Tuesday, April 30, 2013

Dilemma 22: Chicken


Other non-zero-sum games exist. Consider this:
(A,B)            |  nice      mean
nice       |  2,2   |   1,3   |   
A              -|--------|---------|   
mean       |  3,1   |   0,0   |   

This game, “Chicken”, is similar to Dilemma, except that the values of “lose” amd “draw” are interchanged. Now draw is the worst outcome, and lose is the lesser evil. Unlike Dilemma, this game has two Nash equilibria; win/lose and lose/win. This fact weakens exploitation but strengthens intimidation.
This is the general Chicken payoff matrix:

(A,B)            |  nice      mean
           |                     W  =  Win
       ----|--------|---------|       T  =  Truce
nice       |  T,T   |   L,W   |  D  =  Draw
A              -|--------|---------|       L  =  Lose
mean       |  W,L   |   D,D   |      
          -|--------|---------|  where D<L<T<W

It is often presented as a war game; Car Wars, involving irresponsible adolescents in high-powered vehicles. In Car Wars Chicken, two cars approach each other in the same lane at high velocity. The first driver to swerve out is not, as you might expect, lauded for sanity, but jeered for lack of bravado. To “chicken out” is a loss in this stupid game; to chicken others out is to win.
Chicken is a contest of stubbornness, aggression, and intimidation. Like Dilemma, this game rewards mutual cooperation, yet offers tempting opportunities for exploitation. In Dilemma the dialog is between cooperation and competition; in Chicken the dialog is between rationality and irrationality. There is little a loser can do ... except threaten a draw, which is the worst outcome for both players. This is a stubborn, unreasonable bargaining position; but that’s just what Chicken favors.
Dilemma and Chicken both enforce harmony via reciprocity, in the long run. Chicken’s long run is longer and costlier; for Chicken is a metaphor for that most anti-social of games, War.

My father relates this anecdote of wartime Chicken:
During the early 40’s he was trucking high explosives for a certain Army project. It was a hazardous job, made more hazardous by the presence of Chicken players on the road. Many times he found some joker homing in on him; but my father assures me that he never paid the slightest attention to such fools. He just kept on driving as if they weren’t in his lane, and pretty soon they weren’t. That was my Dad!
Those guys didn’t have a chance. After all, he was trucking those explosives for the Manhattan Project!
So my father said to me. From this story I deduce (with horror and awe) that it’s a miracle that I even exist!


Robert Axelrod
The Evolution of Cooperation
Basic Books, New York, 1984
Tim Beardsley
“Never Give A Sucker An Even Break”
Scientific American, Oct. ‘93, p. 22
Bender, Kramer & Stout
“When In Doubt... Cooperation in a noisy Prisoner’s Dilemma”
Journal of Conflict Resolution, v. 35 #4, Dec. ‘91, p. 691-719
Peter Grossmann
“The Dilemma of Prisoners;
Choice during Stalin’s Great Terror, 1936-38”
Journal of Conflict Resolution, v. 38 #1, Mar. ‘94, p. 43
Nathaniel Hellerstein
Diamond, A Paradox Logic
World Scientific Publishing Company, Singapore, 1996
Garrett Hardin
“The Tragedy Of The Commons”
Science, vol. 162, 1968, p. 1243-1248

Peter Kollock
“ ‘An Eye For An Eye Leaves Everyone Blind’:
cooperation and accounting systems”
American Sociological Review,  v. 58, Dec. ‘93, p. 768-786
Multiple authors:
“The Role of Communication in
Interindividual-Intergroup Discontinuity”
Journal of Conflict Resolution, v.37#1, Mar ‘93, p.108-138
“Interindividual-Intergroup Discontinuity
in the Prisoner’s Dilemma Game”
Journal of Conflict Resolution, v.38#1, Mar ‘94, p.87-116
Martin Nowak and Karl Sigmund
“Tit For Tat In Heterogenous Populations”
Nature, vol. 355, 16 January ‘92, p. 250
Martin Nowak and Robert May
“Evolutionary games and spatial chaos”
Nature, vol. 359, 29 Oct. ‘94, p. 826
John Orbell & Robyn Dawes
“Social Welfare, Cooperator’s Advantage,
and the Option of Not Playing the Game”
American Sociological Review,  v. 58, Dec. ‘93, p. 787-806
William Poundstone
Prisoner’s Dilemma
Doubleday, New York, 1992

Anatol Rapoport
Strategy And Conscience
Harper and Row, New York, 1964.
Rudolf Schuessler
“Threshold Effects and the Decline of Cooperation”
Journal of Conflict Resolution, v.34 #3, Sept. ‘90, p. 476
Karl Sigmund
“On Prisoners and Cells”
Nature, vol. 359, 29 Oct. ‘92, p. 77

Monday, April 29, 2013

Dilemma 21: The Unexpected Departure

The Unexpected Departure

The Axelrod upper equilibrium requires certain conditions. One of them is that the expected number of plays be great enough; another is that the play not end at too definite at time. If it does, then a “backwards induction paradox” destroys the Axelrod truce, no matter how long the tournament.
Consider the following scene:
Curly is about to play with Moe in a dilemma tournament sceduled to last exactly 100 rounds. Curly, a Silver Rule player, is optimistic that he can convince Moe (an Iron Rule player) that it’ll be in his own best interest to cooperate.
But Moe said, “What about the 100th round? Won’t that be the last one?”
Curly said, “Yes.”
“There won’t be any after the 100th?”
“Yes,” said Curly.
Moe asked, “So in the very last play, what’s to keep me from defecting?”
“‘Cause I’ll defect the next...” Curly said, then slapped himself on the face. “Alright, nothing will stop you from defecting on the 100th play.”
“So you might as well defect too, right?” Moe said, smiling.
“I guess so,” Curly said reluctantly. “On the 100th play.”
Moe continued, “And what about the 99th play? What’s to keep me from defecting then?”
“‘Cause I’ll defect the next...” Curly said, then slapped himself on the face. “But I’ll defect on the 100th play anyhow.”
“That’s right,” Moe said, smiling.
“So nothing’s keeping you from defecting on the 99th play.”
“That’s right,” Moe said, smiling.
“So I should defect on the 99th play also,” said Curly.
“That’s right,” Moe said. “Now, what about the 98th play?”
And so they continued! Moe whittled down Curly’s proposed truce, one play at a time, starting from the end. By the time the conversation was over, Moe had convinced Curly that the only logical course was for them to defect from each other 100 times, drawing the tournament. And so they did; yet when Curly played with Larry (a Gold Rule player) they cooperated 100 times, for a truce!
Thus we deduce, by mathematical induction, that the prospect of abruptly terminated play, even if in the far future, poisons the relationship at its inception. That is the “backwards induction paradox”.
In dilemma play, cooperation requires continuity to the end. Departure should not be at an expected time lest that light the backwards-induction fuse; departure should be unannounced, at an unexpected time.
We need an unexpected departure; but this yields a paradox. Consider this following story about an Unexpected Exam:

Once upon a time a professor told his students, “Sometime next week I will give you an exam; and that exam will be at an unexpected time. Right up until the moment I give you the exam, you will have no way to deduce when it will happen, or even if it will happen. It will be an Unexpected Exam.”
One of the professor’s students objected, “But then the exam couldn’t happen on Friday; for by then it would be expected!”
The professor said, “True.”
The student continued, “So Friday’s ruled out.”
Another student said, “But if Thursday’s the last possible day for an Unexpected Exam, then it’s ruled out too; for by Thursday the Thursday exam will be expected!”
The professor said, “True.”
And so on; by such steps the students concluded that the Unexpected Exam can’t happen on Friday, Thursday, Wednesday, Tuesday, or Monday; so it can’t happen at all!
“So you don’t expect it?” said the professor.
His students said, “No!”
The professor smiled...
The next Wednesday, he handed out an exam, to the students’ surprise.

That’s the Paradox of the Unexpected Exam. This also is a backwards induction paradox; but this time it is a strangely false result rather than a strangely undesirable result. This match suggests the following fable.

The same professor visited the Dean; he said, “I will depart this school sometime during the next month. To ensure cordial relations between us until that time, my departure will take place on an unexpected day.”
The Dean retorted, “You couldn’t leave on the 31st, for by then your Unexpected Departure would be expected.”
The professor agreed.
The Dean added, “Having ruled out the 31st, the 30th is also ruled out; for it would be expected.”
The professor agreed to that too.
And so the conversation continued; and in the end the Dean concluded, “Your Unexpected Departure can’t happen on any day. Therefore I don’t expect it.” The professor agreed.
On the seventeenth day of the month the professor departed, to the Dean’s astonishment.

This Paradox of the Unexpected Departure is just what the doctor ordered; for here the failure of backwards induction (so puzzling to the reason) is precisely what is needed to defend the Axelrod equilibrium from its backwards induction proof!
Above I insisted that dilemma tournaments use “open bounding”; that is, replay only if a random device permits it. This ensures an Unexpected Departure; play will be finite, but there will be no definite last play during which the Iron player is safe from the danger of Silver retaliation.

The conclusion then is clear; let none of your social relationships end too definitely; let there be some possibility that you might encounter that person again, soon. (And conversely, when you must leave, slip away quietly!)

Friday, April 26, 2013

Dilemma 20: Mutual Profit

Mutual Profit

The “Dilemma Wagers” chapter shows how basic economic interactions involve dilemmas. This has profound implications, both academic and political; for no existing economic ideology has a rational dilemma strategy. All existing economic ideologies center upon money; and money is the one Market element which is necessarily peripheral to dilemma economics.
The Price Parley makes money part of a dilemma wager; but though dilemma economics can involve money, it cannot be based upon money. One cannot parley money for money; for the truce in a money parley would mean an exchange of dollars; but my dollar is worth the same as yours. This is the “fungibility of money”; by definition it precludes mutual gain.
Money is inherently zero-sum, and dilemma is inherently non-zero-sum; so money economics and dilemma economics are mutually exclusive. Dilemma economics is Economics Without Money; a dilemma with which many of us are all too familiar.
Neither Capitalism nor Socialism can account for dilemma. Capitalism assumes, a priori, that any economic interaction under capitalism is zero-sum by nature; this covers the win-lose axis in dilemma. Socialism assumes, a priori, that any economic interaction under socialism is zero-difference by nature; this covers the truce-draw axis in dilemma. Thus both ideologies cover precisely one-half of the puzzle, and between them lose sight of the real question.

Capitalism and Socialism correspond, respectively, to the Iron and Gold rules. (Mixed Government tends to resemble the Random strategy.) The Gold rule would truce with itself; but it is vulnerable to invasion and defeat by the Iron rule; and the former strategy draws against itself. Thus Capitalism describes a world that should not endure; and Socialism describes a world that cannot endure. Neither one is the world which does endure; for both strategies are dead! What endures is what lives, and is thus a conundrum, a mystery, a dilemma.
We should accept such dilemmas, even embrace them; for dilemma makes free-enterprise democracy possible. If there is to be free enterprise, then there must be profit; but if there is to be democracy, then that profit must go to the people. Therefore democratic free-enterprise requires mutual profit; where the people profit from each other!
Mutual profit is, by definition, non-zero-sum. It implies the possibility of mutual loss, along with the win/loss struggle of Capitalist competition; thus full dilemma emerges.
Mutual profit is the Market’s truce. It is economic peace, attained via justice tempered by mercy. Mutual profit flourishes best in communities of mutual aid. It lives by the Silver Rule; value for value.
Mutual profit transcends both Capitalism and Socialism; the first because it is mutual, the second because it is profit. Mutual profit creates social order spontaneously, without coercion; therefore mutual profit is inherently Anarchist. Mutual profit subverts the State.

Thursday, April 25, 2013

Dilemma 19: Predictor’s Paradox

Predictor’s Paradox

This paradox is a failed attempt to resolve Dilemma. It teaches us that not even a confrontation with a Superior Being can make certain people behave themselves.
In the Predictor’s Paradox, you (an ordinary mortal) are shown a pair of boxes. Box A is open; $1 can be seen within. Box B is shut; it contains either $100 or $0. The other player claims to be a Superior Being who can predict your actions. “If you choose to take both boxes”, says the Being, “then you’ll discover that I’ve punished you by putting nothing in box B; but if you have faith in me and take only box B, then you’ll find my reward of $100 there.”
Let us assume that previous experience has shown that the Being can apparently make good on its claim of being able to predict your actions; what should you do? Here’s the game matrix:

payoff for mortal   

            |  rewards    punishes
takes box B |   100    |     0     |
mortal                -|----------|-----------|
takes both  |   101    |     1     |

Meanwhile, what’s in it for the Being? Let us suppose, for the sake of symmetry, that the Being’s game matrix is as follows:

payoff for Being

            |  rewards    punishes
takes box B |   100    |    101    |
mortal                -|----------|-----------|
takes both  |    0     |     1     |

Presumably the Being values your faith one hundred times more than the material profit of punishing you. This way you and the Being are in a Dilemma game, where win = 101, truce = 100, draw = 1 and lose = 0.
Your best move depends on how well the Superior Being can foretell your actions. If the Being can correctly predict your actions with probability exceeding (in this case) 50.5 %, then the expected value of taking one box exceeds that of taking both. In this case, reasoning by expected value favors leaving the $1 alone; yet taking the $1 would still be a dominant strategy!
Two different lines of argument yield two opposite recommendations. How are we to decide? That is the question. The Predictor’s Paradox represents a dispute between the principles of Dominance and Expectation. It shows that not even a confrontation with a Superior Being can make Iron-rule players behave themselves! That $1 sitting there, just begging to be taken... how can they resist such a devilish temptation? If there  really were a Superior Being, then this little test would ensure the self-defeat of all habitual exploiters, whilst humbler folk win riches!

Actually I’m satisfied neither with blind faith, nor with exploitation. Blind faith in the Being is a Gold Rule strategy, and as such is vulnerable if the Being is an Iron Rule player. Similarly the attempt to exploit the Being is an Iron Rule strategy. The optimum long run strategy is the Silver Rule:
Do Unto Others As They Have Done Unto You.
This principle of cosmic justice is so powerful that even a Superior Being must meet us there on equal terms.
Therefore, if I were ever to confront a Superior Being in this fashion, I would form the intention to take either $1 or $100, but not $0 nor $101. After all, why should I ignore the $1 bill if I get nothing otherwise? And conversely, why should I try to cheat a Superior Being of $1 if it already gave me $100?
Thus I, a mere mortal ruled by greed and fear, propose to make myself the equal of a Superior Being! I leave it to you, dear reader, to judge the soundness of my thinking; but note that within this mental context, the Being has every reason to give me $100.

Some may object that there are no Superior Beings in evidence with whom to play this game; to this I reply that the Silver Rule is so powerful that it enables a mere mortal to make a passable imitation of a Superior Being, provided that the shadow of the future is long enough.

For let us iterate this game with open bounds, with replay probability 99/100, so the expected number of plays is 100. Rescale the payoffs accordingly, at 1/100th of the payoffs noted above; i.e. 1 cent in box A, $0 or $1 in box B. In this Predictor’s Tournament, I shall play the Superior Being’s role; my strategy will be tit-for-tat. If you take both boxes on a given round, then on the next round I’ll leave box B empty; and if you take only box B, then on the next round I’ll put $1 in box B. (In a sense, the Superior Being that I emulate is Reciprocity itself!)
If you are rational, and if play is long enough, then we will attain truce; you will always take only box B, and you will always find $1 in it. Though I can’t predict you, I do remember you; so in the long run it will be in your interest to act as if I could predict you. Continuity is the key; if the “shadow of the future” is long enough, then my memory, like the Superior Being’s prophesy, will enforce social harmony. My hindsight equals Reciprocity’s foresight.
A shadow is haunting Earth; the shadow of the future. Will we be or will we not be? That is the question. When the future’s shadow is short, not even a Superior Being can deter the wicked from maximizing profit; but when the shadow of the future extends, then you and I can be like unto Superior Beings, and peace breaks out! Blessed be the shadow of the future!