2d:null 2e:["$","title",null,{"children":"Exponential Woo in Toy Economic Models | unconed | Observable"}] 33:T590,md`# Exponential Woo in Toy Economic Models Recently I posted a [thread about toy social models](https://twitter.com/unconed/status/1210874564764454912) like the Schelling model of segregation. These are typically simple simulations—often on a grid—where people are represented by automatons. The behavior of these "agents" is governed by some simple mathematical or algorithmic law, often applied blindly and at random. The number of parameters is small, sometimes there's even just one. This for instance is considered by many to be a credible model of segregation in society, between two groups, the blues and the yellows: ![](https://pbs.twimg.com/media/EM3j_uTWwAEx2Fz?format=png&name=small) Such models are then used to make sweeping predictions, using their simplicity as a sort of magician's gambit: there's nothing up my sleeve, look how few variables there are! But it does something surprising! Yet this simplicity is exactly why they shouldn't be trusted at all. As I explained, the Schelling model actually has no other option _but_ to create segregation, so the fact that it happens is not surprising at all. If you modify the model to allow for segregation to reverse, it suddenly stops being meaningful, because you can control the effect in either direction. I'm going to demonstrate similar flaws using another common model, this time in economics. --- `34:T4a4,md` You don't need a linear regression to spot that trend. This discovery seems surprising: we started with a completely unbiased model of giving and receiving, and instead we end up with a result where most people give more than they receive. Where did this bias come from? Well, consider what the rules actually are. This isn't actually a simulation of real people, or even agents with any will at all. In this model, an agent always selects a random other person to trade with. They always spend the exact same amount, and they do so regardless of their own accumulated wealth. It's nothing like a real market. There is only one thing that can go wrong: if someone's broke, then they cannot trade away any tokens, and this is usually accounted for in the models by rejecting trades that would incur debt. This is the source of the asymmetry. The fact that people can't go below 0 actually helps them keep their head above water. If you allow for debt, the picture changes entirely. This is even more obvious if you just start everyone off with 0 tokens, so that the graphs are perfectly symmetrical. The cumulative and probability distributions should look exceedingly familiar: `35:T53c,md`Left and right are similar, but they're not the same, with serial skewing more towards lower wealth. One of the most important features of markets is that people have to make simultaneous decisions with imperfect information, and the serial model ignores that by design. It also has a non-zero probability that a person will never be selected in N rounds, while the parallel model does not. All the parameters, random generators and selection rules are the same, only the order in which they are evaluated changes. Yet this is enough to create a different result. Writing serial models is much easier though, it doesn't require complex constraint validation or stability analysis, and as such, it's much more favored by economists and social scientists. --- Doing a curve fitting on a toy model is all but meaningless, unless you can show that all other possible models and implementations, parameters or distributions can be ruled out. As this is pretty much impossible to do in the soft sciences, most of the established wisdom in this regard amounts to little more than masturbation. It's just confusing the aesthetics of simplicity with the elegance of correctness. Sorry but it's true. --- If you enjoyed this presentation and would like more truth than is convenient, come visit my blog at [Acko.net](https://acko.net). `36:T48e,Random = { let rot = (a, b) => ((a|0) << (b|0)) | ((a|0) >>> (32 - (b|0))); let mul = (a, b) => ((a|0) * (b|0))|0; let add = (a, b) => ((a|0) + (b|0))|0; // Murmur3 round // Combine state x with input k let C1 = 0xcc9e2d51; let C2 = 0x1b873593; let C3 = 0xe6546b64; let modmod3 = (x, k) => { k = mul(k, C1); k = rot(k, 15); k = mul(k, C2); x = x ^ k; x = rot(x, 13); x = add(mul(x, C3), 5); return x; } // Murmur3 finalizer let C4 = 0x85ebca6b; let C5 = 0xc2b2ae35; let murmur3 = (x) => { x = x ^ (x >>> 16); x = mul(x, C4); x = x ^ (x >>> 13); x = mul(x, C5); x = x ^ (x >>> 16); return x; } // Seedable PRNGs for predictable testing let C6 = 0x9e23ecb7; let randomInt32 = (seed) => { let f = seed % 1; let i = seed - f; seed = i ^ (f * 0xffffffff); let j = 1; return () => { seed = modmod3(seed, (j = (j + 1) >>> 0)); return murmur3(modmod3(C6 ^ j, seed)) >>> 0; }; } let randomFloat = (seed) => { let rnd = randomInt32(seed); return () => rnd() / 0xffffffff; } return {int32: randomInt32, float: randomFloat}; }37:T6ce,TradeOneToken = (options) => { // Extract options let { allowDebt, allowSelf, rounds, initialBalance: init, people: n, seed, } = options; // Helpers let cmp = (a, b) => a - b; let rnd = Random.int32(seed); // Trade one token let trade = (ppl, j, k) => { ppl[j]--; ppl[k]++; } // Select random person let getRandom = (n) => rnd() % n; // Select random source from list of people let getSource = (n, ppl, limit = 10) => { let j = getRandom(n); if (!allowDebt && ppl[j] <= 0) { // Ensure we always find a source, so serial() matches parallel() return limit ? getSource(n, ppl, limit--) : null; } return j; } // Select suitabl target to give to let getTarget = (n, j) => { let k = getRandom(n); if (!allowSelf && k === j) return null; return k; } // Strategy: Choose 1 person randomly let serial = () => { let ppl = [...Array(n)].map(_ => init); let iterate = () => { let j = getSource(n, ppl); let k = getTarget(n, j); if (j !== null && k !== null) trade(ppl, j, k); } // One parallel round = n serial rounds let r = rounds * n; for (let i = 0; i < r; ++i) iterate(); ppl.sort(cmp); return ppl; } // Strategy: Everyone gives in each round let parallel = () => { let ppl = [...Array(n)].map(_ => init); let iterate = () => { for (let i = 0; i < n; ++i) { if (!allowDebt && ppl[i] <= 0) continue; let j = getTarget(n, i); if (j !== null) trade(ppl, i, j); } } for (let i = 0; i < rounds; ++i) iterate(); ppl.sort(cmp); return ppl; } return { serial, parallel, }; };38:T709,TradePercent = (options) => { // Extract options let { allowDebt, allowSelf, rounds, initialBalance: init, people: n, seed, ratio, } = options; // Helpers let cmp = (a, b) => a - b; let rnd = Random.int32(seed); // Trade one token let trade = (ppl, j, k) => { let amount = ppl[j] * ratio; ppl[j] -= amount; ppl[k] += amount; } // Select random person let getRandom = (n) => rnd() % n; // Select random source from list of people let getSource = (n, ppl, limit = 10) => { let j = getRandom(n); if (!allowDebt && ppl[j] <= 0) { // Ensure we always find a source, so serial() matches parallel() return limit ? getSource(n, ppl, limit--) : null; } return j; } // Select suitabl target to give to let getTarget = (n, j) => { let k = getRandom(n); if (!allowSelf && k === j) return null; return k; } // Strategy: Choose 1 person randomly let serial = () => { let ppl = [...Array(n)].map(_ => init); let iterate = () => { let j = getSource(n, ppl); let k = getTarget(n, j); if (j !== null && k !== null) trade(ppl, j, k); } // One parallel round = n serial rounds let r = rounds * n; for (let i = 0; i < r; ++i) iterate(); ppl.sort(cmp); return ppl; } // Strategy: Everyone gives in each round let parallel = () => { let ppl = [...Array(n)].map(_ => init); let iterate = () => { for (let i = 0; i < n; ++i) { if (!allowDebt && ppl[i] <= 0) continue; let j = getTarget(n, i); if (j !== null) trade(ppl, i, j); } } for (let i = 0; i < rounds; ++i) iterate(); ppl.sort(cmp); return ppl; } return { serial, parallel, }; };39:Tf7e,GraphPopulation = (datasets, options) => { let STYLE = { padding: 20, width: 420, height: 240, backgroundStyle: '#fff', fillStyle: 'rgba(24, 72, 128, 1)', levelStyle: 'rgba(48, 140, 255, 1.0)', zeroStyle: 'rgba(180, 80, 80, 1.0)', medianStyle: '#888', labelStyle: '#444', font: '15px sans-serif', labels: [], overlay: [], }; options = {...STYLE, ...options}; let { backgroundStyle, fillStyle, levelStyle, zeroStyle, medianStyle, labelStyle, referenceLevel, zeroLevel, width, height, transpose, font, padding, titles, overlay, } = options; let graph = (ppl, range) => { let WIDTH = width; let HEIGHT = height; let [min, max] = range; let mapX = (i) => { let f = i / (n - 1); return Math.round(transpose ? ((1 - f) * (HEIGHT - 1)) : (f * (WIDTH - 1))); }; let mapY = (v) => { let f = ((v||0) - min) / (max - min); return Math.round(transpose ? f * WIDTH : (1 - f) * HEIGHT); } let canvas = document.createElement('canvas'); canvas.width = WIDTH + padding; canvas.height = HEIGHT + padding; canvas.style.maxWidth = '100%'; let ctx = canvas.getContext('2d'); ctx.fillStyle = backgroundStyle; ctx.fillRect(0, 0, canvas.width, canvas.height); ctx.fillStyle = fillStyle; let i = 0; let n = ppl.length; for (let v of ppl) { let x1, y1, x2, y2; if (transpose) { y1 = mapX(i++); y2 = mapX(i); x1 = mapY(v); x2 = mapY(zeroLevel); } else { x1 = mapX(i++); x2 = mapX(i); y1 = mapY(v); y2 = mapY(zeroLevel); } ctx.fillRect( Math.min(x1, x2) + padding, Math.min(y1, y2), Math.max(1, Math.abs(x2 - x1)), Math.max(1, Math.abs(y2 - y1)) ); } let markX = (level, style) => { ctx.beginPath(); ctx.strokeStyle = style; if (transpose) { ctx.moveTo(padding, level); ctx.lineTo(padding + WIDTH, level); } else { ctx.moveTo(level + padding, 0); ctx.lineTo(level + padding, HEIGHT); } ctx.stroke(); } let markY = (level, style) => { ctx.beginPath(); ctx.strokeStyle = style; if (transpose) { ctx.moveTo(level + padding, 0); ctx.lineTo(level + padding, HEIGHT); } else { ctx.moveTo(padding, level); ctx.lineTo(padding + WIDTH, level); } ctx.stroke(); } let k = ppl.findIndex(x => x >= referenceLevel); ctx.lineWidth = 3; markX(mapX(k), medianStyle); markY(mapY(referenceLevel), levelStyle); markY(mapY(zeroLevel), zeroStyle); ctx.font = font; let labels = ['People', 'Wealth']; if (transpose) labels.reverse(); ctx.fillStyle = labelStyle; ctx.fillText(labels[0], WIDTH / 2 - 25 + padding, HEIGHT + padding - 3); ctx.save(); ctx.rotate(-Math.PI / 2); ctx.fillText(labels[1], -HEIGHT / 2, 15); ctx.restore(); return canvas; } let measure = (data) => { let min = Infinity, max = -Infinity; for (let x of data) { min = Math.min(min, x); max = Math.max(max, x); } return [min, max]; } let {unify} = options; let range = [Infinity, -Infinity]; if (unify) { for (let dataset of datasets) { for (let data of dataset) { let [min, max] = measure(data); range[0] = Math.min(min, range[0]); range[1] = Math.max(max, range[1]); } } } let graphs = datasets.map((dataset, i) => { let g = Blend.average(dataset.map((data) => { if (!unify) range = measure(data); return graph(data, range); })); if (overlay[i]) g = Blend.over([g, ...overlay[i]]); return g; }); let boxes = graphs.map((graph, i) => HTML.box(titles[i] || '', graph)); return HTML.flex(boxes); } 3a:Tbf7,GraphPDF = (datasets, options) => { let STYLE = { padding: 20, width: 420, height: 240, backgroundStyle: '#fff', strokeStyle: 'rgba(24, 72, 128, 1)', levelStyle: 'rgba(48, 140, 255, 1.0)', zeroStyle: 'rgba(180, 80, 80, 1.0)', labelStyle: '#444', font: '15px sans-serif', labels: ['Wealth', 'Probability'], log: false, unify: false, merge: false, }; options = {...STYLE, ...options}; let { backgroundStyle, strokeStyle, levelStyle, zeroStyle, labelStyle, referenceLevel, zeroLevel, width, height, font, padding, labels, } = options; let graph = (points, rangeX, rangeY) => { let WIDTH = width; let HEIGHT = height; let mapRange = (x, [min, max]) => (x - min) / (Math.max(max, min + 1e-5) - min); let mapX = (x) => mapRange(x, rangeX) * WIDTH + padding; let mapY = (y) => (1 - mapRange(y, rangeY)) * HEIGHT; let canvas = document.createElement('canvas'); canvas.width = WIDTH + padding; canvas.height = HEIGHT + padding; canvas.style.maxWidth = '100%'; let ctx = canvas.getContext('2d'); ctx.fillStyle = backgroundStyle; ctx.fillRect(0, 0, canvas.width, canvas.height); ctx.strokeStyle = strokeStyle; ctx.lineWidth = 2; let i = 0; ctx.beginPath(); for (let [x, y] of points) { ctx[i++ ? 'lineTo' : 'moveTo'](mapX(x), mapY(y)); } ctx.stroke(); ctx.font = font; ctx.fillStyle = labelStyle; ctx.fillText(labels[0], WIDTH / 2 - 25, HEIGHT + padding - 3); ctx.save(); ctx.rotate(-Math.PI / 2); ctx.fillText(labels[1], -HEIGHT / 2, 15); ctx.restore(); return canvas; } let measure = (data) => { let minX = Infinity, maxX = -Infinity, minY = Infinity, maxY = -Infinity; for (let [x, y] of data) { minX = Math.min(minX, x); maxX = Math.max(maxX, x); minY = Math.min(minY, y); maxY = Math.max(maxY, y); } return [[minX, maxX], [minY, maxY]]; } let {unify, merge, log} = options; let diffsets = merge ? datasets.map((dataset) => [Prob.pdf(dataset.flatMap(_=>_))]) : datasets.map((dataset) => dataset.map(d => Prob.pdf(d))); if (log) { diffsets = diffsets.map( (dataset) => dataset.map( (data) => data .map(([x, y]) => [x, Math.log(y)]) .filter(([x, y]) => 1/y))); } let rangeX = [Infinity, -Infinity]; let rangeY = [Infinity, -Infinity]; if (unify) { for (let dataset of diffsets) { for (let data of dataset) { let [[minX, maxX], [minY, maxY]] = measure(data); rangeX[0] = Math.min(minX, rangeX[0]); rangeX[1] = Math.max(maxX, rangeX[1]); rangeY[0] = Math.min(minY, rangeY[0]); rangeY[1] = Math.max(maxY, rangeY[1]); } } } let graphs = diffsets.map((dataset) => Blend.average(dataset.map((data) => { if (!unify) [rangeX, rangeY] = measure(data); return graph(data, rangeX, rangeY); }))); return graphs; }3b:[] 31:{"id":"ec018dd31132c22b","slug":"exponential-woo-in-toy-economic-models","title":"Exponential Woo in Toy Economic Models","thumbnail":"fe98afe5acb3186e573b982bac7eb86ec8b7d2b207aec31ced2ee170436b4509","access_level":"public","create_time":"$D2020-01-28T22:56:55.083Z","update_time":"$D2020-01-29T23:36:28.336Z","view_time":null,"first_public_time":"$D2020-01-29T07:42:24.442Z","has_unpublished_edits":null,"fork":false,"stars":14,"starred":false,"owner":{"type":"team","id":"80a61746f6a22aaa","login":"unconed","name":"","avatar_url":"https://avatars-next.observableusercontent.com/0e7f3c06495f9792cdb6e24c2e0228e26b88eb3591d874c51b0e56339b278e8d","tier":"starter_2024"},"version":942,"publish_version":942,"role":null,"authored":false,"authors":[{"type":"individual","id":"1a2aae7ef5a46fe1","login":"unconed","name":"","avatar_url":"https://avatars-next.observableusercontent.com/0e7f3c06495f9792cdb6e24c2e0228e26b88eb3591d874c51b0e56339b278e8d","tier":null}],"members":[{"type":"individual","id":"1a2aae7ef5a46fe1","login":"unconed","name":"","avatar_url":"https://avatars-next.observableusercontent.com/0e7f3c06495f9792cdb6e24c2e0228e26b88eb3591d874c51b0e56339b278e8d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Exponential Woo in Toy Economic Models","cells":[{"id":0,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":249,"value":"md`It goes like this.\n\n**You start with a group of people who all have the same amount of money or tokens. You select two people at random, and you make them trade a token from one to the other. If you keep doing this, your equal distribution will instead turn into an exponential distribution of wealth.**\n\n(Note: You can re-run the snippet below to re-generate the example with a different random seed)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":279,"value":"rerun1 = Math.random();","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":262,"value":"plot1_1 = {\n return GraphPopulation([experiment1[0]], {\n titles: [`Wealth after ${parameters1.rounds * parameters1.people} random trades between ${parameters1.people} people (${parameters1.runs} runs)`],\n unify: true,\n referenceLevel: parameters1.initialBalance,\n overlay: [\n [GraphText(\"▼\", 223, 200)],\n ],\n })\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":272,"value":"md`\nThis graph is what you get if you line everyone up from poorest to richest, and plot their wealth vertically. We average ${parameters1.runs} random runs to get a smoother picture.\n\nThe blue line indicates the starting balance of ${parameters1.initialBalance} tokens. The wealthiest people tower above the rest, and the median person is now poorer than they started. The gray lines indicate the break even point in each run, which less than half the group exceeds.\n\nThis is more obvious if you plot the cumulative wealth distribution, which is actually just the previous graph with the two axes swapped and the color inverted. It shows what % of people (Y) have less than a certain amount (X).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":93,"value":"plot1_2 = {\n return GraphPopulation([experiment1[0]], {\n titles: ['Cumulative wealth distribution'],\n unify: true,\n transpose: true,\n referenceLevel: parameters1.initialBalance,\n zeroLevel: 1e5,\n })\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":489,"value":"md`\n\nBut is it exponential?\n\nWe can plot the probability density function to find out:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":171,"value":"plot_1_3 = {\n return HTML.flex([\n HTML.box(\"Probability distribution\", Blend.average([\n ...GraphPDF([experiment1[0]], {\n unify: true,\n merge: true,\n })\n ])),\n HTML.box(\"Logarithm of probability distribution\", ...GraphPDF([experiment1[0]], {\n unify: true,\n merge: true,\n log: true,\n labels: ['Wealth', 'log(Probability)'],\n })\n ),\n\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":504,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":618,"value":"rerun2 = Math.random();","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":584,"value":"plot2_1 = {\n return HTML.flex([\n GraphPopulation([experiment2[0]], {\n titles: [`Wealth and debt after ${parameters2.rounds * parameters2.people} random trades`],\n unify: true,\n referenceLevel: parameters2.initialBalance,\n }),\n GraphPopulation([experiment2[0]], {\n titles: ['Cumulative wealth distribution'],\n unify: true,\n transpose: true,\n referenceLevel: parameters2.initialBalance,\n zeroLevel: 1e5,\n })\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":631,"value":"plot_2_3 = {\n return HTML.flex([\n HTML.box(\"Probability distribution\", Blend.average([\n ...GraphPDF([experiment2[0]], {\n unify: true,\n merge: true,\n })\n ])),\n HTML.box(\"Logarithm of probability distribution\", ...GraphPDF([experiment2[0]], {\n unify: true,\n merge: true,\n log: true,\n labels: ['Wealth', 'log(Probability)'],\n })\n ),\n\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":655,"value":"md`Despite the noise, you can see a pretty good bell curve, as well as the characteristic upside-down parabola on the logarithmic axis. This is an approximate Gaussian ${tex`e^{-x^2}`}. When people trade randomly, the result is perfectly symmetrical, exactly what you'd expect, if people can't run out of money.\n\nThis implies that as long as the size of the transactions is small relative to the starting funds, and the number of rounds small enough so nobody can ever reach bankruptcy, the same thing will happen even if you disallow debt.\n\nHere you can see runs tuned so that the first bankruptcy is scheduled to happen just as they end. It looks just the same as above.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":680,"value":"rerun3 = Math.random();","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":683,"value":"plot3_1 = {\n return HTML.flex([\n GraphPopulation([experiment3[0]], {\n titles: [`Wealth after ${parameters3.rounds * parameters3.people} random trades between ${parameters3.people} people`],\n unify: true,\n referenceLevel: parameters3.initialBalance,\n }),\n GraphPopulation([experiment3[0]], {\n titles: ['Cumulative wealth distribution'],\n unify: true,\n transpose: true,\n referenceLevel: parameters3.initialBalance,\n zeroLevel: 1e5,\n })\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":693,"value":"plot_3_2 = {\n return HTML.flex([\n HTML.box(\"Probability distribution\", Blend.average([\n ...GraphPDF([experiment3[0]], {\n unify: true,\n merge: true,\n }),\n ])),\n HTML.box(\"Logarithm of probability distribution\", ...GraphPDF([experiment3[0]], {\n unify: true,\n merge: true,\n log: true,\n labels: ['Wealth', 'log(Probability)'],\n })\n ),\n\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":706,"value":"md`\nSo basically, in this model there are three key numbers:\n- The starting funds ${tex`f`}\n- The transaction size ${tex`t`}\n- The average number of losses ${tex`n`} by the unluckiest person.\n\nIf ${tex`n \\times t < f`} it will lean towards gaussian because it's an undisturbed diffusion process. If ${tex`n \\times t > f`} the result will lean towards exponential, because bankruptcy is likely. This determines how much of the lower half you're folding back towards the upper half.\n\nJust try this slider to change the starting funds:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":783,"value":"rerun4 = Math.random();","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":744,"value":"startingBalance","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":737,"value":"viewof startingBalance = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":730,"value":"plot4_1 = {\n return HTML.flex([\n GraphPopulation([experiment4[0]], {\n titles: [`Wealth after ${parameters4.rounds * parameters4.people} random trades between ${parameters4.people} people`],\n unify: true,\n referenceLevel: parameters4.initialBalance,\n }),\n GraphPopulation([experiment4[0]], {\n titles: ['Cumulative Wealth Distribution'],\n unify: true,\n transpose: true,\n referenceLevel: parameters4.initialBalance,\n zeroLevel: 1e5,\n })\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":722,"value":"plot_4_2 = {\n return HTML.flex([\n HTML.box(\"Probability distribution\", Blend.average([\n ...GraphPDF([experiment4[0]], {\n unify: true,\n merge: true,\n }),\n ...GraphPDF([experiment4[0]], {\n unify: true, \n })\n ])),\n HTML.box(\"Logarithm of probability distribution\", ...GraphPDF([experiment4[0]], {\n unify: true,\n merge: true,\n log: true,\n labels: ['Wealth', 'log(Probability)'],\n })\n ),\n\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":824,"value":"md`You can imagine numerous variants of the model that are equally ill-justified. For example, instead of trading 1 token, everyone trades X% of their money away. A similar weird dynamic exists, where at large percentages, super-hoarders are created by pure chance, even though they also give away the most in absolute numbers when selected.\n\nBut I'd argue that this too is a contrived scenario that doesn't match any dynamics in the real world except by accident.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":787,"value":"rerun5 = Math.random();","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":805,"value":"viewof generosity = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":790,"value":"plot5_1 = {\n return HTML.flex([\n GraphPopulation([experiment5[0]], {\n titles: [`Wealth after ${parameters5.rounds * parameters5.people} random trades between ${parameters5.people} people`],\n unify: true,\n referenceLevel: parameters5.initialBalance,\n }),\n GraphPopulation([experiment5[0]], {\n titles: ['Cumulative Wealth Distribution'],\n unify: true,\n transpose: true,\n referenceLevel: parameters5.initialBalance,\n zeroLevel: 1e5,\n })\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":797,"value":"plot_5_2 = {\n return HTML.flex([\n HTML.box(\"Probability Distribution\", Blend.average([\n ...GraphPDF([experiment5[0]], {\n unify: true,\n merge: true,\n }),\n ])),\n HTML.box(\"Logarithm of Probability Distribution\", ...GraphPDF([experiment5[0]], {\n unify: true,\n merge: true,\n log: true,\n labels: ['Wealth', 'log(Probability)'],\n })\n ),\n\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":812,"value":"md`One final point I'd like to make.\n\nIn this model, we randomly select pairs of people to trade, running the algorithm serially. But is that really justified? What if, instead of randomly selecting a donor and recipient, we actually iterate over every person, choose a receiving trading partner for each, and execute one trade per person in each round?\n\nLet's compare that with our previous approach, using the X% generosity rule:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":859,"value":"rerun6 = Math.random();","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":846,"value":"viewof generosity2 = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":832,"value":"plot5_2 = {\n return HTML.flex([\n GraphPopulation([experiment6s[0], experiment6p[0]], {\n titles: [`Serial selection`, `Parallel selection`],\n unify: true,\n referenceLevel: parameters6.initialBalance,\n }),\n GraphPopulation([experiment6s[0], experiment6p[0]], {\n titles: ['Cumulative (serial)', 'Cumulative (parallel)'],\n unify: true,\n transpose: true,\n referenceLevel: parameters6.initialBalance,\n zeroLevel: 1e5,\n })\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":841,"value":"plot_5_3 = {\n return HTML.flex([\n HTML.box(\"Probability distribution (serial)\", Blend.average([\n ...GraphPDF([experiment6s[0]], {\n unify: true,\n merge: true,\n }),\n ])),\n HTML.box(\"Probability distribution (parallel)\", Blend.average([\n ...GraphPDF([experiment6p[0]], {\n unify: true,\n merge: true,\n }),\n ])),\n ]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":866,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":874,"value":"md`---\n\n---\n\n---\n\n---\n\n---`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":350,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":297,"value":"HTML = {\n let box = (text, child) => {\n let div = document.createElement('div');\n div.style.marginRight = \"12px\";\n div.style.marginTop = \"48px\";\n div.style.marginBottom = \"48px\";\n\n let label = document.createElement('div');\n label.innerText = text;\n label.style.padding = '5px';\n label.style.font = 'bold 16px sans-serif';\n \n div.appendChild(label);\n div.appendChild(child);\n\n return div;\n }\n \n let flex = (children) => {\n let div = document.createElement('div');\n div.style.display = 'flex';\n div.style.flexWrap = 'wrap';\n\n\n for (let child of children) div.appendChild(child);\n\n return div;\n }\n \n return {box, flex};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":292,"value":"Blend = {\n let average = (canvases) => {\n let {width, height} = canvases[0];\n let canvas = document.createElement('canvas');\n canvas.width = width;\n canvas.height = height;\n canvas.style.maxWidth = '100%';\n\n let ctx = canvas.getContext('2d');\n let n = canvases.length;\n canvases.forEach((c, i) => {\n ctx.globalAlpha = 1/(i+1);\n ctx.drawImage(c, 0, 0);\n });\n \n return canvas;\n }\n\n let over = (canvases) => {\n let {width, height} = canvases[0];\n let canvas = document.createElement('canvas');\n canvas.width = width;\n canvas.height = height;\n canvas.style.maxWidth = '100%';\n \n let ctx = canvas.getContext('2d');\n let n = canvases.length;\n canvases.forEach((c, i) => {\n ctx.drawImage(c, 0, 0);\n });\n \n return canvas;\n }\n\n return {average, over};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":403,"value":"Prob = {\n let pdf = (data, bucket = 1) => {\n let keyFor = (x) => Math.round(x / bucket);\n \n let pdf = {};\n for (let x of data) {\n let k = keyFor(x);\n pdf[k] = (pdf[k]||0) + 1;\n }\n let ks = Object.keys(pdf).map(k => +k);\n ks.sort((a, b) => a - b);\n\n let min = ks[0];\n let max = ks[ks.length - 1];\n let out = [];\n\n let i = 0;\n for (let x = min; x < max; x += bucket) {\n out.push([x, pdf[keyFor(x)] || 0]);\n if (i++ > 0xffff) break;\n }\n return out;\n };\n \n return {pdf};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":773,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":303,"value":"GraphText = (text, x, y, options) => {\n let STYLE = {\n padding: 20,\n width: 420,\n height: 240,\n font: '15px sans-serif',\n fillStyle: '#C04',\n };\n\n options = {...STYLE, ...options};\n let {padding, width, height, font, fillStyle} = options;\n\n let canvas = document.createElement('canvas');\n canvas.width = width + padding;\n canvas.height = height + padding;\n canvas.style.maxWidth = '100%';\n \n let ctx = canvas.getContext('2d');\n ctx.font = font;\n ctx.fillStyle = fillStyle;\n ctx.fillText(text, x, y);\n \n return canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":21,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":185,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":111,"value":"parameters1 = ({\n allowDebt: false,\n allowSelf: false,\n initialBalance: 10,\n rounds: 400,\n people: 200,\n runs: 10,\n seed: rerun1,\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"experiment1 = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradeOneToken(parameters1);\n return [\n seq(parameters1.runs).map(_ => run.serial()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":607,"value":"parameters2 = ({\n allowDebt: true,\n allowSelf: false,\n initialBalance: 0,\n rounds: 500,\n people: 500,\n runs: 5,\n seed: rerun2,\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":235,"value":"experiment2 = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradeOneToken(parameters2);\n return [\n seq(parameters2.runs).map(_ => run.serial()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":678,"value":"parameters3 = ({\n allowDebt: false,\n allowSelf: false,\n initialBalance: 105,\n rounds: 500,\n people: 500,\n runs: 5,\n seed: rerun3,\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":686,"value":"experiment3 = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradeOneToken(parameters3);\n return [\n seq(parameters3.runs).map(_ => run.serial()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":719,"value":"parameters4 = ({\n allowDebt: false,\n allowSelf: false,\n initialBalance: startingBalance,\n rounds: 300,\n people: 500,\n runs: 1,\n seed: rerun4,\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":725,"value":"experiment4 = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradeOneToken(parameters4);\n return [\n seq(parameters4.runs).map(_ => run.serial()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":777,"value":"parameters5 = ({\n allowDebt: false,\n allowSelf: false,\n initialBalance: 100,\n ratio: generosity / 100,\n rounds: 400,\n people: 200,\n runs: 5,\n seed: rerun5,\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":780,"value":"experiment5 = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradePercent(parameters5);\n return [\n seq(parameters5.runs).map(_ => run.serial()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":852,"value":"parameters6 = ({\n allowDebt: false,\n allowSelf: false,\n initialBalance: 100,\n ratio: generosity2 / 100,\n rounds: 400,\n people: 200,\n runs: 5,\n seed: rerun6,\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":850,"value":"experiment6s = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradePercent(parameters6);\n return [\n seq(parameters6.runs).map(_ => run.serial()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":828,"value":"experiment6p = {\n let seq = (n, v = 0, d = 1) => [...Array(n)].map((_, i) => v + d * i);\n let run = TradePercent(parameters6);\n return [\n seq(parameters6.runs).map(_ => run.parallel()),\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3b","stdlib":"1","titling":"auto"}} 32:"$31:body"