From: John Conover <john@email.johncon.com>
Subject: Re: NASA, TQM, and SPK/Kerr
Date: Tue, 15 Oct 1996 09:01:54 -0700
Donald Kerr writes:
>
> What do you mean by the only stable state of a zero-sum game being a
> defection strategy? Please expand on this idea.
>
As an over simplified illustration of the budget process being a two
person, zero-sum game, consider a company with two divisions, A and B,
with the executives of each division deliberating the fraction of the
budget that will go to each division. The fiscal budget payoff table
might look like:
B Cooperates B Defects
+------------------------+
A Cooperates | 2,2 | 0,3 |
+--------------+---------+
A Defects | 3,0 | 1,1 |
+--------------+---------+
Which might be interpreted as:
1) if A and B cooperate, they each get 2 million dollars, 1.5
million from the corporate budget, and 0.5 million each for
synergistic (ie., cooperative,) operations. Presumably, this is
the "best" alternative for the corporation, as a whole.
2) If A or B, but not both, opt for the total budget of 3 million,
(ie., one or the other plays a defection strategy,) then the
defectionist will get the 3 million, and the other will get 0,
(since, presumably, one can not operate without the other, the
defectionist will not get 4 million, as in 1), above.)
3) If they both defect, then they each get 1 million, which is the
1.5 million from the budget, and a loss of 0.5 million for not
operating in a synergistic fashion.
1) Operating in a cooperative, synergistic fashion, both A and B
get 2 million each, the highest cumulative "score" available for
the game.
2) There is an incentive for both players to defect from this
strategy.
Which happens to be the requirements of a zero-sum game, (with two
players, in this simple case.) The reasons that both players will
choose a defection strategy can be found by looking at A's logical
process in determining which strategy (cooperation, or defection,)
would be most beneficial.
1) If A cooperates, and B cooperates, A gets 2 million.
2) If A cooperates, and B defects, A gets 0 million, the lowest
"score in the game."
3) If A defects, and B cooperates, A gets 3 million, the highest
"score" in the game.
4) If A defects, and B defects, A gets 1 million, and avoids
getting the lowest "score" in the game.
Likewise for player B.
Note that A playing a defection strategy is the "rational" choice. No
matter what B chooses to do, A is better off choosing a strategy of
defection. So, player B, using the same rationale, decides to defect,
also, making the 1,1 solution the "rational" and inevitable outcome of
the budget process, (at least in the non-iterated version of the
"game.") It is the only stable solution.
The above table is from "Prisoner's Dilemma," William Poundstone,
Doubleday, New York, New York, 1992, pp. 120. For a more formal
presentation, see "Games and Decisions," R. Duncan Luce and Howard
Raiffa, John Wiley & Sons, New York, New York, 1957, pp. 56. For a
general presentation and history, see,
http://william-king.www.drexel.edu/top/class/histf.html.
--
John Conover, john@email.johncon.com, http://www.johncon.com/
Consider the following "game." It is, although simple in operation, a
rather complex game, and I will wade through the optimal logic. The
rules, of this very simplified rendition, of the "game" is as follows:
There are two players, and each player has only two choices for
each iteration of the "game," and those choices are to chose
either "A" or "B." If both players pick "A," then each wins 3
dollars. If one picks "A," and the other "B," then the player
picking "B" wins 6 dollars, and the other player gets
nothing. However, if both players pick "B," then both win 1
dollar.
It is not a zero-sum game-no player can ever loose any money. So there
is an incentive to always play. Or, to put it in clear, concise form,
the payoff matrix would look like:
Player One
A B
+-----+-----+
A | 3,3 | 6,0 |
Player two +-----+-----+
B | 6,0 | 1,1 |
+-----+-----+
This "game" is what is known to game theorists as the "classic
iterated prisoner's dilemma." The choice "A" is known as a
"cooperation strategy," and the choice "B" is known as the "defection
strategy" for each player. It is a very subtile and devious game. Here
is why, and the logic you would go through. Just before you played,
you would think:
1) If I pick "A," there are two possible scenarios:
a) If he picks "A," I would get 3 dollars, and he would get 3
dollars.
b) If he picks "B," I would get 1 dollar, and he would get 6
dollars.
2) If I pick "B," there are also two possible scenarios:
a) If he picks "A," I would get 6 dollars, and he would get
nothing.
b) If he picks "B," I would get one dollar, and he would get
one dollar.
Note that by picking "A," the best you could do is to win 3 dollars,
and the worst is to win nothing. But, by picking "B," the best you
could make is 6 dollars, and the worst is one dollar. It would appear,
at least initially, that "B," is the best choice.
But, here is where it gets subtile. You opponent, unless he is stupid,
(correction, "Thinking Challenged," in politically correct
vernacular,) will determine exactly the same strategy, and will never
play "A." So you both make one dollar every time you play the "game."
But you could make 3 dollars-if you cooperated, by both playing "A."
But if you do that, there is an incentive for either player to play
"B," if he knows the other player is going to play "A," and thus make
6 dollars. And we are right back where we started. Indeed, a very
diabolical "game."
As a simple, empirical, application of the "game," consider the case
of marital dilemma:
Suppose that marital bliss is upset by a grouchy mate that just
happens to get up on the wrong side of the bed, (ie., the mate is
playing a defection strategy-you were playing a cooperation
strategy, in the interest of preserving the marital bliss.) What
do you do? You get grouchy right back, right? And, when the mate
"comes around," and starts cooperating, you start cooperating, and
marital bliss is restored. It turns out that this "tit-for-tat"
strategy is one of the most effective known, and has been recorded
as a political solution to many dilemmas throughout history, (like
a "tooth-for-tooth" statement in the Old Testament-there are
similar statements in Zen, Hinduism, etc.)
This is a key point. Some games have explicit (or statistical,)
solutions. Most don't, and will require a strategy, like
"tit-for-tat." (There are other solutions, and other variants of this
"game" that are far more sophisticated.) How do you separate the
"games" into those that have explicit solutions, and those, like the
iterated prisoner's dilemma that don't? It turns out that, 1) there
must be an incentive to play the cooperation strategy, and 2) there
must be a larger incentive to play the defection strategy, and 3) if
the defection strategy wins, it must be at the expense of your
opponent, ie., most political situations. For example, it is in a
nation's interest to have a large nuclear stockpile. Naturally, if all
nations adopt this defection strategy, we have a prescription for the
nuclear arms race. So, we have international conferences, that
"equalize" the destructive power of consenting nations (cooperation
strategy,) which is immediately broken by Iraq, etc., (defection
strategy.)
So, is nuclear arms races rational and logical? That depends on your
point of view, but in a strict, logical sense, yes they are. It is a
rational, logical strategic solution to the iterated prisoner's
dilemma "game." We have to be very carful about making statements that
politics is irrational, when it apparently it has a formal, logical
basis.
Note that all wars in recorded history have been fought to resolve a
question of authority, ie., at least two players, both choosing a
defection strategy. (Note that if only one player chose a defection
strategy, and the other one chose a cooperation strategy, there would
have been no war-and one less nation.)
It should be clear that politics is an essential ingredient of
civilization, if not its defining attribute, (the Greeks considered
"man the political animal,") and is the dynamic operation of
cooperation and defection strategies between players. Politics and
conflict are inevitable, and unavoidable.
One of the poets in Oruk, the first city in civilization (circa
3000BC, in what is now Iraq,) telling the story of how the gods
delivered civilization to man:
"I offer you civilization. It will contain farming, commerce, and
prosperity. But it will also contain war, greed and famine. And
you must either take it in whole, or nothing. And once taken, it
can never be given back."
The rudiments of game-theory are not new.
John
References:
For "light" reading, see "Prisoner's Dilemma", William Poundstone,
Doubleday, "New York, New York", 1992.
For more rigor, see "Games and Decisions", R. Duncan Luce and Howard
Raiffa, John Wiley & Sons, New York, New York, 1957.
For the relationship of game-theory and human conflict in
civilization, see "The Ascent of Man," J. Bronowski, Little, Brown and
Company, Toronto, 1973.