The Encyclopedia of Economic Principals

Chapter 14

Heuristics, Anchors, and Biased Beliefs

Put the base rate, the anchor weight, the evidence weights and the compounding back in.

Four of the chapter's worked examples, made interactive: a hiring label read as a posterior, an asking price that anchors a valuation, balanced evidence that polarizes, and a linear forecast of a compounding balance. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Resemblance is not the posterior

When an applicant fits the prototype, how likely is a high performer, and is the hire worth it?

The label is informative, but a high likelihood P(E | H) is not the posterior P(H | E). The posterior also depends on the base rate and on how often the label appears among everyone else. The firm should hire only when the posterior clears the break-even probability cost / value.

Equation, written in LaTeX: P(H\mid E)=\frac{P(E\mid H)P(H)}{P(E\mid H)P(H)+P(E\mid\neg H)P(\neg H)}.

Equation, written in LaTeX: 0.80(\$150{,}000)-\$50{,}000=\$70{,}000.

Scroll sideways for the whole equation

H is the event that the applicant becomes a high performer and E is the sales-like label. The base rate P(H) is the share of high performers. The label appears for 80 percent of high performers and for the chosen share of everyone else. A high performer is worth $150,000, any other hire $0, and hiring costs $50,000.

Predict first. If the share of high performers rises from 0.10 to 0.20, does the Bayesian firm hire?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Resemblance is not the posterior. Two bars: the Bayesian posterior 0.2286 and the likelihood 0.80 used as if it were the posterior, against a break-even line at 0.3333. The posterior is below the line.
Share of high performers: 0.1, Label rate among others: 0.3
Constructed example: the chapter's hypothetical applicant pool (base rate 0.10, label rates 0.80 and 0.30, value $150,000, cost $50,000); base rates 0.05, 0.20 and 0.30 and label rates 0.10 and 0.50 among others are added for comparison.

Calculated values

Posterior P(H | label)
0.2286
Bayesian expected net value
-$15,714.29
Perceived net value
$70,000.00
Bayesian decision
Do not hire
Resemblance decision
Hire

P(H | E) = 0.80 x 0.10 / (0.80 x 0.10 + 0.30 x 0.90) = 0.0800 / 0.3500 = 0.2286. Expected net value is 150,000 x 0.0800 / 0.3500 - 50,000 = $34,285.71 - $50,000 = -$15,714.29, so the Bayesian firm does not hire. Treating the 0.80 likelihood as the posterior gives 0.80 x 150,000 - 50,000 = $70,000, so the evaluator hires. The book rounds the posterior to 0.229 before multiplying and prints -$15,650; the exact posterior 0.228571 gives -$15,714.29. The decision is the same.

Worked steps

  1. 0.80 x 0.10 = 0.0800
  2. 0.30 x 0.90 = 0.2700
  3. P(H | E) = 0.0800 / (0.0800 + 0.2700) = 0.0800 / 0.3500 = 0.2286
  4. Bayesian net = 150,000 x 0.0800 / 0.3500 - 50,000 = 34,285.71 - 50,000 = -15,714.29
  5. Perceived net = 0.80 x 150,000 - 50,000 = 120,000 - 50,000 = 70,000
  6. Break-even probability = 50,000 / 150,000 = 0.3333

Use the idea

Before acting on a profile that looks like a winner, write down the base rate and the false-positive rate and compute the posterior.

Where the conclusion applies

A single binary label, a known base rate and fixed payoffs: $150,000 for a high performer and nothing for anyone else. Real assessments give graded scores and other hires create some value.

Check your understanding: With base rate 0.10 and label rate among others 0.10, is hiring worth it?
P = 0.08 / (0.08 + 0.09) = 0.4706; net = 150,000 x 0.08 / 0.17 - 50,000 = 70,588.24 - 50,000 = $20,588.24 > 0, so hire.

Chapter 14 source: section "Representativeness heuristic".

Demonstration 2 of 4

How far an anchor pulls a valuation

With identical evidence, how much do two different asking prices move the valuation and the bid?

The judgment starts at the anchor and adjusts only a fraction 1 - lambda of the way to the evidence. Two anchors with the same evidence therefore leave a gap of lambda times the anchor difference. An independent appraisal that sets lambda to zero removes the gap.

Equation, written in LaTeX: J=(1-\lambda)m(I)+\lambda a, 0<\lambda<1.

Equation, written in LaTeX: J_H=0.75(\$24{,}000)+0.25(\$32{,}000)=\$26{,}000.

Equation, written in LaTeX: J_L=0.75(\$24{,}000)+0.25(\$20{,}000)=\$23{,}000.

Scroll sideways for the whole equation

m(I) = $24,000 is the value the evidence supports, a is the asking price used as an anchor and lambda is the residual weight left on the anchor. J is the judgment. The low anchor is $20,000; the bid is 95 percent of the judgment.

Predict first. If the residual anchor weight doubles from 0.25 to 0.50, does the $3,000 judgment gap double?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: How far an anchor pulls a valuation. A value line from $12,000 to $48,000. Anchors at $20,000 and $32,000 pull judgments toward them from the evidence value $24,000: J_L = $23,000 and J_H = $26,000, a gap of $3,000.
Residual anchor weight lambda: 0.25, High asking price ($): $32,000
Constructed example: the chapter's hypothetical van (evidence $24,000, lambda 0.25 and 0, anchors $32,000 and $20,000, bids at 95 percent); weights 0.10 and 0.50 and high anchors $28,000 and $40,000 are added for comparison.

Calculated values

Judgment after high anchor J_H
$26,000
Judgment after low anchor J_L
$23,000
Judgment gap
$3,000
Bid after high anchor
$24,700
Bid after low anchor
$21,850
Bid gap
$2,850

J_H = 0.75 x 24,000 + 0.25 x 32,000 = 18,000 + 8,000 = $26,000 and J_L = 0.75 x 24,000 + 0.25 x 20,000 = 18,000 + 5,000 = $23,000. The $12,000 anchor difference becomes a $3,000 valuation difference, 0.25 of it. Bidding 95 percent of each judgment gives $24,700 and $21,850.

Worked steps

  1. J_H = 0.75 x 24,000 + 0.25 x 32,000 = 18,000 + 8,000 = 26,000
  2. J_L = 0.75 x 24,000 + 0.25 x 20,000 = 18,000 + 5,000 = 23,000
  3. Gap = 0.25 x (32,000 - 20,000) = 3,000
  4. Bids = 0.95 x 26,000 = 24,700 and 0.95 x 23,000 = 21,850

Use the idea

Write your own valuation from the evidence before you see the asking price, and compare bids made after high and low anchors.

Where the conclusion applies

One common lambda for both anchors and a fixed bidding rule. An implausibly extreme anchor may receive no weight at all.

Check your understanding: With lambda = 0.25 and a high asking price of $40,000, what is the bid gap?
J_H = 18,000 + 10,000 = 28,000 and J_L = 23,000, a gap of 5,000; bids are 26,600 and 21,850, a gap of $4,750.

Chapter 14 source: section "Anchoring and insufficient adjustment".

Demonstration 3 of 4

Same evidence, wider disagreement

Can two people who see the same balanced evidence end up further apart?

A Bayesian multiplies the odds by both likelihood ratios, and balanced signals cancel. Discounting only the contradicting signal leaves each person's favored signal partly uncancelled, so priors on opposite sides drift apart.

Equation, written in LaTeX: \log O_{t+1}=\log O_t+w(s_t,p_t)\log LR(s_t),

Equation, written in LaTeX: 3(3)(\frac{1}{3})^{0.5}=5.196,

Equation, written in LaTeX: p_A'=\frac{5.196}{1+5.196}=0.839.

Scroll sideways for the whole equation

O is the odds that the startup is high quality. Each person sees one favorable signal with likelihood ratio LR and one unfavorable signal with ratio 1/LR. Evidence that confirms a person's view gets weight 1 and contradicting evidence gets the chosen weight. Ava starts at 0.75 and Ben at 0.25.

Predict first. If both give full weight to contradicting evidence, what happens to the gap?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Same evidence, wider disagreement. Ava moves from prior 0.75 to 0.8386 and Ben from 0.25 to 0.1614 after one favorable and one unfavorable signal of likelihood ratio 3; the gap is 0.6772.
Weight on contradicting evidence: 0.5, Signal strength LR(F): 3
Constructed example: the chapter's hypothetical startup (priors 0.75 and 0.25, LR 3 and 1/3, weights 1 and 0.5); weight 0 and likelihood ratios 2 and 5 are added for comparison.

Calculated values

Ava's final odds
5.1962
Ava's posterior
0.8386
Ben's final odds
0.1925
Ben's posterior
0.1614
Gap before
0.50
Gap after
0.6772

Ava's odds are 3 x 3 x (1/3)^0.5 = 5.1962, so her posterior is 5.1962 / 6.1962 = 0.8386. Ben's odds are (1/3) x (1/3) x 3^0.5 = 0.1925, so his posterior is 0.1614. Identical evidence widens the disagreement from 0.50 to 0.6772. The book says about 0.678 because it subtracts the rounded posteriors 0.839 and 0.161; the exact gap is 0.6772.

Worked steps

  1. Ava: odds = 3 x 3 x (1/3)^0.5 = 5.1962
  2. Ava: posterior = 5.1962 / (1 + 5.1962) = 0.8386
  3. Ben: odds = (1/3) x (1/3) x 3^0.5 = 0.1925
  4. Ben: posterior = 0.1925 / (1 + 0.1925) = 0.1614
  5. Gap = 0.8386 - 0.1614 = 0.6772

Use the idea

When a group disagrees more after reviewing the same file, check whether each side discounted the evidence against it; a scoring rule that rewards calibrated forecasts can restore equal weights.

Where the conclusion applies

Two signals of equal strength, one each way, and a fixed weight on contradicting evidence. The initial disagreement remains even with equal weights.

Check your understanding: With weight 0 on contradiction and LR 3, what is Ava's posterior?
Odds 3 x 3 x 1 = 9, so the posterior is 9 / 10 = 0.90.

Chapter 14 source: section "Confirmation bias".

Demonstration 4 of 4

Linear vs compound growth

How much does adding the first year's interest each year understate a compounding balance?

Compounding earns interest on past interest, so the balance curve bends upward while the linear forecast stays a straight line. The two agree after one year and drift apart as rT grows.

Equation, written in LaTeX: B_T=P(1+r)^T,

Equation, written in LaTeX: \widetilde B_T=P(1+rT).

Equation, written in LaTeX: \widetilde B_{30}=\$10{,}000+30(\$700)=\$31{,}000.

Equation, written in LaTeX: B_{30}=\$10{,}000(1.07)^{30}\approx \$76{,}123.

Scroll sideways for the whole equation

P = $10,000 is a one-time contribution, r the annual return and T the number of years, with all returns reinvested. B_T is the compound balance and the tilde marks the linear forecast. The saver contributes only if the balance reaches $60,000.

Predict first. At 5 percent for 30 years, does the compound balance clear $60,000?

Your prediction

Choose an example

Scroll sideways for the whole figure

Figure: Linear vs compound growth. Balance paths over 30 years at 7 percent: the compound curve ends at $76,123 and the straight linear forecast at $31,000, against a $60,000 requirement.
Annual return: 7%, Years: 30
Constructed example: the chapter's hypothetical $10,000 contribution at 7 percent for 30 years with a $60,000 requirement; returns of 3, 5 and 9 percent and horizons of 10 and 20 years are added for comparison.

Calculated values

Linear forecast
$31,000
Compound balance
$76,123
Understatement
$45,123
Decision on the linear forecast
Reject
Decision on the compound balance
Accept

Linear reasoning adds the first year's $700 30 times: 10,000 + 30 x 700 = 31,000. Compounding gives 10,000 x 7.6123 = 76,123 (1.07^30 = 7.6123), $45,123 more. Only the compound balance clears $60,000, so the linear forecast wrongly rejects the contribution.

Worked steps

  1. First-year interest = 0.07 x 10,000 = 700
  2. Linear = 10,000 + 30 x 700 = 31,000
  3. 1.07^30 = 7.6123
  4. Compound = 10,000 x 7.6123 = 76,123
  5. Understatement = 76,122.55 - 31,000 = 45,122.55

Use the idea

Compare a saving or debt forecast with both P(1 + r)^T and P(1 + rT); a large gap means the linear shortcut can reverse the decision.

Where the conclusion applies

One deposit, a constant annual return, annual compounding and no fees, withdrawals or inflation.

Check your understanding: At 9 percent for 20 years, what do the two formulas give?
Linear: 10,000 + 20 x 900 = $28,000. Compound: 10,000 x 1.09^20 = $56,044.

Chapter 14 source: section "Exponential-growth bias".