A bird in the hand is worth two in the bush.

"Plus valet in manibus avis unica fronde duabus" Werner, Latin Sprichwörter, c. AD 1400http://www.youtube.com...http://www.youtube.com..."The things we already have are more valuable than the things we only hope to get.": http://dictionary.reference.com..."It's better to have a small real advantage than the possibility of a greater one.": http://www.phrases.org.uk..."...it is better to have something that is certain than take a risk to get more, where you might lose everything.": http://www.usingenglish.com...In real life, these "birds" correspond to measures of real value, and the whole proverb is a measurement of opportunity cost. For convenience, I am going to argue this in terms of x and y. X equals the bird in hand, and y equals the potential gain over the current value of bird you possess. The proverb can now be restated as follows: It is better to retain x than to sacrifice x for the potential to gain x+y.Obviously the validity of this proverb will vary in the real world; this is a philosophical debate, and cannot be held to quite the same level of proof that a matter of absolute fact can be. I shall argue that in on balance, it is better to retain x than to sacrifice x for the potential to gain x+y; or "a bird in the hand is worth two in the bush."~~~~~~~~~~~~~~~~~~~~~A little logic can help sort out the potential outcomes. The graph involved is simple, assuming even odds of gaining x+y; these are the labels:Keep = those cases in which x is keptRisk = those cases in which x is risked to gain x+yNo = those cases in which x+y would not be gained by risking xYes = those cases in which x+y would be gained by risking x.............No...YesKeep.....x .....x-y Risk ..... 0 ....x+y In the "No" column, we would not gain x+y by risking x. We stand to lose the value of x. In the "Yes" column, if we sacrifice x, we will gain x+y. Keeping x carries an opportunity cost of y.The value for x is immaterial, as long as it is greater than zero: were it a negative number, it would be in our best interest to take any choice that nullified it, whether we gained y or not. .............No...... YesKeep...-|x|.....y-xRisk........0....... y-x Similarly, if it were zero, we would have no incentive to hold on to it and would always risk it on the chance of gaining y.TL,DR TRANSLATION: The bird you hold must have a real non-zero value.The value for y is crucial; both in absolute terms, and as a ratio to x. If y is negative, the graph looks like this:...........No....YesKeep....x.......xRisk.....0.....x-|y|In this case, keeping x incurs no opportunity cost; risking x either loses it all or the remainder of x -|y|. If y is negative, under no circumstances is in our benefit to risk x. Similarly if y = 0, the reward for risking x is gone. In all cases where y is 0 or has a negative value, then, the proverb is completely accurate. TL,DR TRANSLATION: The proverb is correct in every case the birds in the bush have a value equal to or less than zero. For the purposes of debate we will assign y a positive value in all subsequent examples.To make the results easier to see, in every subsequent graph x will be assigned a value of 10. It can be any positive non-zero value, but 10 is convenient.There are three main possibilities to consider:1. when y is significantly smaller than x2. when y is close to or equal to x3. when y is significantly bigger than xIn cases where y is significantly smaller:.............No....YesKeep....10.......9Risk......0.......11The opportunity cost for keeping x is very small, while the potential cost of risking x is large with a small potential payoff. In this case, it would not be reasonable to risk x, and the proverb is affirmed. In cases where y is close to or equal to x:...........No...YesKeep..10......0Risk....0......20If you keep x when you would have gained x + y, you incur an opportunity cost roughly equal to x. Risking x at worst loses x; at best gains x2. It would be in your best interest to risk x, all other things being equal.In cases where Y is significantly greater than x:............No...YesKeep....10.....-5Risk......0......25Again, risking x would be in your best interest, all other things being equal. Keeping x runs the risk of a negative payout, while the worst that can happen by risking x is losing it, and the reward is great.Note that, while these last two graphs seem to argue my opponent's side, these are for the cases in which the odds are 1 in 2 of gaining x + y. Life vary rarely offers odds nearly this good.TL,DR TRANSLATION: As long as the value of the birds in the bush is small, and it's even odds getting them, it isn't worth risking the bird you hold. If their value is greater than the one you hold, and the odds are even, it's worth trying.So, in all cases where y is smaller than x, the proverb holds. With even odds, if y is larger, you should risk it.This seems to argue that the proverb is wrong at least as many times as it is right, but life seldom offers us even odds of success. I do not know how to go about finding the average odds for life's opportunities, but those that we might be tempted to undertake typically range from 1 in 3 to 1 in 5; I doubt anyone but the most hardened gamblers would (knowingly) try for any odds longer than that. If y is close to or equal to x:...........No..No.YesKeep...10...10....0Risk.....0.....0....20..........No..No..No..No..YesKeep..10...10...10...10.....0Risk....0.....0......0....0......20In neither case is the risk justified. As the odds grow longer, the fact that you already have x becomes more significant. At 1 in 3, you are two times as likely to lose x as you are to gain 2x. At best, all you can hope for is to break even. But, since you already have x, it is more reasonable to keep it.At 1 in 5, you are four times as likely to lose x as you are to gain 2x. This is not a reasonable risk to take.As the odds grow longer, the value of y must increase in order to tempt us to risk x. Note that we do not necessarily know the value of y when we risk x, or even the precise odds; this uncertainty adds to the risk in real life, since we might be wagering x (whose value we know) on 15 to 1 odds for a reward less than x.Two examples: y = 20 at 1 in 3 odds and y = 20 at 1 in 5 odds...........No..No...YesKeep..10...10....-10Risk....0.....0......30..........No..No..No..No..YesKeep..10...10...10...10....-10Risk....0.....0.....0.....0......30The first example is barely better than before. Y is twice x, and x + y is three times x, but playing these odds indefinitely yields no net gain.The second is still unjustifiable. Y must be much larger to make these odds worth the risk. Playing these odds indefinitely will result in a net loss.To balance increasing odds, the value of y must increase proportionately to cover the increased risk. Keeping x becomes a better proposition as the odds grow longer, no matter the value of y.TL,DR TRANSLATION: At more realistic odds than 1 in 2, the birds in the bush must be of significantly greater value than the one you hold to be worth the risk.Conclusion: Only if y is significantly larger than the potential losses is risking x reasonable at odds greater than 1 in 2. If y is equal to x or smaller, the reward is greater to retain x than to risk it. As the odds increase, the necessary value of y increases proportionately.In real life, one usually cannot be certain of the exact value of y.In real life, the odds are usually greater than 1 in 2.:. On balance, it is better to retain x than to risk it for x + y. A bird in the hand really is better than two in the bush.