30:T518,From [The Law of Oath](https://cdn.shopify.com/s/files/1/0106/0162/7706/files/Oath_Law_of_Oath_web_Oct_20_2020.pdf?v=1603222270): > __5.5.4 Step 4: Roll Defense.__ The defender rolls their defense pool, then calculates their defense by adding the... - __Shields they rolled.__ - __Warbands at targeted sites.__ If you’re attacking the bandits, add the bandits at targeted sites instead. - __Warbands on their board__ if their pawn is at your site or any site you targeted. > > Each ×2 roll doubles the total number of shields rolled. _(Multiple ×2 rolls stack exponentially—×4, ×8, etc.)_ > __5.5.5 Step 5: Roll Attack.__ The attacker rolls their attack pool. For each skull rolled, the attacker immediately kills one warband in their force (generally on their board). > Then, the attacker calculates their __ATTACK__ by adding the... - __Swords they rolled.__ - __Warbands from their force that they sacrifice,__ adding one per warband. _(If they choose to do so, they must sacrifice exactly as many as needed to be victorious [9.5].)_ > > Two hollow swords count as one sword, but a single hollow sword adds nothing. If the attack is higher than the defense, the attacker is __VICTORIOUS__, and the defender is __DEFEATED__. Otherwise, the defender is victorious and the attacker is defeated.31:[{"id":"f7ccc1135216176ca6bb36ff1e20270fad9bdc6fca70dcc4c084e14e59f04bfe9c6cfba262efdd16bb569ff80104fef62b0e8eb14651ab54794b938065bfab6c","href":"https://static.observableusercontent.com/files/f7ccc1135216176ca6bb36ff1e20270fad9bdc6fca70dcc4c084e14e59f04bfe9c6cfba262efdd16bb569ff80104fef62b0e8eb14651ab54794b938065bfab6c","name":"oath dice faces.png","type":"image/png","lastModified":1626105902286.304,"size":110252},{"id":"1ef2e5e8d95de2d31bd1037d633a4686b34a6f7ba7036abf3284c0c096e06b64b4e10d2b06bc163a681205d1c1982d0c50db0791c68b46ca9ecd79de2a4259f8","href":"https://static.observableusercontent.com/files/1ef2e5e8d95de2d31bd1037d633a4686b34a6f7ba7036abf3284c0c096e06b64b4e10d2b06bc163a681205d1c1982d0c50db0791c68b46ca9ecd79de2a4259f8","name":"17-Oath-MeepleinGame-DiceinBattle_1024x1024.jpg.webp","type":"image/webp","lastModified":1626118853980.051,"size":223206},{"id":"f84a458003cfda07f08e4cfd613c2b182c483b8005fd0a1b4be0352c386c6795962305dd9e7210ce5229aa4c3c138d4f33f97198c87763b269d8ff40973ce352","href":"https://static.observableusercontent.com/files/f84a458003cfda07f08e4cfd613c2b182c483b8005fd0a1b4be0352c386c6795962305dd9e7210ce5229aa4c3c138d4f33f97198c87763b269d8ff40973ce352","name":"oath dice.png","type":"image/png","lastModified":1626120310758.782,"size":305775},{"id":"b1480765ef48e81870b3e1ef1aa0d2dc85d177fb3b41c6e6365e25e89d6ad4f34838c4c56f7e9b51f3a822a753c979b183a8951cd5289258bbee1706e0896134","href":"https://static.observableusercontent.com/files/b1480765ef48e81870b3e1ef1aa0d2dc85d177fb3b41c6e6365e25e89d6ad4f34838c4c56f7e9b51f3a822a753c979b183a8951cd5289258bbee1706e0896134","name":"Screen Shot 2021-07-12 at 1.29.36 PM.png","type":"image/png","lastModified":1626121847255.571,"size":104582},{"id":"98d4d8d379b1405ff96d56d6e93aa0471d0f6d8818f733181595ca56305929be981f38651361e6c8d7b77d61ae17c848e42eef52fe7d611deb731b5e2a826fa9","href":"https://static.observableusercontent.com/files/98d4d8d379b1405ff96d56d6e93aa0471d0f6d8818f733181595ca56305929be981f38651361e6c8d7b77d61ae17c848e42eef52fe7d611deb731b5e2a826fa9","name":"zoomed in.png","type":"image/png","lastModified":1626122704986.38,"size":88508}] 2e:{"title":"Oath dice margin odds","cells":[{"id":0,"value":"# Oath dice margin odds\n\n

\n\nThe campaign dice odds in [Oath](https://ledergames.com/collections/full-catalog/products/oath-chronicles-of-empire-exile) are a bit non-obvious.\n\nSee the [Board Game Geek thread](https://boardgamegeek.com/thread/2691538/dice-odds-visualization) for more discussion.\n\n","pinned":false,"hidden":false,"mode":"md"},{"id":604,"value":"Let’s start with a view for 1-6 defense dice, 1-8 attacking warbands and 0-3 defending warbands. Each square shows the __${aggregation} margin__ from ${n} simulations. Positive values mean the attack was victorious. Zero and negative values mean the attack was defended against. ","pinned":false,"hidden":false,"mode":"md"},{"id":593,"value":"Plot.plot({\n marks: [\n Plot.cellX(\n results,\n Plot.groupX(\n {\n fill: aggregation\n },\n {\n x: \"defenseWarbands\",\n fill: \"net\",\n inset: -0.3\n }\n )\n ),\n Plot.text(\n results,\n Plot.groupX(\n {\n text: aggregation\n },\n {\n x: \"defenseWarbands\",\n text: \"net\"\n }\n )\n )\n ],\n x: { domain: d3.range(0, 3), label: \"Defense warbands\" },\n color: { type: \"diverging\", reverse: true, domain: [-20, 20], pivot: 0.5 },\n facet: {\n data: results,\n x: \"defenseDice\",\n y: \"attackWarbands\",\n marginTop: 40\n },\n fy: {\n domain: d3.range(1, 9),\n label: \"← Attack warbands\",\n labelAnchor: \"center\"\n },\n fx: {\n domain: d3.range(1, 7),\n label: \"Defense dice →\",\n axis: \"top\",\n labelOffset: 35\n },\n width,\n marginRight: 0,\n marginBottom: 50\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":650,"value":"A more zoomed out view for much larger numbers: 1-${dimension} attack warbands, 1-${dimension} defense dice, and 0-${lessDimension} defense warbands. Individual numbers have been removed for density and the color contrast increased.","pinned":false,"hidden":false,"mode":"md"},{"id":219,"value":"Plot.plot({\n marks: [\n Plot.cellX(results, Plot.groupX({\n fill: aggregation,\n title: aggregation,\n }, {\n x: 'defenseWarbands',\n fill: 'net',\n title: 'net',\n inset: -.3,\n tip: true\n }))\n ],\n x: { axis: false },\n color: { type: 'diverging', reverse: true, domain: [-15, 15], pivot: .5 },\n facet: { data: results, x: 'defenseDice', y: 'attackWarbands', marginTop: 40 },\n fy: { label: '← Attack warbands', labelAnchor: 'center', domain: d3.range(1, dimension+1) },\n fx: { label: 'Defense dice →', axis: 'top', labelOffset: 35, domain: d3.range(1, dimension+1) },\n width,\n height: width / 1.1,\n marginRight: 0\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":427,"value":"Plot.plot({\n marks: [\n Plot.cellX(\n { length: 31 },\n { x: (d, i) => i - 15, fill: (d, i) => i - 15, inset: -0.5 }\n ),\n Plot.text([-15], { x: (d) => d, text: (d) => d, dx: 10, textAnchor: \"start\" }),\n Plot.text([15], { x: (d) => d, text: (d) => d, dx: -10, textAnchor: \"end\" })\n ],\n color: { type: \"diverging\", reverse: true },\n x: { axis: null, reverse: true },\n height: 20,\n width: 300,\n marginLeft: 36,\n marginRight: 20\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":327,"value":"viewof aggregation = Inputs.select(['median', 'mean'], {label: 'aggregation'})","pinned":false,"hidden":false,"mode":"ojs"},{"id":418,"value":"viewof n = Inputs.range([0, 50], {step: 10, value: 40, label: 'simulations'})","pinned":false,"hidden":false,"mode":"ojs"},{"id":200,"value":"results = Array.from(simulations(n))","pinned":true,"hidden":false,"mode":"ojs"},{"id":699,"value":"dimension = Math.floor(width / 30)","pinned":true,"hidden":false,"mode":"ojs"},{"id":710,"value":"lessDimension = Math.floor(dimension / 3)","pinned":true,"hidden":false,"mode":"ojs"},{"id":169,"value":"function* simulations(n) {\n for (let attackWarbands = 1; attackWarbands <= 30; attackWarbands++)\n for (let defenseDice = 1; defenseDice <= 30; defenseDice++)\n for (let defenseWarbands = 0; defenseWarbands <= 10; defenseWarbands++)\n for (const net of simulate(attackWarbands, defenseDice, defenseWarbands, n))\n yield {attackWarbands, defenseDice, defenseWarbands, net}\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":524,"value":"$30","pinned":false,"hidden":false,"mode":"md"},{"id":87,"value":"function* simulate(attackWarbands, defenseDice, defenseWarbands, n) {\n for (let i = 0; i < n; i++) {\n const defense = Die.defense.total(defenseDice)\n const attack = Die.attack.total(attackWarbands)\n yield attack.value - attack.skulls + attackWarbands + defense.value - defenseWarbands\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":136,"value":"

","pinned":false,"hidden":false,"mode":"html"},{"id":4,"value":"Die = {\n class Die {\n constructor(faces) {\n this.faces = faces\n }\n\n * roll(n = 1) {\n for (let i = 0; i < n; i++)\n yield this.faces[Math.floor(Math.random() * 6)]\n }\n\n total(n = 1) {\n let value = 0, doubles = 0, skulls = 0\n for (const r of this.roll(n)) {\n if (typeof r === 'number')\n value += r\n else {\n value += r.value || 0\n doubles += r.doubles || 0\n skulls += r.skulls || 0\n }\n }\n value *= Math.pow(2, doubles)\n return {value: Math.floor(value), skulls, doubles}\n }\n }\n\n Die.defense = new Die([0, 0, -1, -1, -2, {doubles: 1}])\n Die.attack = new Die([.5, .5, .5, 1, 1, {value: 2, skulls: 1}])\n\n return Die\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":584,"value":"---\nStatic images for sharing\n\n![oath dice.png](${await FileAttachment(\"oath dice.png\").url()})\n\n![zoomed in.png](${await FileAttachment(\"zoomed in.png\").url()})","pinned":false,"hidden":false,"mode":"md"},{"id":589,"value":"Example 13 vs. 13 square\n\n![Screen Shot 2021-07-12 at 1.29.36 PM.png](${await FileAttachment(\"Screen Shot 2021-07-12 at 1.29.36 PM.png\").url()})","pinned":false,"hidden":false,"mode":"md"}],"files":"$W31","stdlib":"1","titling":"auto"}