Inefficiency of voting in Parrondo games

We study a modification of the so-called Parrondo's paradox where a large number of individuals choose the game they want to play by voting. We show that it can be better for the players to vote randomly than to vote according to their own benefit in one turn. The former yields a winning tendency while the latter results in steady losses. An explanation of this behaviour is given by noting that selfish voting prevents the switching between games that is essential for the total capital to grow. Results for both finite and infinite number of players are presented. It is shown that the extension of the model to the history-dependent Parrondo's paradox also displays the same effect.