Expected Value Pieces

Build Expected Value

Expected value is a probability-weighted average of exact payoff leaves.

highlighted = computed this step

High case

If demand is High, Build pays $120. Motivation: write the payoff before averaging.

High payoff=$120\text{High payoff}=\$120
Build expected valueFold back the Build branch by multiplying each payoff by its probability.Build branch EVHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50

Low case

If demand is Low, Build pays $-20. Interpretation: a negative payoff is a loss in the same units.

Low payoff=$−20\text{Low payoff}=\$-20
Build expected valueFold back the Build branch by multiplying each payoff by its probability.Build branch EVHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50

Weighted average

Expected value multiplies each payoff by probability 1/2 and adds the pieces.

1/2($120)+1/2($−20)1/2(\$120)+ 1/2(\$-20)
Build expected valueFold back the Build branch by multiplying each payoff by its probability.Build branch EVHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50

Build EV

The Build branch folds back to $50. Honesty note: this is exact arithmetic from the assumed probabilities and payoffs.

EV(Build)=$50EV(\text{Build})=\$50
Build expected valueFold back the Build branch by multiplying each payoff by its probability.Build branch EVHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50