Algorithmic Game Theory and Internet Computing
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Transcript of Algorithmic Game Theory and Internet Computing
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Algorithmic Game Theoryand Internet Computing
Vijay V. Vazirani
Nash Bargaining via
Flexible Budget Markets
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The new platform for computing
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Internet
Massive computational power available
Sellers (programs) can negotiate with
individual buyers!
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Internet
Massive computational power available
Sellers (programs) can negotiate with
individual buyers!
Back to bargaining!
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Internet
Massive computational power available
Sellers (programs) can negotiate with
individual buyers!
Algorithmic Game Theory
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Bargaining and Game Theory
Nash (1950): First formalization of bargaining.
von Neumann & Morgenstern (1947):
Theory of Games and Economic Behavior
Game Theory: Studies solution concepts for
negotiating in situations of conflict of interest.
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Bargaining and Game Theory
Nash (1950): First formalization of bargaining.
von Neumann & Morgenstern (1947):
Theory of Games and Economic Behavior
Game Theory: Studies solution concepts for
negotiating in situations of conflict of interest.
Theory of Bargaining: Central!
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Nash bargaining
Captures the main idea that both players
gain if they agree on a solution.
Else, they go back to status quo.
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Example
Two players, 1 and 2, have vacation homes:
1: in the mountains
2: on the beach
Consider all possible ways of sharing.
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Utilities derived jointly
1v
2v
S : convex + compact
feasible set
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Disagreement point = status quo utilities
1v
2v
1c
2c
S
Disagreement point = 1 2( , )c c
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Nash bargaining problem = (S, c)
1v
2v
1c
2c
S
Disagreement point = 1 2( , )c c
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Nash bargaining
Q: Which solution is the “right” one?
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Solution must satisfy 4 axioms:
Paretto optimality
Invariance under affine transforms
Symmetry
Independence of irrelevant alternatives
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1v
2v
1c
2c
v
S
( , ),
& ( , )
v N S c
T S v T v N T c
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1v
2v
1c
2c
v
S
T
( , ),
& ( , )
v N S c
T S v T v N T c
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Thm: Unique solution satisfying 4 axioms
1 2( , ) 1 1 2 2( , ) max {( )( )}v v SN S c v c v c
1v
2v
1c
2c
S
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Generalizes to n-players
Theorem: Unique solution
1 1( , ) max {( ) ... ( )}v S n nN S c v c v c
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Generalizes to n-players
Theorem: Unique solution
1 1( , ) max {( ) ... ( )}v S n nN S c v c v c
(S, c) is feasible if S contains a point that makes each i strictly happier than ci
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Bargaining theory studies promise problem
Restrict to instances (S, c) which are feasible.
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Linear Nash Bargaining (LNB)
Feasible set is a polytope defined by
linear packing constraints
Nash bargaining solution is
optimal solution to convex program:
max log( )
. .
i ii
v c
s t
packing constraints
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Q: Compute solution combinatoriallyin polynomial time?
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Study promise problem?
Decision problem reduces to promise problem
Therefore, study decision and search problems.
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Linear utilities
B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players
0i ij ij ijj G
v u x x
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e.g., ci = i’s utility for initial endowment
B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players
0i ij ij ijj G
v u x x
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Convex program giving NB solution
max log( )
. .
:
: 1
: 0
i ii
i ij ijj
iji
ij
v c
s t
i v
j
ij
u xx
x
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Theorem
If instance is feasible,
Nash bargaining solution is rational! Polynomially many bits in size of instance
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Theorem
If instance is feasible,
Nash bargaining solution is rational! Polynomially many bits in size of instance
Decision and search problems
can be solved in polynomial time.
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Resource Allocation Nash Bargaining Problems
Players use “goods” to build “objects”
Player’s utility = number of objects
Bound on amount of goods available
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2s
1s
2t
1t
( )cap e
Goods = edges Objects = flow paths
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2s
1s
2t
1t
( )cap e
Given disagreement point, find NB soln.
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Theorem:
Strongly polynomial, combinatorial algorithm
for single-source multiple-sink case.Solution is again rational.
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Insights into game-theoretic properties of Nash bargaining problems
Chakrabarty, Goel, V. , Wang & Yu:
Efficiency (Price of bargaining)Fairness Full competitiveness
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Linear utilities
B: n players with disagreement points, ci
G: g goods, unit amount each
S = utility vectors obtained by distributing
goods among players
0i ij ij ijj G
v u x x
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Game plan
Use KKT conditions to
transform Nash bargaining problem to
computing the equilibrium in a certain market.
Find equilibrium using primal-dual paradigm.
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Game plan
Use KKT conditions to
transform Nash bargaining problem to
computing the equilibrium in a certain market.
Find equilibrium using primal-dual paradigm.
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Crown jewel of mathematicaleconomics for over a century!
General Equilibrium Theory
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A central tenet
Prices are such that demand equals supply, i.e.,
equilibrium prices.
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A central tenet
Prices are such that demand equals supply, i.e.,
equilibrium prices.
Easy if only one good
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Supply-demand curves
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Irving Fisher, 1891
Defined a fundamental
market model
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Fisher’s Model
B = n buyers, money mi for buyer i
G = g goods, w.l.o.g. unit amount of each good : utility derived by i
on obtaining one unit of j Total utility of i,
i ij ijj
U u xiju
[0,1]
i ij ijj
ij
v u xx
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Fisher’s Model
B = n buyers, money mi for buyer i
G = g goods, w.l.o.g. unit amount of each good : utility derived by i
on obtaining one unit of j Total utility of i,
Find market clearing prices.
i ij ijj
U u xiju
[0,1]
i ij ijj
ij
v u xx
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An almost entirely non-algorithmic theory!
General Equilibrium Theory
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Flexible budget market,only difference:
Buyers don’t spend a fixed amount of money.
Instead, they know how much utility they desire.
At any given prices, they spend just enough
money to accrue utility desired.
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Most cost-effective goods
At prices p, for buyer i: Si =
Define
arg min jj
ij
p
u
( ) min jj
ij
pcost i
u
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Flexible budget market
Agent i wants utility
At prices p, must spend to get utility
ic
ic. ( )ic cost i
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Flexible budget market
Agent i wants utility
At prices p, must spend to get utility
Define
Find market clearing prices.
ic
ic
1 . ( )i im c cost i
. ( )ic cost i
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Flexible budget market
Agent i wants utility
At prices p, must spend to get utility
Define
Find market clearing prices -- may not exist!!
ic
ic
1 . ( )i im c cost i
. ( )ic cost i
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Flexible budget market
Agent i wants utility
At prices p, must spend to get utility
Define
Find market clearing prices -- may not exist!!
feasible/infeasible
ic
ic
1 . ( )i im c cost i
. ( )ic cost i
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Theorem: Nash Bargaining for linear utilities
reduces to
Equilibrium for flexible budget markets
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Theorem: Nash Bargaining for linear utilitiesreduces to
Equilibrium for flexible budget markets
(S(u), c) M(u, c)
(S, c) is feasible iff M is feasible.
If feasible, x is Nash bargaining solution iff x is equilibrium allocation.
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Primal-Dual Paradigm
Usual framework: LP-duality theory
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Primal-Dual Paradigm
Usual framework: LP-duality theory
Extension to convex programs and
KKT conditions.
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Yin & Yang
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Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual paradigm
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Combinatorial Algorithm for Linear Case of Fisher’s Model
Devanur, Papadimitriou, Saberi & V., 2002
Using primal-dual paradigm
Solves Eisenberg-Gale convex program
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Eisenberg-Gale Program, 1959
max log
. .
:
: 1
: 0
i ii
i ij ijj
iji
ij
m v
s t
i v
j
ij
u xx
x
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Eisenberg-Gale Program, 1959
max log
. .
:
: 1
: 0
i ii
i ij ijj
iji
ij
m v
s t
i v
j
ij
u xx
x
prices pj
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Why remarkable?
Equilibrium simultaneously optimizes
for all agents.
How is this done via a single objective function?
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Idea of algorithm
primal variables: allocations dual variables: prices of goods iterations:
execute primal & dual improvements
Allocations Prices (Money)
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Flexible budget market
Main differences:
mi ’s change as prices change.
problem is not total.
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Allocations Prices (Money)
Allocations Prices (Money)
?
InfeasibleFeasible
Decision
Search
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An easier question
Given prices p, are they equilibrium prices?
If so, find equilibrium allocations.
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An easier question
Given prices p, are they equilibrium prices?
If so, find equilibrium allocations.
For each i, 1 . ( )i im c cost i
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m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
For each i, most cost-effective goods
arg min ji j
ij
pS
u
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Network N(p)
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
infinite capacities
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Max flow in N(p)
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
p: equilibrium prices iff both cuts saturated
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Two important considerations
The price of a good never exceeds
its equilibrium priceInvariant: s is a min-cut
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Max flow
m(1)
m(2)
m(3)
m(4)
p(1)
p(2)
p(3)
p(4)
p: low prices
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Two important considerations
The price of a good never exceeds
its equilibrium priceInvariant: s is a min-cut
Rapid progress is madeBalanced flows
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Max-flow in N
m p
W.r.t. max-flow f, surplus(i) = m(i) – f(i,t)
i
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Balanced flow
surplus vector: vector of surpluses w.r.t. f.
A max-flow that
minimizes l2 norm of surplus vector.
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Allocations Prices (Money)
Allocations Prices (Money)
?
InfeasibleFeasible
Decision
Search
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Balanced flow helps Decision as well!
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Proof of infeasibility: dual solution to
max
. .
:
: 1
, : 0
ij ij ij G
iji B
ij
t
s t
i u x c t
j x
i j x
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Theorem: Algorithm runs in polynomial time.
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Theorem: Algorithm runs in polynomial time.
Q: Find strongly polynomial algorithm!
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Nonlinear programs with rational solutions!
Open
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Nonlinear programs with rational solutions!
Solvable combinatorially!!
Open
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Primal-Dual Paradigm
Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
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Primal-Dual Paradigm
Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
![Page 90: Algorithmic Game Theory and Internet Computing](https://reader036.fdocuments.net/reader036/viewer/2022081603/56813c61550346895da5eaee/html5/thumbnails/90.jpg)
Primal-Dual Paradigm
Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
Algorithmic Game Theory (New Millennium):
Rational solutions to nonlinear convex programs
![Page 91: Algorithmic Game Theory and Internet Computing](https://reader036.fdocuments.net/reader036/viewer/2022081603/56813c61550346895da5eaee/html5/thumbnails/91.jpg)
Primal-Dual Paradigm
Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
Algorithmic Game Theory (New Millennium):
Rational solutions to nonlinear convex programs
![Page 92: Algorithmic Game Theory and Internet Computing](https://reader036.fdocuments.net/reader036/viewer/2022081603/56813c61550346895da5eaee/html5/thumbnails/92.jpg)
Primal-Dual Paradigm
Combinatorial Optimization (1960’s & 70’s):
Integral optimal solutions to LP’s
Approximation Algorithms (1990’s):
Near-optimal integral solutions to LP’s
Algorithmic Game Theory (New Millennium):
Rational solutions to nonlinear convex programs
Approximation algorithms for convex programs?!
![Page 93: Algorithmic Game Theory and Internet Computing](https://reader036.fdocuments.net/reader036/viewer/2022081603/56813c61550346895da5eaee/html5/thumbnails/93.jpg)
Open
Can Nash bargaining problem
for linear utilities case
be captured via an LP?
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