Differential Games and Control Theory by Emilio O. Roxin

By Emilio O. Roxin

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Pure Nash equilibria). In finite games, it may happen that there is no pure Nash equilibrium and therefore only mixed or correlated equilibria are relevant. Mixed strategies have many useful interpretations (Osborne and Rubinstein, 1994), but they may not be implementable in some scenarios; correlated equilibria can be seen as an interesting concept in wireless games where some additional signaling is available to the terminals. Correlated equilibria can be more applicable in terms of a modeling situation where some terminals access signals which may correlate their decisions.

However, note that the sum of quasi-concave functions is not necessarily quasi-concave and quasi-concavity is generally not preserved by adding an affine function. 3 A sigmoidal or S-shaped function is a function which is convex up to a point and then becomes concave from this point on. 45 46 CHAPTER 2 Playing with Equilibria in Wireless Non-Cooperative Games sub-modular games (Topkis, 1979) and super-modular games (Milgrom and Roberts, 1990; Topkis, 1998). The following discussion will give the definition, an existence theorem, and an example of a game for these types of games.

The common knowledge of rationality implies that every player is motivated by maximizing his own utility. If every player always maximizes his utility, and we are thus able to perfectly calculate the possible probabilistic result of every action, we have hyper-rationality. Players can be said to be with limited or bounded rationality because the information they have may be limited, because of the presence of noise, because they may have a finite amount of time to make decisions, or because of intrinsic limitations in terms of computational capacity (Simon, 1972).

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