Chapter 12 & Module E Decision Analysis & Game Theory

Components of Decision Making (D.M.) F Decision alternatives - for managers to choose from. F States of nature - that may actually occur in the future regardless of the decision. F Payoffs - payoff of a decision alternative in a state of nature. The components are given in Payoff Tables.

A Payoff Table (It shows payoffs of different decisions at different states of nature) InvestmentStates of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000

Types of Decision Making (D.M.) - 1 F Deterministic D.M. (D.M. under certainty): –Only one “state of nature”, –Payoff of an alternative is known, –Examples: u Problems for LP, IP, transportation, and network flows.

Types of Decision Making (D.M.) - 2 F D.M. without probabilities (D.M. under uncertainty): –More than one states of nature; –Payoff of an alternative is not known at the time of making decision; –Probabilities of states of nature are not known.

Types of Decision Making (D.M.) - 3 F D.M. with probabilities (D.M. under risk) –More than one states of nature; –Payoff of an alternative is not known at the time of making decision; –Probabilities of states of nature are known or given.

Types of Decision Making (D.M.) - 4 F D.M. in competition (Game theory) –Making decision against a human competitor.

Decision Making without Probabilities No information about possibilities of states of nature. Five criteria (approaches) for a decision maker to choose from, depending on his/her preference.

Criterion 1: Maximax F Pick the maximum of the maximums of payoffs of decision alternatives. (Best of the bests) Investment States of Nature max decision EconomyEconomypayoffs alternatives good bad(bests) Apartment$ 50,000$ 30,000$50,000 Office 100, ,000100,000 Warehouse 30,000 10,000 30,000 F Decision:

Whom Is MaxiMax for? F MaxiMax method is for optimistic decision makers who tend to grasp every chance of making money, who tend to take risk, who tend to focus on the most fortunate outcome of an alternative and overlook the possible catastrophic outcomes of an alternative.

Criterion 2: Maximin F Pick the maximum of the minimums of payoffs of decision alternatives. (Best of the worsts) Investment States of Nature min decision EconomyEconomypayoff alternatives good bad(worsts) Apartment$ 50,000$ 30,000$30,000 Office 100, , ,000 Warehouse 30,000 10,000 10,000 F Decision:

Whom Is MaxiMin for? F MaxiMin method is for pessimistic decision makers who tend to be conservative, who tend to avoid risks, who tend to be more concerned about being hurt by the most unfortunate outcome than the opportunity of being fortunate.

Criterion 3: Minimax Regret F Pick the minimum of the maximums of regrets of decision alternatives. (Best of the worst regrets) F Need to construct a regret table first. Regret of a decision under a state of nature = (the best payoff under the state of nature) – (payoff of the decision under the state of nature)

Investment States of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000 Payoffs Investment States of Nature max decision EconomyEconomy regret alternatives good bad Apartment$ 50,000$ 0$50,000 Office 0 70,000 70,000 Warehouse 70,000 20,000 70,000 Regrets Decision:

Whom Is MiniMax Regret for? F MiniMax regret method is for a decision maker who is afraid of being hurt by the feeling of regret and tries to reduce the future regret on his/her current decision to minimum. “I concern more about the regret I’ll have than how much I’ll make or lose.”

Criterion 4: Hurwicz F Pick the maximum of Hurwicz values of decision alternatives. (Best of the weighted averages of the best and the worst) F Hurwicz value of a decision alternative = (its max payoff)(  ) + (its min payoff)(1-  ) where  (0  1) is called coefficient of optimism.

Investment States of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000 Payoffs Investment decision Hurwicz Values alternatives Apartment 50,000(0.4)+30,000(0.6) =38,000 Office100,000(0.4)  40,000(0.6) = 16,000 Warehouse 30,000(0.4)+10,000(0.6) = 18,000 Hurwicz Values with  =0.4 Decision:

Whom Is Hurwicz Method for? F Hurwicz method is for an extreme risk taker (  =1), an extreme risk averter (  =0), and a person between the two extremes (  is somewhere between 1 and 0).

Criterion 5: Equal Likelihood F Pick the maximum of the average payoffs of decision alternatives. (Best of the plain averages) F Average payoff of a decision alternative

Investment States of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000 Payoffs Investment decision Average Payoffs alternatives Apartment (50,000+30,000) / 2 = 40,000 Office(100,000  40,000) / 2 = 30,000 Warehouse (30,000+10,000) / 2 = 20,000 Average Payoffs Decision:

Whom Is Equally Likelihood for? F Equally likelihood method is for a decision maker who tends to simply use the average payoff to judge an alternative.

Dominated Alternative F If alternative A’s payoffs are lower than alternative B’s payoffs under all states of nature, then A is called a dominated alternative by B. F A dominated alternative can be removed from the payoff table to simplify the problem. Investment States of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000

Decision Making with Probabilities F The probability that each state of nature will actually occur is known. States of Nature Investment EconomyEconomy decision good bad alternatives Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000

Criterion:Expected Payoff F Select the alternative that has the largest expected payoff. F Expected payoff of an alternative: n=number of states of nature P i =probability of the i-th state of nature V i =payoff of the alternative under the i-th state of nature

Example Decision Alt’s Econ Good 0.6 Econ Bad 0.4Expected payoff Apartment 50,00030,000 Office 100,000-40,000 Warehouse 30,00010,000

Expected Opportunity Loss (EOL) F Each decision alternative has an EOL which is the expected value of the opportunity costs (regrets). F The alternative with minimum EOL has the highest expected payoff.

Investment States of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000 Payoffs Investment States of Nature decision EconomyEconomy alternatives good bad Apartment$ 50,000$ 0 Office 0 70,000 Warehouse 70,000 20,000 Opp Loss Table

Example (cont.) F EOL (apartment) = 50,000*0.6 +0*0.4 = 30,000 F EOL (office) =0*0.6+70,000*0.4 = 28,000 F EOL (warehouse) = 70,000*0.6+20,000*0.4 = 50,000 Minimum EOL = 28,000 that is associated with Office.

(Max Exp. Payoff) vs. (Min EOL) F The alternative with minimum EOL has the highest expected payoff. F The alternative selected by (Max expected payoff) and by (Min EOL) are always same.

Expected Value of Perfect Information (EVPI) F It is a measure of the value of additional information on states of nature. F It tells up to how much you would pay for additional information.

An Example If a consulting firm offers to provide “perfect information about the future with $5,000, would you take the offer? States of Nature Investment EconomyEconomy decision good bad alternatives Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000

Another Example F You can play the game for many times. F What is your rational strategy of “guessing”? F Someone offers you perfect information about “landing” at $65 per time. Do you take it? If not, how much you would pay? Land on ‘Head’Land on ‘Tail’ Guess ‘Head’$100- $60 Guess ‘Tail’- $80$150

Calculating EVPI F EVPI F = EV w PI – EV w/o PI = (Exp. payoff with perfect information) – (Exp. payoff without perfect information)

Expected payoff with Perfect Information F EV w PI where n=number of states of nature h i =highest payoff of i-th state of nature P i =probability of i-th state of nature

Example for Expected payoff with Perfect Information States of Nature Investment EconomyEconomy decision good bad alternatives Apartment$ 50,000$ 30,000 Office 100, ,000 Warehouse 30,000 10,000 h i 100,000 30,000 Expected payoff with perfect information = 100,000*0.6+30,000*0.4 = 72,000

Expected payoff without Perfect Information F Expected payoff of the best alternative selected without using additional information. i.e., EV w/o PI = Max Exp. Payoff

Example for Expected payoff without Perfect Information Decision Alt’s Econ Good 0.6 Econ Bad 0.4Expected payoff Apartment 50,00030,000 42,000 *Office 100,000-40,000 *44,000 Warehouse 30,00010,000 22,000

Expected Value of Perfect Information (EVPI) in above Example F EVPI = EV w PI – EV w/o PI = 72, ,000 = $28,000

Example Revisit 0.5 Land on “Head”Land on “Tail” Guess “Head”$100-$60 Guess “Tail”-$80$150 Up to how much would you pay for a piece of information about result of “landing”?

EVPI is equal to (Min EOL) F EVPI is the expected opportunity loss (EOL) for the selected decision alternative.

Maximum average payoff per game Alt. 1, Guess “Head” Alt. 2, Guess “Tail” EMV regret average payoff average payoff EOL Alternatives $

EVPI is a Benchmark in Bargain F EVPI is the maximum $ amount the decision maker would pay to purchase perfect information.

Value of Imperfect Information Expected value of imperfect information = (discounted EV w PI) – EV w/o PI = (EVwPI * (% of perfection)) – EV w/o PI

Decision Tree F Decision tree is used to help make a series of decisions. F A decision tree is composed of decision nodes (square), chance nodes (circle), and payoff nodes (final or tip nodes). F A decision tree reflects the decision making process and the possible payoffs with different decisions under different states of nature.

Making Decision on a Decision Tree F It is actually a process of marking numbers on nodes. F Mark numbers from right to left. F For a chance (circle) node, mark it with its expected value. F For a decision (square) node, select a decision and mark the node with the number associated with the decision.

Example p.551

Example p

Game Theory F Game theory is for decision making under competition. F Two or more decision makers are involved, who have conflicting interests.

Two-Person Zero-Sum Game F Two decision makers’ benefits are completely opposite i.e., one person’s gain is another person’s loss F Payoff/penalty table (zero-sum table): –shows “offensive” strategies (in rows) versus “defensive” strategies (in columns); –gives the gain of row player (loss of column player), of each possible strategy encounter.

Example 1 (payoff/penalty table) Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1$50,000$35,000$30,000 2$60,000$40,000$20,000

Two-Person Constant-Sum Game F For any strategy encounter, the row player’s payoff and the column player’s payoff add up to a constant C. F It can be converted to a two-person zero- sum game by subtracting half of the constant (i.e. 0.5C) from each payoff.

Example 2 (2-person, constant- sum) During the 8-9pm time slot, two broadcasting networks are vying for an audience of 100 million viewers, who would watch either of the two networks.

Payoffs of nw1 for the constant- sum of 100(million) Network 2 Network 1westernSoap Comedy western soap comedy

An equivalent zero-sum table Network 2 Network 1westernSoap Comedy western soap comedy

Equilibrium Point F In a two-person zero-sum game, if there is a payoff value P such that P=max{row minimums} =min{column maximums} then P is called the equilibrium point, or saddle point, of the game.

Example 3 (equilibrium point) Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1$50,000$35,000$30,000 2$60,000$40,000$20,000

Game with an Equilibrium Point: Pure Strategy F The equilibrium point is the only rational outcome of this game; and its corresponding strategies for the two sides are their best choices, called pure strategy. F The value at the equilibrium point is called the value of the game. F At the equilibrium point, neither side can benefit from a unilateral change in strategy.

Pure Strategy of Example 3 Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1$50,000$35,000$30,000 2$60,000$40,000$20,000

Example 4 (2-person, 0-sum) Row Players Column Player Strategies Strategies

Mixed Strategy F If a game does not have an equilibrium, the best strategy would be a mixed strategy.

Game without an Equilibrium Point F A player may benefit from unilateral change for any pure strategy. Therefore, the game would get into a loop. F To break loop, a mixed strategy is applied.

Example: Company ICompany II Strategies Strategies BC

Mixed Strategy F A mixed strategy for a player is a set of probabilities each for an alternative of the player. F The expected payoff of row player (or the expected loss of column player) is called the value of the game.

Example: Company ICompany II Strategies Strategies BC Let mixed strategy for company I be {0.6, 0.4}; and for Company II be {0.3, 0.7}.

Equilibrium Mixed Strategy F An equilibrium mixed strategy makes expected values of any player’s individual strategies identical. F Every game contains one equilibrium mixed strategy. F The equilibrium mixed strategy is the best strategy.

How to Find Equilibrium Mixed Strategy F By linear programming (as introduced in book) F By QM for Windows, – we use this approach.

Both Are Better Off at Equilibrium F At equilibrium, both players are better off, compared to maximin strategy for row player and minimax strategy for column player. F No player would benefit from unilaterally changing the strategy.

A Care-Free Strategy F The row player’s expected gain remains constant as far as he stays with his mixed strategy (no matter what strategy the column player uses). F The column player’s expected loss remains constant as far as he stays with his mixed strategy (no matter what strategy the row player uses).

Unilateral Change from Equilibrium by Column Player probability B C 0.6Strat Strat 3 1 7

Unilateral Change from Equilibrium by Column Player probability B C 0.6Strat Strat 3 1 7

Unilateral Change from Equilibrium by Row Player probability B C 0.2Strat Strat 3 1 7

A Double-Secure Strategy F At the equilibrium, the expected gain or loss will not change unless both players give up their equilibrium strategies. –Note: Expected gain of row player is always equal to expected loss of column player, even not at the equilibrium, since 0-sum)

Both Leave Their Equilibrium Strategies probability B C 0.5Strat Strat 3 1 7

Both Leave Their Equilibrium Strategies probability 0 1 B C 0.2Strat Strat 3 1 7

Penalty for Leaving Equilibrium F It is equilibrium because it discourages any unilateral change. F If a player unilaterally leaves the equilibrium strategy, then –his expected gain or loss would not change, and –once the change is identified by the competitor, the competitor can easily beat the non-equilibrium strategy.

Find the Equilibrium Mixed Strategy F Method 1: As on p of our text book. The method is limited to 2X2 payoff tables. F Method 2: Linear programming. A general method. F Method we use: Software QM.

Implementation of a Mixed Strategy F Applied in the situations where the mixed strategy would be used many times. F Randomly select a strategy each time according to the probabilities in the strategy. F If you had good information about the payoff table, you could figure out not only your best strategy, but also the best strategy of your competitor (!).

Dominating Strategy vs. Dominated Strategy F For row strategies A and B: If A has a better (larger) payoff than B for any column strategy, then B is dominated by A. F For column strategies X and Y: if X has a better (smaller) payoff than Y for any row strategy, then Y is dominated by X. F A dominated decision can be removed from the payoff table to simplify the problem.

Example: Company ICompany II Strategies Strategies A BC

Find the Optimal Mixed Strategy in 2X2 Table F Suppose row player has two strategies, 1 and 2, and column player has two strategies, A and B.

For row player: F Let p be probability of selecting row strategy 1. Then the probability of selecting row strategy 2 is (1-p). F Represent E A and E B by p, where E A (E B ) is the expected payoff of the row player if the column player chose column strategy A (B). F Set E A = E B, and solve p from the equation.

For column player: F Let p be probability of selecting column strategy A. Then the probability of selecting column strategy B is (1-p). F Represent E 1 and E 2 by p, where E 1 (E 2 ) is the expected payoff of the row player if the column player chose column strategy A (B). F Set E 1 = E 2, and solve p from the equation.