Decision Making Under Uncertainty: Pay Off Table and Decision Tree

Decision Making Under Uncertainty A set of quantitative decision-making techniques for decision situations where uncertainty exists

Decision Making States of nature – events that may occur in the future – decision maker is uncertain which state of nature will occur – decision maker has no control over the states of nature

Payoff Table l A method of organizing & illustrating the payoffs from different decisions given various states of nature l A payoff is the outcome of the decision

Payoff Table States Of Nature Decisionab 1Payoff 1aPayoff 1b 2Payoff 2aPayoff 2b

Decision-Making Models Under Uncertainty l Maximax choose decision with the maximum of the maximum payoffs l Maximin choose decision with the maximum of the minimum payoffs l Minimax regret choose decision with the minimum of the maximum regrets for each alternative

l Hurwicz – choose decision in which decision payoffs are weighted by a coefficient of optimism,  – coefficient of optimism (  ) is a measure of a decision maker’s optimism, from 0 (completely pessimistic) to 1 (completely optimistic) l Equal likelihood (La Place) – choose decision in which each state of nature is weighted equally

Decision Making Under Uncertainty Example Expand$ 800,000$ 500,000 Maintain status quo1,300, ,000 Sell now320,000320,000 States Of Nature Good ForeignPoor Foreign DecisionCompetitive Conditions Competitive Conditions

Maximax Solution Expand: $ 800,000 Status quo: 1,300,000 Maximum Sell: 320,000 Decision: Maintain status quo

Maximin Solution Expand: $ 500,000 Maximum Status quo: -150,000 Sell: 320,000 Decision: Expand

Minimax Regret Solution $ 1,300, ,000 = 500,000$ 500,000 - $500,000 = 0 1,300, ,300,000 = 0500,000 - (-150,000) = 650,000 1,300, ,000 = 980,000500, ,000 = 180,000 Good ForeignPoor Foreign Competitive Conditions Expand:$ 500,000 Minimum Status quo:650,000 Sell:980,000 Decision: Expand Regret Value

Hurwicz Solution  = 0.3, 1-  = 0.7 Expand: $ 800,000 (0.3) + 500,000 (0.7) = $590,000 ** Status quo: 1,300,000 (0.3) -150,000 (0.7) = 285,000 Sell: 320,000 (0.3) + 320,000 (0.7) = 320,000 Decision: Expand ** Maximum

Equal Likelihood Solution Two decisions, weight = 0.50 for each state of nature Expand: $ 800,000 (0.50) + 500,000 (0.50) = $590,000 ** Status quo: 1,300,000 (0.50) -150,000 (0.50) = 285,000 Sell: 320,000 (0.50) + 320,000 (0.50) = 320,000 Decision: Expand **Maximum

Decision Making With Probabilities l Risk involves assigning probabilities to states of nature l Expected value is a weighted average of decision outcomes in which each future state of nature is assigned a probability of occurrence

Expected Value

Expected Value Example 70% probability of good foreign competition 30% probability of poor foreign competition EV(expand) $ 800,000 (0.70) + 500,000 (0.30) = $710,000 EV(status quo) $1,300,000 (0.70) - 150,000 (0.30) = 865,000 Maximum EV(sell) $ 320,000 (0.70) + 320,000 (0.30) = 320,000 Decision : Maintain status quo

Case of Pay off Table application An ICT (Information and communication technology) company wants to analyze the future of its business. There are 4 decision alternatives: expand the company, maintain status quo, decrease the business size up to 50% of the current size and sell the company. From the business analysis there will be two possibilities: good economic condition and bad economic condition. If the economic condition is good the profit of the expansion will be Rp. 900 million and only Rp. 400 million when the economic condition is bad. If the economic condition is good the profit of maintain status quo will be Rp million and only Rp. 50 million when the economic condition is bad. If the economic condition is good the profit of decrease the business will be Rp. 600 million and only Rp. 300 million when the economic condition is bad. When the company is sold the current price is Rp. 350 million. Solve this decision problem by using maximax, maximin, minimax, hurwicz (with alpha = 0.3) and Equal likelihood. Based on the analysis provide your best suggestion.

Sequential Decision Trees l A graphical method for analyzing decision situations that require a sequence of decisions over time l Decision tree consists of Square nodes - indicating decision points Circles nodes - indicating states of nature Arcs - connecting nodes

Decision tree basics: begin with no uncertainty l Basic setup: Trees run left to right chronologically. Decision nodes are represented as squares. Possible choices are represented as lines (also called branches). The value associated with each choice is at the end of the branch. North Side South Side Japanese Greek Vietnam Thai Example: deciding where to eat dinner

Assigning values to the nodes involves defining goals. Example: deciding where to eat dinner Taste versus Speed North Side South Side Japanese Greek Vietnam Thai

To solve a tree, work backwards, i.e. right to left. Example: deciding where to eat dinner Speed North Side South Side Japanese Greek Vietnam Thai Value =4 Value =2

Decision making under uncertainty Chance nodes are represented by circles. Probabilities along each branch of a chance node must sum to 1. Example: a company deciding whether to go to trial or settle a lawsuit Go to trial Settle Win [p=0.6] Lose [p= ]

Solving a tree with uncertainty: The expected value (EV) is the probability-weighted sum of the possible outcomes: p win x win payoff + p lose x lose payoff In this tree, “Go to trial” has a cost associated with it that “Settle” does not. We’re assuming the decision- maker is maximizing expected values. Go to trial Settle Win [p=0.6] Lose [p=0.4] -$4M -$8M $0 -$.5M EV= -$3.2M EV= -$3.7M -$3.7M

Decision tree notation Go to trial Settle Win [p=0.6] Lose [p=0.4] -$4M -$8M $0 -$.5M -$4m -$8.5M -$.5M EV= -$3.2M EV= -$3.7M Value of optimal decision Chance nodes (circles) Terminal values corresponding to each branch (the sum of payoffs along the branch). Probabilities (above the branch) Payoffs (below the branch) Decision nodes (squares) -$3.7M -$4M Running total of net expected payoffs (below the branch) Expected value of chance node (or certainty equivalent)

Example of a Decision Tree Problem A glass factory specializing in crystal is experiencing a substantial backlog, and the firm's management is considering three courses of action: A) Arrange for subcontracting B) Construct new facilities C) Do nothing (no change) The correct choice depends largely upon demand, which may be low, medium, or high. By consensus, management estimates the respective demand probabilities as 0.1, 0.5, and 0.4. A glass factory specializing in crystal is experiencing a substantial backlog, and the firm's management is considering three courses of action: A) Arrange for subcontracting B) Construct new facilities C) Do nothing (no change) The correct choice depends largely upon demand, which may be low, medium, or high. By consensus, management estimates the respective demand probabilities as 0.1, 0.5, and 0.4.

Example of a Decision Tree Problem (Continued): The Payoff Table The management also estimates the profits when choosing from the three alternatives (A, B, and C) under the differing probable levels of demand. These profits, in thousands of dollars are presented in the table below:

Step 1. We start by drawing the three decisions A B C

Step 2. Add our possible states of nature, probabilities, and payoffs A B C High demand (0.4) Medium demand (0.5) Low demand (0.1) $90 $50 $10 High demand (0.4) Medium demand (0.5) Low demand (0.1) $200 $25 -$120 High demand (0.4) Medium demand (0.5) Low demand (0.1) $60 $40 $20

Step 3. Determine the expected value of each decision High demand (0.4) Medium demand (0.5) Low demand (0.1) A A $90 $50 $10 EV A =0.4(90)+0.5(50)+0.1(10)=$62 $62

Step 4. Make decision High demand (0.4) Medium demand (0.5) Low demand (0.1) High demand (0.4) Medium demand (0.5) Low demand (0.1) A B C High demand (0.4) Medium demand (0.5) Low demand (0.1) $90 $50 $10 $200 $25 -$120 $60 $40 $20 $62 $80.5 $46 Alternative B generates the greatest expected profit, so our choice is B or to construct a new facility

Format of a Decision Tree State of nature 1 B Payoff 1 State of nature 2 Payoff 2 Payoff 3 2 Choose A’ 1 Choose A’ 2 Payoff 6 State of nature 2 2 Payoff 4 Payoff 5 Choose A’ 3 Choose A’ 4 State of nature 1 Choose A’ Choose A’ 2 1 Decision Point Chance Event

Case of Decision Tree application See Attached Problem