Strategy space

Figure 12.30 Illustration of the calibration strategy space, where any calibration strategy can be mapped by its relevance (x-axis) and it accuracy (y-axis). In this figure, several calibration strategies are mapped onto this...

Theorem 1 (Debreu 1952). Suppose that for each player the strategy space is compact and convex and the payoff function is continuous and quasi-concave with respect to each player s own strategy. Then there exists at least... 

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

Theorem 6. Suppose the strategy space of the game is convex and all equilibria are interior. Then if the determinant H is negative quasi-definite (i.e., if the matrix H + is negative definite) on the players strategy set, there is a unique NE. 

Interior equilibrium is the one in which first-order conditions hold for each player. The alternative is boundary equilibrium in which at least one of the players select the strategy on the boundary of his strategy space. 

This concept is wider than the Sub-game Perfect Nash Equilibrium (SPNE), since the allocation of each firm in the strategy space does not need to be optimal given other players allocation. A direct implication is that this approach encompasses the concept of SPNE. 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

In a systematic search there is a defined endpoint to the procedure, which is reached whe all possible combinations of bond rotations have been considered. In a random search, ther is no natural endpoint one can never be absolutely sure that all of the minimum energ conformations have been found. The usual strategy is to generate conformations until n new structures can be obtained. This usually requires each structure to be generate many times and so the random methods inevitably explore each region of the conformc tional space a large number of times. 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Detailed reaction dynamics not only require that reagents be simple but also that these remain isolated from random external perturbations. Theory can accommodate that condition easily. Experiments have used one of three strategies. (/) Molecules ia a gas at low pressure can be taken to be isolated for the short time between coUisions. Unimolecular reactions such as photodissociation or isomerization iaduced by photon absorption can sometimes be studied between coUisions. (2) Molecular beams can be produced so that motion is not random. Molecules have a nonzero velocity ia one direction and almost zero velocity ia perpendicular directions. Not only does this reduce coUisions, it also aUows bimolecular iateractions to be studied ia intersecting beams and iacreases the detail with which unimolecular processes that can be studied, because beams facUitate dozens of refined measurement techniques. (J) Means have been found to trap molecules, isolate them, and keep them motionless at a predetermined position ia space (11). Thus far, effort has been directed toward just manipulating the molecules, but the future is bright for exploiting the isolated molecules for kinetic and dynamic studies. 

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