Social Learning Model With Parrondo's Paradox

Abstract

Parrondo’s game is invented by J. M. R. Parrondo, by the inspiration of flashing ratchet. Parrondo’s game includes two losing games named game A and game B. Game A is a coin-tossing game with winning probability $p_a = 1/2 - \epsilon$. Game B is also a coin-tossing game with winning probability $p_b = 1/10 - \epsilon$ when the capital is divisible by a integer parameter $M$, winning probability $p_g = 0.75 - \epsilon$ otherwise. Parrondo’s study shows that the long-term capital is decreasing either playing one of the games, but the combination of two games may give the result that the long-term capital is increasing. This phenomenon is called Parrondo’s effect. We reviewed the Parrondo’s effect with both mathematical description and numerical simulation. Two way of combining the Parrondo’s games: deterministic switching and stochastic mixing are introduced to demonstrate the Parrondo’s effect. We have studied the Parrondo’s game in a group of players with the condition that the games’ properties are blinded to the players. Players are interacted with one of the strategies: ‘Follow the winner’ and ‘Avoid the loser’. The two strategies show the phenomenon of Parrondo’s effect, the capital is decreasing for either one of the strategy and mixture of them may give a winning result. We also studied the situation that minimal information can be transferred among players. Players can adopt one of learning scheme: ‘Follow the winner mapping’ and ‘Avoid the loser mapping’. The opinion of games mapping among the players may converge to correct mapping when the first learning scheme is adopted. Mixture of learning schemes does not show the Parrondo’s effect, the time to convergence does not decrease. The convergence time is shortened when one of the rewiring schemes: ‘Rewiring from loser to winner’ and ‘Rewiring from the poorest to the richest’ is applied. We found that adding more game (such that there are three games, or four games) will not increase the difficultly of learning significantly, the time to convergence are close for both cases two to four games.