Best response mapping

Taken together, the two best response functions form a best response mapping or in the more general case RP RP, Clearly, the best response is the best player i can hope for given the decisions of other players. Naturally, an outcome in which all players choose their best responses is a candidate for the non-cooperative solution. Such an outcome is called a Nash Equilibrium (hereafter NE) of the game. 

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

One can think of a contraction mapping in terms of iterative play player 1 selects some strategy, then player 2 selects a strategy based on the decision by player 1, etc. If the best response mapping is a contraction, the NE obtained as a result of such iterative play is stable but the opposite is not necessarily true, i.e., no matter where the game starts, the final outcome is the same. See also Moulin (1986) for an extensive treatment of stable equilibria. 

Hence, the best response mapping in the newsvendor game is a contraction and the game has a unique and stable NE. 

Method 3. Univalent mapping argument. Another method for demonstrating uniqueness of equilibrium is based on verifying that the best response mapping is one-to-one that is, if f x) is 2i RT map-... 

The open-loop strategy implies that each players control is only a function of time, Ui = Ui t). A feedback strategy implies that each players control is also a function of state variables, ui = Ui t Xi t) Xj(t)). As in the static games, NE is obtained as a fixed point of the best response mapping by simultaneously solving a system of first-order optimality conditions for the players. Recall that to find the optimal control we first need to form a Hamiltonian. If we were to solve two individual non-competitive optimization problems, the Hamiltonians would be Hi = fi XiQi, i = 1,2, where Xi t) is an adjoint multiplier. However, with two players we also have to account for the state variable of the opponent so that the Hamiltonian becomes... 

Analysis of down-stream single element profile maps indicates the very fine, -150 mesh, size fraction provides the best response to Tameapa copper-molybdenum mineralization and thus is the optimal size fraction for use in the... 

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... 

Such a vertex may be close to the optimum conditions. In that case, a repeated experiment will still give the best result in this vertex. If a satisfactory result is obtained, then stop and use these conditions as the preferred conditions. It is, of course, also possible to continue and more precisely locate the optimum conditions by reducing the size of simplex, or by determining a response surface model and using it as a map to locate the optimum conditions. 

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