31:T42e,### Hiring Decisions Then we need to account for the other Random Variable here, ${tex`D`}. Looking at the **demographic parity** constraint, since ${tex`D`} is binary, we have that ${tex.block` \mathbb{E}[D \mid A = a] = 0 \cdot \Pr(D = 0 \mid A = a) + 1 \cdot \Pr(D = 1 \mid A = a) = \Pr(D = 1 \mid A = a) `} Meaning that we need a distribution of the **hiring decision** ${tex`D`} for a given value ${tex`a`} of the random variable ${tex`A`}. We'll encode it, following the notion that capitalist firms hire employees whose **marginal product of labor** is greater than the **wage** being paid for the job, as ${tex.block` D = \begin{cases} 1 &\text{if }M > w, \\ 0 &\text{otherwise} \end{cases} `} where ${tex`w`} is the "going wage" in the sector. (We'll use ${tex`w = 1`} as our default, which means that any individual ${tex`i`} whose marginal product ${tex`m_i`} exceeds 1 will be hired). Thus, given the normal distribution below, * ${tex`\Pr(D = 1 \mid A = 0) \approx ${pr_d1a0.toFixed(2)}`} * ${tex`\Pr(D = 1 \mid A = 1) \approx ${pr_d1a1.toFixed(2)}`}32:T48c,function genInputGrid(views, names, args) { const spaceOnLeft = "margin-left: 12px;"; // const spaceOnRight = "margin-right: 20px;"; const cellPadding = "padding-left: 1px; padding-right: 1px;"; const cellBorder = false; const borderStyle = "border: 2px solid darkgrey;"; const widthSettings = `min-width: ${globs.gridCellWidth}; max-width: ${globs.gridCellWidth};`; const form = html`

${views.map( (row, rowIndex) => html`

${row.map( (input, inputIndex) => html`

${input}

` )}

`; form.oninput = () => { const values = views.map(row => row.map(input => input.value)); form.value = values; if (args && args.objOnly) form.value = {}; if (names) { names.forEach((row, i) => row.forEach( (c, j) => values[i][j] != null && (form.value[c] = values[i][j]) ) ); } }; form.oninput(); return form; }33:T424,columns = (args, borders = false) => { const form = html`` form.value = {} let cols = 0 for (const key in args) { let cur_child = args[key]; console.log(Object.keys(args).length); if (Object.keys(args).length > 1) { cur_child.style = `border: 1px solid; padding: 5px;` } else { cur_child.style = `border: 1px solid; padding: 5px; grid-column:1/span 2; width:96%; padding: 2.5px; justify-content:center; align-text: center; display: flex;` } form.appendChild(cur_child); cols++ } //form.style = `display: grid; grid-gap: 10px 15px; grid-template-columns: repeat(${cols}, auto); grid-auto-flow: dense; width:100%` form.style = `display: grid; grid-gap: 5px 5px; grid-template-columns: repeat(${cols}, 48%); grid-auto-flow: dense; width:100%; height:100%; padding:5px;` form.oninput = () => { form.value = Object.keys(args).reduce((result, key) => { result[key] = args[key].value return result }, Array.isArray(args) ? [] : {}) } form.oninput() return form }34:[] 2f:{"title":"Fairness Definitions and Welfare","cells":[{"id":4,"value":"# Fairness Definitions and Welfare","pinned":false,"hidden":false,"mode":"md"},{"id":17,"value":"### Potential Outcomes\n\nLet's use the setup in Kasy (2023), pg. 541, where we evaluate the compatibility of fairness and welfare by seeing how the **demographic parity** constraint\n\n${tex.block`\n\\mathbb{E}[D \\mid A = 1] = \\mathbb{E}[D \\mid A = 0]\n`}\n\nin turn constrains welfare.\n\nIf we model the welfare of an individual, ${tex`Y`}, as the potential outcome\n\n${tex.block`\nY^d = d - \\rho A,\n`}\n\nwhere ${tex`\\rho \\in [0,1]`} is a measure of the **disadvantage** faced by members of a marginalized group, then we get the following **table** of potential outcome values ${tex`Y^d`}:\n\n| | ${tex`D = 0`} (Not Hired) | ${tex`D = 1`} (Hired) |\n| - | - | - |\n| ${tex`A = 0`} **(White)** | ${tex`0`} | ${tex`1`} |\n| ${tex`A = 1`} **(Black)** | ${tex`0 - \\rho`} | ${tex`1 - \\rho`} |\n\nThe way in which hiring decisions **cause** these levels of welfare for individuals will then determine social welfare (as a function of ${tex`n`} individual ${tex`Y`} values), as well as whether demographic parity is satisfied for this society.","pinned":false,"hidden":false,"mode":"md"},{"id":200,"value":"### Population Base Rates\n\nFirst, to account for how the actual **base rates** of white and black individuals in the population might affect our measures, let's introduce a \"prior\", ${tex`p_1`}, on the probability that a given person in the population is Black. We'll choose this specific parameterization since this is the value leading to ${tex`A = 1`}, so that this gives us equivalence between **expected values** and **probabilities**:\n\n${tex.block`\n\\mathbb{E}[A] = 0 \\cdot \\Pr(A = 0) + 1 \\cdot \\Pr(A = 1) = \\Pr(A = 1) = p_1\n`}\n\nWe'll start with ${tex`p_1 = 0.5`}, to see how issues may already arise even if White and Black applicants are equally likely to apply, but it's good to note that across the entire US, for example, ${tex`p`} would be about [0.14](https://www.census.gov/quickfacts/fact/table/US/RHI225224).","pinned":false,"hidden":false,"mode":"md"},{"id":206,"value":"viewof p1 = genSlider(\"p_1\", [0,1], 0.5)","pinned":false,"hidden":false,"mode":"ojs"},{"id":383,"value":"### Level of Disadvantage\n\nThe default value for ${tex`\\rho`} here will be 0.5","pinned":false,"hidden":false,"mode":"md"},{"id":385,"value":"viewof rho = genSlider(\"\\\\rho\", [0, 1], 0.5)","pinned":false,"hidden":false,"mode":"ojs"},{"id":50,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":240,"value":"viewof w = genSlider(\"w\", [0, 3], 1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":393,"value":"### The \"Merit\" Distribution","pinned":false,"hidden":false,"mode":"md"},{"id":59,"value":"// https://observablehq.com/@observablehq/plot-psr-b1919-21\nmeritDistPlot = Plot.plot({\n height: 300,\n x: {\n axis: \"bottom\",\n label: \"Merit Value (M) →\",\n domain: [-3, 3],\n },\n y: {\n domain: [0, d3.max(meritCurve, d => d.y)]\n },\n color: {\n legend: true,\n },\n facet: {\n data: meritCurve, y: \"a\"\n },\n // marks: [Plot.lineY(pulsar, {x: \"0\", y: \"1\", fy: \"2\", fill: \"white\", stroke: \"currentColor\", strokeWidth: 1})]\n marks: [\n Plot.lineY(meritCurve, {x: \"x\", y: \"y\", stroke: \"black\", strokeWidth: 1}),\n Plot.areaY(meritCurve, {x: \"x\", y2: \"y\", fill: (d => d.x > w ? \"Hired\" : \"Not Hired\"), fillOpacity: 0.5}),\n Plot.ruleX([w]),\n ],\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":44,"value":"Here we can **\"encode\"** our antecedents about this setup, and see whether or not demographic parity \"aligns\" with them. First, the distribution of **merit** ${tex`M`}:","pinned":false,"hidden":false,"mode":"md"},{"id":288,"value":"meritGrid = columns({\n col0: Inputs.form({\n mu0input: viewof mu0,\n s0input: viewof sigma20,\n }),\n col1: Inputs.form({\n m1input: viewof mu1,\n s1input: viewof sigma21,\n }),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":84,"value":"viewof mu0 = genSlider(\"\\\\mu_0\", [-globs.absMMax, globs.absMMax], globs.default_mu0)","pinned":false,"hidden":false,"mode":"ojs"},{"id":296,"value":"viewof mu1 = genSlider(\"\\\\mu_1\", [-globs.absMMax, globs.absMMax], globs.default_mu1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"viewof sigma20 = genSlider(\"\\\\sigma^2_0\", [0.5, 3], globs.default_sigma20)","pinned":false,"hidden":false,"mode":"ojs"},{"id":300,"value":"viewof sigma21 = genSlider(\"\\\\sigma^2_1\", [0.5, 3], globs.default_sigma21)","pinned":false,"hidden":false,"mode":"ojs"},{"id":210,"value":"### Demographic Parity vs. Welfare\n\nWe've now established enough antecedents to determine **social welfare**, as well as whether **demographic parity** is satisfied:\n\n**Demographic Parity:**\n\n${tex.block`\n\\mathbb{E}[D \\mid A = 1] \\approx ${pr_d1a1.toFixed(2)} \\overset{?~${dp_sat ? \"✅\" : \"❌\"}}{=} ${pr_d1a0.toFixed(2)} \\approx \\mathbb{E}[D \\mid A = 0]\n`}\n\n**Welfare Between Groups:**\n\n${tex.block`\n\\mathbb{E}[Y \\mid A = 0] \\approx ${y_a0.toFixed(2)} \\overset{?~${eow_sat ? \"✅\" : \"❌\"}}{${eow_sym}} ${y_a1.toFixed(2)} \\approx \\mathbb{E}[Y \\mid A = 1]\n`}\n\nThe welfare gap (expected difference in welfare) between White and Black individuals in this model is: ${y_diff.toFixed(2)}\n\n(The more written-out math is:)\n\n${tex.block`\n\\begin{aligned}\n\\mathbb{E}[Y \\mid A = 0] &= 1 \\cdot \\Pr(D = 1 \\mid A = 0) + 0 \\cdot \\Pr(D = 0 \\mid A = 0) \\approx ${y_a0.toFixed(2)} \\\\\n\\mathbb{E}[Y \\mid A = 1] &= (1 - \\rho) \\cdot \\Pr(D = 1 \\mid A = 1) + (0 - \\rho) \\cdot \\Pr(D = 0 \\mid A = 1) \\approx ${y_a1.toFixed(2)}\n\\end{aligned}\n`}","pinned":false,"hidden":false,"mode":"md"},{"id":396,"value":"### Compensating for Disadvantage?\n\nSo, this raises the question... Is it possible to compensate for any amount of disadvantage (any scenario where ${tex`\\rho > 0`}), by adjusting the hiring decision process, while still satisfying demographic parity? The answer is **no**, as Kasy (2023) shows.\n\nTo see this for yourself, set ${tex`\\rho`} above to be any non-zero value, and then try to adjust the \"merit\" distributions: you'll see that any movement towards more equal **welfare** between groups will lead to a violation of this constraint.","pinned":false,"hidden":false,"mode":"md"},{"id":264,"value":"## Model Computation Pieces","pinned":false,"hidden":false,"mode":"md"},{"id":40,"value":"potentialOutcome = (d, a) => d - rho * a","pinned":false,"hidden":false,"mode":"ojs"},{"id":266,"value":"pr_d1a0 = 1 - ncdf(w, mu0, sigma20)","pinned":false,"hidden":false,"mode":"ojs"},{"id":310,"value":"pr_d1a1 = 1 - ncdf(w, mu1, sigma21)","pinned":false,"hidden":false,"mode":"ojs"},{"id":275,"value":"dp_sat = pr_d1a1 - pr_d1a0 < 0.001","pinned":false,"hidden":false,"mode":"ojs"},{"id":349,"value":"y_a0 = pr_d1a0","pinned":false,"hidden":false,"mode":"ojs"},{"id":351,"value":"y_a1 = (0 - rho) * (1 - pr_d1a1) + (1 - rho) * (pr_d1a1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":400,"value":"y_diff = Math.abs(y_a0 - y_a1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":357,"value":"eow_sat = y_diff < 0.01","pinned":false,"hidden":false,"mode":"ojs"},{"id":361,"value":"eow_sym = eow_sat ? \"=\" : (y_a0 < y_a1 ? \"<\" : \">\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":82,"value":"## JS Pieces","pinned":false,"hidden":false,"mode":"md"},{"id":144,"value":"function dnorm(x, mu, sigma2) {\n return 1 / Math.sqrt(2 * Math.PI * sigma2) * Math.exp(-0.5 * Math.pow((x-mu)/sigma2, 2));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":56,"value":"// https://observablehq.com/@toja/histogram-with-normal-distribution\nmeritCurve = {\n const points = [];\n for (let i = -globs.absMMax; i < globs.absMMax; i = i + globs.plotResolution) {\n points.push({\n \"x\": i,\n \"a\": 0,\n \"y\": dnorm(i, mu0, sigma20),\n })\n points.push({\n \"x\": i,\n \"a\": 1,\n \"y\": dnorm(i, mu1, sigma21),\n })\n }\n return points;\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":90,"value":"function genSlider(paramName, range, defaultVal) {\n var sliderElt = Inputs.range(range, {\n value: defaultVal,\n label: makeLarger(paramName),\n width: globs.sliderWidth,\n format: x => x.toFixed(2)\n });\n var styleStr = `--label-width: ${globs.labelWidth}`;\n Object.assign(sliderElt, {style: styleStr});\n return(sliderElt);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":112,"value":"makeLarger = (raw) => tex`\\Large{ ${raw} }`","pinned":false,"hidden":false,"mode":"ojs"},{"id":178,"value":"resetParams = () => {\n set(viewof mu0, globs.default_mu0);\n set(viewof mu1, globs.default_mu1);\n set(viewof sigma20, globs.default_sigma20);\n set(viewof sigma21, globs.default_sigma21);\n set(viewof w, globs.default_w);\n set(viewof p1, globs.default_p1);\n};","pinned":false,"hidden":false,"mode":"ojs"},{"id":285,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":183,"value":"function set(input, value) {\n input.value = value;\n input.dispatchEvent(new Event(\"input\"));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":75,"value":"globs = ({\n default_mu0: 0,\n default_mu1: 0,\n default_sigma20: 1,\n default_sigma21: 1,\n default_p1: 0.5,\n default_w: 1,\n absMMax: 3,\n labelWidth: \"40px\",\n plotResolution: 0.005,\n margin: {\n top: 20, right: 30, bottom: 20, left: 30\n }\n});","pinned":true,"hidden":false,"mode":"ojs"},{"id":319,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":432,"value":"// https://observablehq.com/d/e10b5239b369bb08\nstdlib = require( 'https://unpkg.com/@stdlib/stdlib@0.0.90' )","pinned":false,"hidden":false,"mode":"ojs"},{"id":443,"value":"stdlib.base.dists.normal.cdf(0, 0, 1)","pinned":true,"hidden":false,"mode":"ojs"},{"id":455,"value":"ncdf = (x, mu, sigma) => stdlib.base.dists.normal.cdf(x, mu, sigma)","pinned":true,"hidden":false,"mode":"ojs"},{"id":457,"value":"ncdf(0, 0, 1)","pinned":true,"hidden":false,"mode":"ojs"},{"id":176,"value":"viewof resetButton = Inputs.button(\"Reset\", {reduce: resetParams});","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}