PowerPoint Presentation

Juan-Camilo CrdenasUniversidad de los Andes

Jim MurphyUniversity of Alaska AnchorageExperimental Methods in Social Ecological SystemsAgenda Day 1Noon 12:15Welcome, introductions12:15 1:15Play Game #1 (CPR: 1 species vs. 4 species)1:15 2:00Debrief game #1 and other results from the field2:00 2:15Break2:15 3:15Game #2 (Beans game)3:15 4:00Debrief Game #24:00 4:15Break4:15 5:00Basics of Experimental designHomework for Day 2: Think of an interesting question or problem to be worked in groups tomorrow

Agenda Day 28:30 9:15Designing and running experiments in the field9:15 10:15Classwork: work in groups solving experimental design problems10:15 10:30Break10:30 11:15Discussion on group solutions11:15 noonBegin design your own experiment(form groups based on best ideas proposed) Noon 1:00 Lunch1:00 1:30Continue design your own experiment (work in groups)1:30 2:30Present designs2:30 3:00Feedback: how could we make this workshop better?

Materials onlineWe will create a web site with materials from the workshop. Please give us your email address (write neatly!!) and we will send you a link when it is ready.Why run experiments?

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Source: John List NBER presentation7Types of experiments1. Speaking to TheoristsTest a theory or discriminate between theoriesCompare theoretical predictions with experimental observationsDoes non-cooperative game theory accurately predict aggregate behavior in an unregulated CPR?Explore the causes of a theorys failureIf what you observe in the lab differs from theory, try to figure out why.Communication increases cooperation in a CPR even though it is cheap talkWhy?Is my experiment designed correctly?What caused the failure?Theory stress tests (boundary experiments)8Types of experiments (cont.)2. Searching for FactsEstablish empirical regularities as a basis for new theoryIn most sciences, new theories are often preceded by much observation.I keep noticing this. Whats going on here?The Double AuctionYears of experimental data showed its efficiency even though no formal models had been developed to explain why this was the case.Behavioral EconomicsMany experiments identifying anomalies, but have not yet developed a theory to explain.

9Types of experiments (cont.)3. Whispering in the Ears of PrincesEvaluate policy proposalsAlternative institutions for auctioning emissions permitsAllocating space shuttle resourcesTest bed for new institutionsElectric power marketsWater marketsPollution permitsFCC spectrum licenses10Basics of Experimental DesignBaseline static CPR gameCommon pool resource experimentSocial dilemmaIndividual vs group interestsBenefits to cooperationIncentives to not cooperateField experiments in rural ColombiaGroups of 5 peopleDecide how much to extract/harvest from a shared natural resource

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Subjects choose a level of extraction 0 8 Low harvest levels(conservative)High harvest levels13

Payoffs also depend on choices of other 4 group members 14

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Group earnings largest if all choose 116

Strong incentives to harvest more than 117

Nash equilibrium:All choose 6Social optimum:All choose 118Comment on payoff tablesThe early CPR experiments typically used payoff tables.We dont live in a world of payoff tablesFrames how a person should think about the gameA lot of numbers, hard to readToo abstract??More recent CPR experiments using richer ecological contextse.g., managing a fishery is different than an irrigation systemObjectiveTo explore interaction between: Formal regulations imposed on a community to conserve local natural resourcesInformal non-binding verbal agreements to do the same.

20Possible 2x3 factorial designExternal EnforcementNoneLowMediumCommunicationNoBaselineLowMediumYesComm OnlyLow + CommMedium + CommGroups of N=5 participantsPlay 10 rounds of one of the 6 treatmentsEnforcementIndividual harvest quota = 1 (Social optimum)Exogenous probability of auditFine (per unit violation) if caught exceeding quota Participants paid based on cumulative earnings in all 10 roundsThese 2 treatments have been conducted ad nauseum. Are they necessary?Baselines and replicationReplicationIn any experimental science, it is important for key results to be replicated to test robustnessLink to previous research. Is your sample unique?Baseline or control groupThe baseline treatment also gives us a basis for evaluating what the effects are of each treatmentIn any experimental study, it is crucial to think carefully about the relevant control!Alternative designStage 1 Baseline CPR (5 rounds)Stage 2 one of the 5 remaining treatments (5 rounds)Comm onlyLowLow + CommMedMed + CommAdvantage Having all groups play Stage 1 baseline facilitates a clean comparison across groups.Disadvantage fewer rounds of the Stage 2 treatments. Enough time to converge??Disadvantage(?) All stage 2 decisions conditioned upon having already played a baselineOptimal sample sizeExternal EnforcementNoneLowMediumCommunicationNoBaselineLowMediumYesComm OnlyLow + CommMedium + CommGroups of N=5 participantsHow many groups per treatment cell?John Lists notes on sample sizeAlso see:John A. List Sally Sadoff Mathis WagnerSo you want to run an experiment, now what? Some simple rules of thumb for optimal experimental designExperimental Economics (2011). 14:439-457SSome Design InsightsA. 0 (control) / 1 (treatment), equal outcome variances

B. 0/1 treatment, unequal outcome variances

C. Treatment Intensityno longer binary

D. Clusters

26Some Design Rules of Thumb for Differences in between-subject experiments

Assume that X0 is N(0,02) and X1 is N(1, 12); and the minimum detectable effect 1 0= . H0: 0= 1 and H1: 1 0= . We need the difference in sample means X1 X0 to satisfy:1.Significance level (probability of Type I error) = :

2. Power (1 probability of Type II error) = 1-:

27Standard Case

28PowerA. Our usual approach stems from the standard regression model: under a true null what is the probability of observing the coefficient that we observed?

B. Power calculations are quite different, exploring if the alternative hypothesis is true, then what is the probability that the estimated coefficient lies outside the 95% CI defined under the null.29Sample Sizes for Differences in Means (Equal Variances)Solving equations 1 and 2 assuming equal variances 12 = 22:

Note that the necessary sample sizeIncreases rapidly with the desired significance level (ta/2) and power (tb).Increases proportionally with the variance of outcomes (s).Decreases inversely proportionally with the square of the minimum detectable effect size (d).

Sample size depends on the ratio of effect size to standard deviation. Hence, effect sizes can just as easily be expressed in standard deviations.

30Standard is to use =0.05 and have power of 0.80 (=0.20).

So if we want to detect a one-standard deviation change using the standard approach, we would need: n = 2(1.96 + 0.84)2*(1)2 = 15.68 observations in each cell

std. dev. change is detectable with 4*15.68 ~ 64 observations per cell

n=30 seems to be the magic number in many experimental studies: ~ 0.70 std. dev. change.

31Sample Size Rules of Thumb: Assuming =0.05 and = 0.20 requires n subjects:

= 0.05 and = 0.05 1.65 n = 0.01 and = 0.20 1.49 n = 0.01 and = 0.05 2.27 n

32Example from a recent undergrad research projectLocal homeless shelter was conducting a fundraising campaign. They asked us to replicate Lists study about the effects of matching contributions.The shelter wanted the same 4 treatments as in List:No match, 1:1, 2:1, and 3:1 to test whether high match ratios would increase contributions. Local oil company agreed to donate up to $5000 to provide a match for money donated.Fundraising exampleThe shelter had funds to send out 16,000 letters to high income women in Anchorage who had never donated before.Expected response rate was about 3 to 4% (n480-640)Question: How many treatments should we run, if we expect about 500 responses?They said a meaningful treatment effect would be ~$25.Standard deviation from previous campaigns was ~$100.

Sample sizeWith only 500 expected responses, we could only conduct 2 treatments.

Sample Sizes for Differences in Means (unequal variances)

Another Rule of Thumbif the outcome variances are not equal then:

The ratio of the optimal proportions of the total sample in control and treatment groups is equal to the ratio of the standard deviations.

Example: Communication tends to reduce the variance, so perhaps groups in this treatment.36Treatment levelsExternal EnforcementNoneLowMediumHighCommunicationNoBaselineLowMediumHighYesComm OnlyLow + CommMedium + CommHigh + CommHow many levels of enforcement do we need?Do we need 3 levels of enforcement?What about Treatment Levels?Assume that you are interested in understanding the intensity of treatment :Level of enforcement (e.g., audit probability)Assume that the outcome variance is equal across various cells.How should you allocate the sample if audit probability could be between 0-1?For simplicity, say X=25%, 50%, or 75%Assume that you have 1000 subjects available.38Reconsider what we are doing:Y = XB + eOne goal in this case is to derive the most precise estimate of B by using exogenousvariation in X.

Recall that the standard error of B is =var(e)/n*var(X) 39Rules of ThumbLinear sample @ X=25%0 @ X=50% @ X=75%Quadratic@X=25%@X=50%@X=75%

Intuition:The test for a quadratic effect compares the mean of the outcomes at the extremes to the mean of the outcome at the midpoint40Intra-cluster CorrelationWhat happens when the level of randomization differs from the unit of observation? Think of randomization at the village level, or at the store level, and outcomes are observed at the individual level. Classic example: comparing two textbooks. Randomization over classroomsObservations at individual level

Another Example:To test robustness of results, you may want to conduct the experiments in multiple communities. How do you allocate treatments across communities, especially if number of participants per village is small?In our Colombian enforcement study, we replicated the entire design in three regions.In a separate CPR experiment in Russia, we visited 3 communities in one region. Each treatment was conducted 1x in each community. We are assuming that the differences across communities are small.Cannot make cross-community comparison

41Intracluster CorrelationReal Sample Size (RSS) = mk/CEm = number of subjects in a clusterk = number of clustersCE = 1 + (m-1)

= intracluster correlation coefficient= s2B/(s2B + s2w)s2B = variance between clusterss2w = variance within clusters42Intracluster CorrelationWhat does 0 mean? No correlation of responses within a clusterNo need to adjust optimal sample sizes

What does 1 mean?All responses within a cluster are identicalLarge adjustment needed: RSS is reduced to the number of clusters 43ExamplePilot testing confirms our suspicion, yielding = 0.04.

They wish to detect a 1/10 std. dev. change.

Using the standard approach, what should the sample size equal?

44 0:What is n? Sample Size Formula:n = 2*(ta + tB )2 * [/]2

n = 1568 at each level; 3136 total.

45ExampleRSS = mk/CE=784*4/(1+.04(784-1))~97!

What is the required sample size?= 2*(ta + tB )2 * 100(1+783(0.04))

= 15.68*3232(note that 0: 15.68*100)

=50,678 at each incentive level!46Randomized factorial designAdvantagesIndependence among the factor variablesCan explore interactions between factorsDisadvantagesNumber of treatments grows quickly with increase in number of factors or levels within a factorExample: Conduct experiment in multiple communities and use community as a treatment variableFractional factorial designSay we want to add informal sanctions with a 3:1 ratioI can pay $3 to reduce your earnings by $11 new factor with 2 levelsTo run all combinations would require 2x2x2 = 8 treatmentsAssume optimal sample size per cell is 6 groups of 5 people (30 total per cell)8 treatments x 30 people/cell = 240 peopleAssume you can only recruit about half that (~120)You could run only 3 groups per cell (15 people) lose power/significanceSolution: conduct a balanced subset of treatmentsExternal EnforcementLowMediumCommunicationNoLowMediumYesLow + CommMedium + CommFractional factorial designIf you are considering this approach, there are a few different design options depending upon the effects you want to capture, number of treatments, etc.This is just one example!

CommunicationExternalEnforcementSanctionshttp://www.itl.nist.gov/div898/handbook/pri/section3/pri3341.htm49Fractional factorial designAdvantage: dramatically reduces the number of trials

Disadvantage: achieves balance by systematically confounding some direct effects with some interactions. It may not be serious, but you will lose the ability to analyze all of the different possible interactions.Nuisance VariablesOther factors of little or no primary interest that can also affect decisions. These nuisance effects could be significant.Common examplesGender, age, nationality (most socio-economic vbls)Selection biasRecruitment -- open to whoever shows up vs random selection ExperienceParticipated in previous experimentsLearningConcern in multi-round experimentsNon-experiment interactionsPeople talking before an experiment while waiting to startIn a community, people may hear about experiment from others

Confounded variablesConfounding occurs when the effects of two independent variables are intertwined so that you cannot determine which of the variables is responsible for the observed effect.Example:

What are some potential confounds when comparing the Baseline with Low? External EnforcementNoneLowMediumCommunicationNoBaselineLowMediumYesComm OnlyLow + CommMedium + CommAnother design approachIf trying to identify factors that influence decisions, try adding them one at a time.Imposing a fine for non-compliance differs from the baseline CPR in multiple ways. Possible confounds:FRAMEThe simple existence of a quota may send a signal about expected behavior, independent of any audits or fines.GUILT = FRAME + auditGetting audited may generate feelings of guilt because the individual is privately reminded about anti-social choicesFINE = FRAME + GUILT (audit) + fine for violationsAre people responding to the expected penalty? Or are they responding to the frame from the quota?3 Sources of variabilityconditions of interest (wanted)measurement error (unwanted)People can make mistakes, misunderstand instructions, typosexperimental material and process (unwanted)No two people are identical, and their responses to the same situation may not be the same, even if your theory predicts otherwise.

Design in a nutshell

Isolate the effects of interestControl what you canRandomize the rest

Some Practical AdviceSome thoughts in no particular orderThink carefully about your research questionFormulate testable hypotheses grounded in theoryHow does your idea contribute to the literature?Think carefully about possible results and how they would be interpretedWhat if results are consistent with theory/expectations?What if they are not?Be prepared for either possibilityPrepare code for data analysis BEFORE running experimentsForces you to think carefully about what your data will look like, and what you want to get out of it.Some thoughts on data analysisAre your data discrete, binary or continuous?Multinomial logit, ordered probit, logit, Poission, linearRepeated observations or one-shot decisionsRandom effects, hierarchical mixed models, nonparametrics

More thoughtsSubject payments and salienceOne distinguishing feature of economic experiments is that subjects are paid based on their decisions and possibly the decisions of othersMust pay enough for subjects to take experiment seriouslyAvoid tournamentsE.g., giving a bonus to person who earns the most moneyTypically pay in cash, in some field experiments may use another mediumNever use deception!Keep earnings and decisions privateInstructionsThink carefully about every word in your instructionsFraming effectspartner in the UG or your opponentCould frame UG as an offer to sell at a priceUsing examplesI used the example of $14/$6 split. Does that suggest proposers should take more than half?What if I used a 10/10 split? Or 6/14?Could give multiple examplesExperiment lengthBe aware that people get tired and boredOther stuffStrategy methodHot vs cold decisionsPaying for just one round in multi-round gameAB-BA designs for within-subject comparisonsPlaying multiple games and paying for just oneFactor levels should allow for enough distance between hypothesesSocial optimum is people will harvest 10% of the fishNash equilibrium predicts 15%.Nash equilibrium & social optimum should be farther apartParticipants relabelMi nivel de extraccionNivel de extraction de ellosPromedio12345678941.0900996108711721252132613951458151651.3882976106411461223129513611421147661.5864955104011201194126313261384143671.8846934101710941165123112921347139682.082991499410681137120012581310135792.3811893970104211081168122312731317102.5793873947101610791137118912361277112.877585292398910501105115411981237123.075783190096310211073112011611197133.37398118779379921042108611241157143.57217908539119631010105110871117153.8703769830885934978101710501077164.068674980785990694798310131038174.3668728783833877915948976998184.5650708760807848884914939958194.8632687736780819852879901918205.0614666713754790820845864878215.3596646690728761789811827838225.5578625666702732757776790798235.8560604643676703725742753758246.0543584620650675694708716719256.3525563596624646662673679679266.5507543573598617631639642639276.8489522549571588599604604599287.0471501526545559567570567559297.3453481503519530536536530519307.5435460479493501504501493479317.8417439456467472472467456439328.0400419433441444441433419400338.3382398409415415409398382360348.5364378386389386378364345320358.8346357362362357346329307280369.0328336339336328314295270240

participants'payofftableMy Level of Extractiona116.875012345678b0Total Level of Extraction by Others09009961087117212521326139514581516c17.87518829761064114612231295136114211476d2.7528649551040112011941263132613841436384693410171094116512311292134713960.00482991499410681137120012581310135758118939701042110811681223127313176793873947101610791137118912361277xn6.0777585292398910501105115411981237x*1.0875783190096310211073112011611197973981187793799210421086112411571072179085391196310101051108711171170376983088593497810171050107712686749807859906947983101310381366872878383387791594897699814650708760807848884914939958156326877367808198528799019181661466671375479082084586487817596646690728761789811827838185786256667027327577767907981956060464367670372574275375820543584620650675694708716719215255635966246466626736796792250754357359861763163964263923489522549571588599604604599244715015265455595675705675592545348150351953053653653051926435460479493501504501493479274174394564674724724674564392840041943344144444143341940029382398409415415409398382360303643783863893863783643453203134635736236235734632930728032328336339336328314295270240

&LParticipant's PayoffTable without relabelling

target1Lowpenaltya116.875xj012345678b009009961071113912031260131313591401c17.87518829761047111311741229127813221361d2.7528649551024108711451197124412851321384693410001061111611651209124812810.0048299149771035108711341175121112415811893954100910581102114111741201679387393098310291071110611371161xn6.0777585290795610001039107210991121x*1.0875783188493097210071038106210829739811860904943976100310251042LowExpPenalty16.5107217908378789149449699881002HigherExpPenalty3311703769813852885912934951962Penalty16512686749790826856881900914922LowAudit0.113668728767800827849866877882HighAudit0.2146507087437747988188318408421563268772074776978679780280216614666697721741754763765763nash equilibriumexpected penalty175966466736957127237287287234.787878787920185786256506696836916946916833.5757575758401956060462664365465965965464320543584603617625628625617603200215255635805915965965915805632002250754355656556756555654352323489522533538538533522505483244715015105125105014884684442545348148648648147045343140426435460463460452438419394364274174394394344234063843573242840041941640839437535032028429382398393382365343316283244303643783693563363122812462043134635734632930728024720816432328336323303279248213171125

&LLow Expected Penalty Target = 1

target1highpenaltya116.875xj012345678b009009961054110611531194123012601285c17.87518829761031108011241163119612231245d2.752864955100710541095113111611186120538469349841028106610991127114911650.00482991496110021038106810931112112658118939379761009103610581075108667938739149509801005102410381046xn6.0777585289092395197398910001006x*1.087578318678979229419559639669739811844871893910921926926LowExpPenalty16.510721790820845864878886889886HigherExpPenalty3311703769797819835846852852846Penalty16512686749774793807815818815807Highpenalty0.213668728750767778783783778767146507087277417497527497417271563268770371472072071470368716614666680688691688680666647175966466576626626576466296071857862563363663362561159256719560604610610604593577555527205435845875845765625435184882152556356355854753050848144822507543540532518499474444408234895225165054894674394063682447150149347946043540536932825453481470453431404371332288264354604464274023723362952482741743942340137334030225820828400419400375345309268221169293823983763493162772331841293036437835332328724619914789313463573292962582141641094932328336306270229182130729

&LHigh Expected PenaltyTarget = 1

Participants relabelMi nivel de extraccionNivel de extraction de ellosPromedio12345678941.0900996108711721252132613951458151651.3882976106411461223129513611421147661.5864955104011201194126313261384143671.8846934101710941165123112921347139682.082991499410681137120012581310135792.3811893970104211081168122312731317102.5793873947101610791137118912361277112.877585292398910501105115411981237123.075783190096310211073112011611197133.37398118779379921042108611241157143.57217908539119631010105110871117153.8703769830885934978101710501077164.068674980785990694798310131038174.3668728783833877915948976998184.5650708760807848884914939958194.8632687736780819852879901918205.0614666713754790820845864878215.3596646690728761789811827838225.5578625666702732757776790798235.8560604643676703725742753758246.0543584620650675694708716719256.3525563596624646662673679679266.5507543573598617631639642639276.8489522549571588599604604599287.0471501526545559567570567559297.3453481503519530536536530519307.5435460479493501504501493479317.8417439456467472472467456439328.0400419433441444441433419400338.3382398409415415409398382360348.5364378386389386378364345320358.8346357362362357346329307280369.0328336339336328314295270240

participants'payofftableMy Level of Extractiona116.875012345678b0Total Level of Extraction by Others09009961087117212521326139514581516c17.87518829761064114612231295136114211476d2.7528649551040112011941263132613841436384693410171094116512311292134713960.00482991499410681137120012581310135758118939701042110811681223127313176793873947101610791137118912361277xn6.0777585292398910501105115411981237x*1.0875783190096310211073112011611197973981187793799210421086112411571072179085391196310101051108711171170376983088593497810171050107712686749807859906947983101310381366872878383387791594897699814650708760807848884914939958156326877367808198528799019181661466671375479082084586487817596646690728761789811827838185786256667027327577767907981956060464367670372574275375820543584620650675694708716719215255635966246466626736796792250754357359861763163964263923489522549571588599604604599244715015265455595675705675592545348150351953053653653051926435460479493501504501493479274174394564674724724674564392840041943344144444143341940029382398409415415409398382360303643783863893863783643453203134635736236235734632930728032328336339336328314295270240

&LParticipant's PayoffTable without relabelling

target1Lowpenaltya116.875xj012345678b009009961071113912031260131313591401c17.87518829761047111311741229127813221361d2.7528649551024108711451197124412851321384693410001061111611651209124812810.0048299149771035108711341175121112415811893954100910581102114111741201679387393098310291071110611371161xn6.0777585290795610001039107210991121x*1.0875783188493097210071038106210829739811860904943976100310251042LowExpPenalty16.5107217908378789149449699881002HigherExpPenalty3311703769813852885912934951962Penalty16512686749790826856881900914922LowAudit0.113668728767800827849866877882HighAudit0.2146507087437747988188318408421563268772074776978679780280216614666697721741754763765763nash equilibriumexpected penalty175966466736957127237287287234.787878787920185786256506696836916946916833.5757575758401956060462664365465965965464320543584603617625628625617603200215255635805915965965915805632002250754355656556756555654352323489522533538538533522505483244715015105125105014884684442545348148648648147045343140426435460463460452438419394364274174394394344234063843573242840041941640839437535032028429382398393382365343316283244303643783693563363122812462043134635734632930728024720816432328336323303279248213171125

&LLow Expected Penalty Target = 1

target1highpenaltya116.875xj012345678b009009961054110611531194123012601285c17.87518829761031108011241163119612231245d2.752864955100710541095113111611186120538469349841028106610991127114911650.00482991496110021038106810931112112658118939379761009103610581075108667938739149509801005102410381046xn6.0777585289092395197398910001006x*1.087578318678979229419559639669739811844871893910921926926LowExpPenalty16.510721790820845864878886889886HigherExpPenalty3311703769797819835846852852846Penalty16512686749774793807815818815807Highpenalty0.213668728750767778783783778767146507087277417497527497417271563268770371472072071470368716614666680688691688680666647175966466576626626576466296071857862563363663362561159256719560604610610604593577555527205435845875845765625435184882152556356355854753050848144822507543540532518499474444408234895225165054894674394063682447150149347946043540536932825453481470453431404371332288264354604464274023723362952482741743942340137334030225820828400419400375345309268221169293823983763493162772331841293036437835332328724619914789313463573292962582141641094932328336306270229182130729

&LHigh Expected PenaltyTarget = 1

Participants relabelMi nivel de extraccionNivel de extraction de ellosPromedio12345678941.0900996108711721252132613951458151651.3882976106411461223129513611421147661.5864955104011201194126313261384143671.8846934101710941165123112921347139682.082991499410681137120012581310135792.3811893970104211081168122312731317102.5793873947101610791137118912361277112.877585292398910501105115411981237123.075783190096310211073112011611197133.37398118779379921042108611241157143.57217908539119631010105110871117153.8703769830885934978101710501077164.068674980785990694798310131038174.3668728783833877915948976998184.5650708760807848884914939958194.8632687736780819852879901918205.0614666713754790820845864878215.3596646690728761789811827838225.5578625666702732757776790798235.8560604643676703725742753758246.0543584620650675694708716719256.3525563596624646662673679679266.5507543573598617631639642639276.8489522549571588599604604599287.0471501526545559567570567559297.3453481503519530536536530519307.5435460479493501504501493479317.8417439456467472472467456439328.0400419433441444441433419400338.3382398409415415409398382360348.5364378386389386378364345320358.8346357362362357346329307280369.0328336339336328314295270240

participants'payofftableMy Level of Extractiona116.875012345678b0Total Level of Extraction by Others09009961087117212521326139514581516c17.87518829761064114612231295136114211476d2.7528649551040112011941263132613841436384693410171094116512311292134713960.00482991499410681137120012581310135758118939701042110811681223127313176793873947101610791137118912361277xn6.0777585292398910501105115411981237x*1.0875783190096310211073112011611197973981187793799210421086112411571072179085391196310101051108711171170376983088593497810171050107712686749807859906947983101310381366872878383387791594897699814650708760807848884914939958156326877367808198528799019181661466671375479082084586487817596646690728761789811827838185786256667027327577767907981956060464367670372574275375820543584620650675694708716719215255635966246466626736796792250754357359861763163964263923489522549571588599604604599244715015265455595675705675592545348150351953053653653051926435460479493501504501493479274174394564674724724674564392840041943344144444143341940029382398409415415409398382360303643783863893863783643453203134635736236235734632930728032328336339336328314295270240

&LParticipant's PayoffTable without relabelling

target1Lowpenaltya116.875xj012345678b009009961071113912031260131313591401c17.87518829761047111311741229127813221361d2.7528649551024108711451197124412851321384693410001061111611651209124812810.0048299149771035108711341175121112415811893954100910581102114111741201679387393098310291071110611371161xn6.0777585290795610001039107210991121x*1.0875783188493097210071038106210829739811860904943976100310251042LowExpPenalty16.5107217908378789149449699881002HigherExpPenalty3311703769813852885912934951962Penalty16512686749790826856881900914922LowAudit0.113668728767800827849866877882HighAudit0.2146507087437747988188318408421563268772074776978679780280216614666697721741754763765763nash equilibriumexpected penalty175966466736957127237287287234.787878787920185786256506696836916946916833.5757575758401956060462664365465965965464320543584603617625628625617603200215255635805915965965915805632002250754355656556756555654352323489522533538538533522505483244715015105125105014884684442545348148648648147045343140426435460463460452438419394364274174394394344234063843573242840041941640839437535032028429382398393382365343316283244303643783693563363122812462043134635734632930728024720816432328336323303279248213171125

&LLow Expected Penalty Target = 1

target1highpenaltya116.875xj012345678b009009961054110611531194123012601285c17.87518829761031108011241163119612231245d2.752864955100710541095113111611186120538469349841028106610991127114911650.00482991496110021038106810931112112658118939379761009103610581075108667938739149509801005102410381046xn6.0777585289092395197398910001006x*1.087578318678979229419559639669739811844871893910921926926LowExpPenalty16.510721790820845864878886889886HigherExpPenalty3311703769797819835846852852846Penalty16512686749774793807815818815807Highpenalty0.213668728750767778783783778767146507087277417497527497417271563268770371472072071470368716614666680688691688680666647175966466576626626576466296071857862563363663362561159256719560604610610604593577555527205435845875845765625435184882152556356355854753050848144822507543540532518499474444408234895225165054894674394063682447150149347946043540536932825453481470453431404371332288264354604464274023723362952482741743942340137334030225820828400419400375345309268221169293823983763493162772331841293036437835332328724619914789313463573292962582141641094932328336306270229182130729

&LHigh Expected PenaltyTarget = 1

Participants relabelMi nivel de extraccionNivel de extraction de ellosPromedio12345678941.0900996108711721252132613951458151651.3882976106411461223129513611421147661.5864955104011201194126313261384143671.8846934101710941165123112921347139682.082991499410681137120012581310135792.3811893970104211081168122312731317102.5793873947101610791137118912361277112.877585292398910501105115411981237123.075783190096310211073112011611197133.37398118779379921042108611241157143.57217908539119631010105110871117153.8703769830885934978101710501077164.068674980785990694798310131038174.3668728783833877915948976998184.5650708760807848884914939958194.8632687736780819852879901918205.0614666713754790820845864878215.3596646690728761789811827838225.5578625666702732757776790798235.8560604643676703725742753758246.0543584620650675694708716719256.3525563596624646662673679679266.5507543573598617631639642639276.8489522549571588599604604599287.0471501526545559567570567559297.3453481503519530536536530519307.5435460479493501504501493479317.8417439456467472472467456439328.0400419433441444441433419400338.3382398409415415409398382360348.5364378386389386378364345320358.8346357362362357346329307280369.0328336339336328314295270240

participants'payofftableMy Level of Extractiona116.875012345678b0Total Level of Extraction by Others09009961087117212521326139514581516c17.87518829761064114612231295136114211476d2.7528649551040112011941263132613841436384693410171094116512311292134713960.00482991499410681137120012581310135758118939701042110811681223127313176793873947101610791137118912361277xn6.0777585292398910501105115411981237x*1.0875783190096310211073112011611197973981187793799210421086112411571072179085391196310101051108711171170376983088593497810171050107712686749807859906947983101310381366872878383387791594897699814650708760807848884914939958156326877367808198528799019181661466671375479082084586487817596646690728761789811827838185786256667027327577767907981956060464367670372574275375820543584620650675694708716719215255635966246466626736796792250754357359861763163964263923489522549571588599604604599244715015265455595675705675592545348150351953053653653051926435460479493501504501493479274174394564674724724674564392840041943344144444143341940029382398409415415409398382360303643783863893863783643453203134635736236235734632930728032328336339336328314295270240

&LParticipant's PayoffTable without relabelling

target1Lowpenaltya116.875xj012345678b009009961071113912031260131313591401c17.87518829761047111311741229127813221361d2.7528649551024108711451197124412851321384693410001061111611651209124812810.0048299149771035108711341175121112415811893954100910581102114111741201679387393098310291071110611371161xn6.0777585290795610001039107210991121x*1.0875783188493097210071038106210829739811860904943976100310251042LowExpPenalty16.5107217908378789149449699881002HigherExpPenalty3311703769813852885912934951962Penalty16512686749790826856881900914922LowAudit0.113668728767800827849866877882HighAudit0.2146507087437747988188318408421563268772074776978679780280216614666697721741754763765763nash equilibriumexpected penalty175966466736957127237287287234.787878787920185786256506696836916946916833.5757575758401956060462664365465965965464320543584603617625628625617603200215255635805915965965915805632002250754355656556756555654352323489522533538538533522505483244715015105125105014884684442545348148648648147045343140426435460463460452438419394364274174394394344234063843573242840041941640839437535032028429382398393382365343316283244303643783693563363122812462043134635734632930728024720816432328336323303279248213171125

&LLow Expected Penalty Target = 1

target1highpenaltya116.875xj012345678b009009961054110611531194123012601285c17.87518829761031108011241163119612231245d2.752864955100710541095113111611186120538469349841028106610991127114911650.00482991496110021038106810931112112658118939379761009103610581075108667938739149509801005102410381046xn6.0777585289092395197398910001006x*1.087578318678979229419559639669739811844871893910921926926LowExpPenalty16.510721790820845864878886889886HigherExpPenalty3311703769797819835846852852846Penalty16512686749774793807815818815807Highpenalty0.213668728750767778783783778767146507087277417497527497417271563268770371472072071470368716614666680688691688680666647175966466576626626576466296071857862563363663362561159256719560604610610604593577555527205435845875845765625435184882152556356355854753050848144822507543540532518499474444408234895225165054894674394063682447150149347946043540536932825453481470453431404371332288264354604464274023723362952482741743942340137334030225820828400419400375345309268221169293823983763493162772331841293036437835332328724619914789313463573292962582141641094932328336306270229182130729

&LHigh Expected PenaltyTarget = 1

Participants relabelMi nivel de extraccionNivel de extraction de ellosPromedio12345678941.0900996108711721252132613951458151651.3882976106411461223129513611421147661.5864955104011201194126313261384143671.8846934101710941165123112921347139682.082991499410681137120012581310135792.3811893970104211081168122312731317102.5793873947101610791137118912361277112.877585292398910501105115411981237123.075783190096310211073112011611197133.37398118779379921042108611241157143.57217908539119631010105110871117153.8703769830885934978101710501077164.068674980785990694798310131038174.3668728783833877915948976998184.5650708760807848884914939958194.8632687736780819852879901918205.0614666713754790820845864878215.3596646690728761789811827838225.5578625666702732757776790798235.8560604643676703725742753758246.0543584620650675694708716719256.3525563596624646662673679679266.5507543573598617631639642639276.8489522549571588599604604599287.0471501526545559567570567559297.3453481503519530536536530519307.5435460479493501504501493479317.8417439456467472472467456439328.0400419433441444441433419400338.3382398409415415409398382360348.5364378386389386378364345320358.8346357362362357346329307280369.0328336339336328314295270240

participants'payofftableMy Level of Extractiona116.875012345678b0Total Level of Extraction by Others09009961087117212521326139514581516c17.87518829761064114612231295136114211476d2.7528649551040112011941263132613841436384693410171094116512311292134713960.00482991499410681137120012581310135758118939701042110811681223127313176793873947101610791137118912361277xn6.0777585292398910501105115411981237x*1.0875783190096310211073112011611197973981187793799210421086112411571072179085391196310101051108711171170376983088593497810171050107712686749807859906947983101310381366872878383387791594897699814650708760807848884914939958156326877367808198528799019181661466671375479082084586487817596646690728761789811827838185786256667027327577767907981956060464367670372574275375820543584620650675694708716719215255635966246466626736796792250754357359861763163964263923489522549571588599604604599244715015265455595675705675592545348150351953053653653051926435460479493501504501493479274174394564674724724674564392840041943344144444143341940029382398409415415409398382360303643783863893863783643453203134635736236235734632930728032328336339336328314295270240

&LParticipant's PayoffTable without relabelling

target1Lowpenaltya116.875xj012345678b009009961071113912031260131313591401c17.87518829761047111311741229127813221361d2.7528649551024108711451197124412851321384693410001061111611651209124812810.0048299149771035108711341175121112415811893954100910581102114111741201679387393098310291071110611371161xn6.0777585290795610001039107210991121x*1.0875783188493097210071038106210829739811860904943976100310251042LowExpPenalty16.5107217908378789149449699881002HigherExpPenalty3311703769813852885912934951962Penalty16512686749790826856881900914922LowAudit0.113668728767800827849866877882HighAudit0.2146507087437747988188318408421563268772074776978679780280216614666697721741754763765763nash equilibriumexpected penalty175966466736957127237287287234.787878787920185786256506696836916946916833.5757575758401956060462664365465965965464320543584603617625628625617603200215255635805915965965915805632002250754355656556756555654352323489522533538538533522505483244715015105125105014884684442545348148648648147045343140426435460463460452438419394364274174394394344234063843573242840041941640839437535032028429382398393382365343316283244303643783693563363122812462043134635734632930728024720816432328336323303279248213171125

&LLow Expected Penalty Target = 1

target1highpenaltya116.875xj012345678b009009961054110611531194123012601285c17.87518829761031108011241163119612231245d2.752864955100710541095113111611186120538469349841028106610991127114911650.00482991496110021038106810931112112658118939379761009103610581075108667938739149509801005102410381046xn6.0777585289092395197398910001006x*1.087578318678979229419559639669739811844871893910921926926LowExpPenalty16.510721790820845864878886889886HigherExpPenalty3311703769797819835846852852846Penalty16512686749774793807815818815807Highpenalty0.213668728750767778783783778767146507087277417497527497417271563268770371472072071470368716614666680688691688680666647175966466576626626576466296071857862563363663362561159256719560604610610604593577555527205435845875845765625435184882152556356355854753050848144822507543540532518499474444408234895225165054894674394063682447150149347946043540536932825453481470453431404371332288264354604464274023723362952482741743942340137334030225820828400419400375345309268221169293823983763493162772331841293036437835332328724619914789313463573292962582141641094932328336306270229182130729

&LHigh Expected PenaltyTarget = 1

Participants relabelMi nivel de extraccionNivel de extraction de ellosPromedio12345678941.0900996108711721252132613951458151651.3882976106411461223129513611421147661.5864955104011201194126313261384143671.8846934101710941165123112921347139682.082991499410681137120012581310135792.3811893970104211081168122312731317102.5793873947101610791137118912361277112.877585292398910501105115411981237123.075783190096310211073112011611197133.37398118779379921042108611241157143.57217908539119631010105110871117153.8703769830885934978101710501077164.068674980785990694798310131038174.3668728783833877915948976998184.5650708760807848884914939958194.8632687736780819852879901918205.0614666713754790820845864878215.3596646690728761789811827838225.5578625666702732757776790798235.8560604643676703725742753758246.0543584620650675694708716719256.3525563596624646662673679679266.5507543573598617631639642639276.8489522549571588599604604599287.0471501526545559567570567559297.3453481503519530536536530519307.5435460479493501504501493479317.8417439456467472472467456439328.0400419433441444441433419400338.3382398409415415409398382360348.5364378386389386378364345320358.8346357362362357346329307280369.0328336339336328314295270240

participants'payofftableMy Level of Extractiona116.875012345678b0Total Level of Extraction by Others09009961087117212521326139514581516c17.87518829761064114612231295136114211476d2.7528649551040112011941263132613841436384693410171094116512311292134713960.00482991499410681137120012581310135758118939701042110811681223127313176793873947101610791137118912361277xn6.0777585292398910501105115411981237x*1.0875783190096310211073112011611197973981187793799210421086112411571072179085391196310101051108711171170376983088593497810171050107712686749807859906947983101310381366872878383387791594897699814650708760807848884914939958156326877367808198528799019181661466671375479082084586487817596646690728761789811827838185786256667027327577767907981956060464367670372574275375820543584620650675694708716719215255635966246466626736796792250754357359861763163964263923489522549571588599604604599244715015265455595675705675592545348150351953053653653051926435460479493501504501493479274174394564674724724674564392840041943344144444143341940029382398409415415409398382360303643783863893863783643453203134635736236235734632930728032328336339336328314295270240

&LParticipant's PayoffTable without relabelling

target1Lowpenaltya116.875xj012345678b009009961071113912031260131313591401c17.87518829761047111311741229127813221361d2.7528649551024108711451197124412851321384693410001061111611651209124812810.0048299149771035108711341175121112415811893954100910581102114111741201679387393098310291071110611371161xn6.0777585290795610001039107210991121x*1.0875783188493097210071038106210829739811860904943976100310251042LowExpPenalty16.5107217908378789149449699881002HigherExpPenalty3311703769813852885912934951962Penalty16512686749790826856881900914922LowAudit0.113668728767800827849866877882HighAudit0.2146507087437747988188318408421563268772074776978679780280216614666697721741754763765763nash equilibriumexpected penalty175966466736957127237287287234.787878787920185786256506696836916946916833.5757575758401956060462664365465965965464320543584603617625628625617603200215255635805915965965915805632002250754355656556756555654352323489522533538538533522505483244715015105125105014884684442545348148648648147045343140426435460463460452438419394364274174394394344234063843573242840041941640839437535032028429382398393382365343316283244303643783693563363122812462043134635734632930728024720816432328336323303279248213171125

&LLow Expected Penalty Target = 1

target1highpenaltya116.875xj012345678b009009961054110611531194123012601285c17.87518829761031108011241163119612231245d2.752864955100710541095113111611186120538469349841028106610991127114911650.00482991496110021038106810931112112658118939379761009103610581075108667938739149509801005102410381046xn6.0777585289092395197398910001006x*1.087578318678979229419559639669739811844871893910921926926LowExpPenalty16.510721790820845864878886889886HigherExpPenalty3311703769797819835846852852846Penalty16512686749774793807815818815807Highpenalty0.213668728750767778783783778767146507087277417497527497417271563268770371472072071470368716614666680688691688680666647175966466576626626576466296071857862563363663362561159256719560604610610604593577555527205435845875845765625435184882152556356355854753050848144822507543540532518499474444408234895225165054894674394063682447150149347946043540536932825453481470453431404371332288264354604464274023723362952482741743942340137334030225820828400419400375345309268221169293823983763493162772331841293036437835332328724619914789313463573292962582141641094932328336306270229182130729

&LHigh Expected PenaltyTarget = 1