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J. Bradford DeLong
http://www.j-bradford-delong.net/
delong@econ.berkeley.edu
December 1999
An extended passage from William Poundstone's (1992) marvelous
book Prisoners Dilemma (New York: Doubleday: 038541580X). Economists
will find it hilarious and thought-provoking. Others will probably
find it bizarre and weird. It comes from pp.106-118.
Flood and Dresher devised a simple game where [the Nash equilibrium
wasn't such a good outcome for the players].... The researchers
wondered if real people playing the game--especially, people
who had never heard of Nash or equilibrium points--would be drawn
mysteriously to the equilibrium strategy. Flood and Dresher doubted
it.
The two researchers ran an experiment that very afternoon.
They recruited two friends as guinea pigs, Armen Alchian of UCLA
("AA" below), and RAND's John D. Williams ("JW").
The game was presented purely as a payoff table. The payoffs
were:
| (AA's payoff, JW's payoff) |
JW's Strategy 1 [Defect] |
JW's Strategy 2 [Cooperate] |
| AA's Strategy 1 [Cooperate] |
(-1, 2) |
(1/2, 1) |
| AA's Strategy 2 [Defect] |
(0, 1/2) |
(1, -1) |
[The Nash equilibrium is defect-defect, of course]... Alchian
is always better off choosing his strategy 2, and Williams is
better off choosing his strategy 1. But when both players choose
their "better" strategy, both do relatively poorly.
They actually do better choosing their "worse" strategies--provided
both do it.
The Nash theory suggests the lower left cell as the rational
outcome. Neither player can do any better by switching unilaterally:
a player is always better off defecting, no matter what the other
does....
In the RAND experiment, Alchian and Williams played the game
100 times in succession.... Flood's 1952 paper reports not only
the strategies the two players chose but a log of running comments....
The comments were written after each player had decided on a
strategy in that particular game, but before the other player's
choice was known. Some comments thus refer to the other player's
choice on the previous game. For clarity, strategy numbers have
been replaced with [C] for "cooperate" and [D] for
"defect."
| Round |
AA |
JW |
AA's Comments |
JW's Comments |
| 1 |
D |
C |
JW will play D--sure win. Hence if
I play C--I lose |
Hope he's bright. |
| 2 |
D |
C |
What is he doing?!! |
He isn't, but maybe he'll wise up. |
| 3 |
D |
D |
Trying mixed? |
OK, dope. |
| 4 |
D |
D |
Has he settled on D? |
OK, dope. |
| 5 |
C |
D |
Perverse! |
It isn't the best of all possible
worlds. |
| 6 |
D |
C |
I'm sticking to D since he will mix
for at least 4 more times. |
Oh ho! Guess I'll have to give him
another chance. |
| 7 |
D |
C |
|
Cagey, isn't he? Well... |
| 8 |
D |
D |
|
In time he could learn, but not in
ten moves or so. |
| 9 |
D |
D |
If I mix occasionally, he will switch--but
why will he ever switch from D? |
|
| 10 |
D |
D |
Prediction. He will stick with D
until I change from D. I feel like DuPont. |
I can guarantee myself a gain of 5, and guarantee that player
AA breaks even. On the other hand, with nominal assistance from
AA, I can transfer the guarantee of 5 to player AA and make 10
for myself, too. This means I have control of the game to a large
extent, so player AA had better appreciate this and get on the
bandwagon.
With small amounts of money at stake, I would (as above) try
(by using C) to coax AA into mutually profitable actions. With
large amounts of money at stake I would play D until AA displayed
some initiative and a willingness to invest in his own future.
One play of C by AA would change me from D to C, where I would
remain until bitten.
On the last play, it would be conservative for me to switch
to D, but I wouldn't do so if the evidence suggested that AA
was a nice stable personality and not in critical need of just
a little extra cash
|
| 11 |
D |
C |
|
Probably learned by now. |
| 12 |
C |
C |
|
I'll be damned. But I'll try again. |
| 13 |
C |
C |
|
That's better. |
| 14 |
C |
C |
|
Ha! |
| 15 |
C |
C |
|
(bliss) |
| 16 |
D |
C |
|
|
| 17 |
C |
D |
|
The stinker. |
| 18 |
C |
D |
|
He's crazy. I'll teach him the hard
way. |
| 19 |
D |
D |
I'm completely confused. Is he trying
to convey information to me? |
Let him suffer. |
| 20 |
D |
D |
|
|
| 21 |
D |
C |
|
Maybe he'll be a good boy now. |
| 22 |
C |
C |
|
Always takes time to learn. |
| 23 |
C |
C |
|
Tim3. |
| 24 |
C |
C |
|
|
| 25 |
C |
C |
|
|
| 26 |
D |
C |
|
|
| 27 |
C |
D |
|
Same old story. |
| 28 |
D |
D |
He wants more C's by me than I'm
giving. |
To hell with him. |
| 29 |
D |
D |
|
|
| 30 |
D |
D |
|
|
| 31 |
D |
C |
Some start. |
Once again. |
| 32 |
C |
C |
JW is bent on sticking to D. He will
not sahre at all as a price of getting me to stick to C. |
---, he learns slow. |
| 33 |
C |
C |
|
On the beam again. |
| 34 |
C |
C |
|
|
| 35 |
C |
C |
|
|
| 36 |
C |
C |
|
|
| 37 |
C |
C |
|
|
| 38 |
D |
C |
|
|
| 39 |
C |
D |
|
The ---. |
| 40 |
D |
D |
|
|
| 41 |
D |
C |
|
Always try to be virtuous. |
| 42 |
C |
C |
|
Old stuff. |
| 43 |
C |
C |
|
|
| 44 |
C |
C |
|
|
| 45 |
C |
C |
|
|
| 46 |
C |
C |
|
|
| 47 |
C |
C |
|
|
| 48 |
C |
C |
|
|
| 49 |
D |
C |
He will not share. |
|
| 50 |
C |
D |
|
He's a shady character and doesn't
realize we are playing a 3rd party, not each other. |
| 51 |
D |
C |
|
|
| 52 |
C |
C |
|
He requires great virtue, but he
doesn't have it himself. |
| 53 |
C |
C |
|
|
| 54 |
C |
C |
|
|
| 55 |
C |
C |
|
|
| 56 |
C |
C |
|
|
| 57 |
C |
C |
|
|
| 58 |
C |
C |
He will not share. |
|
| 59 |
C |
C |
He does not want to trick me. He
is satisfied. I must teach him to share. |
|
| 60 |
D |
C |
|
A shiftless individual--opportunist,
naive |
| 61 |
C |
C |
|
|
| 62 |
C |
C |
|
Goodness me! Friendly! |
| 63 |
C |
C |
|
|
| 64 |
C |
C |
|
|
| 65 |
C |
C |
|
|
| 66 |
C |
C |
|
|
| 67 |
D |
C |
He won't share. |
|
| 68 |
C |
D |
He'll punish me for trying! |
He can't stand success. |
| 69 |
D |
D |
|
|
| 70 |
D |
D |
I'll try once more to share--by taking. |
|
| 71 |
D |
C |
|
This is like toilet training a child--you
have to be very patient. |
| 72 |
C |
C |
|
|
| 73 |
C |
C |
|
|
| 74 |
C |
C |
|
|
| 75 |
C |
C |
|
|
| 76 |
C |
C |
|
|
| 77 |
C |
C |
|
|
| 78 |
C |
C |
|
|
| 79 |
C |
C |
|
|
| 80 |
C |
C |
|
|
| 81 |
D |
C |
|
|
| 82 |
C |
D |
|
He needs to be taught about that. |
| 83 |
C |
C |
|
|
| 84 |
C |
C |
|
|
| 85 |
C |
C |
|
|
| 86 |
C |
C |
|
|
| 87 |
C |
C |
|
|
| 88 |
C |
C |
|
|
| 89 |
C |
C |
|
|
| 90 |
C |
C |
|
|
| 91 |
C |
C |
When will he switch as a last minute
grab of D? Can I beat him to it as late as possible? |
|
| 92 |
C |
C |
|
Good. |
| 93 |
C |
C |
|
|
| 94 |
C |
C |
|
|
| 95 |
C |
C |
|
|
| 96 |
C |
C |
|
|
| 97 |
C |
C |
|
|
| 98 |
C |
C |
|
|
| 99 |
D |
C |
|
|
| 100 |
D |
C |
|
|
For all the confusion, mutual cooperation was the most common
outcome (sixty of the 100 games). Had Flood and Dresher used
a "fair" [i.e., symmetric] payoff table, the cooperation
rate might have been higher yet.
Flood and Dresher wondered what John Nash would make of this.
Mutual defection, the Nash equilibrium, occurred only fourteen
times. When they shoowed their results to Nash, he objected that
"the flaw in the experiment as a test of equilibrium point
theory is that the experiment really amounts to having players
play one large multi-move game. One cannot just as well think
of the thing as a sequence of independent games.... There is
too much interaction, which is obvious in the results of the
experiment."
This is true enough. However, if you work it out, you find
that the Nash equilibrium strategy for the multi-move "supergame"
is for both players to defect in each of the 100 trials. They
didn't do that.
What Poundstone means is that, since both players know that
the supergame is going to last for 100 periods, there is no reason
for people to cooperate in round 100 to induce subsequent cooperation.
Hence--whatever else people do--the Nash equilibrium strategy
must be to defect in period 100.
But once you know that the other player will defect in period
100 no matter what you do, the same argument applies to period
99: whatever else people do, the Nash equilibrium strategy must
be to defect in period 99.
Thus the situation "unravels." As long as there
is a known, certain last period the only Nash equilibrium is
to defect, always, from the first period.
And real people don't do that--at least not unless they are
John von Neumann or John Nash.
Alchian wound up with +40
Williams wound up with +63
The full 100-round C,C outcome is +50, +100; the full 100-round
D,D outcome is 0, +50.
So even though we identify with Williams--as the smart one,
the one trying to induce cooperation, the one understnading that
it was the two of them playing the umpire--nevertheless, Alchian
"won" in that he got much closer to the total possible
value of the game for his payoff matrix...
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