Chaos in the Cobweb Model with a New Learning Dynamic George Waters Illinois State University

Cobweb Model stability supply & demand slopes Dynamic switching between forecasting strategies Brock and Hommes (1997) –MNL Sethi and Franke (1995) –EGT Branch and McGough (2005) Chaos

BNN Brown, von Neumann and Nash (1950) Well behaved positive correlation inventiveness Nash stationarity Dynamics under BNN and MNL interpretation of steady states stability analysis / bifurcations

Notation q k,t – fraction using strategy k  k,t – payoff to strategy k population average payoff excess payoff

Excess Payoff Dynamics (Sandholm 2006)

BNN

Desiderata Positive Correlation –weak monotonicity condition Inventiveness –positive excess payoff implies a positive fraction of followers

Selection Dynamics and Desiderata Proposition: BNN satisfies Positive Correlation and Inventiveness. MNL does not satisfy Positive Correlation. Imitative Dynamics (replicator) do not satisfy Inventiveness.

MNL  is search intensity. q>0 for all k

Imitative Dynamics  is the weighting function linear  – replicator Strategies cannot be reborn.

Alternatives Infinite search intensity in MNL Imitative dynamics with drift combining imitative and excess payoff dynamics Best response dynamics –Gilboa and Matsui (1991) Pairwise comparison dynamics –Sandholm (2006) –agents compare their payoff with payoff of a random strategy –scarcity of data –2 strategies – equivalent to BNN

Nash stationarity Steady states correspond to Nash equilibria BNN satisfies MNL does not Imitative dynamics –Lyapunov stable steady states correspond to NE Cobweb model doesn’t have NE.

Cobweb Model Quantity supplied by firms determined by price expectations and cost. Rational and Naïve predictors Linear demand determines price. Payoffs are firm profits –adjusted for cost of the predictor

Predictors Rational - cost C - used by q Naïve - costless - used by 1-q S&D:

Price dynamics Ratio < 1 implies a stable steady state at p t = 0.

Payoffs

Excess Payoffs

Payoff Difference Function

Evolution Function (p t+1,q t+1 ) =F(p t,q t )

Steady States

Stability If C>0, (p t,q t ) = (0,0) is stable iff In a stable market, no incentive to incur the cost of the rational forecast. Under MNL, this steady state only exists for infinite search intensity.

Stability of the 2-cycle

Eigenvalue condition does not guarantee (in)stability(!). The 2-cycle is exponentially unstable.

Chaos Bifurcations in Different than MNL –no saddles under BNN –unstable two cycle has two eigenvalues passing through -1 Periodic attractors Strange attractors

 - BNN Weibull (1994)

 - BNN For the 2-cycle is stable but not asymptotically unstable For MNL, dynamics depend on search intensity.

Summary BNN is well behaved –PC, I, NS –The edges of the simplex are not problematic. Cobweb model –Stable steady states are easy to interpret –Dynamics don’t depend strongly on  

Imitative Dynamics Similar steady states Another steady state at q=1 –all edges of the simplex are steady states –lacking Inventiveness –if C>0, hard to justify

Stability If C=0, (p t,q t ) = (0,1) is the unique stable steady state. No reason to use the naïve forecast. Under MNL, both predictors have equal population shares.

Counterexample