パロンドのパラドックスを生成するパラメータ空間の凸領域

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書誌事項

タイトル別名
  • A Convex Domain in the Parameter Space that Generates Parrondo’s Paradox
公開日
2014-03-28
資源種別
departmental bulletin paper
公開者
浜松医科大学

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説明

The author proposed, in his classes for medical students at Hamamatsu, a study of Parrondo’s paradox described in these books [1], [2] and [3]. This paradox interested them and their naïve discussions in the classes stimulated him to investigate the class of all Parrondo’s games from a viewpoint of generalized random walks ([4]. The game consists of two basic games A and B of the same type. Suppose that a player repeats the game and observes the partial sum S X n k k n = = Σ1 at time n. Then, if Sn is a multiple of 3, she plays game A to get the next result Xn+1 that takes on one of the two values, +1(win) and –1(loss); otherwise she plays game B to get the result ′+ Xn 1 in like manner. We can therefore parametrize such a game by means of the expectations of games A and B, which we put E(Xn+1 ) = –α,E(Xn′+1 ) = β (–1 <α,β < 1). Our main result is Theorem 4 which asserts that Sn /n converges (in the weak sense) to the value δ1=3{2β –α(1+β2)}/(9+β2–2αβ) as n→∞. Hence we see that the game is favorable or unfavorable for a player according to δ1 > 0 or δ1 < 0. Since the domain given by α < 2β / (1+β2), 0 ≤ β <1, is convex, we now explain how one can generate Parrondo’s paradox. Indeed, noting that the mixed strategy of game I with α1, β 1 and game II with α2, β 2 becomes the game having the parameter α = (α +α ) / ,β = (β + β ) / 1 2 1 2 2 2, we generate various examples of α i, βi (i =1,2) in the final section, such that the inequality αi > 2βi / (1+β i2) holds for each i, whereas we have the inverse inequality α < 2β /(1+ β 2 ) satisfied by the mixed strategy. Applying the theory of generalized random walks developed in [4], we can also solve the ruin problem for every Parrondo’s game; we thus understand his paradox from another point of view.

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