Robert Axelrod’s
Tournaments,
as reported in
Axelrod, Robert. 1980a. “Effective Choice in the Prisoner’s Dilemma.”
Journal of Conflict Resolution 24: 3-25.
Axelrod, Robert. 1980b. “More Effective Choice in the Prisoner’s
Dilemma.” Journal of Conflict Resolution 24 (3): 379-403.
Axelrod, Robert. 1984. Evolution of Cooperation.
Tournament Num. 1
(1980)
-non-zero sum setting, given payoff matrix (R=3, T=5, S=0, P=1)
-round robin tournament (play all other entrants, twin, and RANDOM)
-each entrant told to write a program to select C or D choice every
move, can use history of the game so far in this decision making
-sent copies of preliminary tournament in which TFT scored second,
so known to be powerful competitor, also told RANDOM was
somewhere in the competition  tried to improve on TFT principle
-known number of moves per game: 200
-entire round robin run 5 times  total 120,000 moves and 240,000
choices
14 Entrants
-3 countries, 5 disciplines (psychology, math, economics, sociology,
political sciences)
-scores range from 0 to 1000, but “useful benchmark for very good
performance is 600,” attained if both always cooperate together
-“very poor performance [benchmark] is 200 points” (if both always D)
-winner Tit for Tat (TFT) scored 504 (but if change P=2, does not win)
-top 8 entries were nice (defined as not first to defect), rest were not
-nice entries’ scores scored from 472 to 504, while best of mean
entries only scored 401 points (huge disparity!)
-logically, because nice ones cooperate together, this is how TFT wins!
(though it cannot get a score higher than its opponent’s)
14 Entrants
-important to be nice and forgiving
-2 kingmakers (defined as players who do not do well themselves but
“LARGELY determine the rankings among the top contenders”):
GRAASKAMP and DOWNING
-DOWNING most important kingmaker since it had the largest range
of scores achieved with the nice rules, important to note DOWNING
was not based on TFT principle
-now to look at the actual results!, then to examen the strategies,
since strategies aside from TFT are just denoted by name of creator
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STRATEGIES!
1. Tit for Tat (TFT)- winner with 504.5 points, from Toronto
(psychology), as we all know- cooperates on first move, then does
what opponent did last move, “eye for eye” style, 4 lines FORTRAN
2. TIDEMAN and CHIERUZZI- 500.4 points, from US (Economics),
begins with cooperation/ TFT, but after opponent finishes second run
of D, institutes extra punishment  increases number of punishments
(D) by 1 with each run of opponent’s defections, then decides whether
to give opponent a fresh start and begin with TFT again based on- if it
has 10+ points more than opponent, opponent has not started
another run of D’s, been 20+ moves since last fresh start, are 10+
moves left, number of opponent’s D’s “differs from 50-50 generator
by at least 3 standard deviations,” 41 lines of code
STRATEGIES!
3. NYDEGGER- 485.5 points, starts with TFT for first 3 moves unless
it was only one to C on first move and only one to D on second move,
then it will D on third move, after third move- it chooses based on a
complex weighted sum (2 points for opponent’s D, 1 point for own D,
then weight this sum for past three terms- 16 for last term, then 4,
then 1; if sum = 63, i.e. three turns of mutual defection  it will C)
4. GROFMAN- 481.9 points, always cooperates unless players did not
do the same thing on the last move, then cooperates with prob 2/7
5. SHUBIK- 480.7 pts, cooperates until opponent plays D, then it
defects once, if other defects again- it begins again with cooperation,
in general- “length of retaliation is increased by one for each
departure from mutual cooperation”
STRATEGIES!
6. STEIN- 477.8 pts, TFT except it cooperates always first four moves
and defects on last 2 moves (move 199 and 200 of game), every 15
moves checks to see if opponent is RANDOM with chi-squared test of
opponent’s transition probabilities and alternating CD/DC moves
7. FRIEDMAN- 473.4 pts, cooperates until opponent defects, then it
defects forever
8. DAVIS- 471.9 pts, last of the nice guys, cooperates first 10 moves,
then if there is a defection, it will defect forever
9. GRAASKAMP- 400.7 pts, one of kingmakers, TFT for 50 moves,
defects on move 51, then plays 5 more TFT, check to see if opponent
is RANDOM, if so- D from then on (also checks for TFT, ANALOGY,
CLONE), otherwise- randomly defects every 5-15 moves, enough trust
STRATEGIES!
10. DOWNING- 390.6, main kingmaker, starts with D since assumes
opponent is unresponsive (i.e. initially assumes 1/2 for conditional
probabilities, its downfall!), from then on- assesses and updates
probabilities (that opponent cooperates if DOWNING defects, etc) to
calculate choice to maximize its long-term expected payoff, if the 2
conditional probabilities have similar values- DOWNING determines
pays to D, conversely- if opponent is responsive (much more likely to
play C after DOWNING plays C than after D), then it will cooperate
11. FELD- 327.6 pts, starts with TFT, gradually lowers probability of C
following the other plays C to 1/2 by the 200th move
12. JOSS- 304.4, cooperates 90% after opponent’s C, always D after D
13. TULLOCK- 300.5, cooperates first 11 moves, then cooperates 10%
less than opponent has on preceding 10 moves
Last of STRATEGIES!
14. GRADUATE STUDENT NAME WITHHELD- 282.2 pts, starts with
probability of C of 30%, which is updated every 10 moves if opponent
seems very cooperative, very uncooperative, or random, after 130
moves if losing- probability is adjusted, this complex process kept P
between 30% and 70%, making it seem random to most opponents
15. RANDOM- 276.3 pts, C with probability 1/2 and D with
probability 1/2 (C and D with equal probabilities)
Tournament Num. 2
(1980)
-same non-zero sum setting, again round robin tournament (play all)
-each entrant was sent report of first tournament, given same task
-instead of known number of moves per game, “length of the game
was determined probabilistically with .00346 chance of ending with
each given move” (one way to include w), w chosen so expected
median length = 200 moves (w = .99654 in second tournament)
-average length turned out to be shorter: closer to 150 moves
-endgame effects successfully avoided this time
-features of entries do not relate to success (length of program, type,
nationality, type of program, etc)
63 Entrants
-6 countries, contests largely recruited via journals, etc
-everyone from first tournament re-invited, entrants ranged from 11
year-old Steve Newman to professors from many disciplines, including
computer science and evolutionary biology this time
-more than half of entries were nice, Tit for Tat (TFT) won again
-Tit for Two Tats- too forgiving, suggested post-Tourney 1, submitted
Tourney 2 by evolutionary biologist, ended up in bottom half of group
-5 representative rules can predict how a given rule did with the 63
rules- GRAASKAMP & KATZEN (S6), PINKLEY (S30), ADAMS (S35),
GLADSTEIN (S46), and FEATHERS (S27)  predicted tournament score T
= 120 + (.202) S6 + (.198) S30 + (.110) S35 + (.072) S46 + (.086) S27
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.