Choice Rules and WARP

In this scene you pick bundles yourself under two different budgets.

A chosen bundle reveals what you like. A bundle you could afford but skipped counts as worse. If two situations flip each other's picks, that is a contradiction. The formula below, WARP, is the rule that catches it.

For example, you have 💰 1,000 won and pick 🍎 apples, though 🍌 bananas cost no more. The next day you pick bananas, though apples cost no more. Your two picks now contradict each other.

x chosen at (p,w), p⋅y≤w⇒y not chosen at (p′,w′) when p′⋅x≤w′x \text{ chosen at } (p,w),\ p\cdot y \le w \Rightarrow y \text{ not chosen at } (p',w') \text{ when } p'\cdot x \le w'

Part of these learning paths

Examples & Exercises

Counterexample

The consumer chose x=(10,50)x = (10, 50) in situation 1 (p,w)=((1,1),60)(p, w) = ((1,1), 60) and y=(30,30)y = (30, 30) in situation 2 (p′,w′)=((2,1),90)(p', w') = ((2,1), 90). Check that both bundles are inside their budgets and decide whether WARP is violated.

p⋅y=60≤60=p⋅x⇒x≿∗yp \cdot y = 60 \le 60 = p \cdot x \quad\Rightarrow\quad x \succsim^* y
p′⋅x=70≤90=p′⋅y⇒y≿∗xp' \cdot x = 70 \le 90 = p' \cdot y \quad\Rightarrow\quad y \succsim^* x
Verdict

The claim is false — this choice pair violates WARP, so no preference relation rationalizes it.

x=(10,50)x = (10, 50), y=(30,30)y = (30, 30) — revealed preference both ways, WARP violated.
Show proof

Counterexample and verification

  1. Budget check: p⋅x=10+50=60≤60p \cdot x = 10 + 50 = 60 \le 60 and p′⋅y=60+30=90≤90p' \cdot y = 60 + 30 = 90 \le 90 — both sit on their budget lines, so no clamping occurs.

  2. In situation 1, yy costs p⋅y=30+30=60≤60=p⋅xp \cdot y = 30 + 30 = 60 \le 60 = p \cdot x. Since yy was affordable but xx was chosen, xx is revealed preferred to yy: x≿∗yx \succsim^* y.

  3. In situation 2, xx costs p′⋅x=20+50=70≤90=p′⋅yp' \cdot x = 20 + 50 = 70 \le 90 = p' \cdot y. Since xx was affordable but yy was chosen, y≿∗xy \succsim^* x. Revealed preference runs both ways between two distinct bundles, so WARP is violated — no preference relation can rationalize this pair of choices.

The consumer chose x=(30,30)x = (30, 30) in situation 1 and y=(10,70)y = (10, 70) in situation 2. Show in which single direction revealed preference holds, and verify that WARP holds.

p⋅y=80>60=p⋅xp \cdot y = 80 > 60 = p \cdot x
p′⋅x=90≤90=p′⋅y⇒y≿∗xp' \cdot x = 90 \le 90 = p' \cdot y \quad\Rightarrow\quad y \succsim^* x
x=(30,30)x = (30, 30), y=(10,70)y = (10, 70) — revealed preference one way only, WARP holds.
Show proof

  1. Budget check: p⋅x=60≤60p \cdot x = 60 \le 60 and p′⋅y=20+70=90≤90p' \cdot y = 20 + 70 = 90 \le 90 — no clamping.

  2. In situation 1, yy costs p⋅y=10+70=80>60=p⋅xp \cdot y = 10 + 70 = 80 > 60 = p \cdot x. Since yy was not affordable at all, there is no revealed preference for xx.

  3. In situation 2, xx costs p′⋅x=60+30=90≤90=p′⋅yp' \cdot x = 60 + 30 = 90 \le 90 = p' \cdot y, so y≿∗xy \succsim^* x. Revealed preference holds in only one direction, so WARP holds — this data does not contradict utility maximization.

2. Preference and Choice