregate ResultsOwn payoff EffectQuantal Response and Risk AversionThe learning cycleIndividual BehaviourHeterogeneityMovers and ShakersConclusions

Experimental Design:Computerized Experiment at FEELE (the Finance and Economics Experimental Laboratory at Exeter University).Programmed by Tim Miller in z-tree.A session lasted approx. 1:30 h, on average a student earned 18.120 subjects

Written instructions, summaryTest questions2 trial rounds50 paid roundsQuestionnaire, paymentNo storyEither row- or column player throughoutEach player makes 10 decisions per period

Six Sessions:H1: High penaltyH2: High penalty, row interchangedH3: High penalty, column interchangedL4: Low penaltyL5: Low penalty, row interchangedL6: Low penalty, column interchanged(Data adjusted here)

Row players decision screenScreen shot:

Screen shot:Column players decision screen

Row players waiting screenScreen shot:

Row players result screen Screen shot:

Column players result screenScreen shot:

Results 1, Watchmen (col)The average proportion of watching (R) is lower in the high-punishment treatments.The observation is robust over time.High Punishment: Too close to 50%?

Results 2, Thieves (row)average proportion of stealing (B)(Own-payoff effect): In the high-penalty sessions the average proportion of stealing is 10-15% below equilibrium while it is close to equilibrium in the other session.

Results 3The noise in the aggregate per-period data is large. The data fit roughly in a circle of diameter 0.5.There is no convergence to equilibrium.

Chart1

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Turn

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11.4114257402-9.36532430154.321263533410.523796776417.914534588-4.2274679854

1.8161848143-1.49053765620.68775045171.67491427712.8511867329-0.6728224266

59.4387757403-48.781232385922.508196602654.8153763493.3115658028-22.0196430534

TurnSmooth

from10to2020.0897538418-16.459492920525.95594698959.92863652640.216989947121.9738022667

period50503period505111period50512period50510period505112period505113

RTd1d2ldpiRTd1d2ldpiRTd1d2ldpiRTd1d2ldpiRTd1d2ldpiRTd1d2ldpi

1skcossksincossinarc11skcossksincossinarc11skcossksincossinarc01skcossksincossinarc11skcossksincossinarc01skcossksincossinarc0

222222

333333

444444

555555

666666

777777

888888

999999

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442.355.590.07-0.260.26925824040.0020.0340.0587220220.99827437321.51204050410443.055.990.09-0.140.16643316980.00520.00730.58018227970.81448666190.95184385690443.755.85-0.02-0.010.02236067980.0010.0060.16439898730.98639392381.40564764940445.645.310.090.070.11401754250.0075-0.01150.5462677805-0.8376105968-0.99289438980446.485.040.240.010.2402082430.0464-0.01730.9369914089-0.3493524003-0.35687986660445.644.84-0.090.210.2284731932-0.0036-0.0264-0.1351132047-0.990830168-1.70632404080

452.455.490.1-0.10.14142135620.0330.0190.86662245690.49896444490.52240343170453.115.710.06-0.280.28635642130.0446-0.01680.9358107892-0.3525027188-0.36024414660453.626.08-0.130.230.26419689630.0003-0.00590.0507818527-0.9987097694-1.51999262270455.55.37-0.140.060.1523154621-0.00840.0152-0.48368611170.87524153542.07565769850456.564.990.08-0.050.09433981130.0187-0.01280.8251990069-0.564842101-0.60024189440455.524.82-0.12-0.020.12165525060.00660.0270.23745309050.97139900651.33105321790

462.525.340.07-0.150.16552945360.022-0.0080.9397934235-0.3417430631-0.34877100360463.145.920.030.210.2121320344-0.0570.021-0.93834311680.34570535882.78860226580463.356.09-0.270.010.27018512170.03740.06080.52394109190.85175450231.01932489790465.75.40.20.030.2022374842-0.0262-0.0162-0.8505417666-0.5259075045-2.58781091970466.695.010.130.020.13152946440.00940.00810.75754737690.65278018640.71124864370465.524.8500.030.03-0.0006-0.0036-0.1643989873-0.9863939238-1.73594500420

472.555.250.03-0.090.09486832980.0156-0.00180.9934089359-0.114624108-0.11487660540472.935.84-0.21-0.080.2247220505-0.02310.0417-0.48457391720.8747503182.07667233790473.325.87-0.03-0.220.22203603310.00590.05970.09834835990.99515204871.47228872870475.785.370.08-0.030.08544003750.0151-0.00840.8738841516-0.48613423-0.50766061790476.744.830.05-0.180.18681541690.0029-0.02440.1180217997-0.9930110044-1.45249880490475.494.92-0.030.070.07615773110.00210.00090.919145030.39391929860.40489178630

482.625.180.07-0.070.09899494940.00840.00420.8944271910.44721359550.4636476090482.965.690.03-0.150.15297058540.00570.03390.1658140160.98615704231.40421293010483.525.850.2-0.020.2009975124-0.00160.0446-0.0358513770.99935713271.60665538830485.625.43-0.160.060.1708800749-0.01460-103.14159265361487.044.710.3-0.120.32310988840.03660.0480.60634265710.79520348470.91934309930485.374.89-0.12-0.030.12369316880.00150.00930.15923243880.98724112071.41088320360

492.785.180.1600.160.01120.01120.70710678120.70710678120.78539816340493.155.610.19-0.080.20615528130.01770.02610.56126876310.82763359980.97487832130493.615.80.09-0.050.10295630140.019-0.00820.9181422776-0.3962508777-0.40742985190495.535.37-0.09-0.060.10816653830.01080.0150.58430472580.81153434150.94677327380496.984.84-0.060.130.1431782106-0.03360.0318-0.72629339980.68738482482.38371047950495.434.640.06-0.250.25709920260.00030.03180.00943354250.99995550311.56136264440

503.195.250.410.070.41593268690.06560.01120.98573642550.16829646290.1691012260503.265.830.110.220.24596747750.00330.05060.06507913730.9978801061.50567116350503.635.890.020.090.0921954446-0.00270.0091-0.28444700720.95869176491.85922588530505.515.23-0.02-0.140.14142135620.01020.01140.66679485950.74524131350.84089666860506.954.83-0.03-0.010.03162277660.00050.00450.11043152610.99388373471.46013910560505.284.51-0.15-0.130.19849433240.0235-0.04530.4604887468-0.8876655418-1.09225060890

3.856955855-3.15999579144.98318408981.90616137377.7211077884.2186671861

0.6138535896-0.50292895030.79309838020.3033750051.22885247060.6714217359

20.0897538418-16.459492920525.95594698959.92863652640.216989947121.9738022667

Timeseries

50503

505111

50512

50510

505112

50513

Timeseries

0.750.17

0.620.16

0.640.38

0.580.36

0.730.17

0.570.22

0.530.26

0.560.08

0.540.43

0.710.23

0.520.26

0.50.29

0.570.38

0.530.18

0.580.26

0.490.2

0.520.35

0.540.48

0.620.26

0.540.29

0.550.31

0.710.15

0.480.25

0.540.34

0.540.21

0.540.51

0.610.5

0.760.24

0.460.46

0.660.27

0.770.24

0.550.3

0.620.34

0.640.24

0.70.15

0.620.12

0.680.31

0.580.21

0.50.14

0.490.23

0.410.27

0.650.39

0.580.22

0.380.31

0.60.25

0.470.19

0.590.34

0.510.28

0.50.3

0.560.64

DT

0.890.39

0.810.24

0.540.32

0.640.27

0.590.55

0.650.41

0.740.35

0.590.23

0.770.31

0.660.36

0.710.31

0.60.37

0.630.31

0.630.25

0.730.47

0.650.26

0.730.32

0.620.28

0.670.33

0.430.24

0.50.4

0.460.44

0.70.46

0.740.42

0.640.34

0.80.28

0.590.28

0.730.29

0.780.24

0.680.24

0.590.25

0.80.25

0.630.23

0.680.23

0.710.24

0.330.29

0.580.48

0.680.29

0.670.27

0.470.35

0.710.28

0.720.32

0.580.21

0.540.32

0.430.3

0.540.32

0.50.27

0.530.32

0.590.46

0.690.46

DW

0.680.33

0.780.35

0.720.21

0.450.19

0.580.43

0.750.37

0.690.28

0.630.34

0.530.37

0.610.28

0.710.41

0.650.32

0.620.3

0.660.29

0.520.19

0.620.29

0.310.29

0.430.48

0.570.54

0.580.44

0.640.33

0.750.28

0.660.28

0.730.24

0.550.23

0.460.25

0.470.41

0.630.44

0.650.27

0.590.27

0.480.27

0.510.3

0.530.32

0.490.22

0.410.38

0.360.47

0.580.49

0.580.37

0.510.31

0.590.49

0.720.46

0.870.42

0.750.16

0.480.2

0.640.25

0.370.2

0.360.46

0.560.57

0.460.4

0.680.51

Sheet1

0.410.53

0.460.56

0.370.66

0.540.51

0.40.6

0.560.53

0.520.67

0.430.36

0.340.51

0.40.73

0.590.71

0.60.52

0.470.45

0.40.53

0.420.58

0.520.7

0.440.4

0.480.56

0.540.67

0.490.59

0.570.54

0.480.5

0.470.55

0.430.59

0.540.75

0.650.76

0.620.74

0.760.68

0.740.57

0.730.58

0.570.48

0.570.49

0.410.53

0.520.53

0.490.53

0.430.55

0.570.53

0.480.51

0.560.57

0.590.55

0.420.63

0.650.63

0.530.52

0.590.62

0.550.39

0.460.75

0.540.61

0.540.35

0.50.48

0.450.53

Sheet2

0.580.58

0.430.59

0.480.44

0.260.54

0.480.68

0.480.58

0.450.6

0.530.72

0.350.47

0.360.71

0.290.74

0.470.8

0.460.68

0.350.75

0.60.7

0.460.57

0.420.79

0.450.69

0.630.79

0.720.51

0.360.55

0.480.68

0.480.59

0.490.73

0.630.65

0.550.72

0.610.68

0.520.75

0.550.7

0.60.55

0.430.65

0.510.64

0.450.55

0.490.48

0.480.52

0.460.52

0.590.71

0.650.46

0.380.59

0.450.73

0.50.7

0.50.79

0.530.74

0.50.72

0.430.6

0.480.65

0.410.76

0.530.76

0.510.53

0.440.7

Sheet3

0.60.48

0.640.33

0.540.53

0.540.52

0.680.53

0.620.4

0.590.36

0.530.42

0.520.38

0.580.42

0.420.49

0.510.42

0.510.69

0.660.46

0.410.29

0.480.4

0.510.46

0.410.46

0.490.38

0.30.55

0.620.44

0.330.54

0.340.52

0.490.52

0.450.53

0.310.54

0.380.6

0.370.66

0.460.8

0.580.62

0.570.38

0.320.45

0.60.65

0.40.53

0.50.6

0.480.5

0.330.51

0.460.57

0.60.58

0.420.63

0.420.64

0.480.62

0.540.55

0.610.44

0.480.48

0.510.5

0.40.48

0.430.45

0.350.64

0.290.48

50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10

Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910

111111

201000111112000001-10-102-10-1-1-1000-1120-10-10000012-1-100111010200001-10-1-11

3100-100-11-1-1301010001003001010000030111-1000-1-130-10000-110-13100011110-1

4010-11-1000-14-1-1-1-100-111041101-10001140-1001-10000401100111114-100-110-10-10

50-10000-1101501-110010-1050-1-101-100015010-1010101500-1000-11-105-1000-1-1100-1

61000-1010016-10100000-10600-1-10000006000-1-1-1000-1600-1000100-16-100-10001-11

7010100-101070000-10-10007-100-1100001701-11-1000007011001101-170001-10001-1

8-1-1000000008-10-110001008010000000-1800-10-11-11-108000010-111081101110-1-11

9-11000100019-10-100-11-110900-1001000-1901-11001100900-1000-110-19-1-10-111111-1

10-10-100000-10101-1-1-10100101000-100-100001010-1-1-1-1-100-11001-10000010100011-1-10-101

1101-11000001110000000000111-1-1000100011-100001-1-10011000000-100011001-110010-1

120-11001001-1120-10000111012001-100000-11200-10-100-10-112-1-1-10-1-1000-11200-1-101-1-10-1

13-10-1-11-1-100-113000000100013-110-1010011131001001-11-1130000-1000-1013001-1001-101

14-1-1000001001401010-10-1-1014-1001-1-1100014-111-110000114-10-1000-100114000-100000-1

15-1-101-111000151-10-100010015010000000-1151-1-11-1-1110-1150-1000-11000151101-110101

16110-10-10001160-1000000-10160-10-101000016000-11110001600-100-1010-1160-1-1011010-1

1710001-100001700000000-10170-11100000-117-10-1-1-1-10-10-11700-100110001700101-1-1-10-1

180-1000-10000180000001100180-10000000018011-101-1-10-11810000000-1018001001-1101

19000000000019000000-1-11019-10-10-1000001900-1-10-1-10011900000-1-1-1101900-10-110000

200110100101200010000-1-102000-1000000020001-101000020-10-1000000020111-11-1110-1

210000000-1102110-11011000210-10000000-12100-1-10-1000-12111001110112100-11-10-1-100

22-1-1000000012200-1001000022000000000022001-101100-122-101000-10-1022110011-1101

23111-101001123-100000000023001000000-12300-110-1-110-12300000-111-1023-1000111100

2400-110100012400000000-1024010000010124001-1-110-10024-1000-10-101124-1-11-100-1-101

2500-1-1-1-10000250-1-100-11-100250-1-11100-10025001-100000125-10100000002500-1101110-1

2601-11010-1112600-10000-1002600000100-10260000-101-10-12600001-10000261-1000-1010-1

27000000-1-1002701-110000102701-10000010270000-1-1-101027000000-10002710100-1-1-101

280000-10000-1280-10000-1000280-1000-1000-12800-1000010128-11-110000-1-1281000-100-101

291101-10001129001-101-1-1-1029001001000-12900-100-1-10002900-10000000290-10-1-110-100

30000000-1000300010011000301001-1-10000300010-1001-103000-1001000030000-10-1-1000

31-1-10-10101103101-1-10-10-1003101-10011000310010-1100113111-1110101-1310-110-1-10-11-1

3211100-1001-1320-10-10-10000320-1100-100003200111-1-11-1-132-10-100000013211110-10101

330000010-1-1-13300010101003311-1-100000-13300-110110-1133000000100-133-100001-1-10-1

340-1100011103400-1-1000-1-1034000000000134000-1000-10134000001-100034-100-1-11110-1

3500000-10-101350110-11-1-1-10350-1011100003500011101-1-135100000-100035-100101-1-10-1

360-110111-1103610011-1011036-1-1-100-100003600110-1001036000000-100036-1-1-10110-100

3700-10-1-1-11-1037-100-1001-10037-1-10000000-137000-1-1-100-1037-11-1000-10-1037-1-101-1-11-101

380001010-1013810000000003800-100000003800001-100-113810-1000-1000380-1-1-1-1-10-10-1

390-11001111039100000-1000390101000000390010-1-1000039100101101139-1-1-1-1-1-1-1001

4001-101-1-110-1401011000110401-1-100100014000-110100-1140000-10-1-1000401000-11-110-1

410010-1010114100-1000000041-1-1000000004100001100114100-1-10000004100-1101-1-10-1

420-1-1-10100-1-142-100-10010104200-1-110000042000-1-1-1000-142000-101-100042-1-10-100011-1

4301100-1000-1430-1010000-1043-1-10000000-1430001-110011430000000000430-1-1-11-10-111

44001100111-144-10100000-104410-111100114400110-1010-14400-100-1000044001-100110-1

450-1-1000000-14510000001-10450-10-10-100004500-100000114500-1000-1000450101-111-101

4600100101104600-100001-1046-101101000-14600110-100-1-146000001-100-1461-110-110-10-1

470-1-10-1-11-1-1-147-101-100-101047000-10-110014700000100004710-1000-100-147000-11-1-1101

4801001-1-101-1480-1010001-1048000-1000000480000000-11048000000000048-10011-11-101

49001001100-14900100011004900-100-100004900001001-1149000000-100049110-1-11-110-1

5000000-100105000000010-1050-1000000001500011110-11-15010-100110005000101-11-10-1

-1298469584106555971469100566121152601185200810913658106970123151415553101181112121216013105

026141322211220151412169202315172921191515312052114162025182929281821831301112187222016101771727172527201229212221713161512125113286121

1381054106813976637716676049454647212540011294113671164742336515440848871481231284

20313131313131313131313103131313131313131313131031313131313131313131310313131313131313131313103131313131313131313131031313131313131313131310

50

0.20967741940.18064516130.18064516130.22258064520.17096774190.3387096774

0.54516129030.66129032260.66129032260.57096774190.70.3903225806

0.24516129030.15806451610.15806451610.20645161290.12903225810.2709677419

50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10

Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910

111111

200-1000000020000000-1002-1000-1-10-1002-11000001012-101000000020000000000

3-10011-10-10030-1-100-1-100030-100000000300111-1-1-110300-10010-10030100010-10-1

401000000-1-14100010-10004-1000000-100400-10-11000-140010010-10041010001000

501010000005-1-1-1-10000-1-151-1-1011111150010-10-110051110010-1105000-1110000

6-10-1-100-10-106000000-100060-1000-1000061100-100-1-116-1000-1-10000600-10000000

70100-100001700001000-1-17-100-10000007-10-1-1-1011-10711000101007-100-10-10001

8-10-1000100080-1000-100-108000000-1-1-1081000-100000800001001-10800-110-10000

9-111000-1-11190010001-100911000-100009-111-100101190000100000911-100-11000

100-100000000100-100-1000001000100001-1010111110-1001101000-11011010-100-110-1-101

1100100000-10110000000-110110-1-10-10010-111-10-100010001100101-1000011-10000-1100-1

120-1-1100010012-10-10-1-1-100-1120000-100000120-10-1000-10-1121-100-1101001200110-10100

130-11-1-1000111300-10000001130-100-1-1-1000130-100-1110001301000000-101301110-11100

140-1000000-1014000000-1100140-1-10000-10-1140-1000101101400100000-10140000000-100

15000-100100115-1101-1-10-10015-1000000000151011100-1-1115-1-1-10100-10015000000-100-1

1600000101101600-100000-10160-10-100010116000001-1010160000000-10016011-11-11000

1710-110-10-10017-1000-100-10-1170000-11000017-1-11000000017111000011117-111110-10-1-1

18010001100-11800-10000-1-10181-1110-1111118-1000101110180-1-10000000181010-101010

19-100-11000-1-119-1-10000000019-11-1101-110019010000-111-11910-1000000119-10-10-10100-1

20-100001000-1200100-10-1-1002010000010002000100000002000000-100012000000-10100

210001101010210-111-110-111210000-1-100-11210-11000-1-1-1021-100-10-100-102100-1-11-10-101

2200000000102200-110-1011022-1-1-100000002200010000-102200101-111002210-1-1010000

230-10-100000-1231000101101230000-1-1000023010-1000-1102300000-100-11230001000-101

240000011011240-100010000240-100000000240-11001100124010001001124011-1-1-11100

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Session H1

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Data 050503

Thief

50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum

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50503Wm 1Wm 2Wm 3Wm 4Wm 5Wm 6Wm 7Wm 8Wm 9Wm 10Sum505111Wm 1Wm 2Wm 3Wm 4Wm 5Wm 6Wm 7Wm 8Wm 9Wm 10Sum50512Wm 1Wm 2Wm 3Wm 4Wm 5Wm 6Wm 7Wm 8Wm 9Wm 10Sum50510Wm 1Wm 2Wm 3Wm 4Wm 5Wm 6Wm 7Wm 8Wm 9Wm 10Sum505112Wm 1Wm 2Wm 3Wm 4Wm 5Wm 6Wm 7Wm 8Wm 9Wm 10Sum50513Wm 1Wm 2Wm 3Wm 4Wm 5Wm 6Wm 7Wm 8Wm 9Wm 10Sum

Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910

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50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum

Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910Period12345678910

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