a tribute to prof. lloyd stowell shapley and prof. alvin elliot roth
DESCRIPTION
Expository talk on the major contribution of Prof. L. S. Shapley.TRANSCRIPT
A Tribute to Prof. Lloyd Stowell Shapley andProf. Alvin Elliot Roth
2012 Nobel Memorial Prize in Economic Sciences ”for the theory ofstable allocations and the practice of market design”
October 19, 2012
Contribution of ShapleyStable allocation
Overview
1 Contribution of ShapleyNon-cooperative GamesCo-operative Games
2 Stable allocationProblem with examplesTheorem
L. Shapley
Contribution of ShapleyStable allocation
Overview
1 Contribution of ShapleyNon-cooperative GamesCo-operative Games
2 Stable allocationProblem with examplesTheorem
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Necessity
Figure: John von Neumann
In their book The Theory ofGames and Economic Behavior(1944), von Neumann andMorgenstern asserted that themathematics developed forthe physical sciences, whichdescribes the workings of adisinterested nature, was apoor model for economics.
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Philosophy
Figure: Non-cooperative DynamicGame
Game theory does notattempt to state what aplayer’s goal should be,instead, it shows how a playercan best achieve his goal,whatever that goal is.
It is assumed that Players of agame are rational in theirchoices, and each assumesrationality of opponent, andhence can reconstructopponent’s rational moves.
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Philosophy
Figure: Non-cooperative DynamicGame
Game theory does notattempt to state what aplayer’s goal should be,instead, it shows how a playercan best achieve his goal,whatever that goal is.
It is assumed that Players of agame are rational in theirchoices, and each assumesrationality of opponent, andhence can reconstructopponent’s rational moves.
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Solution
Figure: Example: A Static Gamemax
qmin
pV (p, q) = min
pmax
qV (p, q)
= V (p∗, q∗),(p∗, q∗) = (0.1, 0.9)
Safe I contains Rs. 1 Crore
Safe II contains Rs. 9 Crore
Safes are in separate locations
Only one Guard to protect
Only one thief to steal
Guard protects according toimportance
Thief attempts, according toavailability
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Solution
Figure: Example: A Static Gamemax
qmin
pV (p, q) = min
pmax
qV (p, q)
= V (p∗, q∗),(p∗, q∗) = (0.1, 0.9)
Safe I contains Rs. 1 Crore
Safe II contains Rs. 9 Crore
Safes are in separate locations
Only one Guard to protect
Only one thief to steal
Guard protects according toimportance
Thief attempts, according toavailability
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Generalization
Figure: John Forbes Nash
Every finite, two-personconstant-sum static gamehas a saddle pointequilibrium in mixedstrategies [John vonNeumann 1928].
For every finite staticgame, there exists amixed-strategy NE [Nash1950].
A SE exists for a class oftwo-person constant-summulti-stage (stochastic)games [Shapley 1953].
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Generalization
Figure: John Forbes Nash
Every finite, two-personconstant-sum static gamehas a saddle pointequilibrium in mixedstrategies [John vonNeumann 1928].
For every finite staticgame, there exists amixed-strategy NE [Nash1950].
A SE exists for a class oftwo-person constant-summulti-stage (stochastic)games [Shapley 1953].
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Generalization
Figure: John Forbes Nash
Every finite, two-personconstant-sum static gamehas a saddle pointequilibrium in mixedstrategies [John vonNeumann 1928].
For every finite staticgame, there exists amixed-strategy NE [Nash1950].
A SE exists for a class oftwo-person constant-summulti-stage (stochastic)games [Shapley 1953].
L. Shapley
Contribution of ShapleyStable allocation
Non-cooperative GamesCo-operative Games
Some of the ground breaking works
Shapley value (1953)To each cooperative game, it assigns a unique distribution(among the players) of a total surplus generated by thecoalition of all players.
Shapley-Shubik power index (1954)It measures the powers of players in a voting game.
Bondareva-Shapley theorem (1960)It describes a necessary and sufficient condition for thenon-emptiness of the core of a cooperative game.
Gale-Shapley algorithm (1962)Existence of a stable allocation for marriage problem.
L. Shapley
Contribution of ShapleyStable allocation
Problem with examplesTheorem
Stable allocation
Figure: Diagram of preferences
Consider a community of n menand n women. Each personranks those of the opposite sexin accordance with his or herpreferences for a marriagepartner. Is there a satisfactoryway of marrying off all membersof the community?Definition: A set of marriagesis called unstable if under itthere are a man and a womanwho are not married to eachother but prefer each other totheir actual mates.
L. Shapley
Contribution of ShapleyStable allocation
Problem with examplesTheorem
Stable allocation
A B C
α 1,3 2,2 3,1β 3,1 1,3 2,2γ 2,2 3,1 1,3
Table: Ranking matrix for threemen and three women
Stable sets(α,A), (β,B) and (γ,C )(α,C ), (β,A) and (γ,B)(α,B), (β,C ) and (γ,A)All other arrangements areunstable.
Figure: Ranking matrix for fourmen and four women
One can check, there is only onestable set of marriages for thisexample.
L. Shapley
Contribution of ShapleyStable allocation
Problem with examplesTheorem
Stable allocation
Figure: L. Shapley
THEOREM [GALE &SHAPLEY (1962)]:For any finite marriage problem,there always exists a stable setof marriages.
Born June 2, 1923 (age 89)Cambridge, Massachusetts
Nationality AmericanAffiliation University of California,
Los Angeles (since 1981)Fields Mathematics, Economics
L. Shapley
Contribution of ShapleyStable allocation
Problem with examplesTheorem
Concluding Remark
L. Shapley
Thank You