Expository talk on the major contribution of Prof. L. S. Shapley.

1. A Tribute to Prof. Lloyd Stowell Shapley and
Prof. Alvin Elliot Roth
2012 Nobel Memorial Prize in Economic Sciences ”for the theory of
stable allocations and the practice of market design”
October 19, 2012

2. Contribution of Shapley
Stable allocation
Overview
1 Contribution of Shapley
Non-cooperative Games
Co-operative Games
2 Stable allocation
Problem with examples
Theorem
L. Shapley

3. Contribution of Shapley
Stable allocation
Overview
1 Contribution of Shapley
Non-cooperative Games
Co-operative Games
2 Stable allocation
Problem with examples
Theorem
L. Shapley

4. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Necessity
In their book The Theory of
Games and Economic Behavior
(1944), von Neumann and
Morgenstern asserted that the
mathematics developed for
the physical sciences, which
describes the workings of a
disinterested nature, was a
poor model for economics.
Figure: John von Neumann
L. Shapley

5. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Philosophy
Game theory does not
attempt to state what a
player’s goal should be,
instead, it shows how a player
can best achieve his goal,
whatever that goal is.
It is assumed that Players of a
game are rational in their
Figure: Non-cooperative Dynamic choices, and each assumes
Game rationality of opponent, and
hence can reconstruct
opponent’s rational moves.
L. Shapley

6. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Philosophy
Game theory does not
attempt to state what a
player’s goal should be,
instead, it shows how a player
can best achieve his goal,
whatever that goal is.
It is assumed that Players of a
game are rational in their
Figure: Non-cooperative Dynamic choices, and each assumes
Game rationality of opponent, and
hence can reconstruct
opponent’s rational moves.
L. Shapley

7. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Solution
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
Figure: Example: A Static Game Guard protects according to
max min V (p, q) = min max V (p, q) importance
q p p q
= V (p ∗ , q ∗ ), Thief attempts, according to
(p ∗ , q ∗ ) = (0.1, 0.9) availability
L. Shapley

8. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Solution
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
Figure: Example: A Static Game Guard protects according to
max min V (p, q) = min max V (p, q) importance
q p p q
= V (p ∗ , q ∗ ), Thief attempts, according to
(p ∗ , q ∗ ) = (0.1, 0.9) availability
L. Shapley

9. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Generalization
Every finite, two-person
constant-sum static game
has a saddle point
equilibrium in mixed
strategies [John von
Neumann 1928].
For every finite static
game, there exists a
mixed-strategy NE [Nash
1950].
A SE exists for a class of
two-person constant-sum
Figure: John Forbes Nash
multi-stage (stochastic)
games [Shapley 1953].
L. Shapley

10. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Generalization
Every finite, two-person
constant-sum static game
has a saddle point
equilibrium in mixed
strategies [John von
Neumann 1928].
For every finite static
game, there exists a
mixed-strategy NE [Nash
1950].
A SE exists for a class of
two-person constant-sum
Figure: John Forbes Nash
multi-stage (stochastic)
games [Shapley 1953].
L. Shapley

11. Contribution of Shapley Non-cooperative Games
Stable allocation Co-operative Games
Generalization
Every finite, two-person
constant-sum static game
has a saddle point
equilibrium in mixed
strategies [John von
Neumann 1928].
For every finite static
game, there exists a
mixed-strategy NE [Nash
1950].
A SE exists for a class of
two-person constant-sum
Figure: John Forbes Nash
multi-stage (stochastic)
games [Shapley 1953].
L. Shapley

12. Contribution of Shapley Non-cooperative Games
Stable allocation Co-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 the
coalition 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 the
non-emptiness of the core of a cooperative game.
Gale-Shapley algorithm (1962)
Existence of a stable allocation for marriage problem.
L. Shapley

13. Contribution of Shapley Problem with examples
Stable allocation Theorem
Stable allocation
Consider a community of n men
and n women. Each person
ranks those of the opposite sex
in accordance with his or her
preferences for a marriage
partner. Is there a satisfactory
way of marrying off all members
of the community?
Definition: A set of marriages
is called unstable if under it
there are a man and a woman
who are not married to each
Figure: Diagram of preferences other but prefer each other to
their actual mates.
L. Shapley

14. Contribution of Shapley Problem with examples
Stable allocation Theorem
Stable allocation
A B C
α 1,3 2,2 3,1
β 3,1 1,3 2,2
γ 2,2 3,1 1,3
Table: Ranking matrix for three
men and three women
Stable sets Figure: Ranking matrix for four
men and four women
(α, A), (β, B) and (γ, C )
(α, C ), (β, A) and (γ, B)
(α, B), (β, C ) and (γ, A) One can check, there is only one
All other arrangements are stable set of marriages for this
unstable. example.
L. Shapley

15. Contribution of Shapley Problem with examples
Stable allocation Theorem
Stable allocation
THEOREM [GALE &
SHAPLEY (1962)]:
For any finite marriage problem,
there always exists a stable set
of marriages.
Born June 2, 1923 (age 89)
Cambridge, Massachusetts
Nationality American
Affiliation University of California,
Los Angeles (since 1981)
Fields Mathematics, Economics
Figure: L. Shapley
L. Shapley

16. Contribution of Shapley Problem with examples
Stable allocation Theorem
Concluding Remark
L. Shapley

17. Thank You