Contraction mapping theorem

The following convergence theorem (sometimes called the contraction mapping theorem) will provide this information. 

There is no guarantee of convergence (the contracting mapping theorem must be applied, but it is conservative). 

We will also use the theorem on contraction mappings. A mapping S y —> y is called a contraction mapping if it is Lipschitz continuous,... 

Theorem 1.21. If S is a contraction mapping in a Hilbert space V then there exists a fixed point u such that Su = u and solutions u of the equation... 

A challenge associated with the contraction mapping argument is finding best response functions because in most SC models best responses cannot be found explicitly. Fortunately, Theorem 5 only requires the derivatives of the best response functions, which can be done using the Implicit Function Theorem (from now on, IFT, see Bertsekas 1999). Using the IFT, Theorem 5 can... 

As with the contraction mapping approach, with two players the Theorem becomes easy to visualize. Suppose we have found best response functions X = fi x2) and X2 = /2( i) as in Figure 2,2. Find an inverse function X2 = fi xi) and construct an auxiliary function g xi) = f xi) — f2 xi) that measures the distance between two best responses. It remains to show that g x ) crosses zero only once since this would directly imply a single crossing point of fi xi) and f2 x2)- Suppose we could show that every time crosses zero, it does so Jrom below. If that is the case, we are assured there is only a single crossing it is impossible for a continuous function to cross zero more than once from below because it would also have to cross zero from above somewhere. It can be shown that the function g xi) crosses zero only from below if the slope of g xi) at the crossing point is positive as follows... 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

Theorem 4. If the best response mapping is a contraction on the entire strategy space, there is a unique NE in the game. 

While Theorem 4 is a starting point towards a method for demonstrating uniqueness, it does not actually explain how to validate that a best reply mapping is a contraction. Suppose we have a game with n players each endowed... 

In either case. Van Kampen s theorem can be applied. The space A is contractible, the intersection A fl B is homeomorphic to a circle, and the space B deformation retracts to a circle. To see the last of these facts, simply retract the punctured unit disk to its boundary, and note that the antipodal self-identification of the boundary again produces a circle. We have nfiA) = 0 and 7Ti(B) = Tri(AnB) = Z. The fundamental group homomorphisms induced by the inclusion maps i AC B A and An B B are the following ... 

Proof. Assume that a = E,B,p) is a fiber bundle, and assume that B is contractible. Let q B B he map that takes the whole space B to some point b B. By our assumptions, the maps q and idg are homotopic. It follows by Theorem 8.8 that the pullbacks q a and idgo are isomorphic. On the other hand, we see that (7 0 is a trivial brmdle, and idgo = E. ... 

Remark 15.13. It may be worthwhile to explicitly mention the following special case when all the spaces v) in the diagram are contractible, then we have a diagram map from V to the point diagram defined in Example 15.9(0). It follows by Theorem 15.12 that the homotopy colimit of our diagram is homotopy equivalent to the base trisp, with the homotopy equivalence given by Pfc. 

Note that the specific topological structure of is not yet sufficient for realizing a blue sky catastrophe there exists also a quantitative condition in Theorem 12.9 which is needed to ensure contraction. If this condition is violated, i.e. if fo 1 at some then infinitely many bifurcations occur in the region /i > 0, just like the cases considered in the preceding sections. Indeed, consider the lift of the map (12.4.1) onto... 

---