Transcript Economics of Gambling - Northern Illinois University

Game Theory
“ If it’s true that we are here to help others,
then what exactly are the others here for? ”
- George Carlin
What is Game Theory?
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Game Theory: The study of situations involving
competing interests, modeled in terms of the
strategies, probabilities, actions, gains, and losses of
opposing players in a game. A general theory of
strategic behavior with a common feature of
Interdependence.
In other Words: The study of games to determine
the probability of winning, given various strategies.
Example: Six people go to a restaurant.
- Each person pays for their own meal – a simple decision
problem
- Before the meal, every person agrees to split the bill evenly
among them – a game
A Little History on Game Theory
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John von Neumann and Oskar Morgenstern
- Theory of Games and Economic Behaviors
John Nash
- "Equilibrium points in N-Person Games", 1950,
Proceedings of NAS.
"The Bargaining Problem", 1950, Econometrica.
"Non-Cooperative Games", 1951, Annals of
Mathematics.
Howard W. Kuhn – Games with Imperfect information
Reinhard Selten (1965) -“Sub-game Perfect Equilibrium"
(SPE) (i.e. elimination by backward induction)
John C. Harsanyi - "Bayesian Nash Equilibrium"
Some Definitions for Understanding
Game theory
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Players-Participants of a given game or games.
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Rules-Are the guidelines and restrictions of who can do what and
when they can do it within a given game or games.
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Payoff-is the amount of utility (usually money) a player wins or
loses at a specific stage of a game.
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Strategy- A strategy defines a set of moves or actions a player
will follow in a given game. A strategy must be complete, defining
an action in every contingency, including those that may not be
attainable in equilibrium
Dominant Strategy -A strategy is dominant if, regardless of
what any other players do, the strategy earns a player a larger
payoff than any other. Hence, a strategy is dominant if it is always
better than any other strategy, regardless of what opponents may
do.
Important Review Questions for
Game Theory
 Strategy
• Who are the players?
• What strategies are available?
• What are the payoffs?
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What are the Rules of the game
•
•
•
•
What
What
What
What
is the time-frame for decisions?
is the nature of the conflict?
is the nature of interaction?
information is available?
Five Assumptions Made to
Understand Game Theory
1. Each decision maker ("PLAYER“) has available to him two or
more well-specified choices or sequences of choices (called
"PLAYS").
2. Every possible combination of plays available to the players
leads to a well-defined end-state (win, loss, or draw) that
terminates the game.
3. A specified payoff for each player is associated with each endstate (a ZERO-SUM game means that the sum of payoffs to all
players is zero in each end-state).
4. Each decision maker has perfect knowledge of the game and of
his opposition; that is, he knows in full detail the rules of the
game as well as the payoffs of all other players.
5. All decision makers are rational; that is, each player, given two
alternatives, will select the one that yields him the greater
payoff.
Cooperative Vs. Non-Cooperative
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Cooperative Game theory has perfect
communication and perfect contract
enforcement.
A non-cooperative game is one in which
players are unable to make enforceable
contracts outside of those specifically
modeled in the game. Hence, it is not
defined as games in which players do not
cooperate, but as games in which any
cooperation must be self-enforcing.
Interdependence of Player
Strategies
1) Sequential – Here the players move
in sequence, knowing the other
players’ previous moves.
- To look ahead and reason Back
2) Simultaneous – Here the players act
at the same time, not knowing the
other players’ moves.
- Use Nash Equilibrium to solve
Simultaneous-move Games of Complete
Information
 A set of players (at least two players)
S1 S2 ... Sn
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For each player, a set of strategies/actions
{Player 1, S1, Player 2,S2 ... Player Sn}
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Payoffs received by each player for the
combinations of the strategies, or for each
player, preferences over the combinations of
the strategies
ui(s1, s2, ...sn), for all s1S1, s2S2, ... snSn
Nash’s Equilibrium
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This equilibrium occurs when each player’s strategy
is optimal, knowing the strategy's of the other
players.
A player’s best strategy is that strategy that
maximizes that player’s payoff (utility), knowing the
strategy's of the other players.
So when each player within a game follows their best
strategy, a Nash equilibrium will occur.
Logic
Logic
Definition: Nash Equilibrium
In the normal-form game {S1 , S2 , ..., Sn , u1 , u2 , ...,
un}, a combination of strategies ( s1* ,..., sn* ) is a Nash
equilibrium if, for every player i,
ui ( s1* ,..., si*1, si* , si*1,..., sn* )
 ui ( s1* ,..., si*1, si , si*1,..., sn* )
for all si  Si . That is, si* solves
Maximize
Subject to
ui ( s1*,..., si*1, si , si*1,..., sn* )
si  Si
Given others’
choices, player i
cannot be better-off
if she deviates from
si*
Nash’s Equilibrium cont.:
Bayesian Nash Equilibrium
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The Nash Equilibrium of the imperfectinformation game
• A Bayesian Equilibrium is a set of strategies such
that each player is playing a best response, given
a particular set of beliefs about the move by
nature.
• All players have the same prior beliefs about the
probability distribution on nature’s moves.
–
So for example, all players think the odds of player 1
being of a particular type is p, and the probability of her
being the other type is 1-p
Bayes’ Rule
• A mathematical rule of logic explaining
how you should change your beliefs in
light of new information.
• Bayes’ Rule:
P(A|B) = P(B|A)*P(A)/P(B)
• To use Bayes’ Rule, you need to know a few
things:
– You need to know P(B|A)
– You also need to know the probabilities of A and
B
Examples of Where Game
Theory Can Be Applied
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Zero-Sum Games
Prisoner’s Dilemma
Non-Dominant Strategy moves
Mixing Moves
Strategic Moves
Bargaining
Concealing and Revealing
Information
Zero-Sum Games
Penny Matching:
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Each of the two players has a penny.
Two players must simultaneously choose whether to show the
Head or the Tail.
Both players know the following rules:
-If two pennies match (both heads or both tails) then player
2 wins player 1’s penny.
-Otherwise, player 1 wins player 2’s penny.
Player 2
Head
Head
Player 1
Tail
-1 ,
Tail
1
1 , -1
1 , -1
-1 ,
1
Prisoner’s Dilemma
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No communication:
- Strategies must be undertaken
without the full knowledge of what
the other players (prisoners) will do.
 Players (prisoners) develop dominant
strategies but are not necessarily the
best one.
Payoff Matrix for Prisoner’s
Dilemma
Ted
Confess
Confess
Both get 5
years
1 year for
Bill
10 years for
Ted
10 years for
Bill
1 year for
Ted
Both get 3
years
Bill
Not Confess
Not Confess
Solving Prisoners’ Dilemma
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Confess is the dominant strategy for both Bill
and Ted.
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Dominated strategy
-There exists another strategy which always does better
regardless of other players’ choices
-(Confess, Confess) is a Nash equilibrium but is not
always the best option
Players
Ted
Strategies
Bill
Confess
Not Confess
Payoffs
Confess
-5, -5
-10,-1
Not Confess
-1,-10
-3,-3
Non-Dominant strategy games
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There are many games when players do
not have dominant strategies
- A player’s strategy will sometimes
depend on the other player's strategy
- According to the definition of
Dominant strategy, if a player depends
on the other player’s strategy, he has
no dominant strategy.
Non-Dominant strategy games
Ted
Confess
Confess
Bill
Not Confess
Not Confess
7 years for Bill
6 years for Bill
2 years for Ted
4 years for Ted
9 years for Bill
5 years for Bill
0 years for Ted
3 years for Ted
Solution to Non-Dominant
strategy games
Ted Confesses
Bill
Confesses
7 years
Ted doesn’t confess
Bill
Not confess
Confesses
9 years
6 years
Not confess
5 years
Best Strategies
There is not always a dominant strategy and
sometimes your best strategy will depend
on the other players move.
Examples of Where Game
Theory Can Be Applied
Mixing Moves
Examples in Sports (Football & Tennis)
Strategic Moves
War –Cortes Burning His Own Ships
Bargaining
Splitting a Pie
Concealing and Revealing Information
Bluffing in Poker
Applying Game Theory to NFL
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Solving a problem within the Salary Cap.
How should each team allocate their
Salary cap. (Which position should get
more money than the other)
The Best strategy is the most effective
allocation of the team’s money to obtain
the most wins.
Correlation can be used to find the best
way to allocate the team’s money.
What is a correlation?
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A correlation examines the relationship
between two measured variables.
- No manipulation by the experimenter/just
observed.
- E.g., Look at relationship between height and
weight.
You can correlate any two variables as long as
they are numerical (no nominal variables)
Is there a relationship between the height and
weight of the students in this room?
- Of course! Taller students tend to weigh more.
Salaries vs. Points scored/Allowed
Position
Correlation
T-test
Position
Correlation
T-test
RB
.27
2.67
DE
.25
2.52
k
.25
2.52
CB
.15
1.48
TE
.17
1.74
S
.06
.61
OL
.04
.34
LB
.05
.52
QB
.03
.32
DT
.04
.34
WR
-.03
-.30
P
0
0
Running Backs edge out Kickers for best correlation of position
spending to team points scored. Tight Ends also show some
modest relationship between spending and points.
The Defensive Linemen are the top salary correlators, with
cornerbacks in the second spot
Total Position spending vs. Wins
Position
Wins
Points Scored
Points Allowed
Total Position
Correlation
Correlation
Correlation
Spending
K
0.27
0.27
0.17
0.27
CB
0.17
0.12
0.12
0.23
TE
0.16
0.2
0.15
0.17
OL
0.15
0.02
0.2
0.08
RB
0.11
0.11
-0.03
0.26
QB
0.1
0.08
0.08
0.04
DE
0.08
-0.14
0.17
0.16
P
0.08
0.01
0.03
0.04
LB
0.05
-0.08
0.15
-0.02
S
0.03
0.02
0.05
0.04
DT
-0.02
-0.01
0.02
-0.04
WR
-0.08
-0.01
-0.04
0.01
Note: Kicker has highest correlation also OL is ranked high also.
What this means
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NFL teams are not very successful at
delivering results for the big money
spent on individual players.
There's high risk in general, but
more so at some positions over
others in spending large chunks of
your salary cap space.
Future Study
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Increase the Sample size.
Cluster Analysis
Correspondence analysis
Exploratory Factor Analysis
Conclusion
 There are many advances to this
theory to help describe and prescribe
the right strategies in many different
situations.
 Although the theory is not complete,
it has helped and will continue to
help many people, in solving
strategic games.
References
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Nasar, Sylvia (1998), A Beautiful Mind: A Biography of John
Forbes Nash, Jr., Winner of the Nobel Prize in Economics, 1994.
Simon and Schuster, New York.
Rasmusen, Eric (2001), Games and Information: An Introduction
to Game Theory, 3rd ed. Blackwell, Oxford.
Gibbons, Robert (1992), Game Theory for Applied Economists.
Princeton University Press, Princeton, NJ.
Mehlmann, Alexander. The Games Afoot! Game Theory in Myth
and Paradox. AMS, 2000.
Wiens, Elmer G. Reduction of Games Using Dominant Strategies.
Vancouver: UBC M.Sc. Thesis, 1969.
H. Scott Bierman and Luis Fernandez (1993) Game Theory with
Economic Applications, 2nd ed. (1998), Addison-Wesley Publishing
Co.
D. Blackwell and M. A. Girshick (1954) Theory of Games and
Statistical Decisions, John Wiley & Sons, New York.
NFL Official, 2004 NFL Record and Fact Book; Time Inc. Home
Entertainment, New York, New York.
Questions?
Comments?