29:null 2a:["$","title",null,{"children":"Risk-Dominant Equilibria in Assurance Games | Jeff Jacobs | Observable"}] 2f:T696,The "Palanpur Assurance Game", given on page 43 of [Bowles (2004)](https://www.google.com/books/edition/Microeconomics/HAiMDU4qv0IC?hl=en&gbpv=0) (and described on page 42), models the following scenario: * Two farmers, Ana and Bina, are deciding whether to plant their crops **Early** or **Late** * Planting **Early** gives best results, such that if both plant **Early** then both can sell their resulting crops on the market for 4 rupees each * Planting **Late** gives worse results, such that if both plant **Late** then both can only sell their resulting crops on the market for **2 rupees** each * However, there is a **"first-mover effect"** here, wherein if only **one** of the two farmers plants Early while the other plants Late, the early-planter's seeds are all **eaten up by birds**: > *The farmers do not doubt that earlier planting would give them larger harvests, but no one, the farmer explained, is willing to be the first to plant, as the seeds on any lone plot would be quickly eaten by birds...* > *[What if you all organized to plant on the same day, to reap rewards of earlier planting while minimizing bird losses (dividing by ${tex`N`} instead of 1)?]* > *"If we knew how to do that", he said, looking up from his hoe at me, "we would not be poor."* This scenario can be represented in the form of a ${tex`2 \times 2`} normal-form game as follows. The left side contains the Normal-form game matrix, showing the **two Nash equilibria** (as the bolded cells), while the right side is a plot of the key insight from this model for our purposes: the fact that the both-plant-late equilibrium is **risk-dominant** relative to the both-plant-early equilibrium. 30:T47c,pointPriorPlot = Plot.plot({ width: 420, height: 200, insetTop: 25, x: { domain: [0, 1], label: 'v', }, y: { // interval: 1, ticks: 1, tickFormat: (d) => d == 1 ? '∞' : 0, label: `Probability Density δ(v)`, }, marks: [ Plot.ruleX([0]), Plot.ruleX([1]), Plot.line(diracData, { x: 'x', y: 'density', stroke: 'black', fill: '#56b4e9', }), Plot.ruleX( diracData, Plot.pointerX({ x: "x", y1: 0, y2: 1, // py: 'density', stroke: '#e69f00', strokeWidth: 1.5, }) ), Plot.text( diracData, Plot.pointerX({ // x: 0.5, px: "x", py: 'density', // dy: -17, frameAnchor: "top", fontVariant: "tabular-nums", // If anyone looks at this... note that it only works because of quantization! // In general we need to integrate a small region around the pointer's // x coordinate text: (d) => `Ana thinks the probability that p is near ${d.x.toFixed(2)} is ${d.density}`, fontSize: 12, }) ), ], })31:T5e3,### Dirac Delta Sidebar This weird-looking Dirac delta measure is just a contrived (but immensely useful!) pseudo-probability-distribution that one would arrive at by thinking through the problem of constructing a pdf for a "corner case" Random Variable ${tex`V`}: * Since all pdfs must sum to 1, this one should sum to one, but... * It should assign **all** of its probability density to the single point ${tex`\hat{p}`}, to represent the **certainty** that the value of ${tex`V`} will be a specific constant ${tex`v`}. Thus, my favorite way to think of the Dirac delta measure is just to imagine taking the pdf of ${tex`V`} to be a Continuous Uniform Distribution centered at the specific value ${tex`v`}, like ${tex`V \sim \textrm{U}(v - \varepsilon, v + \varepsilon)`}, and then letting ${tex`\varepsilon`} get smaller and smaller. The **area under the curve** of this pdf would **remain exactly 1**, but its **width** would asymptotically approach **0**, meaning that its **height** would technically asymptotically approach **infinity**. Nonetheless, since a computer actually only needs the single piece of information that this Dirac delta measure should be "an infinite density placed at ${tex`v`}" (meaning, numerically, it only needs to **store** the value ${tex`v`}), this is very doable, and is in fact how some symbolic programming languages like Wolfram Mathematica are able to represent both **discrete** distributions and **continuous** distributions using the same underlying structure.32:T416,Despite how strange the Dirac delta measure is, in fact it has a straightforward interpretation in this case (literally just, a vertical line on the risk-dominance plot) that makes it easy to derive what Ana should do given this prior: > *If the expected utility from planting **Late** when ${tex`p = \hat{p}`} is greater than the expected utility from planting **Early** when ${tex`p = \hat{p}`}, then Ana should plant **Late**. Otherwise Ana should plant **Early*** You can quickly visualize Ana's optimization in this case by hovering your mouse over the plot on the right: the tooltip that pops up will show Ana's **expected utilities at this value of ${tex`\hat{p}`}**, one expected utility if she plants **Early** (the ${tex`y`}-coordinate of the *blue* line at this value) and another expected utility if she plants **Late** (the value of the *green* line at this value). Whichever of these two values is greater corresponds to the action that she should take to achieve the best possible outcome for herself in the face of this beliefs.33:Tcb9,**(3) Some level of (non-certain) confidence in a point ${tex`\hat{p}`}** We can generalize from the integrals at the end of the previous section to any other valid probability distribution! In fact, this will allow us to represent any degree of uncertainty "in between" the two extremes of full certainty (Example (1)) and full uncertainty (Example (2)). In the previous case the pdf of ${tex`P`} was just the pdf of a Continuous Uniform Distribution with endpoints 0 and 1, which is just the constant function ${tex`f_X(x) = \frac{1}{a-b} = \frac{1}{1-0} = 1`}, so that it didn't appear in the integral at all (it would just multiply everything above by 1). But in general these integrals would include a ${tex`f_P(p)`} term, so, let's see how we can "put this term to work", in the sense of modeling Ana's own inference process based on the following three assumptions: * (i) Ana has a certain **guess** ${tex`\hat{p} \in [0,1]`} for the value of ${tex`p`}, * (ii) Ana has a certain **degree of certainty** ${tex`\eta \geq 0`} about this guess, and * (iii) Ana would like to **update** this guess and this degree of certainty in an "optimal" ^[1] way as she acquires new information (for example, she might start to visit other villagers and ask them whether they intend to plant early or late). The The beauty of using a distribution called the **Beta Distribution** to model Ana's beliefs here is that, although the "standard" parameterization of this distribution just tells you to use some magical pair of numbers ${tex`(\alpha, \beta)`}, in fact these two parameter choices encode precisely this "(Guess, Uncertainty about guess)" pair in the following way: * ${tex`\frac{\alpha}{\alpha + \beta} = \hat{p}`}, meaning that a Beta distribution ${tex`\text{Beta}(\alpha, \beta)`} "encodes" the information that, starting out (without any additional info on other villagers' actions), Ana's "best guess" is ${tex`\hat{p}`}, and * ${tex`\frac{\alpha + \beta}{2} = \eta`}, meaning that a Beta distribution ${tex`\text{Beta}(\alpha, \beta)`} also "encodes" the **starting** confidence level ${tex`\eta`}, which will increase as Ana collects more information. The phrasing in those two points is weird only because the goal is to emphasize how these starting values for the ${tex`(\alpha, \beta)`} can then be **updated** to take additional samples into account. The simplest final modeling assumption we'll make, just to illustrate the connection between ${tex`(\alpha, \beta)`} and Bayesian updating ^[2], that we'll interpret Ana's confidence level ${tex`\eta`} as **the number of villagers, including herself, whose information she is using to arrive at her guess ${tex`\hat{p}`}**. Thus, if she hasn't obtained any information from anyone besides herself so that ${tex`\eta = 1`}, and her initial guess is ${tex`\hat{p} = 0.5`}, what does this give us? "Converting" these values into ${tex`(\alpha, \beta)`} values as described above gives ${tex.block` \frac{\alpha}{\alpha + \beta} = 0.5, \frac{\alpha + \beta}{2} = 1, `} which has ${tex`\alpha = \beta = 1`} as its unique solution. And what does the ${tex`\text{Beta}(1, 1)`} distribution look like? 34:T5ce,That's right, a ${tex`\text{Beta}(1,1)`} distribution is exactly the Continuous Uniform Distribution on ${tex`[0,1]`}! But, we already looked at that case, so let's now consider how we can **update** this distribution to handle Ana obtaining some new information and updating her guess and her confidence level accordingly. If we imagine that Ana asked *one* other villager who said they would plant **Early**, but then a *second* villager said they would plant **Late**, her best guess remains ${tex`\hat{p} = 0.5`} (we could do this step by step, incorporating the two pieces of information one-by-one, and we would arrive at this result – see the extra writeup linked in Footnote 2 for examples!). However, her **degree of confidence** ${tex`\eta`} should now be **increased**, as she now has information from 2 (as opposed to 0) villagers. It turns out that the update equation for ${tex`\text{Beta}(\alpha, \beta)`}, after observing ${tex`a`} "negative" cases and ${tex`b`} "positive" cases (here we'll call **Early** the "negative" case and **Late** the "positive" case), is just ${tex.block` \text{Beta}'(\alpha, \beta \mid a, b) = \text{Beta}(\alpha + a, \beta + b) `} meaning that we literally just add 1 to our original ${tex`\alpha`} and our original ${tex`\beta`} to obtain the **posterior** distribution (the distribution after incorporating the two new pieces of information) ${tex`\text{Beta}(2, 2)`}. What does a ${tex`\text{Beta}(2, 2)`} distribution look like? 35:T4bb,beta22Plot = Plot.plot({ width: 420, height: 200, insetTop: 25, // clip: true, x: { domain: [0, 1], label: 'v', }, y: { domain: [0,2], label: `Probability Density δ(v)`, }, marks: [ Plot.ruleX([0]), Plot.ruleX([1]), Plot.ruleY([0]), Plot.area(beta22Data, { y1: 0, y2: 'density', x1: 'x', fill: '#56b4e9', // stroke: 'black', fillOpacity: 0.333, }), Plot.line(beta22Data, { x: 'x', y: 'density', stroke: 'black', }), Plot.ruleX( beta22Data, Plot.pointerX({ x: "x", y1: 0, y2: 2, // py: 'density', stroke: '#e69f00', strokeWidth: 2, }) ), Plot.text( beta22Data, Plot.pointerX({ // x: 0.5, px: "x", py: 'density', // dy: -17, frameAnchor: "top", fontVariant: "tabular-nums", // Here we bite the bullet and actually compute the quantized probability as density / sum(density) text: (d) => `Ana thinks the probability that p is near ${d.x.toFixed(2)} is about ${(d.density / beta22Sum).toFixed(4)}`, fontSize: 12, }) ), ], })36:T7f1, 1\. What "optimality" means in this context is outside the scope of this course sadly, but it *is* within the scope of the summer class 😉. Hence [this writeup from that class](https://jjacobs.me/dsan5650/writeups/informative-priors/FlatPriors_MiniLab.html), which describes (through example, though it is provable as a consequence of [De Finetti's Theorem](https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture1.pdf)) how the Beta distribution gives us the unique "optimal" update rule – in the sense of optimally incorporating information to improve future predictions – for the simple case of trying to determine an unknown ${tex`p \in [0,1]`}, like the bias of a coin, from known binary observations. ↩︎ 2\. This final "updating process assumption" isn't included as a third bulleted assumption here because, whereas the two bullet points above represent "the" unique way to convert ${tex`(\alpha, \beta)`} values into ${tex`(\hat{p}, \eta)`} values given this parameterization, there are actually many different Bayesian updating rules that would "work" here, and choosing between those is part of the modeling process itself! For example, the update rule assumed here gives **equal weight** to the information from each villager, but in reality most people probably have a "meta-model" they use to keep track of how much they **trust** information from each villager, which then feeds back into these information weights... If the idea of a constant feedback loop between a model and a meta-model doesn't make you run away, you may love Avner Greif's book [*Institutions and the Path to the Modern Economy* (2006)](https://www.dropbox.com/scl/fi/vjzjamjc0qngosn4htwaf/Avner_Greif-_Institutions_and_the_Path_to_the_Mod-Bokos-Z1.pdf?rlkey=um0vhm4n7taj11jvdy1joyq77&dl=1)! It's also a model we'll use at the veery end of the course, in the last few slides of the [Week 12 lecture](https://jjacobs.me/dsan5450/w12/). ↩︎37:T4c4,betaABPlot = Plot.plot({ width: 600, height: 250, insetTop: 25, // clip: true, x: { domain: [0, 1], label: 'v', }, y: { label: `Probability Density δ(v)`, }, marks: [ Plot.ruleX([0]), Plot.ruleX([1]), Plot.ruleY([0]), Plot.area(betaABData, { y1: 0, y2: 'density', x1: 'x', fill: '#56b4e9', // stroke: 'black', fillOpacity: 0.333, }), Plot.line(betaABData, { x: 'x', y: 'density', stroke: 'black', }), Plot.ruleX( betaABData, Plot.pointerX({ x: "x", // y1: 0, // y2: 2, // py: 'density', stroke: '#e69f00', strokeWidth: 2, insetTop: 15, }) ), Plot.text( betaABData, Plot.pointerX({ // x: 0.5, px: "x", py: 'density', // dy: -17, frameAnchor: "top", fontVariant: "tabular-nums", // Here we bite the bullet and actually compute the quantized probability as density / sum(density) text: (d) => `Ana thinks the probability that p is near ${d.x.toFixed(2)} is about ${(d.density / beta22Sum).toFixed(4)}`, fontSize: 12, }) ), ], })38:T508,unifPriorPlotBuilder = () => Plot.plot({ width: 420, height: 200, insetTop: 25, // clip: true, x: { domain: [0, 1], label: 'v', }, y: { domain: [0,2], label: `Probability Density δ(v)`, }, marks: [ Plot.ruleX([0]), Plot.ruleX([1]), Plot.ruleY([0]), Plot.area(unifData, { y1: 0, y2: 'density', x1: 'x', fill: '#56b4e9', // stroke: 'black', fillOpacity: 0.333, }), Plot.line(unifData, { x: 'x', y: 'density', fill: '#56b4e9', stroke: 'black', }), Plot.ruleX( unifData, Plot.pointerX({ x: "x", y1: 0, y2: 2, // py: 'density', stroke: '#e69f00', strokeWidth: 2, }) ), Plot.text( unifData, Plot.pointerX({ // x: 0.5, px: "x", py: 'density', // dy: -17, frameAnchor: "top", fontVariant: "tabular-nums", // Here we use the 0.00:1.00 quantization to... simulate the probability as the // integral of the density across a small region around the pointer's x coordinate text: (d) => `Ana thinks the probability that p is near ${d.x.toFixed(2)} is ${d.density} / 100`, fontSize: 12, }) ), ], })39:Tab2,viewof rdPlot = Plot.plot({ // aspectRatio: 6, x: { domain: [0,1], label: "Proportion of others planting early", }, y: { domain: [0,4], label: "Aram's Expected Utility", axis: "left", }, // color: { // legend: true, // type: 'categorical', // domain: ['early','late'], // range: ['#194381','#198c46'], // }, marks: [ Plot.ruleX([0], { strokeWidth: 1.5, }), Plot.ruleX([1], { strokeWidth: 1.5, }), Plot.ruleY([0], { strokeWidth: 1.5, }), Plot.axisY({ anchor: "right", // label: "Expected utility | High", }), // Early Plot.line( [[0,0],[1,uHigh]], { stroke: '#194381', strokeWidth: 3, }, ), // Late Plot.line( [[0, uLow], [1, uMid]], { stroke: '#198c46', strokeWidth: 3, }, ), // Lines meet when y = uLow + uMid*x intersects y = 0 + uHigh*x // => uLow + uMid*x = uHigh*x => uLow = (uHigh - uMid)*x => // => x = uLow / (uHigh - uMid) // Value at this point is uLow + uMid * x = 8/3 Plot.dot([[intersection.x, intersection.y]], { stroke: 'black', fill: 'black', r: 5, }), // Now plot the "pure" equilibria Plot.dot([ [0,0], [0,uLow], [1,uMid], [1,uHigh], ], { stroke: 'black', fill: 'black', r: 3, }), Plot.ruleX([intersection.x], { strokeWidth: 2, strokeDasharray: [5], // y1: 0.275, y2: intersection.y, }), Plot.text([[intersection.x + 0.01,0.075]], { text: [`p* = ${intersection.x.toFixed(2)}`], fontSize: 16, fontWeight: 600, // // fill: 'black', // stroke: 'white', strokeWidth: 1.5, lineAnchor: 'bottom', textAnchor: 'start', // frameAnchor: 'left', }), // Plot.text([[0.05,0.05*uHigh + 0.1]], { // text: ['Early'], // fontSize: 16, // fontWeight: 600, // rotate: -Math.atan(4)*(180/Math.PI) // }), curvedText("Early", { path: [[0,0+0.1],[1,uHigh+0.1]], // anchor: 'start', offset: '5%', fontSize: 16, fontWeight: 600, stroke: '#194381', }), curvedText("Late", { path: [[0,uLow+0.1],[1,uMid+0.1]], // anchor: 'start', offset: '5%', fontSize: 16, fontWeight: 600, stroke: '#194381', }), Plot.ruleX( xVals, Plot.pointerX({ y1: (x,y) => eUtil(x), y2: (x,y) => lUtil(x), strokeWidth: 3, fontSize: 14, tip: true, }), ), // hairK("x",{ // text: (x,y) => getUtils(x), // fontSize: 15, // fill: 'black', // interval: 0.01, // }), ], })3a:T952,// https://observablehq.com/@jwolondon/curved-text-mark function curvedText(txt, opts = {}) { const { path = [ [0, 0], [1, 0] ], showPath = false, fitToPath = false, leftToRight = false, // anchor = "middle", offset = "50%", baseline = "auto", fontSize = 10, fontWeight = 400, fontFamily = "system-ui", fill = "#000000", stroke, strokeWidth = 0.2, letterSpacing = 1, wordSpacing = 1 } = opts; const path2 = leftToRight && path[0][0] > path.at(-1)[0] ? [...path].reverse() : path; const SVG_NS = "http://www.w3.org/2000/svg"; const TEXT_STYLE = "white-space:normal; stroke-linejoin:round; paint-order:stroke"; const smooth = d3.line().curve(d3.curveCatmullRom); const getPathLength = (pathData) => { const tempSvg = document.createElementNS(SVG_NS, "svg"); const tempPath = document.createElementNS(SVG_NS, "path"); tempPath.setAttribute("d", pathData); tempSvg.appendChild(tempPath); document.body.appendChild(tempSvg); const pathLength = tempPath.getTotalLength(); document.body.removeChild(tempSvg); return pathLength; }; return Plot.text(path2, { render: (index, scales, { x: X, y: Y }) => { const pathData = smooth([...index].map((i) => [X[i], Y[i]])); const pathId = DOM.uid("curve").id; const length = fitToPath ? getPathLength(pathData) : null; // const textAnchor = fitToPath ? "start" : anchor; // const offset = // fitToPath || anchor === "start" // ? "0%" // : anchor === "end" // ? "100%" // : "50%"; return htl.svg` ${txt} `; } }); }3b:[] 2d:{"id":"647d5da28bfc7688","slug":"assurance-games-and-risk-dominance","title":"Risk-Dominant Equilibria in Assurance Games","thumbnail":"ddecfff961608cae2d1777e8fba432e6d1ff2c59343681dba007007695506ce0","access_level":"public","create_time":"$D2026-03-11T02:01:20.088Z","update_time":"$D2026-03-21T07:07:03.344Z","view_time":null,"first_public_time":"$D2026-03-16T10:45:13.185Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"38d71e6f34a30993","login":"jpowerj","name":"Jeff Jacobs","avatar_url":"https://avatars-next.observableusercontent.com/cf34f4139c8c0841a9701629ca7d66e8ddcc88b7623f62f05e243b8980480f1c","tier":"starter_2024"},"version":1023,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"3d6a884a9195123e","login":"jpowerj","name":"Jeff 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Assurance Games","cells":[{"id":0,"value":"# Risk-Dominant Equilibria in Assurance Games","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"$2f","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":68,"value":"inputsGroup(\n [\n [viewof gameTableForm, viewof rdPlot],\n ],\n [\n ['game-table-form', 'rd-plot'],\n ],\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":380,"value":"We could interpret this plot on its own, by making some assumptions, but to me the way to understand this model in \"full\" detail is to consider it relative to Ana's **prior distribution** over others' planting decisions:\n\n**(1) Complete certainty that ${tex`p = \\hat{p}`}**\n\nThe easiest case is, if Ana is **certain** that the proportion of others who will plant early is exactly ${tex`\\hat{p}`}, then Ana has what's called a \"point-density prior\", which is a strange thing to represent probabilistically: it corresponds to the [Dirac measure](https://en.wikipedia.org/wiki/Dirac_measure) with ${tex`X = [0, 1]`} and ${tex`A = \\{\\hat{p}\\}`}), and... technically it can be \"plotted\" in a sense as follows (here we assume that the specific value ${tex`\\hat{p}`} that Ana is fully confident about is ${tex`\\hat{p} = 0.5`}:\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":559,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":589,"value":"$31","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":613,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":555,"value":"**(2) Complete *un*certainty about the value of ${tex`p = \\hat{p}`}**\n\nIn this case, we can model Ana's prior as a **uniform distribution** over all possible values of ${tex`p`}, meaning that we can model her uncertainty by defining a random variable ${tex`P \\sim \\mathscr{U}[0,1]`}, such that the realized value of ${tex`p`} is the proportion of others in the village who decide to plant early.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":845,"value":"unifPriorPlotBuilder()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":709,"value":"\nWith the prior defined in this way, we can compute Ana's overall expected utility if she plants **Early** the same way we would compute the expected value of any function of a random variable (via the definition of expected value):\n\n${tex.block`\n\\mathbb{E}_P[u_A(\\textsf{Early}, p)] = \\int_{0}^{1}${uHigh}p~\\mathrm{d}p = \\left[\\frac{${uHigh}p^2}{2}\\right]_{0}^{1} = ${(1/2)*uHigh}\n`}\n\nand similarly for her overall expected utility if she plants **Late**:\n\n${tex.block`\n\\mathbb{E}_P[u_A(\\textsf{Late}, p)] = \\int_{0}^{1}(${uLow} + (${uMid - uLow})p)~\\mathrm{d}p = \\left[${uLow}p + ${uMid-uLow}\\cdot \\frac{p^2}{2} \\right]_{0}^{1} = ${uLow} + \\frac{${uMid-uLow}}{2} = ${uLow + (1/2)*(uMid-uLow)}\n`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":696,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":828,"value":"unifPriorPlotBuilder()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":826,"value":"$34","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":867,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":863,"value":"We can now see the beauty of this Beta-updating process! A Random Variable with ${tex`\\text{Beta}(2,2)`} as its pdf has greatest density when ${tex`v = 0.5`} and this density decreases symmetrically as we consider values further above and further below 0.5.\n\nAnd, if Ana was to continue going around the village collecting information about the planting behaviors of neighbors, this distribution could be updated easily with each new piece of information using the above updating rule. You can use the sliders above the final plot below to see this in action! For example, if more and more neighbors started to tell Ana that they plan to plant **Early**, you'll see the distribution update accordingly, to place more and more density on the left side of the ${tex`[0,1]`} interval:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":896,"value":"betaABPanel = Inputs.form({\n r1: abSliders,\n r2: betaABPlot,\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":749,"value":"## Footnotes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":751,"value":"$36","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":916,"value":"## Global Values / Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":894,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":902,"value":"viewof betaSlider = genSlider(tex`\\Large\\beta`, [0, 15], 1, 2)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":920,"value":"globs = ({\n defaultAlpha: 2,\n defaultBeta: 2,\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":898,"value":"viewof alphaSlider = genSlider(tex`\\Large\\alpha`, [0, 15], 1, 2)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":912,"value":"resetBetaParams = () => {\n set(viewof alphaSlider, globs.defaultAlpha)\n set(viewof betaSlider, globs.defaultBeta)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":910,"value":"viewof abResetButton = Inputs.button(\"Reset\", {reduce: resetBetaParams})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":701,"value":"## Computation Cells","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":905,"value":"abSliders = inputsGroup(\n [[viewof betaSlider, viewof alphaSlider, viewof abResetButton]],\n [['beta-slider','alpha-slider','ab-reset-button']],\n 50,\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":927,"value":"betaABData = xVals.map(x => ({x: x, density: stdlib.base.dists.beta.pdf(x, alphaSlider, betaSlider)}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":869,"value":"beta22Data = xVals.map(x => ({x: x, density: 6*x*(1-x)}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":875,"value":"beta22Sum = beta22Data.map(x => x.density).reduce((a, c) => a + c, 0); ","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":711,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":717,"value":"unifData = xVals.map(x => ({x: x, density: 1}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":563,"value":"diracData = xVals.map(x => ({x: x, density: x == 0.5 ? 1 : 0}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":141,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":411,"value":"xVals = d3.range(0, 101).map(x => x/100)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":493,"value":"xUtils = xVals.map(x => ({x: x/100, ulx: lUtil(x/100), uex: eUtil(x/100)}))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"utilsAtIndex = (i) => {\n const xAtIndex = xVals[i]\n const eAtIndex = eUtil(xAtIndex)\n const lAtIndex = lUtil(xAtIndex)\n return `${i}: ${eAtIndex}, ${lAtIndex}`\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":397,"value":"eUtil = (x) => {\n return 0 + uHigh*x\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":432,"value":"lUtil = (x) => {\n return uLow + (uMid - uLow)*x\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":392,"value":"utilPoints = [\n {x: 0, y: 0, strategy: 'Early', label: 'EL'},\n {x: 0, y: uLow, strategy: 'Late', label: 'LL'},\n {x: 1, y: uMid, strategy: 'Late', label: 'LE'},\n {x: 1, y: uHigh, strategy: 'Early', label: 'EE'},\n]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":201,"value":"intersection = ({\n x: uLow / (uHigh - uMid + uLow),\n y: (uHigh * uLow) / (uHigh - uMid + uLow),\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"viewof gameTable = html`\n\n\n\n\n\n\n\n\n \n \n \n\n\n \n \n \n \n\n\n \n \n \n \n\n\n \n \n \n\n\n

		Bina
		Early	Late
Ana	Early	${uHigh >= uMid ? \"\" + uHigh + \"\" : uHigh}, ${uHigh >= uMid ? \"\" + uHigh + \"\" : uHigh}	0, ${uMid >= uHigh ? \"\" + uMid + \"\" : uMid}
Late	${uMid >= uHigh ? \"\" + uMid + \"\" : uMid}, 0	${uLow > 0 ? \"\" + uLow + \"\" : uLow}, ${uLow > 0 ? \"\" + uLow + \"\" : uLow}

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":106,"value":"viewof paramForm = Inputs.form({\n r1: viewof uHigh,\n r2: viewof uMid,\n r3: viewof uLow,\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":156,"value":"viewof gameTableForm = Inputs.form({\n gt1: viewof gameTable,\n gt2: viewof paramForm,\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":83,"value":"viewof uHigh = genSlider(tex`\\Large\\overline{u}`, [0,4], defaults.stepSize, defaults.uHigh)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":102,"value":"viewof uMid = genSlider(tex`\\sout{\\Large u}`, [0,4], defaults.stepSize, defaults.uMid)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"viewof uLow = genSlider(tex`\\Large\\underline{u}`, [0,4], defaults.stepSize, defaults.uLow)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":51,"value":"defaults = ({\n uHigh: 4,\n uLow: 2,\n uMid: 3,\n stepSize: 0.5,\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":133,"value":"## Descriptions from the Book","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":136,"value":"> *Crop yields would be higher if they all were to plant earlier in the year. But if a single farmer were to plant early, the seeds\nwould be taken by the birds that would flock to his plot. Suppose there are just two farmers who interact noncooperatively for a single period with the payoffs in table 1.5. I'll assume that planting late gives a higher return if the other farmer planted early than if both plant late.* (42)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":70,"value":"## JS Pieces","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":72,"value":"// https://observablehq.com/@kin-fung-negawatt/inputs-in-grid\nfunction inputsGroup(views,names, minWidth = 300){\n const form = html`

${\n views.map(row => html`

${\n row.map(input => html`

${input}

`)\n }

`;\n \n form.oninput = () => {\n form.value = views.map(row => row.map(input => input.value))\n if(names){\n names.forEach((row,i)=>row.forEach((c,j)=> form.value[i][j] && (form.value[c]=form.value[i][j])))\n }\n };\n form.oninput();\n return form;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":93,"value":"function genSlider(varLabel, varRange, varStep, defaultVal) {\n var sliderElt = Inputs.range(varRange, {\n value: defaultVal,\n label: varLabel,\n width: 20,\n step: varStep,\n format: x => x.toFixed(1)\n });\n var styleStr = `--label-width: 35px`;\n Object.assign(sliderElt, {style: styleStr});\n return(sliderElt);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":207,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":399,"value":"import { hairK } from '@observablehq/plot-continuous-crosshair'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":602,"value":"// https://observablehq.com/@davidrleonard/callout\nfunction calloutOpen(strings, ...values) {\n const markdown = md`${String.raw({ raw: strings }, ...values)}`;\n const titleEl = markdown.querySelector('h1');\n const titleText = titleEl.textContent;\n titleEl.parentElement.removeChild(titleEl);\n console.log(titleText);\n return htl.html`\n

${titleText}

\n ${markdown}\n

\n `;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":914,"value":"function set(input, value) {\n input.value = value;\n input.dispatchEvent(new Event(\"input\"));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":960,"value":"stdlib = require( 'https://unpkg.com/@stdlib/stdlib@0.0.90' )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":964,"value":"stdlib.base.dists.beta.pdf(0.5, 1, 1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":929,"value":"// stdLib = import('https://cdn.skypack.dev/@stdlib/stats-base-dists-beta-pdf@0.3.1?min').then(x => x.default)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3b","stdlib":"1","titling":"auto"}} 2e:"$2d:body"