HARRY MARKOWITZ should be accepted when offered and"fur- plies the desirability of playing conserva ther(written) instructions. "The "further tively when losing moderately and playi instructionsare revealing; for example, liberally when winning moderately A-II--Play till you drop to 75 cents then This implication holds true whatever be you get low, play only the level of customary wealth of the individ very good odds;"C-I--If you are ahead, ual. In the F-S analysis a person with wealth you may play the four 4's for as low as $3; equal to D in Figure 3 would play liberally C-II-If player finds that he is winning, when losing, conservatively when winning, he shall go ahead and bet at his discretion; so as to attain negative skewness of the fre- C-IV--If his winnings exceed $2. 50, he may quency distribution. This, I should say, is play any and every hand as he so desires, another one of those peculiar implications but, if his amount should drop below 6o which flow from the F-S analys cents, he should use discretion in regard to Now let us consider the effect of wins or the odds and hands that come up. No one losses on the liberality of betting when we do gave instructions to play more liberally not have the strategic considerations which when losing than when winning. The tend- were central in the previous discussion. For ency to play liberally when winning, con- example, suppose that the"evening"is over servatively when losing, can be explained in The question arises as to whether or not the two different ways. These two explanations game should be continued into the morning apply to somewhat different situations whether or not a new series of games a bet which a person makes during a se- should be initiated ). There is also a question ries of games("plays"in the von Neumann of whether or not the stakes should be higher sense)cannot be explained without reference or lower. We abstract from fatigue or loss of to the gains and losses which have occurred interest in the game before and the possibilities open afterward. How do the evenings wins or losses af- What is important is the outcome for the fect the individuals preferences on these whole series of games: the winnings or los- questions? Since his gain or loss is a"wind ening a the evening consists of a series of inde- middle inflection point (presumably by the endent games(say matching pennies); sup- amount of the gain or loss) pose that the probability(frequency) dis- A person who broke even would, by hy tribution of wins and losses for a particular pothesis, have the same preferences as at the game is symmetric about zero. Suppose that beginning of the evening at each particular game the player has a A person who had won moderately would choice of betting liberally or conservatively (by definition of "moderate") be between (i.e,, he can influence the dispersion of the the second and third infecti on point. The wins and losses). If he bet with equal liberal- moderate winner would wish to continue the ity at each game, regardless of previous wins game and increase the stakes or losses, then the frequency distribution of a person who had won very much would final wins and losses(for the evening as a (by the definition of "very much") be to the laying conservatve metric. The effect of right of the third inflection point.He would when winning, is to make the frequency dis- at all. In the vernacular, the heavy winner tribution of final outcomes skewed to the would have made his <killing", and would right. Such skewness is implied as desirable wish to"quit while winning n a large neighborhood of customary in- The moderate loser, between the first and come) by our utility function. In sum, our second inflection points, would wish to play utility function implies the desirability of for lower stakes or not to play at all. some positive skewness of the final outcome A person who lost extremely heavily(to frequency distribution, which in turn im- the left of the first infection point) would