Summary of Business Risk, Information Systems, Big Data

## Introduction Decision trees are a visual and quantitative tool for making decisions under uncertainty. They map choices (decisions) and chance outcomes, assign payoffs and probabilities, and use expected values to identify the financially best route. This material explains how to build and evaluate decision trees, how to compute the value of additional information, and illustrates the methods with worked examples. > Definition: A decision tree is a diagram representing decisions (triangles), chance outcomes (circles), probabilities, and payoffs; evaluation proceeds by calculating expected values and working backwards to choose the best decision. ## Key concepts broken down ### 1. Structure of a decision tree - Decision nodes (triangle): points where the decision maker chooses among alternatives. - Chance nodes (circle): points where probabilistic outcomes occur. - Branches: each choice or outcome has a branch labelled with the option and, for chance branches, the probability. - Payoffs: financial outcomes attached to terminal branches. > Definition: Expected value (EV) is the probability-weighted average payoff of a chance node: if outcomes $v_i$ occur with probabilities $p_i$, then $\mathrm{EV}=\sum_i p_i v_i$. ### 2. Building the tree (left to right) 1. Identify decisions and order them logically. 2. For each decision, list feasible options. 3. For each option, list possible chance outcomes and estimate their probabilities. 4. Attach payoffs to terminal outcomes. 5. Evaluate from right to left to choose the best decisions. ### 3. Evaluating the tree (right to left) - At a chance node compute EV: $\mathrm{EV}=\sum_i p_i v_i$. - At a decision node select the option with the highest EV (subtract any option-specific costs when comparing). ### 4. Value of additional information - Additional (imperfect) information may change decisions and expected payoffs. - The value of information (VOI) is the increase in expected value from using the information: $$\mathrm{VOI}=\mathrm{EV}_{\text{with info}}-\mathrm{EV}_{\text{without info}}$$ - Maximum a decision-maker should pay equals VOI (pay less to obtain net benefit). > Definition: Perfect information predicts outcomes with 100% accuracy; imperfect information improves probabilities but is not guaranteed accurate. ## Worked example: Kris's purchasing decision (condensed and calculated) Scenario summary: Kris currently buys a fixed volume of 1,000,000 packets per month. Historic demand (units '000) with probabilities and contribution if sold are: - Demand and contribution when constrained by 1,000 units bought: - 800 (probability $0.08$, contribution $1{,}600$), 900 ($0.16$, $1{,}900$), 1000 ($0.32$, $2{,}200$), 1100 ($0.32$, $2{,}200$), 1200 ($0.12$, $2{,}200$). Compute EV without extra information (current policy): For each demand scenario use the contribution achievable given purchase of 1,000 units. EV calculation (in $000): $$\mathrm{EV}_{\text{no info}}=0.08\times1600\; +\;0.16\times1900\; +\;0.32\times2200\; +\;0.32\times2200\; +\;0.12\times2200$$ Carrying out the arithmetic gives: $$\mathrm{EV}_{\text{no info}}=128\; +\;304\; +\;704\; +\;704\; +\;264\; =\;2104\quad(\$000)$$ So the expected monthly contribution is $2{,}104{,}000$. Compute EV with additional forecast information (analyst predicts demand and Kris buys exactly predicted volume so there is no waste or shortage): If purchase matches actual demand, contribution per demand level increases (unit contribution applied to full demand). The contribution values (in $000) become: 800 -> 1,760; 900 -> 1,980; 1000 -> 2,200; 1100 -> 2,420; 1200 -> 2,640. Compute EV with info: $$\mathrm{EV}_{\text{info}}=0.08\times1760\; +\;0.16\times1980\; +\;0.32\times2200\; +\;0.32\times2420\; +\;0.12\times2640$$ Which yields: $$\mathrm{EV}_{\text{info}}=140.8\; +\;316.8\; +\;704\; +\;774.4\; +\;316.8\; =\;2252.8\quad(\$000)$$ VOI and maximum payment for the analyst: $$\mathrm{VOI}=2252.8-2104.0=148.8\quad