Quasi-concavity

This section presents the definitions, properties and relationships of quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions. 

Definition 2.3.2 (Quasi-concave function) /(jc) is quasi-concave if... 

Illustration 2.3.1 Figure 2.12 shows a quasi-convex and quasi-concave function. 

Similarly a concave function is also quasi-concave since... 

Figure 2.12 Quasi-convex and quasi-concave functions...

Definition 2.3.4 (Strictly quasi-concave function) f(x) is strictly quasi-concave if... 

Illustration 2.3.2 Figure 2.13 shows a strictly quasi-convex and strictly quasi-concave function. Theorem 2.3.2... 

Let f(x) be a lower (upper) semicontinuousfunction on the convex set S in J n. If f(x) is strictly quasi-convex (strictly quasi-concave) on S, then... 

Properties of Quasi-convex and Quasi-concave Functions... 

Remark 2 Convex and concave functions do not satisfy properties (ii) and (iii) of the quasi-convex and quasi-concave functions. For instance, it is true that the reciprocal of a positive concave function is convex, but the reverse does not hold. As an example consider the function f(x) = ex which is convex and whose reciprocal is also convex. 

For twice differentiable quasi-concave functions /(x) on the open, nonempty convex set S in Sftn, a direction z orthogonal to V/ exhibits the following interesting properties ... 

Remark 1 From property (ii) we observe that the generalization of concavity to quasi-concavity is equivalent to allowing the existence of at most one positive eigenvalue of the Hessian. 

Section 2.3 focuses on the generalizations of convex and concave functions and treats the quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions, and their properties. Further reading in this subject is the excellent book of Avriel et al. (1988). 

Let the individual be represented by a twice differentiable, quasi-concave utility function... 

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... 

To gain some intuition about why non-quasi-concave payoffs may lead to non-existence of NE, suppose that in a two-player game, player 2 has a bi-modal objective function with two local maxima. Furthermore, suppose that a small change in the strategy of player 1 leads to a shift of the global maximum for player 2 from one local maximum to another. To be more specific, let us say that at x l the global maximum is on the left (Figure 2.3 left) and... 

To continue with the newsvendor game analysis, it is easy to verify that the newsvendor s objective function is concave and hence quasi-concave w.r.t. the stocking quantity by taking the second derivative. Hence the conditions of Theorem 1 are satisfied and a NE exists. There are virtually dozens of papers employing Theorem 1. See, for example, Lippman and McCardle (1997)... 

If quasi-concavity of the players payoffs cannot be verified, there is an alternative existence proof that relies on Tarski s (1955) fixed point theorem and involves the notion of supermodular games. The theory of supermodular games is a relatively recent development introduced and advanced by Topkis (1998). 

Hence, increasing best response functions is the only major requirement for an equilibrium to exist players objectives do not have to be quasi-concave or even continuous. However, to describe an existence theorem with non-continuous payoffs requires the introduction of terms and definitions from lattice theory. As a result, we restricted ourselves to the assumption of continuous payoff functions, and in particular, to twice-differentiable payoff functions. 

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