Decision Tree Rollback: Expected Value vs Risk-Adjusted Choice

Strategy Outcome Probabilities

Launch New Product — P(success) 40%

EMV—

Improve Existing — P(success) 62%

EMV—

Hold / Do Nothing — P(success) 50%

EMV—

Risk Preference

Risk tolerance R $250k

low R = risk-aversehigh R ≈ risk-neutral

Recommendation

EMV-optimal strategy

—

Risk-adjusted (CE) optimal

—

Do the two views agree?

—

Launch New Product Improve Existing Hold / Do Nothing

How it works

Each strategy is a chance node: a coin-flip between a "success" payoff and a "failure" payoff (both in thousands of dollars), with an adjustable success probability p. Backward induction (rollback) folds each chance node back to a single number at the decision node, then picks the branch with the best number — the standard operations-research method for decision trees.

Expected Monetary Value:
  EMV = p · payoff_success + (1-p) · payoff_failure

CARA utility (constant absolute risk aversion):
  U(x) = 1 - e^(-x / R)          R = risk tolerance ($k)
  Certainty Equivalent:
  CE = -R · ln( 1 - [p·U(success) + (1-p)·U(failure)] )

EMV is the "risk-neutral" view: it only cares about the long-run average payoff, ignoring how painful a loss feels.
CE is the "risk-adjusted" view: the guaranteed cash amount a decision-maker with risk tolerance R would accept instead of the gamble. A volatile strategy (big upside, real downside) can have a high EMV but a low — even negative — CE.
Lowering R makes the decision-maker more loss-averse: CE drops fastest for the branches with the widest spread between success and failure payoffs.
The animated particles flow backward along the tree, from each outcome leaf into its chance node and from there into the decision node — a visual of the rollback computation itself. The brightest stream marks the branch currently recommended by EMV or CE.

Real-world relevance: this exact EMV / certainty-equivalent rollback is how capital-budgeting and product-launch decisions are formally evaluated in operations research and management science — probabilities and payoffs come from market research, and R is calibrated to the organization's actual risk appetite.

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