Parrondo’s paradox for games with three players and its potential application in combination therapy for type II diabetes

Abstract

Parrondo’s paradox appears in game theory which asserts that playing two losing games, A and B (say) randomly or periodically may result in a winning expectation. In the original paradox the strategy of game B was capital-dependent. Some extended versions of the original Parrondo’s game as history dependent game, cooperative Parrondo’s game and others have been introduced. In all of these methods, games are played by two players. In this paper, we introduce a generalized version of this paradox by considering three players. In our extension, two games are played among three players by throwing a three-sided dice. Each player will be in one of three places in the game. We set up the conditions for parameters under which player 1 is in the third place in two games A and B. Then paradoxical property is obtained by combining these two games periodically and chaotically and (s)he will be in the first place when (s)he plays the games in one of the mentioned fashions. Mathematical analysis of the generalized strategy is presented and the results are also justified by computer simulations. A potential application of the model in treatment of type II diabetes is presented. In this theoretical work, we consider two types of treatments as two games which are played among three different players. The player 1 is considered as the treatment success index, the player 2 is the insulin deficiency index and the player 3 is assumed to be the insulin resistance index. It is shown that certain combinations of two losing games for player 1 (unsuccessful treatments) will result in a win (successful treatment).