chapter 12 & module e decision theory & game theory
TRANSCRIPT
Chapter 12 & Module E
Decision Theory
&
Game Theory
Decision Making
A decision is made for a future action.A decision making process is a process
of “selection” : Selecting one from many options (alternatives) as the decision.
Decision Theory
Decision theory deals with following type of decision making problems:– The outcome for an decision alternative is not
certain, which is affected by some factors that are not controlled by the decision maker.
– Example: Selecting a stock for investment.
Components of Decision Making (D.M.)
Decision alternatives - for managers to choose from.
States of nature - that may actually occur in the future regardless of the decision.
Payoff - outcome of a decision alternative under a state of nature.The components are given in Payoff Tables.
A Decision Table
States of Nature
Investment Economy Economy
decision good bad
alternatives 0.6 0.4
Apartment $ 50,000 $ 30,000
Office 100,000 - 40,000
Warehouse 30,000 10,000
Criterion:Expected Payoff
Select the alternative that has the largest expected value of payoffs.
Expected payoff of an alternative:
n=number of states of nature
Pi=probability of the i-th state of nature
Vi=payoff of the alternative under the i-th state of nature
n
i
ii PV1
*
Example
Decision Alt’s
Econ
Good
0.6
Econ
Bad
0.4 Expected payoff
Apartment 50,000 30,000
Office 100,000 -40,000
Warehouse 30,000 10,000
Expected Value of Perfect Information (EVPI)
It is a measure of the value of additional information on states of nature.
It tells up to how much you would pay for additional information.
An ExampleIf a consulting firm offers to provide “perfect information
about the future with $5,000, would you take the offer?
States of Nature
Investment Economy Economy
decision good bad
alternatives 0.6 0.4
Apartment $ 50,000 $ 30,000
Office 100,000 - 40,000
Warehouse 30,000 10,000
Calculating EVPI
EVPI
= EVwPI – EVw/oPI
= (Exp. payoff with perfect information) –
(Exp. payoff without perfect information)
Expected payoff with Perfect InformationEVwPI
where n=number of states of nature hi=highest payoff of i-th state of nature
Pi=probability of i-th state of nature
n
i
ii Ph1
Example for Expected payoff with Perfect Information
States of Nature
Investment Economy Economy
decision good bad
alternatives 0.6 0.4
Apartment $ 50,000 $ 30,000
Office 100,000 - 40,000
Warehouse 30,000 10,000
hi 100,000 30,000
Expected payoff with perfect information
= 100,000*0.6+30,000*0.4 = 72,000
Expected payoff without Perfect Information
Expected payoff of the best alternative selected without using additional information. i.e.,
EVw/oPI = Max Exp. Payoff
Example for Expected payoff without Perfect Information
Decision Alt’s
Econ
Good
0.6
Econ
Bad
0.4 Expected payoff
Apartment 50,000 30,000 42,000
*Office 100,000 -40,000 *44,000
Warehouse 30,000 10,000 22,000
Expected Value of Perfect Information (EVPI) in above Example
EVPI
= EVwPI – EVw/oPI
= 72,000 - 44,000
= $28,000
EVPI is a Benchmark in Bargain
EVPI is the maximum $ amount the decision maker would pay to purchase perfect information.
Value of Imperfect Information
Expected value of imperfect information
= (discounted EVwPI) – EVw/oPI
= (EVwPI * (% of perfection)) – EVw/oPI
Game Theory
Game theory is for decision making with two decision makers of conflicting interests in competition.
In decision theory: Human vs. God. In game theory: Human vs. Human.
Two-Person Zero-Sum Game
Two decision makers’ benefits are completely oppositei.e., one person’s gain is another person’s loss
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
For any strategy encounter, the row player’s payoff and the column player’s payoff add up to a constant C.
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 1 western Soap Comedy
western 35 15 60
soap 45 58 50
comedy 38 14 70
An equivalent zero-sum table
Network 2
Network 1 western Soap Comedy
western -15 -35 10
soap - 5 8 0
comedy -12 -36 20
Equilibrium Point
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
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.
The value at the equilibrium point is called the value of the game.
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 1 2 3
1 4 4 10
2 2 3 1
3 6 5 7
Mixed Strategy
If a game does not have an equilibrium, the best strategy would be a mixed strategy.
Game without an Equilibrium Point
A player may benefit from unilateral change for any pure strategy. Therefore, the game would get into a loop.
To break loop, a mixed strategy is applied.
Example:
Company I Company II Strategies
Strategies B C
2 8 4
3 1 7
Mixed Strategy
A mixed strategy for a player is a set of probabilities each for an alternative of the player.
The expected payoff of row player (or the expected loss of column player) is called the value of the game.
Example:
Company I Company II Strategies
Strategies B C
2 8 4
3 1 7
Let mixed strategy for company I be
{0.6, 0.4}; and for Company II be
{0.3, 0.7}.
Equilibrium Mixed StrategyAn equilibrium mixed strategy
makes expected values of any player’s individual strategies identical.
Every game contains one equilibrium mixed strategy.
The equilibrium mixed strategy is the best strategy.
How to Find Equilibrium Mixed Strategy
By linear programming (as introduced in book)
By QM for Windows, – we use this approach.
Both Are Better Off at Equilibrium
At equilibrium, both players are better off, compared to maximin strategy for row player and minimax strategy for column player.
No player would benefit from unilaterally changing the strategy.
A Care-Free Strategy
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).
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 Playerprobability 0.1 0.9
B C
0.6 Strat 2 8 4
0.4 Strat 3 1 7
Unilateral Change from Equilibrium by Column Playerprobability 1.0 0
B C
0.6 Strat 2 8 4
0.4 Strat 3 1 7
Unilateral Change from Equilibrium by Row Player
probability 0.3 0.7
B C
0.2 Strat 2 8 4
0.8 Strat 3 1 7
A Double-Secure Strategy
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 0.8 0.2
B C
0.5 Strat 2 8 4
0.5 Strat 3 1 7
Both Leave Their Equilibrium Strategies
probability 0 1
B C
0.2 Strat 2 8 4
0.8 Strat 3 1 7
Penalty for Leaving Equilibrium
It is equilibrium because it discourages any unilateral change.
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
Method 1: As on p.573-574 of our text book. The method is limited to 2X2 payoff tables.
Method 2: Linear programming. A general method.
Method we use: Software QM.
Implementation of a Mixed Strategy Applied in the situations where the mixed
strategy would be used many times. Randomly select a strategy each time
according to the probabilities in the strategy.
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
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.
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.
A dominated decision can be removed from the payoff table to simplify the problem.
Example:
Company I Company II Strategies
Strategies A B C
1 9 7 2
2 11 8 4
3 4 1 7
Find the Optimal Mixed Strategy in 2X2 Table
Suppose row player has two strategies, 1 and 2, and column player has two strategies, A and B.
For row player:
Let p be probability of selecting row strategy 1. Then the probability of selecting row strategy 2 is (1-p).
Represent EA and EB by p, where EA (EB) is the expected payoff of the row player if the column player chose column strategy A (B).
Set EA = EB , and solve p from the equation.
For column player:
Let p be probability of selecting column strategy A. Then the probability of selecting column strategy B is (1-p).
Represent E1 and E2 by p, where E1 (E2) is the expected payoff of the row player if the column player chose column strategy A (B).
Set E1 = E2 , and solve p from the equation.