Nash Equilibria, Basins of Attraction, and Chaos in Economics

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Nash Equilibria, Basins of Attraction, and Chaos
in Economics

• Presented by
• Dr. Douglas Anderson
• Dustin White
• Nicholas Myran

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Discrete Logistic Equation

• xn r xn-1 (1 xn-1)
• Origin
• Recursion relation
• Let h(x) r x (1 x)

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Fixed Points h(x) r x (1 x)

• Definition
• Solve h(x) x
• Example r 3.5
• 3.5 x (1 x) x
• -3.5 x2 2.5 x 0
• x 0 , x .714286
• Fixed points
• x 0 , x 1- (1/r)

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Periodic Points h(x) r x (1 x)

• Definition
• Solve hn(x) x to find period-n points
• Example for period-2 r 3.5
• h2(x) h(h(x)) h(3.5 x (1-x)) x
• -42.875x4 42.875x3 55.125x2 11.25x 0
• x 0, 0.428571, 0.714286, 0.857142
• Since 0 and .714286 are fixed points, 0.428571
  and 0.857142 are the points for our attracting
  2-cycle.

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Stability at the Fixed Points h(x) r x (1 x)

• Theorem Assume x is a fixed point for h and h
  has a continuous 1st derivative. Then,
• Hyperbolic case
• If h(x) lt 1, x is an attractor
• If h(x) gt 1, x is a repellor
• Non-hyperbolic case
• If h(x) 1, this test is inconclusive

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Bifurcations on Logistic Model h(x) r x (1 x)

• When 0 lt r lt 1
• x 0 is an attractor
• x 1- (1/r) is a repellor
• At r 1 Bifurcation
• Fixed point x 0 becomes repellor
• Fixed point x 1 (1/r) becomes attractor
• Continues from 1 lt r lt 3

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Bifurcations (cont.) h(x) r x (1 x)

• At r 3 Bifurcation
• x 1- (1/r) becomes a repellor
• An attracting 2-cycle is born of the form
• (1/2r2)(r r2 ? r ?(r2 2r 3))
• Continues from 3 lt r lt 1?6

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Bifurcations (cont.) h(x) r x (1 x)

• At r 1?6 Bifurcation
• Period-2 cycle becomes a repellor
• An attracting 4 cycle is born
• Continues from 1?6 lt r lt 3.54409
• At r 3.54409 Bifurcation
• Period-4 cycle becomes a repellor
• An attracting 8-cycle is born
• Continues from 3.54409 lt r lt 3.56440

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Bifurcations (cont.) h(x) r x (1 x)

• As r continues to increase, period doubling
  occurs more rapidly.
• At r 1?8 (approx. 3.82843), period-3 cycle
  comes into existence.

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Sarkovskiis Ordering of Natural Numbers

• For all n 1, 2, 3
• 3 gt 5 gt 7 gt 9 gt gt 2n1
• 2(3) gt 2(5) gt 2(7) gt 2(9) gt gt 2(2n1)
• 22(3) gt 22(5) gt 22(7) gt 22(9) gt gt22(2n1)
• .
• .
• .
• 2n(3) gt 2n(5) gt 2n(7) gt 2n(9) gt gt 2n(2n1)
• 2n gt 2n-1 gt 2n-2 gtgt 8 gt 4 gt 2 gt 1

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Bifurcations (cont.) h(x) r x (1 x)

• Thus we can infer from Sarkovskiis Ordering that
  there exists a period-n cycle for all n in the
  natural numbers from 1?8 lt r lt 4
• Finally, at r gt 4, there is no attractor
• This is referred to as chaos

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Cournot Duopoly Model

• A Dynamical System of Two Nonlinear Difference
  Equations
• xn1 (1-A)xn AByn(1-yn)
• yn1 (1-A)yn ABxn(1-xn)

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Basic Economic Market Structures

• Different Market structures
• Pure Competition many competing firms producing
  the same product
• Monopoly market controlled by a single seller
• Oligopoly market dominated by a few large
  suppliers
• Duopoly an oligopoly with exactly two firms

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Duopoly

• Competing firms produce goods that are perfect
  substitutes
• Both firms must consider the actions and
  reactions of the competitor
• Each firm forms expectations of the other firms
  output in order to determine a profit maximizing
  quantity to produce in the next time period.
• this situation is called strategic interdependence

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Cournot Duopoly Map of Kopel

• Balance between the practicality and mathematical
  tractability
• Actual Cournot Duopoly much more complex
• Model is simplified to make mathematical analysis
  possible, but still mimics the behavior of a real
  Cournot duopoly
• Even this simple model has very complex and
  interesting dynamics

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xn1 (1-A)xn AByn(1-yn)yn1 (1-A)yn
ABxn(1-xn)

• The variables x and y represent the quantities
  that each firm produces
• x,y ? 0,1

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xn1 (1-A)xn AByn(1-yn)yn1 (1-A)yn
ABxn(1-xn)

• Cournot Duopoly model consists of two parameters,
  A and B
• A represents how the firms will form their
  beliefs according to adaptive expectations
• We are most concerned with A ? 0,1, but A is
  always greater than or equal to 0
• B measures the intensity of the affect that one
  firms actions has on the other firm
• We are most concerned with B ? 3, but B is always
  greater than or equal to 0.

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Cournot Duopoly Model on the line y x

• Anything that ends up on the line y x, stays on
  the line for all time.
• Cournot duopoly model then becomes just a
  one-dimensional function of x
• f(x) (1-A)xABx(1-x)
• which behaves similarly to our logistic function.

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Equivalence to Logistic

• Since we know the bifurcation values for r in the
  logistic model, we can use the intervals of r in
  which we know the behavior of the model to
  predict the behavior of our Cournot model using
  the relationship r 1 A AB
• Graphical depiction of this behavior

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Cournot Model in the 2-D Case

• In this case, y ? x
• In this model, the term Nash-Equilibrium is used
  interchangeably with the term fixed points
• At Nash Equilibrium, each firms strategy is the
  most optimal reaction to the other firms
  predicted strategy

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Nash-Equilibrium

• There are four fixed points that exist depending
  on the values of A and B
• We call them fp1, fp2, fp3, fp4
• fp1 (0, 0)
• Exists for all A and B
• fp2 ((B-1)/B, (B-1)/B)
• Exists for B ? 0
• Always on line y x

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Nash-Equilibrium

• fp3 ((1B-?(B2-2B-3))/2B, (1B?(B2-2B-3))/2B)
• fp4 ((1B?(B2-2B-3))/2B, (1B-?(B2-2B-3))/2B)
• fp3 and fp4 are reflections of each other over
  the y x line
• fp3 and fp4 dont exists until B ? 3

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Stability of Nash Equilibria

• Behavior patterns similar to the logistic
  equation still exist in 2D, though the analysis
  is more complex
• 3 Types
• Attractors Sinks
• Repellors Sources
• Saddles

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A-B Planes

• fp1
• fp2
• fp3 and fp4

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Phase Portraits

• xn1 (1-A)xn AByn(1-yn)yn1 (1-A)yn
  ABxn(1-xn)
• Simulations (0.01, 0.86) vs. (0.01, 0.87)
• (0.005, 0.003)
• (0.005, 0.004)
• (0.005, 0.005)
• (0.005, 0.006)

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Basins of Attraction

• A0.2, B3.4
• A0.6, B3.4
• A0.7, B3.4
• A1.0, B3.83

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Acknowledgments

• Michael Kopel, University of Technology, Vienna,
  Austria
• Gian Italo Bischi, University of Urbino, Urbino,
  Italy