{"product_id":"expected-utility","title":"Expected Utility Calculator","description":"\u003cstyle\u003e\n.eu-calculator {\n  --ink: #0f172a;\n  --muted: #475569;\n  --border: #e2e8f0;\n  --surface: #ffffff;\n  --tint: #f8fafc;\n  --primary: #1d4ed8;\n  --accent: #c2410c;\n  --accent-hover: #9a3412;\n  --chart-1: #1e40af;\n  --chart-2: #0d9488;\n  --chart-3: #7c3aed;\n  --chart-4: #be185d;\n  --chart-5: #334155;\n  width: 100%;\n  max-width: 1200px;\n  margin: 0 auto;\n  color: var(--ink);\n  background: var(--surface);\n  border: 1px solid var(--border);\n  border-radius: 8px;\n  box-shadow: 0 1px 2px rgba(15, 23, 42, .06);\n  font-family: -apple-system, BlinkMacSystemFont, \"Segoe UI\", Roboto, Helvetica, Arial, sans-serif;\n  font-size: 15px;\n  line-height: 1.55;\n  overflow-wrap: anywhere;\n  container-type: inline-size;\n  container-name: eu-calc;\n}\n.eu-calculator,\n.eu-calculator *,\n.eu-calculator *::before,\n.eu-calculator *::after {\n  box-sizing: border-box;\n}\n.eu-calculator * {\n  min-width: 0;\n}\n.eu-calculator .eu-header {\n  padding: 24px 24px 16px;\n  border-bottom: 1px solid var(--border);\n  background: linear-gradient(180deg, #ffffff 0%, #f8fafc 100%);\n}\n.eu-calculator .eu-header h2 {\n  margin: 0;\n  font-size: 24px;\n  line-height: 1.25;\n  font-weight: 700;\n  letter-spacing: -.02em;\n}\n.eu-calculator .eu-subtitle {\n  margin: 8px 0 0;\n  color: var(--muted);\n  max-width: 760px;\n}\n.eu-calculator .eu-pills {\n  display: flex;\n  flex-wrap: wrap;\n  gap: 8px;\n  margin-top: 16px;\n}\n.eu-calculator .eu-pill {\n  display: inline-flex;\n  align-items: baseline;\n  gap: 6px;\n  padding: 6px 10px;\n  border: 1px solid var(--border);\n  border-radius: 999px;\n  background: var(--surface);\n  color: var(--muted);\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-pill strong {\n  color: var(--ink);\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator .eu-toolbar {\n  display: flex;\n  flex-wrap: wrap;\n  align-items: center;\n  gap: 12px;\n  padding: 16px 24px;\n  border-bottom: 1px solid var(--border);\n  background: var(--surface);\n}\n.eu-calculator .eu-button {\n  min-height: 44px;\n  border-radius: 6px;\n  padding: 11px 18px;\n  border: 1px solid var(--border);\n  background: var(--surface);\n  color: var(--ink);\n  font: inherit;\n  font-weight: 650;\n  cursor: pointer;\n  transition: background-color .15s ease, border-color .15s ease, box-shadow .15s ease, transform .15s ease;\n}\n.eu-calculator .eu-button:hover {\n  border-color: #cbd5e1;\n  background: var(--tint);\n  box-shadow: 0 2px 5px rgba(15, 23, 42, .10);\n}\n.eu-calculator .eu-button:active {\n  transform: translateY(1px);\n}\n.eu-calculator .eu-button:focus-visible,\n.eu-calculator input:focus-visible,\n.eu-calculator select:focus-visible,\n.eu-calculator summary:focus-visible,\n.eu-calculator a:focus-visible {\n  outline: 3px solid rgba(29, 78, 216, .35);\n  outline-offset: 2px;\n}\n.eu-calculator .eu-download {\n  display: inline-flex;\n  align-items: center;\n  gap: 10px;\n  white-space: nowrap;\n  border-color: var(--accent);\n  background: var(--accent);\n  color: #ffffff;\n}\n.eu-calculator .eu-download:hover {\n  border-color: var(--accent-hover);\n  background: var(--accent-hover);\n  color: #ffffff;\n}\n.eu-calculator .eu-download svg {\n  width: 18px;\n  height: 18px;\n  flex: 0 0 auto;\n}\n.eu-calculator .eu-workspace {\n  display: grid;\n  grid-template-columns: minmax(0, 1fr);\n  gap: 24px;\n  padding: 24px;\n  background: var(--tint);\n}\n.eu-calculator .eu-panel,\n.eu-calculator .eu-section-card {\n  min-width: 0;\n  background: var(--surface);\n  border: 1px solid var(--border);\n  border-radius: 8px;\n  box-shadow: 0 1px 2px rgba(15, 23, 42, .06);\n}\n.eu-calculator .eu-panel-header,\n.eu-calculator .eu-section-head {\n  padding: 16px 16px 12px;\n  border-bottom: 1px solid var(--border);\n}\n.eu-calculator .eu-panel-header h3,\n.eu-calculator .eu-section-head h3,\n.eu-calculator .eu-education h2 {\n  margin: 0;\n  font-size: 18px;\n  line-height: 1.35;\n  font-weight: 650;\n}\n.eu-calculator .eu-panel-header p,\n.eu-calculator .eu-section-head p {\n  margin: 4px 0 0;\n  color: var(--muted);\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-inputs-body,\n.eu-calculator .eu-results-body {\n  padding: 16px;\n}\n.eu-calculator .eu-event-grid {\n  display: grid;\n  grid-template-columns: repeat(2, minmax(0, 1fr));\n  gap: 16px;\n}\n.eu-calculator .eu-event {\n  margin: 0;\n  padding: 16px;\n  border: 1px solid var(--border);\n  border-radius: 8px;\n  background: var(--tint);\n}\n.eu-calculator .eu-event legend {\n  padding: 0 6px;\n  font-size: 15px;\n  font-weight: 700;\n}\n.eu-calculator .eu-field {\n  display: grid;\n  grid-template-rows: auto auto minmax(36px, auto);\n  gap: 6px;\n  margin-top: 12px;\n}\n.eu-calculator .eu-field:first-of-type {\n  margin-top: 4px;\n}\n.eu-calculator .eu-field label {\n  font-size: 14px;\n  font-weight: 600;\n}\n.eu-calculator .eu-control-wrap {\n  position: relative;\n}\n.eu-calculator .eu-control-wrap input {\n  width: 100%;\n  min-height: 44px;\n  padding: 10px 44px 10px 12px;\n  border: 1px solid #cbd5e1;\n  border-radius: 6px;\n  background: var(--surface);\n  color: var(--ink);\n  font: inherit;\n  font-size: 15px;\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator .eu-control-wrap input[aria-invalid=\"true\"] {\n  border-color: #b91c1c;\n}\n.eu-calculator .eu-unit {\n  position: absolute;\n  right: 12px;\n  top: 50%;\n  transform: translateY(-50%);\n  color: var(--muted);\n  font-size: 13px;\n  font-weight: 650;\n  pointer-events: none;\n}\n.eu-calculator .eu-help,\n.eu-calculator .eu-error {\n  margin: 0;\n  font-size: 13px;\n  font-weight: 500;\n  line-height: 1.4;\n}\n.eu-calculator .eu-help {\n  color: var(--muted);\n}\n.eu-calculator .eu-error {\n  color: #991b1b;\n}\n.eu-calculator .eu-formula {\n  margin-top: 16px;\n  padding: 12px;\n  border-left: 4px solid var(--primary);\n  border-radius: 0 6px 6px 0;\n  background: #eff6ff;\n  color: #1e3a8a;\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-result-primary {\n  padding: 16px;\n  border: 1px solid #bfdbfe;\n  border-radius: 8px;\n  background: #eff6ff;\n}\n.eu-calculator .eu-result-primary span {\n  display: block;\n  color: #1e3a8a;\n  font-size: 13px;\n  font-weight: 650;\n}\n.eu-calculator .eu-result-primary strong {\n  display: block;\n  margin-top: 4px;\n  color: #172554;\n  font-size: 30px;\n  line-height: 1.2;\n  font-weight: 700;\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator .eu-result-primary p {\n  margin: 8px 0 0;\n  color: #1e3a8a;\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-metrics {\n  display: grid;\n  grid-template-columns: repeat(2, minmax(0, 1fr));\n  gap: 12px;\n  margin-top: 12px;\n}\n.eu-calculator .eu-metric {\n  padding: 12px;\n  border: 1px solid var(--border);\n  border-radius: 8px;\n  background: var(--surface);\n}\n.eu-calculator .eu-metric span {\n  display: block;\n  color: var(--muted);\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-metric strong {\n  display: block;\n  margin-top: 4px;\n  font-size: 20px;\n  line-height: 1.25;\n  font-weight: 700;\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator .eu-status {\n  margin-top: 12px;\n  padding: 10px 12px;\n  border: 1px solid var(--border);\n  border-radius: 6px;\n  background: var(--tint);\n  color: var(--muted);\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-status.eu-status-warn {\n  border-color: #fed7aa;\n  background: #fff7ed;\n  color: #9a3412;\n}\n.eu-calculator .eu-live {\n  position: absolute;\n  width: 1px;\n  height: 1px;\n  padding: 0;\n  margin: -1px;\n  overflow: hidden;\n  clip: rect(0, 0, 0, 0);\n  white-space: nowrap;\n  border: 0;\n}\n.eu-calculator .eu-content {\n  display: grid;\n  gap: 24px;\n  padding: 0 24px 24px;\n  background: var(--tint);\n}\n.eu-calculator .eu-section-body {\n  padding: 16px;\n}\n.eu-calculator .eu-chart-card {\n  min-width: 0;\n}\n.eu-calculator .eu-chart-cluster {\n  display: grid;\n  grid-template-columns: minmax(0, 1fr);\n  align-items: center;\n  justify-content: center;\n  gap: 24px;\n  max-width: 900px;\n  margin: 0 auto;\n}\n.eu-calculator .eu-plot {\n  display: grid;\n  place-items: center;\n  min-width: 0;\n}\n.eu-calculator .eu-chart-svg {\n  display: block;\n  width: 100%;\n  max-width: 580px;\n  height: auto;\n  min-height: 150px;\n}\n.eu-calculator .eu-chart-svg text {\n  font-family: -apple-system, BlinkMacSystemFont, \"Segoe UI\", Roboto, Helvetica, Arial, sans-serif;\n  fill: var(--ink);\n  font-size: 13px;\n  font-weight: 600;\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator .eu-legend {\n  display: grid;\n  gap: 10px;\n  align-content: center;\n  justify-content: start;\n}\n.eu-calculator .eu-legend-row {\n  display: grid;\n  grid-template-columns: 14px minmax(max-content, auto) max-content max-content;\n  align-items: center;\n  justify-content: start;\n  gap: 8px 12px;\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-swatch {\n  width: 12px;\n  height: 12px;\n  border-radius: 3px;\n}\n.eu-calculator .eu-legend-name {\n  color: var(--ink);\n  font-weight: 650;\n}\n.eu-calculator .eu-legend-value,\n.eu-calculator .eu-legend-percent {\n  color: var(--muted);\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator .eu-chart-caption,\n.eu-calculator .eu-table-note {\n  margin-top: 16px;\n  padding: 10px 12px;\n  border: 1px solid var(--border);\n  border-radius: 6px;\n  background: var(--tint);\n  color: var(--muted);\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-empty {\n  padding: 16px;\n  border: 1px dashed #cbd5e1;\n  border-radius: 6px;\n  background: var(--tint);\n  color: var(--muted);\n  text-align: center;\n  font-size: 13px;\n  font-weight: 500;\n}\n.eu-calculator .eu-chart-card.eu-safe-chart-stack .eu-chart-cluster {\n  grid-template-columns: minmax(0, 1fr);\n  row-gap: 24px;\n}\n.eu-calculator .eu-chart-card.eu-safe-chart-stack .eu-legend {\n  justify-content: center;\n}\n.eu-calculator .eu-chart-card.eu-safe-chart-stack .eu-chart-caption {\n  margin-top: 20px;\n}\n.eu-calculator .eu-table-overflow {\n  width: 100%;\n  overflow-x: auto;\n  overflow-y: visible;\n  -webkit-overflow-scrolling: touch;\n  border: 1px solid var(--border);\n  border-radius: 6px;\n}\n.eu-calculator table {\n  width: 100%;\n  min-width: 760px;\n  border-collapse: collapse;\n  background: var(--surface);\n  font-size: 13px;\n}\n.eu-calculator th,\n.eu-calculator td {\n  padding: 10px 12px;\n  border-bottom: 1px solid var(--border);\n  text-align: left;\n  vertical-align: middle;\n}\n.eu-calculator th {\n  background: #172554;\n  color: #ffffff;\n  font-weight: 700;\n  white-space: nowrap;\n}\n.eu-calculator td.eu-num,\n.eu-calculator th.eu-num {\n  text-align: right;\n  font-variant-numeric: tabular-nums;\n}\n.eu-calculator tbody tr:hover {\n  background: #f8fafc;\n}\n.eu-calculator tfoot td {\n  border-bottom: 0;\n  background: #eff6ff;\n  font-weight: 700;\n}\n.eu-calculator .eu-table-card.eu-safe-table-stack .eu-table-note {\n  margin-top: 20px;\n}\n.eu-calculator .eu-education {\n  padding: 24px;\n  background: var(--surface);\n  border-top: 1px solid var(--border);\n}\n.eu-calculator .eu-education-inner {\n  max-width: 900px;\n  margin: 0 auto;\n}\n.eu-calculator .eu-education h2 {\n  margin-top: 28px;\n}\n.eu-calculator .eu-education h2:first-child {\n  margin-top: 0;\n}\n.eu-calculator .eu-education h3 {\n  margin: 20px 0 6px;\n  font-size: 16px;\n  line-height: 1.4;\n  font-weight: 700;\n}\n.eu-calculator .eu-education p,\n.eu-calculator .eu-education ul {\n  margin: 8px 0 0;\n  color: #334155;\n}\n.eu-calculator .eu-education ul {\n  padding-left: 22px;\n}\n.eu-calculator .eu-education li + li {\n  margin-top: 6px;\n}\n.eu-calculator .eu-education a {\n  color: var(--primary);\n  font-weight: 600;\n  text-underline-offset: 2px;\n}\n.eu-calculator .eu-education a:hover {\n  color: #1e40af;\n}\n@container eu-calc (min-width: 640px) {\n  .eu-calculator .eu-chart-cluster {\n    grid-template-columns: minmax(320px, 1.25fr) minmax(240px, .75fr);\n  }\n}\n@container eu-calc (min-width: 900px) {\n  .eu-calculator .eu-workspace {\n    grid-template-columns: minmax(0, .95fr) minmax(0, 1.05fr);\n  }\n}\n@container eu-calc (max-width: 639px) {\n  .eu-calculator .eu-event-grid,\n  .eu-calculator .eu-metrics {\n    grid-template-columns: minmax(0, 1fr);\n  }\n  .eu-calculator .eu-legend {\n    justify-content: center;\n  }\n  .eu-calculator .eu-legend-row {\n    grid-template-columns: 14px minmax(0, auto) max-content;\n  }\n  .eu-calculator .eu-legend-percent {\n    grid-column: 2 \/ 4;\n    padding-left: 0;\n  }\n}\n@media (max-width: 639px) {\n  .eu-calculator .eu-header,\n  .eu-calculator .eu-toolbar,\n  .eu-calculator .eu-workspace,\n  .eu-calculator .eu-education {\n    padding-left: 16px;\n    padding-right: 16px;\n  }\n  .eu-calculator .eu-content {\n    padding-left: 16px;\n    padding-right: 16px;\n  }\n  .eu-calculator .eu-button {\n    width: 100%;\n    justify-content: center;\n  }\n  .eu-calculator .eu-chart-caption,\n  .eu-calculator .eu-table-note {\n    margin-top: 16px;\n  }\n}\n\u003c\/style\u003e\n\u003cdiv class=\"eu-calculator\" data-calculator-root\u003e\n  \u003csection class=\"eu-header\"\u003e\n    \u003ch2\u003eExpected Utility Calculator\u003c\/h2\u003e\n    \u003cp class=\"eu-subtitle\"\u003eCompare two uncertain monetary outcomes using a square-root utility function, then see the expected payoff, certainty equivalent, and risk premium in one live view.\u003c\/p\u003e\n    \u003cdiv class=\"eu-pills\" aria-label=\"Live calculation summary\"\u003e\n      \u003cspan class=\"eu-pill\"\u003eProbability total \u003cstrong data-eu-pill-probability\u003e100.00%\u003c\/strong\u003e\u003c\/span\u003e\n      \u003cspan class=\"eu-pill\"\u003eExpected value \u003cstrong data-eu-pill-ev\u003e$16,000.00\u003c\/strong\u003e\u003c\/span\u003e\n      \u003cspan class=\"eu-pill\"\u003eExpected utility \u003cstrong data-eu-pill-utility\u003e124.85\u003c\/strong\u003e\u003c\/span\u003e\n      \u003cspan class=\"eu-pill\"\u003eCertainty equivalent \u003cstrong data-eu-pill-ce\u003e$15,588.23\u003c\/strong\u003e\u003c\/span\u003e\n    \u003c\/div\u003e\n  \u003c\/section\u003e\n\n  \u003cdiv class=\"eu-toolbar\" aria-label=\"Calculator actions\"\u003e\n    \u003cbutton class=\"eu-button eu-download\" type=\"button\" data-eu-download\u003e\n      \u003csvg viewbox=\"0 0 24 24\" aria-hidden=\"true\"\u003e\u003cpath fill=\"currentColor\" d=\"M12 3a1 1 0 0 1 1 1v8.59l2.3-2.29a1 1 0 1 1 1.4 1.42l-4 3.96a1 1 0 0 1-1.4 0l-4-3.96a1 1 0 1 1 1.4-1.42l2.3 2.29V4a1 1 0 0 1 1-1Zm-7 13a1 1 0 0 1 1 1v2h12v-2a1 1 0 1 1 2 0v3a1 1 0 0 1-1 1H5a1 1 0 0 1-1-1v-3a1 1 0 0 1 1-1Z\"\u003e\u003c\/path\u003e\u003c\/svg\u003e\n      \u003cspan\u003eDownload Excel\u003c\/span\u003e\n    \u003c\/button\u003e\n    \u003cbutton class=\"eu-button\" type=\"button\" data-eu-reset\u003eReset\u003c\/button\u003e\n  \u003c\/div\u003e\n\n  \u003cdiv class=\"eu-workspace\"\u003e\n    \u003csection class=\"eu-panel eu-input-panel\" aria-labelledby=\"eu-input-heading\"\u003e\n      \u003cdiv class=\"eu-panel-header\"\u003e\n        \u003ch3 id=\"eu-input-heading\"\u003eOutcome assumptions\u003c\/h3\u003e\n        \u003cp\u003eEnter each outcome's probability and nonnegative monetary value.\u003c\/p\u003e\n      \u003c\/div\u003e\n      \u003cdiv class=\"eu-inputs-body\"\u003e\n        \u003cdiv class=\"eu-event-grid\"\u003e\n          \u003cfieldset class=\"eu-event\"\u003e\n            \u003clegend\u003eEvent 1\u003c\/legend\u003e\n            \u003cdiv class=\"eu-field\"\u003e\n              \u003clabel for=\"eu-probability-1\"\u003eProbability\u003c\/label\u003e\n              \u003cdiv class=\"eu-control-wrap\"\u003e\n                \u003cinput id=\"eu-probability-1\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\" value=\"40\" aria-describedby=\"eu-probability-1-help eu-probability-1-error\" data-eu-input data-eu-kind=\"percent\" data-eu-field=\"probability1\"\u003e\n                \u003cspan class=\"eu-unit\"\u003e%\u003c\/span\u003e\n              \u003c\/div\u003e\n              \u003cp class=\"eu-help\" id=\"eu-probability-1-help\"\u003eA value from 0% to 100%.\u003c\/p\u003e\n              \u003cp class=\"eu-error\" id=\"eu-probability-1-error\" data-eu-error=\"probability1\"\u003e\u003c\/p\u003e\n            \u003c\/div\u003e\n            \u003cdiv class=\"eu-field\"\u003e\n              \u003clabel for=\"eu-value-1\"\u003eMonetary value\u003c\/label\u003e\n              \u003cdiv class=\"eu-control-wrap\"\u003e\n                \u003cinput id=\"eu-value-1\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\" value=\"10000\" aria-describedby=\"eu-value-1-help eu-value-1-error\" data-eu-input data-eu-kind=\"currency\" data-eu-field=\"value1\"\u003e\n                \u003cspan class=\"eu-unit\"\u003eUSD\u003c\/span\u003e\n              \u003c\/div\u003e\n              \u003cp class=\"eu-help\" id=\"eu-value-1-help\"\u003eThe payoff if Event 1 occurs.\u003c\/p\u003e\n              \u003cp class=\"eu-error\" id=\"eu-value-1-error\" data-eu-error=\"value1\"\u003e\u003c\/p\u003e\n            \u003c\/div\u003e\n          \u003c\/fieldset\u003e\n\n          \u003cfieldset class=\"eu-event\"\u003e\n            \u003clegend\u003eEvent 2\u003c\/legend\u003e\n            \u003cdiv class=\"eu-field\"\u003e\n              \u003clabel for=\"eu-probability-2\"\u003eProbability\u003c\/label\u003e\n              \u003cdiv class=\"eu-control-wrap\"\u003e\n                \u003cinput id=\"eu-probability-2\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\" value=\"60\" aria-describedby=\"eu-probability-2-help eu-probability-2-error\" data-eu-input data-eu-kind=\"percent\" data-eu-field=\"probability2\"\u003e\n                \u003cspan class=\"eu-unit\"\u003e%\u003c\/span\u003e\n              \u003c\/div\u003e\n              \u003cp class=\"eu-help\" id=\"eu-probability-2-help\"\u003eA value from 0% to 100%.\u003c\/p\u003e\n              \u003cp class=\"eu-error\" id=\"eu-probability-2-error\" data-eu-error=\"probability2\"\u003e\u003c\/p\u003e\n            \u003c\/div\u003e\n            \u003cdiv class=\"eu-field\"\u003e\n              \u003clabel for=\"eu-value-2\"\u003eMonetary value\u003c\/label\u003e\n              \u003cdiv class=\"eu-control-wrap\"\u003e\n                \u003cinput id=\"eu-value-2\" type=\"text\" inputmode=\"decimal\" autocomplete=\"off\" value=\"20000\" aria-describedby=\"eu-value-2-help eu-value-2-error\" data-eu-input data-eu-kind=\"currency\" data-eu-field=\"value2\"\u003e\n                \u003cspan class=\"eu-unit\"\u003eUSD\u003c\/span\u003e\n              \u003c\/div\u003e\n              \u003cp class=\"eu-help\" id=\"eu-value-2-help\"\u003eThe payoff if Event 2 occurs.\u003c\/p\u003e\n              \u003cp class=\"eu-error\" id=\"eu-value-2-error\" data-eu-error=\"value2\"\u003e\u003c\/p\u003e\n            \u003c\/div\u003e\n          \u003c\/fieldset\u003e\n        \u003c\/div\u003e\n        \u003cdiv class=\"eu-formula\"\u003eModel: expected utility = probability₁ × √value₁ + probability₂ × √value₂. Probabilities are converted from percentages to decimals.\u003c\/div\u003e\n      \u003c\/div\u003e\n    \u003c\/section\u003e\n\n    \u003csection class=\"eu-panel eu-results-panel\" aria-labelledby=\"eu-results-heading\"\u003e\n      \u003cdiv class=\"eu-panel-header\"\u003e\n        \u003ch3 id=\"eu-results-heading\"\u003eLive results\u003c\/h3\u003e\n        \u003cp\u003eAll outputs update as you edit the assumptions.\u003c\/p\u003e\n      \u003c\/div\u003e\n      \u003cdiv class=\"eu-results-body\"\u003e\n        \u003cdiv class=\"eu-result-primary\"\u003e\n          \u003cspan\u003eExpected utility\u003c\/span\u003e\n          \u003cstrong data-eu-primary\u003e124.85\u003c\/strong\u003e\n          \u003cp data-eu-primary-note\u003eThe probability-weighted utility of the two outcomes.\u003c\/p\u003e\n        \u003c\/div\u003e\n        \u003cdiv class=\"eu-metrics\"\u003e\n          \u003cdiv class=\"eu-metric\"\u003e\n            \u003cspan\u003eExpected monetary value\u003c\/span\u003e\n            \u003cstrong data-eu-ev\u003e$16,000.00\u003c\/strong\u003e\n          \u003c\/div\u003e\n          \u003cdiv class=\"eu-metric\"\u003e\n            \u003cspan\u003eCertainty equivalent\u003c\/span\u003e\n            \u003cstrong data-eu-ce\u003e$15,588.23\u003c\/strong\u003e\n          \u003c\/div\u003e\n          \u003cdiv class=\"eu-metric\"\u003e\n            \u003cspan\u003eRisk premium\u003c\/span\u003e\n            \u003cstrong data-eu-risk\u003e$411.77\u003c\/strong\u003e\n          \u003c\/div\u003e\n          \u003cdiv class=\"eu-metric\"\u003e\n            \u003cspan\u003eProbability total\u003c\/span\u003e\n            \u003cstrong data-eu-probability\u003e100.00%\u003c\/strong\u003e\n          \u003c\/div\u003e\n        \u003c\/div\u003e\n        \u003cdiv class=\"eu-status\" data-eu-status\u003eProbabilities total 100%, so the outcomes form a complete two-event scenario.\u003c\/div\u003e\n        \u003cdiv class=\"eu-live\" aria-live=\"polite\" aria-atomic=\"true\" data-eu-live\u003eExpected utility 124.85. Expected monetary value $16,000.00.\u003c\/div\u003e\n      \u003c\/div\u003e\n    \u003c\/section\u003e\n  \u003c\/div\u003e\n\n  \u003cdiv class=\"eu-content\"\u003e\n    \u003csection class=\"eu-section-card eu-chart-card\" aria-labelledby=\"eu-breakdown-heading\" data-eu-chart-card\u003e\n      \u003cdiv class=\"eu-section-head\"\u003e\n        \u003ch3 id=\"eu-breakdown-heading\"\u003eWeighted utility contribution\u003c\/h3\u003e\n        \u003cp data-eu-chart-interpretation\u003eEvent 2 supplies the larger share of expected utility.\u003c\/p\u003e\n      \u003c\/div\u003e\n      \u003cdiv class=\"eu-section-body\"\u003e\n        \u003cdiv class=\"eu-chart-cluster\" data-eu-chart-cluster\u003e\n          \u003cdiv class=\"eu-plot\" data-eu-plot\u003e\n            \u003csvg class=\"eu-chart-svg\" viewbox=\"0 0 580 170\" role=\"img\" aria-labelledby=\"eu-chart-title eu-chart-desc\" data-eu-chart-svg\u003e\n              \u003ctitle id=\"eu-chart-title\"\u003eWeighted utility contribution by event\u003c\/title\u003e\n              \u003cdesc id=\"eu-chart-desc\" data-eu-chart-desc\u003eEvent 1 contributes 40.00 utility units and Event 2 contributes 84.85 utility units.\u003c\/desc\u003e\n              \u003cg data-eu-chart-marks\u003e\u003c\/g\u003e\n            \u003c\/svg\u003e\n            \u003cdiv class=\"eu-empty\" hidden data-eu-chart-empty\u003eEnter positive probabilities and monetary values to see the utility breakdown.\u003c\/div\u003e\n          \u003c\/div\u003e\n          \u003cdiv class=\"eu-legend\" data-eu-legend aria-label=\"Chart legend\"\u003e\u003c\/div\u003e\n        \u003c\/div\u003e\n        \u003cdiv class=\"eu-chart-caption\" data-eu-chart-caption\u003eThe bar divides total expected utility into the exact contribution made by each event. Contribution depends on both probability and the square-root utility of the payoff.\u003c\/div\u003e\n      \u003c\/div\u003e\n    \u003c\/section\u003e\n\n    \u003csection class=\"eu-section-card eu-table-card\" aria-labelledby=\"eu-table-heading\" data-eu-table-card\u003e\n      \u003cdiv class=\"eu-section-head\"\u003e\n        \u003ch3 id=\"eu-table-heading\"\u003eCalculation detail\u003c\/h3\u003e\n        \u003cp\u003eEach row follows the same data model used by the results, chart, and Excel workbook.\u003c\/p\u003e\n      \u003c\/div\u003e\n      \u003cdiv class=\"eu-section-body\"\u003e\n        \u003cdiv class=\"eu-table-overflow\" data-eu-table-overflow\u003e\n          \u003ctable\u003e\n            \u003cthead\u003e\n              \u003ctr\u003e\n                \u003cth scope=\"col\"\u003eEvent\u003c\/th\u003e\n                \u003cth scope=\"col\" class=\"eu-num\"\u003eProbability\u003c\/th\u003e\n                \u003cth scope=\"col\" class=\"eu-num\"\u003eMonetary value\u003c\/th\u003e\n                \u003cth scope=\"col\" class=\"eu-num\"\u003eUtility √value\u003c\/th\u003e\n                \u003cth scope=\"col\" class=\"eu-num\"\u003eWeighted utility\u003c\/th\u003e\n                \u003cth scope=\"col\" class=\"eu-num\"\u003eUtility share\u003c\/th\u003e\n              \u003c\/tr\u003e\n            \u003c\/thead\u003e\n            \u003ctbody data-eu-table-body\u003e\u003c\/tbody\u003e\n            \u003ctfoot data-eu-table-foot\u003e\u003c\/tfoot\u003e\n          \u003c\/table\u003e\n        \u003c\/div\u003e\n        \u003cdiv class=\"eu-table-note\" data-eu-table-note\u003eThe certainty equivalent is the guaranteed amount with the same square-root utility as the normalized expected utility. The risk premium is expected monetary value minus certainty equivalent.\u003c\/div\u003e\n      \u003c\/div\u003e\n    \u003c\/section\u003e\n  \u003c\/div\u003e\n\n  \u003csection class=\"eu-education\"\u003e\n    \u003cdiv class=\"eu-education-inner\"\u003e\n      \u003ch2\u003eWhat does this expected utility calculator estimate?\u003c\/h2\u003e\n      \u003cp\u003eThis calculator converts two uncertain monetary outcomes into a single expected utility score. It applies a square-root utility function to each monetary value and then weights that utility by the outcome's probability. Square-root utility is concave: doubling money does not double utility. That property represents diminishing marginal utility and a risk-averse preference pattern. The score itself is expressed in utility units rather than dollars, so it is most useful for comparing scenarios that use the same utility function.\u003c\/p\u003e\n      \u003cp\u003eThe page also reports the expected monetary value, a normalized certainty equivalent, and a risk premium. Expected monetary value is the probability-weighted average payoff. The certainty equivalent translates expected utility back into a dollar amount: it is the guaranteed payoff that produces the same utility as the uncertain scenario. The risk premium is the amount by which expected monetary value exceeds that certainty equivalent. These measures help separate the size of a possible payoff from the decision-maker's attitude toward uncertainty.\u003c\/p\u003e\n\n      \u003ch2\u003eHow should each input be used?\u003c\/h2\u003e\n      \u003ch3\u003eEvent 1 probability and Event 2 probability\u003c\/h3\u003e\n      \u003cp\u003eEnter each likelihood as a percentage from 0% to 100%. A higher probability increases that event's contribution to expected utility and expected monetary value. For a complete two-outcome scenario, the probabilities should total 100%. The calculator still shows a raw expected utility when the total differs, but it flags the mismatch and normalizes the dollar comparison metrics by the entered probability total. Common mistakes include typing a decimal such as 0.4 when 40% was intended, assigning overlapping events that can occur together, or omitting a possible outcome.\u003c\/p\u003e\n\n      \u003ch3\u003eEvent 1 monetary value and Event 2 monetary value\u003c\/h3\u003e\n      \u003cp\u003eEnter the nonnegative payoff associated with each event in U.S. dollars. Higher values increase both utility and expected monetary value, but utility rises at a decreasing rate because the model uses the square root. For example, moving from $10,000 to $20,000 adds less utility than moving from $0 to $10,000. Use net outcomes measured on a consistent basis. Do not mix revenue for one event with profit for the other, and do not mix nominal and inflation-adjusted dollars. Negative payoffs are not accepted because the square root of a negative monetary value is not a real number under this model.\u003c\/p\u003e\n\n      \u003ch2\u003eHow are the results calculated?\u003c\/h2\u003e\n      \u003cp\u003eFor each event, the calculator first converts the probability percentage to a decimal. It then computes utility as the square root of the monetary payoff. Weighted utility equals probability multiplied by utility. The expected utility result is the sum of the two weighted utility contributions. With probabilities of 40% and 60% and payoffs of $10,000 and $20,000, the expected utility is approximately 124.85.\u003c\/p\u003e\n      \u003cp\u003eExpected monetary value is the probability-weighted average payoff. When probabilities total 100%, the certainty equivalent equals expected utility squared because the inverse of the square-root utility function is squaring. When probabilities do not total 100%, the calculator first divides expected utility by the entered probability total and then applies the inverse utility function. This normalization keeps the certainty-equivalent comparison interpretable without changing the raw expected utility formula.\u003c\/p\u003e\n\n      \u003ch2\u003eHow should the outputs be interpreted?\u003c\/h2\u003e\n      \u003ch3\u003eExpected utility\u003c\/h3\u003e\n      \u003cp\u003eA larger expected utility is preferable to a smaller one when comparing alternatives under the same square-root utility function. The absolute number has no direct dollar meaning. A zero result means all weighted utilities are zero, usually because probabilities or monetary values are zero. The most important comparison is between complete scenarios built with consistent inputs.\u003c\/p\u003e\n\n      \u003ch3\u003eExpected monetary value\u003c\/h3\u003e\n      \u003cp\u003eThis is the average payoff implied by the probability distribution. It does not incorporate diminishing marginal utility, so it can favor a high-variance alternative even when the certainty equivalent is lower. Expected value is useful as a financial baseline, but it should not be treated as a guaranteed result.\u003c\/p\u003e\n\n      \u003ch3\u003eCertainty equivalent and risk premium\u003c\/h3\u003e\n      \u003cp\u003eThe certainty equivalent converts the utility result back to dollars. Under concave utility it will generally be less than or equal to expected monetary value. The gap is the risk premium: the theoretical amount a person with this utility function would give up to replace the uncertain scenario with a guaranteed payoff. A zero risk premium occurs when outcomes are identical, when only one certain outcome remains, or when the scenario otherwise has no dispersion relevant to the model.\u003c\/p\u003e\n\n      \u003ch3\u003eChart and calculation table\u003c\/h3\u003e\n      \u003cp\u003eThe stacked bar shows how much each event contributes to total expected utility. A large segment can come from a high probability, a high payoff, or both. The detail table exposes the exact probability, payoff, square-root utility, weighted contribution, and share for each event. Use it to verify that an apparently dominant outcome is not merely the result of a data-entry error.\u003c\/p\u003e\n\n      \u003ch2\u003eWhat are the model's benefits and limitations?\u003c\/h2\u003e\n      \u003cp\u003eExpected utility provides a disciplined way to compare uncertain choices while recognizing that money may have diminishing personal value. It is especially useful when alternatives have different payoff distributions. However, the square-root function is only one possible representation of risk preference. Real preferences can vary by wealth, time horizon, liquidity needs, and the nature of the decision. The model also assumes the probabilities are meaningful and that the monetary outcomes capture everything important.\u003c\/p\u003e\n      \u003cp\u003eFor broader background, review the \u003ca href=\"https:\/\/plato.stanford.edu\/entries\/rationality-normative-utility\/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eStanford Encyclopedia of Philosophy discussion of utility and rational choice\u003c\/a\u003e, the \u003ca href=\"https:\/\/www.investopedia.com\/terms\/e\/expectedutility.asp\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eInvestopedia overview of expected utility\u003c\/a\u003e, and the \u003ca href=\"https:\/\/www.investor.gov\/introduction-investing\/investing-basics\/what-risk\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eInvestor.gov introduction to investment risk\u003c\/a\u003e. This calculator is an educational decision-analysis tool, not personalized financial or investment advice.\u003c\/p\u003e\n    \u003c\/div\u003e\n  \u003c\/section\u003e\n\u003c\/div\u003e","brand":"FinancialModelsLab","offers":[{"title":"Default Title","offer_id":49909480292595,"sku":"expected-utility","price":0.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0522\/6191\/2762\/files\/expected-utility.webp?v=1783935367","url":"https:\/\/financialmodelslab.com\/products\/expected-utility","provider":"Financial Models Lab","version":"1.0","type":"link"}