2c:null 2d:["$","title",null,{"children":"Primer's Coin Game Simulation | Lao | Observable"}] 32:T1763,{ // window svg const margin = ({top: 30, right: 15, bottom: 120, left: 35}) const w = width const h = 500 // simulation const colorScheme = 'Cool' const simsPerFrame = settings.sims const maxRun = settings.score const maxFlips = 30 const linesOnScreen = 100 const startMoney = rules.bank const simplify = settings.simplify let max = 0 let tick = 0 let binSize = settings.bin let histogramData = new Array(Math.round(maxRun/binSize)+1) histogramData.fill(0) // random walk funcction let rand, blobIsFair, bank, heads, tails, move, score function walk() { bank = startMoney let values = [[0,bank]]; for (let i = 0; i < maxRun; ++i) { if(Math.random() < rules.populationFair){ blobIsFair = true }else{ blobIsFair = false } // blob player this round heads = 0 tails = 0 for (let j = 0; j < maxFlips; ++j) { move = policy(heads, tails, bank) if(move == 0 ){ // label fair if( blobIsFair==true){ bank += rules.truePositive } // true positive if( blobIsFair==false){ bank += rules.falsePositive } // false positive break } else if(move == 1 || bank < rules.priceOfFlip){ // label cheater if( blobIsFair==true ){ bank += rules.falseNegative } // false negative if( blobIsFair==false ){ bank += rules.trueNegative } // true negative break } else if(move == 2){ //flip coin bank -= rules.priceOfFlip if( blobIsFair ){ if(Math.random() < rules.expectHeadsFair){ heads++ }else{ tails++ } }else{ if(Math.random() < rules.expectHeadsCheat){ heads++ }else{ tails++ } } } } if(bank < rules.priceOfFlip || i == maxRun-1 ){ // ends game values.push([i+1,Math.max(0,bank)]); // record score after labling each blob histogramData[Math.round((i)/binSize)] ++ score = i if( score < histogramData.length-2){ max = Math.max( max, histogramData[Math.round((i)/binSize)] ) } break } else if(tick == 0){ values.push([i+1,Math.max(0,bank)]); // record score after labling each blob } tick = (tick+1)%simplify } return values; } // histogram function let points, scale function graphHistogram( data ){ scale = settings.bank *.26 / Math.max(100, max) points = [[binSize*-.5,0]] for( let i=0; i xScale(d[0])) .y(d => yScale(d[1])) // SVG const svg = d3.select(DOM.svg(w, h)) .style("overflow", "visible"); var linearGradient = svg.append("linearGradient") .attr("id", "linear-gradient") .attr("x1", "0%") .attr("y1", "0%") .attr("x2", "100%") .attr("y2", "0%"); linearGradient.append("stop") .attr("offset", "0%") .attr("stop-color", colorScale(0)); linearGradient.append("stop") .attr("offset", "33%") .attr("stop-color", colorScale(.33)); linearGradient.append("stop") .attr("offset", "66%") .attr("stop-color", colorScale(.66)); linearGradient.append("stop") .attr("offset", "100%") .attr("stop-color", colorScale(1)); const histogram = svg .append('path') .attr("transform", `translate(0,${120})`) .datum( graphHistogram(histogramData) ) // graphHistogram(histogramData)) .attr('d', line) .attr('fill', "url(#linear-gradient)") .attr('stroke', 'none') // numbered axes let xAxis = d3.axisBottom(xScale).ticks(20); svg.append("g") .attr("transform", `translate(0,${h - margin.bottom})`) .call(xAxis) let yAxis = d3.axisLeft(yScale); svg.append("g") .attr("transform", `translate(${margin.left}, 0)`) .call(yAxis) .call(g => g.select(".domain").remove()); //removes line // text labels let textData = [ {text: "Bank", x:margin.left-10, y:margin.top-20, weight:700, anchor:"end"}, {text: "↑ Score Final Score Frequency ↓", x:margin.left + (w-margin.left-margin.right) /2, y: yScale(0)+35, weight:700, anchor:"middle"} ] const text = svg.selectAll("t") .data(textData) .enter() .append('text') .attr("x", (d) => d.x ) .attr("y", (d) => d.y ) .attr("text-anchor", d => d.anchor) .style("font-size", "10px") .attr("font-family", "Arial") .attr("font-weight", (d) => d.weight) .text((d) => d.text); let lineData while (true) { yield svg.node(); histogram .datum(graphHistogram(histogramData)) .attr('d', line) for( let i=0; i colorScale(score/maxRun)) .attr("stroke-width", 0.5) .style("opacity", 1) .transition() .ease(d3.easeCubicOut) .duration(16.7*linesOnScreen) // 60 frames per second .style("opacity", 0) .remove(); /* svg.append('circle') .attr('cx', xScale(lineData[lineData.length-1][0]) ) .attr('cy', yScale(lineData[lineData.length-1][1]) ) .attr('r', 2) .attr("fill", colorScale(score/maxRun)) .attr("stroke", 'none' ) .attr("stroke-width", 0.5) .style("opacity", 1) .transition() .ease(d3.easeCubicOut) .duration(2*16.7*linesOnScreen) // 60 frames per second .attr('r', 5) .style("opacity", 0) .remove(); */ } } }33:T405,viewof settings = { const form = html``; form.oninput = () => form.value = ({ bank: form.bank.valueAsNumber, score: form.score.valueAsNumber, bin: form.bin.valueAsNumber, simplify: form.simplify.valueAsNumber, sims: form.sims.valueAsNumber }); form.oninput(); return form; }34:T112a,beta = { const w = width; const h = 400; const padding = 25; const margin = ({top: 10, right: 10, bottom: 70, left: 10}); const color = [ '#18c97b', '#9022DD'] // display options let a = game.a let b = game.b let B = cephes.beta(a, b) let densityFair = betaPDF(rules.expectHeadsFair, a, b, B) let densityCheat = betaPDF(rules.expectHeadsCheat, a, b, B) let dPoints = [[rules.expectHeadsFair,densityFair], [rules.expectHeadsCheat,densityCheat]] // linear scales let xScale = d3.scaleLinear([0, 1], [margin.left, w - margin.right]); let yScale = d3.scaleLinear([4, 0], [margin.top, h - margin.bottom]); // FUNCTIONS //////////////////////////////////////////////////////////////////////////// // PROBABILITY DENSITY FUNCTIONS function graphCurve() { let x, y; let f = []; f.push([0, 0]); // initial point let resolution = Math.round(width/2-padding)*2 for (let i = 1; i <= resolution; i++) { x = (1 / (resolution-1)) * (i-1); y = betaPDF(x, a, b, B) f.push([x, y]); // final point } f.push([1, 0]); return f; } // SVG //////////////////////////////////////////////////////////////////////////////// let svg = d3 .select(DOM.svg(w, h)) let line = d3 .line() .x((d) => xScale(d[0])) .y((d) => yScale(d[1])) let pdf = svg.selectAll("curve") // graphs the probability density functions .data([1]) // one graph per handles .enter() .append("path") .attr("stroke", 'steelblue') .attr("stroke-width", 1) .attr("fill", 'steelblue') .attr("fill-opacity", .05) .style("mix-blend-mode", "darken") .datum((d) => graphCurve()) // input parameter for curve .attr("d", line) var circle = svg .append("g") .selectAll("circle") .data( dPoints ) .enter() .append("circle") .attr("fill", (d,i) => color[i]) .attr("stroke", "white") .attr("stroke-width", 2) .attr("cx", (d) => xScale(d[0]) ) .attr("cy", (d) => yScale(d[1]) ) .attr("r", (d) => 4) let lineData = [ {color:color[0], weight: 2, line: [[rules.expectHeadsFair,0], dPoints[0]]}, {color:color[1], weight: 2, line: [[rules.expectHeadsCheat,0], dPoints[1]]}, {color:color[0], weight: 10, line: [ [0, -.8] , [ table[game.h][game.t].probabilityFair,-.8] ]}, {color:color[1], weight: 10, line: [ [table[game.h][game.t].probabilityFair,-.8] , [1,-.8] ]} ] var points = svg .append("g") .selectAll("lines") .data( lineData ) .enter() .append("path") .attr("fill", 'none') .attr("stroke", d => d.color) .attr("stroke-width", d => d.weight) .datum((d) => d.line ) // input parameter for curve .attr("d", line) // text labels let textData = [ {text: "Coin Likelihood, Given "+game.h+" Heads and "+game.t+" Tails", x:margin.left + (w-margin.left-margin.right) /2, y: margin.top, size:"13px", weight:700, anchor:"middle", color:'black'}, {text: "↑ Expected Value of Heads Conditional Likelihood ↓", x:margin.left + (w-margin.left-margin.right) /2, y: yScale(0)+40, size:"10px", weight:700, anchor:"middle", color:'black'}, {text: "Fair Density", x:xScale(rules.expectHeadsFair), y: yScale(densityFair)-17, size:"10px", weight:300, anchor:"middle", color:color[0]}, {text: "Cheat Density", x:xScale(rules.expectHeadsCheat), y: yScale(densityCheat)-17, size:"10px", weight:300, anchor:"middle", color:color[1]}, {text: Math.round(100*table[game.h][game.t].probabilityFair)+"% chance blob is fair", x:xScale(0), y: yScale(0)+53, size:"10px", weight:300, anchor:"start", color:color[0]}, {text: Math.round(100*table[game.h][game.t].probabilityCheat)+"% chance blob is a cheater", x:xScale(1), y: yScale(0)+53, size:"10px", weight:300, anchor:"end", color:color[1]}, ] const text = svg.selectAll("t") .data(textData) .enter() .append('text') .attr("x", (d) => d.x ) .attr("y", (d) => d.y ) .attr("fill", d => d.color) .attr("text-anchor", d => d.anchor) .style("font-size", d => d.size) .attr("font-family", "Arial") .attr("font-weight", d => d.weight) .text((d) => d.text); let xAxis = d3.axisBottom(xScale).ticks(20); svg.append("g").attr("transform", `translate(0,${yScale(0)})`).call(xAxis); return svg.node(); }35:T989,viewof game = { const form = html``; // make value look up table let a, b, h, t, B, dFair, dCheat, dFairPlus, dCheatPlus, probabilityFair, probabilityCheat, inFair, inCheat, profitFairPlus let cell form.oninput = () => { h = form.countH.valueAsNumber t = form.countT.valueAsNumber a = rules.priorH + h b = rules.priorT + t B = cephes.beta(a, b) cell = table[h][t] dFair = betaPDF(rules.expectHeadsFair, a, b, B) dCheat = betaPDF(rules.expectHeadsCheat, a, b, B) form.pF.value = cell.probabilityFair form.pC.value = cell.probabilityCheat form.profF.value = Math.round(10*cell.incomeFair)/10 form.profC.value = Math.round(10*cell.incomeCheat)/10 if(decisionMatrix[h][t] == 2){ form.decision.value = 'flip coin'} else if(decisionMatrix[h][t] == 0){ form.decision.value = 'label fair'} else if(decisionMatrix[h][t] == 1){ form.decision.value = 'label cheat'} else{ form.decision.value = ''} form.value = ({ a: a, b: b, h: h, t: t }) } form.oninput(); return form; }36:T4f7,table = { let array = [] let n = 33 let m = 33 let aPrior = 1 let bPrior = 1 let a, b, B, dFair, dCheat, dFairPlus, dCheatPlus, probabilityFair, probabilityCheat, inFair, inCheat, profitFair, profitCheat, profitFlip // make table for(let i=0; i

	Your Labels		Blob Settings
Truth	Fair	Cheater	Proportion	Expect Value
Actually Fair	True Positive Reward	False Negative Penalty	50% play fair	50% heads
Actually Cheater	False Positive Penalty	True Negative Reward	50% cheat	75% heads

Prior Assumptions		Coins
Heads		Starting coins
Tails		Cost of each flip

`; form.oninput = () => form.value = ({ trueNegative: form.trueNeg.valueAsNumber, truePositive: form.truePos.valueAsNumber, falseNegative: form.falseNeg.valueAsNumber, falsePositive: form.falsePos.valueAsNumber, populationCheat: form.popCheat.valueAsNumber, populationFair: form.popFair.valueAsNumber, expectHeadsCheat: form.expectedCheat.valueAsNumber, expectHeadsFair: form.expectedFair.valueAsNumber, priorH: form.priorH.valueAsNumber, priorT: form.priorT.valueAsNumber, bank: form.bank.valueAsNumber, priceOfFlip: form.price.valueAsNumber }); form.oninput(); return form; }38:[] 30:{"id":"fda6401011e5a68b","slug":"primers-coin-game","title":"Primer's Coin Game Simulation","thumbnail":"730fe412729583707cc5b73b7127e16cb650024cc040a2a0bcf9a67d1c217524","access_level":"public","create_time":"$D2022-04-09T18:24:53.669Z","update_time":"$D2025-05-16T14:09:33.629Z","view_time":null,"first_public_time":"$D2022-04-12T05:42:26.156Z","has_unpublished_edits":null,"fork":false,"stars":9,"starred":false,"owner":{"type":"team","id":"cf45be0120eccaad","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/aeddc181df6ca80a347940ba62e17601ac6395e9d2c21142887eca5fd277326a","tier":"starter_2024"},"version":1323,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"5da1d0646b8496ac","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/9c14da488e1a68ca1868ed025eb7edcce3d300c8a9f3fbcdae66a3daa9ccca63","tier":null}],"members":[{"type":"individual","id":"5da1d0646b8496ac","login":"laotzunami","name":"Lao","avatar_url":"https://avatars-next.observableusercontent.com/9c14da488e1a68ca1868ed025eb7edcce3d300c8a9f3fbcdae66a3daa9ccca63","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":{"id":"9e8c57d86af129bc","slug":"finance-and-strategy","title":"Finance and Strategy","owner":{"id":"f35c755083683fe5","login":"observablehq","name":"Observable"}},"type":"notebook","body":{"titling":"auto","title":"Primer's Coin Game Simulation","cells":[{"id":3,"value":"md` # Primer's Coin Game Simulation`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"Primer, one of my all time favorite YouTubers, just released a [probability game][1]! Watch [this video][2] for more information.\n\nIn the game, half of the blobs have fair coins, and the other half cheat with a coin that is biased 75% towards heads. You start the game with 100 coins, and you can pay one coin to have the blob flip. When the blob has flipped enough coins for you to detect a trend, you can label them as fair or a cheater. If you guessed correctly, you get 15 coins, and if you don't, you lose 30 coins. The game goes on until you lose all of your money.\n\nTest your strategy in this simulator, and when you are happy, go for the high score in the game! Top 5 people and 5 random people on the leaderboard get a plushie.\n\n[1]:https://primerlearning.org/\n[2]:https://www.youtube.com/watch?v=QC91Bf8hQVo","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":565,"value":"md` ## Random Walk Simulation`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":570,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1151,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":866,"value":"Write your own policy here to see how it does in the simulation:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":562,"value":"function policy (heads, tails, bank){\n let fair = 0\n let cheat = 1\n let flip = 2\n \n if(bank < rules.priceOfFlip){\n if(table[heads][tails].profitFair > table[heads][tails].profitCheat){ return fair }else{ return cheat }\n }\n \n return decisionMatrix[heads][tails]\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":995,"value":"decisionMatrix = [ // i=heads, j=tails. Solution by Rimbaud \n [2, 2, 0, 0, , , , , , , , , ],\n [2, 2, 2, 0, 0, , , , , , , , ],\n [2, 2, 2, 0, 0, , , , , , , , ],\n [1, 2, 2, 2, 0, 0, , , , , , , ],\n [1, 2, 2, 2, 0, 0, , , , , , , ],\n [1, 1, 2, 2, 2, 0, 0, , , , , , ],\n [ , 1, 2, 2, 2, 2, 0, 0, , , , , ],\n [ , 1, 1, 2, 2, 2, 0, 0, , , , , ],\n [ , , 1, 1, 2, 2, 2, 0, 0, , , , ],\n [ , , , 1, 2, 2, 2, 0, 0, , , , ],\n [ , , , 1, 1, 2, 2, 2, 0, 0, , , ],\n [ , , , , 1, 2, 2, 2, 2, 0, 0, , ],\n [ , , , , 1, 1, 2, 2, 2, 0, 0, , ],\n [ , , , , , 1, 1, 2, 2, 2, 0, 0, ],\n [ , , , , , , 1, 2, 2, 2, 0, 0, ],\n [ , , , , , , 1, 1, 2, 2, 2, 0, 0],\n [ , , , , , , , 1, 2, 2, 2, 0, 0],\n [ , , , , , , , 1, 1, 2, 2, 2, 0],\n [ , , , , , , , , 1, 1, 2, 2, 0],\n [ , , , , , , , , , 1, 1, 1, 0]\n ]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":881,"value":"## Game","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":895,"value":"When playing, enter the current heads and tails to get the likelihood you are dealing with a fair player or cheater, and to calculate the expected income from labeling based on current certainty. Notice that with too little information, you expect to lose money from guessing, but if you pay for too many flips, the cost of the flips out weighs the expected income. To win, you must balance these two factors.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":365,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":216,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":506,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":838,"value":"## Rules","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":872,"value":"Adjust game rules used in the simulation. Will reset other settings.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":16,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":876,"value":"## Libraries and Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":358,"value":"cephes = {\n const c = await require(\"https://bundle.run/cephes@1.2.0\");\n await c.compiled;\n return c;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":361,"value":"function betaPDF(x, alpha, beta, B){\n if(x==0){x=.00000001}\n if(x==1){x=.99999999}\n if(0\n .correct {\n border-radius: 10px;\n background-color: #d4ffdc;\n width: 100px;\n }\n .incorrect {\n border-radius: 10px;\n background-color: #ffe3e3;\n width: 100px;\n }\n .set {\n width: 100px;\n }\n table, td, th, tr {\n border: 1px solid white;\n border-collapse: collapse;\n text-align:center;\n vertical-align:middle;\n }\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W38","stdlib":"1"}} 31:"$30:body"

Count	Action	Likelihood	Expected Income	Decision Matrix
	Label Cheater	50	-15	flip
	Label Fair	50	-15