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On two-point Nash equilibria in bimatrix games with convexity properties

100%
Połowczuk W.
Applicationes Mathematicae
|
2006
|
tom 33
|
nr 1
71-84
EN
This paper considers bimatrix games with matrices having concavity properties. The games described by such payoff matrices well approximate two-person non-zero-sum games on the unit square, with payoff functions F₁(x,y) concave in x for each y, and/or F₂(x,y) concave in y for each x. For these games it is shown that there are Nash equilibria in players' strategies with supports consisting of at most two points. Also a simple search procedure for such Nash equilibria is given.
2
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Nonzero-sum semi-Markov games with countable state spaces

100%
Połowczuk W.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 4
395-402
EN
We consider nonzero-sum semi-Markov games with a countable state space and compact metric action spaces. We assume that the payoff, mean holding time and transition probability functions are continuous on the action spaces. The main results concern the existence of Nash equilibria for nonzero-sum discounted semi-Markov games and a class of ergodic semi-Markov games with the expected average payoff criterion.
3
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Equilibria in constrained concave bimatrix games

63%
Połowczuk W., Radzik T.
Applicationes Mathematicae
|
2013
|
tom 40
|
nr 2
167-182
EN
We study a generalization of bimatrix games in which not all pairs of players' pure strategies are admissible. It is shown that under some additional convexity assumptions such games have equilibria of a very simple structure, consisting of two probability distributions with at most two-element supports. Next this result is used to get a theorem about the existence of Nash equilibria in bimatrix games with a possibility of payoffs equal to -∞. The first of these results is a discrete counterpart of the Debreu Theorem about the existence of pure noncooperative equilibria in n-person constrained infinite games. The second one completes the classical theorem on the existence of Nash equilibria in bimatrix games. A wide discussion of the results is given.
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