Matakuliah
Tahun
: K0442-Metode Kuantitatif
: 2009
Utility and the Game Theory
Pertemuan 3
Material Outline
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The Meaning Of Utility
Utility and Decision Making
Utility : Other Consideration
Introduction to game Theory
The Meaning of Utility
Utility is the measure of the total worth of a
consequence reflecting a decision maker’s attitude
toward considerations such as profit, loss, and risk
Lottery is the hypothetical investment alternative
with a probability p of obtaining the best payoff and
probability of (1-p) of obtaining the worst payoff
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The Meaning of Utility
• Example :Payoff Table for Swofford Inc.
State of Nature
Decision
Alternative
Price Up s1
Price Stable
Price Down
Investment A, d1
$ 30,000
$20,000
-$50,000
Investment B, d2
$50,000
-$20,000
-30,000
0
0
0
Do not Invest
The best estimate of the probability that real estate prices will go up is 0.3; the best
estimate of the probability that prices will remain stable is 0.5; the best estimate of the
probability that prices will go down is 0.2
EV(d1) = 0.3 (30,000) +0.5(20,000) + 0.2 (-50,000) = 9000
EV(d1) = 0.3 (50,000) +0.5(-20,000) + 0.2 (-30,000) = -1000
EV(d1) = 0.3 (0) +0.5(0) + 0.2 (0) = 0
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Utility and Decision Making
$50,000 is the Best payoff and -50,000 is the worst.
Utility of -50,000 = U (-50,000) = 0
Utility of $50,000 = U (50,000) = 10
Determine the utility associated with every other payoff ($30,000,
20,000,…..,…). Assume p=0.95
Lottery : Swofford obtains a payoff of $50,00 with probability p
And payoff -$50,000 with probability (1-p)
U(30,000) = pU(50,000) + (1-p)U(-50,000)
= 0.95(10) + (0.05)(0)
= 9.5
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Utility and Decision Making
• Utility Of Monetary payoffs for Swofford, Inc.
Monetary
Value
Indifference
Value of p
$ 50,000 Does not apply
10.0
30,000
0.95
9.5
20,000
0.90
9.0
0
0.75
7.5
-20,000
0.55
5.5
-30,000
0.40
4.0
-50,000 Does not apply
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Utility value
0
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Utility and Decision Making
• The Expected Utility Approach
EU (d i )   j i P( s j )U ij
N
EU(d1) = 0.3(9.5) + 0.5 (9.0) + 0.2 (0) = 7.35
EU(d2) = 0.3(10) + 0.5 (5.5) + 0.2 (4.0) = 6.55
EU(d3) = 0.3(7.5) + 0.5 (7.5) + 0.2 (7.5) = 7.50
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8
Utility: Other Considerations
• Risk Avoider Versus Risk Taker
Risk Avoider
A decision maker who would choose a guaranteed
payoff over a lottery with a better expected payoff
Risk Taker
A decision maker who would choose a lottery over a
better guaranteed payoff
Risk-neutral Decision Maker
A Decision maker who a is neutral to risk
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Utility: Other Considerations
• Utility Function for Risk, Avoider Risk, Risk taker,
and risk Neutral Decision maker
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Risk Avoider
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Utility
6
Risk Neutral
4
Risk Taker
2
0
-50
Bina Nusantara University
-35
- 10
0
10
35
Monetary Value ($1000)
50
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Introduction to game Theory
• Game theory was developed by John Von Neumann and
Oscar Morgenstern in 1944 – Economists!
• One of the fundamental principles of game theory, the idea of
equilibrium strategies was developed by John F. Nash, Jr. (A
Beautiful Mind), a Bluefield, WV native.
• Game theory is a way of looking at a whole range of human
behaviors as a game.
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Components of a Game
• Games have the following characteristics:
– Players
– Rules
– Payoffs
• Based on Information
– Outcomes
– Strategies
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Types of Games
• We classify games into several types.
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By the number of players:
By the Rules:
By the Payoff Structure:
By the Amount of Information Available to the players
Games as Defined by the Number of Players:
• 1-person (or game against nature, game of chance)
• 2-person
• n-person( 3-person & up)
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Games as Defined by the Rules:
• These determine the number of options/alternatives in
the play of the game.
• The payoff matrix has a structure (independent of value)
that is a function of the rules of the game.
• Thus many games have a 2x2 structure due to 2
alternatives for each player.
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Games as Defined by the Payoff Structure:
• Zero-sum
• Non-zero sum
• (and occasionally Constant sum)
– Examples:
• Zero-sum
– Classic games: Chess, checkers, tennis, poker.
– Political Games: Elections, War
• Non-zero sum
– Classic games: Football (?), D&D, Video games
– Political Games: Policy Process
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Games defined by information
• In games of perfect information, each player
moves sequentially, and knows all previous
moves by the opponent.
– Chess & checkers are perfect information games
– Poker is not
– In a game of complete information, the rules are
known from the beginning, along with all possible
payoffs, but not necessarily chance moves
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Strategies
• We also classify the strategies that we employ:
• It is natural to suppose that one player will attempt to anticipate what the other
player will do. Hence
– Minimax - to minimize the maximum loss - a defensive strategy
– Maximin - to maximize the minimum gain - an offensive strategy.
Iterated Play
• Games can also have sequential play which lends to more complex strategies.
– (Tit-for-tat - always respond in kind.
– Tat-for-tit - always respond conflictually to cooperation and cooperatively towards conflict.
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Game or Nash Equilibria
• Games also often have solutions or equilibrium points.
• These are outcomes which, owing to the selection of
particular reasonable strategies will result in a
determined outcome.
• An equilibrium is that point where it is not to either
players advantage to unilaterally change his or her
mind.
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Saddle points
• The Nash equilibrium is also called a saddle point because of the two curves used
to construct it:
• an upward arching Maximin gain curve
• and a downward arc for minimum loss.
• Draw in 3-d, this has the general shape of a western saddle (or the shape
of the universe; and if you prefer). .
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Some Simple Examples
• Battle of the Bismark Sea
• Prisoner’s Dilemma
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The Battle of the Bismarck Sea
• Simple 2x2 Game
• US WWII Battle
Japanese Options
Sail North Sail
South
US Options
Recon
North
2 Days
2 Days
Recon
South
1 Day
3 Days
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The Battle of the Bismarck Sea
Japanese Options
US Options
Sail North Sail
South
Minima
of Rows
Recon
North
2 Days
2 Days
2
Recon
South
1 Day
3 Days
1
2
3
Maxima of Columns
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The Battle of the Bismarck Sea - examined
• This is an excellent example of a two-person zero-sum game with a
Nash equilibrium point.
• Each side has reason to employ a particular strategy
– Maximin for US
– Minimax for Japanese).
• If both employ these strategies, then the outcome will be Sail
North/Watch North.
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Decision Tree
Decision Tree Version of Battle of Bismark Sea
Search
North
2
Sail North
Search
South
1
Japanese
Search
North
2
Sail South
Search
South
3
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The Prisoners Dilemma
• The Prisoner’s dilemma is also 2-person game but not a zero-sum game.
• It also has an equilibrium point, and that is what makes it interesting.
• The Prisoner's dilemma is best interpreted via a “story.”
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A Simple Prisoner’s Dilemma
Prisoner A
~ Confess Confess
~ Confess
Prisoner B
-1
-1
Confess
0
-10
-10
0
-5
-5
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Alternate Prisoner’s Dilemma Language
Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.
Prisoner A
Cooperate Defect
Cooperate
Prisoner B
-1
-1
Defect
-10
-10
0
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0
-5
-5
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What Characterizes a Prisoner’s Dilemma
Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.
Prisoner A
Cooperate Defect
Prisoner B
Cooperate
Reward
Tempt
Reward
Sucker
Defect
Sucker
Punish
Tempt
Punish
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What makes a Game a Prisoner’s Dilemma?
• We can characterize the set of choices in a PD as:
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–
–
–
Temptation (desire to double-cross other player)
Reward (cooperate with other player)
Punishment (play it safe)
Sucker (the player who is double-crossed)
• A game is a Prisoner’s Dilemma whenever:
– T>R>P>S
– Or Temptation > Reward > Punishment > Sucker
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What is the Outcome of a PD?
• The saddle point is where both Confess
• This is the result of using a Minimax strategy.
• Two aspects of the game can make a difference.
– The game assumes no communication
– The strategies can be altered if there is sufficient trust between the players.
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Solutions to PD?
• The Reward option is the joint optimal payoff.
• Can Prisoner’s reach this?
– Minimax strategies make this impossible
– Are there other strategies?
Iterated Play
• The PD is a single decision game in which the Nash
equilibrium results from a dominant strategy.
• In iterated play (a series of PDs), conditional strategies
can be selected
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The Theory of Metagames
• Metagames step back from the game and look at the other players
strategy
• Strategic choice is based upon opponents choice.
• For instance, we could adopt the following strategies:
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–
–
–
Tit-for-tat
Tat-for-tit
Choose Confess regardless
Choose ~Confess regardless
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A Prisoner’s Dilemma Metagame
Prisoner A
~ Confess Confess
Tit-for-tat Tat-for-tit
Regardless Regardless
~ Confess
Pris
B
Confess
-1
-1
0
-10
-10
0
-1
-1
-5
-5
0
-10
-5
-5
-10
0
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