Expected Value Pieces

Pieces of a Decision Tree

A decision tree separates choices, chances, and payoffs before doing any arithmetic.

highlighted = computed this step

Decision node

Start at a decision node. Motivation: this is where the model asks us to choose Build or Small.

choose one branch\text{choose one branch}
Decision tree piecesA square decision node chooses a branch; a round chance node averages branches.rollback treeHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50

Chance node

After a choice, demand is a chance node. Each demand branch has probability 1/2.

p(High)=p(Low)=1/2p(\text{High})=p(\text{Low})=1/2
Decision tree piecesA square decision node chooses a branch; a round chance node averages branches.rollback treeHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50

Payoff leaves

The end of each branch is a dollar payoff. Interpretation: the tree is a picture of assumed inputs, not a promise about the future.

leaf=payoff\text{leaf}=\text{payoff}
Decision tree piecesA square decision node chooses a branch; a round chance node averages branches.rollback treeHigh p=1/2Low p=1/2High p=1/2Low p=1/2Build chosenSmallPayoff$120Payoff$-20ChanceEV $50Payoff$40Payoff$30ChanceEV $35DecisionEV $50

Payoff table

The same inputs can be read as a table. Visual tree first, table second; both contain the same exact payoffs.

tree inputs=table inputs\text{tree inputs}=\text{table inputs}
Payoff tableRows are choices and columns are demand states.payoffs and expected valueHighLowEVBuild120-2050Small403035