The Mathematics of Tucker: A Sampler
Author(s): A. W. Tucker
Reviewed work(s):
Source: The Two-Year College Mathematics Journal, Vol. 14, No. 3 (Jun., 1983), pp. 228-232
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/3027092 .
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The Mathematicsof Tucker:
A Sampler
A. W. Tucker
A Two-Person Dilemma:
The Prisoner's Dilemma
Two men,chargedwitha joint violationof law, are held separatelyby the police.
Each is told that
ifone confessesand theotherdoes not,theformerwillbe givena rewardof
one unitand the latterwill be finedtwo units,
(2) if both confess,each will be finedone unit.
At the same timeeach has good reason to believethat
(3) if neitherconfesses,both will go clear.
(1)
This situationgivesrise to a simplesymmetric
two-persongame (not zero-sum)
with the followingtable of payoffs,in which each ordered pair representsthe
payoffsto I and II, in thatorder:
confess
confess
not confess
II
not confess
(-1,-1)
(-2, 1)
(1,-2)
(0,0)
Clearly,foreach man thepure strategy"confess"dominatesthe pure strategy"not
confess." Hence, there is a unique equilibriumpoint* given by the two pure
strategies"confess." In contrastwith this non-cooperativesolutionone sees that
both men would profitif theycould forma coalitionbindingeach otherto "not
confess."
The game becomes zero-sumthree-person
by introducingthe State as a third
but receives
player.The Stateexercisesno choice (thatis, has a singlepurestrategy)
payoffsas follows:
confess
confess
not confess
II
not confess
2
1
*See J. Nash, Proc. Nat. Acad. Sci., 36 (1950) 48-49.
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1
0
Changing the Way We Think about the Social World*
Philip D. Straffin,
Jr.
Beloit College
AlbertW. Tucker'snote was the firstwrittendescriptionof what has come to be
knownas the "prisoner'sdilemma."The examplein thatnote,withits accompanying story,has played a major role in social thoughtin the last thirtyyears.It is an
exampleof a simpleidea, originating
in mathematicalanalysis,whichcan be said to
have changedtheway we thinkabout our social world.
In 1944, Johnvon Neumann and Oskar Morgensternpublishedthe Theoryof
Games and EconomicBehaviorand foundedthe theoryof games as a branch of
mathematics.Von Neumann's celebratedminimaxtheoremstatedthateveryfinite
two-personzero-sumgame has an equilibriumoutcome in mixed strategies.By
1950,JohnNash, thena Ph.D. studentunderTucker,had generalizedthisresultto
prove thatfinitetwo-personnon-zero-sum-games
also have equilibria.However,it
was clear thattheequilibriaof non-zero-sum
gamescould have a numberof strange
and undesirableproperties.The payoffmatrixin the note was one of a numberof
examplesdevisedby Melvin Dresherand MerrillFlood at the RAND Corporation
to exhibitsome of these strangeproperties.Tucker recalls that he firstsaw the
matrixin Dresher'sofficeon a visitto RAND in 1950. Somewhatlater,Tuckerwas
asked by thepsychologydepartment
at Stanfordto give a talkon game theory.He
illustrationof the difficulty
of
thoughtthat this example would be an interesting
analyzingnon-zero-sumgames,but that it should be presentedwitha "story"to
accompanyit. The famousstoryof the note is the result.
As the prisoner'sdilemmawas popularizedamong social scientistsby Howard
Raiffa, Duncan Luce, and Anatol Rapoport in the 1950's and early 1960's, it
became apparentthatDresherand Flood's simplegame was a usefulmodel fora
largenumberof social situations.Must an invisiblehand governeconomicsin such
a way that individuallyrational behavior always leads to a socially optimal
outcome?Not always, and the prisoner'sdilemma illustrateswhy not. For two
nationsengagedin an arms race, the payoffsforthe strategies"continueto arm"
and "disarm"may look like thoseof theprisoner'sdilemma,and armsraces persist.
A prisoner'sdilemmagame with a largernumberof playerslies at the heart of
GarrettHardin's influential1968 essay, "The Tragedy of the Commons," which
shows how environmentalpollution and over-exploitation
of resourcescan be
dominantstrategiesthatlead to disastroussocial outcomes.
The prisoner'sdilemmagame became a usefulexperimentaltool forpsychologists interestedin attributesthat govern human behavior in social situations.
Experimentalliteratureon the prisoner'sdilemma grew steadilythroughoutthe
1960's: Rapoportestimatesthat200 experiments
relatedto it werereportedbetween
1965 and 1971.The game has been at leastas fruitful
fortheoreticians.
Anymodern
discussionof themeaningof rationality
in social behaviormustcome to termswith
the prisoner'sdilemma.
What kindsof mathematicalideas can be mostproductiveto social science?A
simple idea may be best. Mathematicalthinking,for instance concentratingon
propertiesof equilibriain non-zero-sumgames, can pare away inessentialsand
reveal a core common to many social situations.It can provide a simple model
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embodyingthatcore,perhapseven a model around whichexperimentalworkcan
be done. It helpsif the model comes witha cleverstoryand an attractivetitle.The
prisoner'sdilemmawas born in mathematicalanalysis,and provedso usefulthatit
of the social sciences.
has become part of the conceptualframework
We are gratefulto ProfessorTucker for his permissionto publish "A Two-PersonDilemma," to
WilliamLucas forkeepinga mimeographcopy of the note in circulation,and to Tucker and Merrill
Flood fortheiraccountsof the eventsof 1950.
* Originally
publishedas "The Prisoner'sDilemma" in theUMAP Journal.Reprintedwithpermission.
The TuckerTableau
Considerthefollowingdisplay,wherethe symbolsinsidethebox representnumerical data as they mightstand in an economic table, and those outside the box
represent
algebraicquantities:
..
Xi
all
vI
. a..
Vm aml
-
-
...
1...**,
X
-1
= _Y1
blan
amn bm
=
Cn
U
...
=
Un
= _Yn
f
=
This is the "Tucker tableau" for the dual linear programsthat can be stated in
matrixnotationas follows,where x = (xl, .. , xn), A = [ai], x > 0 means each
xi > 0, cx is a dot product,and so on.
Row problem:maximizef = cx for
Columnproblem:minimizeg = vb for
Ax-b=-y,
vA -c= u,
x ?O.y
> O.
>
u > Ov? > O.
Any solutionsx, y and u,v of the tableau's row and column equations satisfy
Tucker's"dualityequation":
+
UIX + 1* * * + Unxn
VIy+ i
+ vMYm= gf
So feasiblesolutionsx > 0, y > 0 and u > 0, v > 0 of the tableau equationsmake
feasibleincreasein
f < g. Iff = g, thesefeasiblesolutionsare optimal,since further
f or decreasein g is clearlyblocked.
The Tuckertableau has evolvedfromthe connectionbetweengames and linear
programsseen by John von Neumann and George Dantzig in 1947, the year
Dantzig devised his Simplex Method. For readers familiarwith the theoryof
2-personmatrixgames, the followingwill make the connectionspecific.Take all
a, > 0, bi = l and c1= 1 in the tableau above. Then the 2-person0-sumgame with
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p = v/g and q = x/f and minimax
payoffmatrixA has optimalmixed strategies
value 1/g = 1/f,wherev, x and f = g are determinedby optimalsolutionsof the
by the tableau.
dual linearprogramsrepresented
Tucker'sCubicalLemma
Is it possibleto arrangeplayingcards in a square so thatno two cards of the same
or diagonally-unless they
horizontally
color are adjacent to each other-vertically,
are of the same suit,and so thaton the outsiderowseach pair of oppositecards is
suits?
of the same color but of different
The answeris No. This is the2-dimensionalcase of Tucker'sLemma,in disguise.
by 1, each diamondby - 1, each spade by 2 and
Let each heartcard be represented
each club by -2. Then the situationdescribedis illustratedby the figurebelow.
Tucker's Lemma says that theremust be two numberson the same littlesquare
whichsum to 0. Such a pair is underlinedin the figure.
2
1
2
1
2
1
2
2
2
-1
1
2
-l
-1
1
-2
-1
-2
-1
-2
-1
-2
-1
-2
1
The readercan now figureout whatTucker'sLemma says in the n-dimensional
case.
Despite the whimsicalsetting,thisLemma can be used to prove severalof the
deep existencetheoremsof topologyabout mappingsof antipodalpoints,such as
theorem.These consetheBorsuk-Ulamtheoremand theLusternick-Schnirelmann
quences,and manyothers,as well as thecubical lemmaitself,all appear togetherin
a reviewpaper given by Tucker at the firstmeetingin 1945 of the Canadian
MathematicalCongress[2]. Here in detail is anotherof those consequencesfrom
thatpaper,a coveringtheoremdue to Tuckerhimself.
The Four-Set Covering Theorem. If the surfaceof a sphereis coveredbyfour
closedsets,no one of whichcontainsa pair of antipodalpoints,then(a) any threeof
thesetscontaina pointwhoseantipodebelongsto thefourthset,and (b) any twoofthe
sets containa pointwhoseantipodebelongsto theothertwosets.
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This generalizesto n + 2 closed sets coveringan n-sphere.Then anyp + 1 sets,
where0 ? p ? n, contain a point whose antipode belongs to the othern - p + 1
sets.
Tucker discoveredthe cubical lemma because he wanted to prove topological
theoremsby startingwithdiscreteresultsand usinglimitingarguments.He wanted
to expose the combinatorialfoundationsof such theorems.
The originalproof of the cubical lemma was not constructive.Constructive
proofshave now been found[1].
REFERENCES
1. RobertM. Freund and Michael J. Todd, A constructive
proofof Tucker'scombinatoriallemma,J.
CombinatorialTheory,SeriesA, 30, 1981.
2. AlbertW. Tucker,Some topologicalpropertiesof diskand sphere,Proceedingsof theFirstCanadian
MathematicalCongress,Universityof TorontoPress,Toronto,1946.
Decompositionof a Matrixinto
Rank-Determining
Submatrices
A. W. Tucker
Any matrixA #,0 has an invertiblesubmatrixA * containedin submatricesA c and
A r such that
A =ACA*lAr
(as exemplified
below). If A is m X n and A* is k x k, thenA c is m x k and A r is
k x n. It followsthatA * is a largestinvertible
submatrixof A and thatA c and A r
providecolumnand rowbases forA. So k is therankofA by any definition.
Such a decompositionA = A CA* - IA r resultsreadilyfromGaussian elimination
withk arbitrary
pivots-organizedto preservethe naturaldualitybetweencolumns
and rows by operatingon themjointly.If A has rank one, thenA c can be any
nonzerocolumnand A r any nonzerorow,withA * theentrycommonto A c and A r.
If A has full rank (m or n), thenA * = A c or A r and A =A r or A c. A simple
rankis thefollowing(withentriesofA * underlined):
exampleof intermediate
4
5
2-4
6
A
4
1
1
]=t2
63
Ac
1 2[
A*"
1
2
.3
Ar
This is easilychecked: multiplyA c or A r by A*- I (to get a "reduced" column or
row basis matrix)and thenby A ' or A c to get A. A c and A r use columns 1,3 and
rows 2,4 of A, but any two columns and two rows of A can be used, except
proportionalrows3,4-i.e., theA * used above is just one of 3(6 1) = 15 possibilities.
-
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