Algorithmic Game Theory and Internet Computing Nash Bargaining via Flexible Budget Markets Algorithmic Game Theory and Internet Computing Vijay V. Vazirani

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.

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!

The new platform for computing

Internet Open, distributed platform with enormous numbers of users interacting and sharing scarce resources Rife with situations of conflict of interest Massive computational power available

Internet Open, distributed platform with enormous numbers of users interacting and sharing scarce resources Rife with situations of conflict of interest Massive computational power available Algorithmic Game Theory

Nash bargaining Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo.

Example Two players, 1 and 2, have vacation homes: 1: in the mountains 2: on the beach Consider all possible ways of sharing.

Utilities derived jointly : convex + compact feasible set

Nash bargaining problem = (S, c) Disagreement point =

Typically: assume that S contains a point which makes each player happier Disagreement point =

Nash bargaining Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo. Q: Which solution is the “right” one?

Solution must satisfy 4 axioms: Paretto optimality Invariance under affine transforms Symmetry Independence of irrelevant alternatives

Thm: Unique soln: Disagreement point =

Nash bargaining Captures the main idea that both players gain if they agree on a solution. Else, they go back to status quo. Link between cooperative and non-cooperative games

Generalizes to n-players Theorem: Unique solution

Linear Nash Bargaining (LNB) Feasible set is a polytope defined by linear packing constraints Nash bargaining solution is optimal solution to convex program:

Q: Compute solution combinatorially in polynomial time?

Linear utilities B: n players with disagreement points, ci G: g goods, unit amount each S = utility vectors obtained by distributing goods among players

Convex program giving NB solution

Theorem For linear utilities case, Nash bargaining solution is rational! Polynomially many bits in size of instance

Theorem For linear utilities case, Nash bargaining solution is rational! Polynomially many bits in size of instance Solution can be computed in polynomial time.

Resource Allocation Nash Bargaining Problems Players use “goods” to build “objects” Player’s utility = number of objects. Bound on amount of goods available

Based on Kelly (1997): Mathematical framework for understanding TCP congestion control Jain & V. (2007): Eisenberg-Gale markets

Goods = edges Objects = flow paths

Given disagreement point, find NB soln.

Strongly polynomial, combinatorial algorithm Theorem: Strongly polynomial, combinatorial algorithm for single-source multiple-sink case. Solution is again rational.

Insights into Nash bargaining problems Chakrabarty, Goel, V., & Wang: Fairness Efficiency Competition monotonicity

Primal-Dual Paradigm Normal framework: LP-duality theory Extension to convex programs and KKT conditions.

Game plan Use KKT conditions to transform Nash bargaining problem to computing the equilibrium in a certain market. Find equilibrium using primal-dual paradigm.

Game plan Use KKT conditions to transform Nash bargaining problem to computing the equilibrium in a certain market. Find equilibrium using primal-dual paradigm.

Irving Fisher, 1891 Defined a fundamental market model

Fisher’s Model B = n buyers, money m(i) 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,

Fisher’s Model B = n buyers, money m(i) 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.

Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 Using primal-dual paradigm

Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 Using primal-dual paradigm Solves Eisenberg-Gale convex program

Flexible budget market 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.

Most cost-effective goods At prices p, for buyer i: Si =

Flexible budget market Lower bound on utility = At prices p, must spend to get utility

Flexible budget market Lower bound on utility = At prices p, must spend to get utility Define Find market clearing prices.

prices pj Theorem: x and p are optimal primal and dual solutions iff they are equilibrium allocations and prices. Also, equilibrium prices are unique.

Flexible budget market Several goods, fixed amount of each good Several buyers, with individual utilities and lower bound Find equilibrium prices and allocations, i.e., prices s.t., Each buyer gets an optimal bundle No deficiency or surplus of any good

An easier question Given prices p, are they equilibrium prices? If so, find equilibrium allocations.

An easier question Given prices p, are they equilibrium prices? If so, find equilibrium allocations. For each i,

For each i, most cost-effective goods p(1) p(2) p(3) p(4)

Network N(p) infinite capacities p(1) m(1) p(2) m(2) p(3) m(3) m(4)

p: equilibrium prices iff both cuts saturated Max flow in N(p) p(1) m(1) p(2) m(2) p(3) m(3) m(4) p(4) p: equilibrium prices iff both cuts saturated

Idea of algorithm “primal” variables: allocations “dual” variables: prices of goods Approach equilibrium prices from below: start with very low prices; buyers have surplus money iteratively keep raising prices, until outcome is clear.

Algorithm figures out if: A subset of players must get disagreement utility. There exists Nash bargaining solution (each player is happier than disagreement utility).

Finds dual feasible solution to:

Idea of algorithm Iterations: execute primal & dual improvements Allocations Prices (Money)

Two important considerations The price of a good never exceeds its equilibrium price Invariant: s is a min-cut

Max flow p(1) m(1) p(2) m(2) p(3) m(3) m(4) p(4) p: low prices

Two important considerations The price of a good never exceeds its equilibrium price Invariant: s is a min-cut Rapid progress is made Balanced flows

W.r.t. max-flow f, surplus(i) = m(i) – f(i,t) Max-flow in N p m i W.r.t. max-flow f, surplus(i) = m(i) – f(i,t)

Balanced flow surplus vector: vector of surpluses w.r.t. f. A max-flow that minimizes l2 norm of surplus vector.

Balanced flow surplus vector: vector of surpluses w.r.t. f. A max-flow that minimizes l2 norm of surplus vector. makes surpluses as equal as possible.

Running time? Proof technique of DPSV very difficult – impossible to carry over to the more complex algorithm.

DPSV proof idea Algorithm reduces l2 norm of surplus vector by inverse polynomial fraction in each iteration.

“Natural” approach Total surplus money = l1 norm of surplus vector Show: l1 norm of surplus vector reduces by inverse polynomial fraction in each iteration.

Simpler proof l1 norm of surplus vector decreases by a factor of (1 – 1/n2) in each iteration. Gives poly run time for more complex algorithm

Simpler proof l1 norm of surplus vector decreases by a factor of (1 – 1/n2) in each iteration. Gives poly run time for more complex algorithm Q: Find strongly poly algorithm!!

Resource Allocation Nash Bargaining Problems Players use “goods” to build “objects” Player’s utility = number of objects. Bound on amount of goods available

Goods = edges Objects = flow paths

Given disagreement point, find NB soln.

Theorem: Strongly polynomial, combinatorial algorithm for single-source multiple-sink case.

Other resource allocation problems k source-sink pairs (directed/undirected)

Other resource allocation problems k source-sink pairs (directed/undirected) Branchings rooted at sources (agents) Spanning trees Network coding

Branching construction (for broadcasting)

Sources want to obtain rooted branchings

Convex program for branching problem s.t. packing of branchings

Other resource allocation problems k source-sink pairs (directed/undirected)

JV: Irrational for 3 source-sink pairs

Irrational for 3 source-sink pairs Dual variables of convex program

Other resource allocation problems 2 k source-sink pairs (directed/undirected)

Max-flow min-cut theorem!

Theorem: Strongly polynomial algs for following NB problems: 2 source-sink pairs, undirected (Hu, 1963) spanning tree (Nash-William & Tutte, 1961) 2 sources branching (Edmonds, 1967 + JV, 2005) JV: 3 sources branching: irrational

Theorem: Strongly polynomial algs for following NB problems: 2 source-sink pairs, undirected (Hu, 1963) spanning tree (Nash-William & Tutte, 1961) 2 sources branching (Edmonds, 1967 + JV, 2005) JV: 3 sources branching: irrational No max-min theorems. Use full power of LP-duality theorem: 2 source-sink pairs, directed 2 sources, network coding

Nash Bargaining vs. Market Equilibrium

Linear Case of Arrow-Debreu Model Instead of money, each agent has an initial endowment of goods. Find market clearing prices, i.e., prices s.t. If each agent sells all her goods Buys optimal bundle using this money No surplus or deficiency of any good

Initial endowment of goods Agents Goods

Prices = $25 = $15 = $10 Agents Goods

Incomes Agents $50 $60 Goods Prices =$25 =$15 =$10 $40 $40

Maximize utility Agents $50 $60 Goods Prices =$25 =$15 =$10 $40 $40

Find prices s.t. market clears Agents $50 $60 Goods Prices =$25 =$15 =$10 $40 Maximize utility $40

Natural transformation: AD to NB ci = ui (initial-endowment(i))

Initial endowments

Utilities

AD solution: gets NB solution: gets ~ ½ unit of gets rest.

Open Solve Nash bargaining problems with other utility functions s.t. solution is still captured by a convex program!

Nonlinear programs with rational solutions! Open Nonlinear programs with rational solutions!

Nonlinear programs with rational solutions! Solvable combinatorially!! Open Nonlinear programs with rational solutions! Solvable combinatorially!!

Primal-Dual Paradigm Combinatorial Optimization (1960’s & 70’s): Integral optimal solutions to LP’s

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

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

Can Nash bargaining problem for linear utilities case Open Can Nash bargaining problem for linear utilities case be captured via an LP?

Eisenberg-Gale Program, 1959