Introduction to Algorithmic aspect of Market Equlibra

IntroductionAlgorithms

Discussion and Open Problems

Introduction to Algorithmic Aspect of MarketEquilibria

Abner Chih Yi Huang

March 26, 2009

CSBB Lab, CS, NTHU

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Materials

Vijay V. Vazirani, chapter 5, Algorithmic Game Theory,Cambridge University Press, 2008.[5]

Devanur, et al., Market Equilibrium via a Primal-Dual-TypeAlgorithm, FOCS’02, 389-395.[1]

Devanur, et al., Market equilibrium via a primal–dualalgorithm for a convex program, J. ACM, 2008, v. 55,pp.1-18[2]

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Terminology

Market?Fisher’s model, 1891

Arrow-Debreu model, 1954

Efficiency?

Equilibria? Existence?

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Fisher’s model

n buyers, with specified money, m(i) for buyer i

k goods (unit amount of each good)

Linear utilities (效用/爽度): ui ,j is utility derived by i onobtaining one unit of j . Total utility of buyer i ,

ui =∑

j

ui ,jxi ,j

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Arrow-Debreu model

n buyers, k goods (unit amount of each good)

Linear utilities : ui ,j is utility derived by i on obtaining oneunit of j . Total utility of buyer i ,

ui =∑

j

ui ,jxi ,j

The Arrow-Debreu model is also known as the Walrasianmodel or the exchange model (以物易物), and it generalizesFisher’s model.

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Pareto improvement, 1886

Definition

Given a set of alternative allocations of, say, goods or income for aset of individuals, a movement from one allocation to another thatcan make at least one individual better off without making anyother individual worse off is called a Pareto improvement.

Definition

An allocation is Pareto efficient or Pareto optimal when no furtherPareto improvements can be made. This is often called a strongPareto optimum (SPO).

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Market-Clearing Price

Definition

The price at Market equilibrium is often called the equilibriumprice or market clearing price and will tend not to change unlessdemand or supply change.

Figure: The price at Market equilibrium

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Market Models Have Equilibria

Definition

A competitive equilibrium is a market, for Fisher’s model orArrow-Debreu model, with the following assumptions.

complete markets

No Uncertainty, i.e., we can calculate the futuredeterministically.

perfect competition

All information are available, i.e., preferences and budgets, arepublic.Zero transaction cost

local nonsatiation of preferences

preference function are continuous and strictly increasing in allgoods, i.e., more is better.

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First Fundamental Theorem of Welfare Economics

Theorem (Arrow-Debreu, 1954)

Any competitive equilibrium or Walrasian equilibrium leads to an

Pareto efficient allocation of resources.

In other words,

Theorem (Adam Smith’s invisible hand)

That competitive markets tend toward the efficient allocation of

resources.

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Combinatorial Algorithm for Fisher’s Model

Problem

Given a market of Fisher’s model, what is the market equilibria? In

the other words, what is the price such that the total utility of all

people in this market is maximum?

INPUT: Linear utility functions of buyers, integral money of

buyers, divisible goods.

OUTPUT: The prices of goods.

Measurement: Total utilities.

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No LP’s known for capturing equilibriumallocations for Fisher’s model, although thismodel doesn’t require integer constraints.

max∑

i

ui , s.t.,∑

j

pjxi ,j ≤ mi , ui =∑

j

ui ,jxi ,j

Devanur, Papadimitriou, Saberi, and Vazirani,2002[1]

Use primal-dual schema. It is highly successful algorithmdesign technique from exact and approximation algorithms

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Ask firstly: Given prices p,

are they equilibrium prices?

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Reduce It to Assignment Problem

Figure: An edge exists if and only if it produce maximum utilities ratio.

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Example of Building Equality Graph N(p)

Figure: The 4-tuple of each buyer means the linear coefficients of utilityfunction, i.e., (ui ,1, ui ,2, ui ,3, ui ,4).

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Figure: For buyer 1, we haveui,1

p1= 1

25 ,ui,2

p2= 1

5 ,ui,3

p3= 1

2 ,ui,4

p4= 2

9 . Hencewe can connect 1, 3 by an edge.

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Figure: Build N(p) by similar way.

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Reduce It to Maximal Flow Problem

Figure: Hence, we can construct the network of p, which is denoted byN(p), if p is given.

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A Verifying Algorithm of Market Clearing Prices

Theorem

The price p is equilibrium prices iff in the network N(p) the two

cuts (s,A∪B ∪ t) and (s ∪A∪B , t) are min-cuts. If so, allocations

corresponding to any max-flow in N are equilibrium allocations.

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Computing Market Clearing Prices for Linear Fisher Model

Now we have a verifier.

Could we devise an algorithm to compute

Market Clearing Prices for Linear Fisher

Model?

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Idea of Algorithm

”primal” variables: allocations

”dual” variables: prices of goodsApproach equilibrium prices from below:

start with very low prices;buyers have surplus money iteratively.Keep raising prices and decreasing surplus.

i.e., iterations of executing primal & dualimprovements

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Recall: maximum-flow problem

FORD-FULKERSON Method costs O(|E ||f ∗|).

Find augmenting path repeatedly.

Augmenting path with better property is critical!Edmonds-Karp algorithm runs in O(|V ||E |2) time.

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Invariant 1

Invariant 1

Let A, B denote goods and buyers respectively. Theprices p are such that (s, A ∪ B ∪ t) is a min-cut in

N(p).

The Invariant ensures that, at current prices, allgoods can be sold. The only eventuality is thatbuyers may be left with surplus money. When thesurplus vanishes, market clearing prices have beenattained.

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Invariant 1

Definition

Let A,B denote goods and buyers respectively. For S ⊆ B , defineits money flow mf (S) =

∑

i∈S m(i). For S ⊆ A, define its moneyflow mf (S) =

∑

i∈S p(i).

Definition

Let Γ(S) denote the neighborhood in the bipartite graphsG (A,B ,E ).

Lemma

For given prices p network N(p) satisfies the Invariant 1 iff.

∀S ⊆ A,mf (S) ≤ mf (Γ(S))

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An Example

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Invariant 1

If the Invariant 1 holds, it is easy to see thatthere is a unique maximal set S ⊆ A such thatmf (S) = mf (Γ(S)).

(S , Γ(S)) is the tight set w.r.t. prices p.The subgraph induced by (S , Γ(S)) is named

frozen subgraph.The active subgraph is the induced

subgraph of G by (A − S , B − Γ(S)).

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Lemma

LemmaIf the Invariant 1 holds and S ⊆ A is the tight set,

then each good j ∈ (A − S) has an edge, in the

graph N(p), to some buyer i ∈ (B − Γ(S)).

Consider a buyer j ∈ (A − S). If all edges of jincident to Γ(S), thenmf (S ∪ j) > mf (S) = mf (Γ(S)) = mf (Γ(S ∪ j)).

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Physical Meaning of Invariant 1

Consider the set S such that mf (S) = mf (Γ(S)).

Tight set S means p(Γ(S)) are equilibrium prices of thesub-market. It is Pareto optimum, since the prices of goods inthe tight set cannot be increased without violating theInvariant 1.

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Physical Meaning of Invariant 1

The active subgraph implies that the prices of goods 3, 4 aretoo high to buyer 4.

Hence sellers need to reduce prices.

Note that we can avoid this case by starting with very lowprices.

Another case is the prices of goods 3, 4 are too low to buyer4. Seller just raise the prices.

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Basic Algorithm

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Basic Algorithm

We would like to raise prices of goods in the activesubgraph in such a way that the equality edges in itare retained. The algorithm raises x , starting with

x = 1, until one of the following happens:

Event 1: A set R 6= ∅ goes tight in the activesubgraph.

Event 2: An new equality edge from goods in S

to buyers outside the tight set.

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An Example of Basic Algorithm

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Basic Algorithm

Hence, there is an intuitive approach to solve

this problem by raising the prices of goods inA − S until arriving the goal.

Like the idea of FORD-FULKERSON Method, itwill increase the utilities by finding tight sets

until the only tight set is (A, B).

We skip the details of finding the tight set. (p.

61)

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Time Complexity of Basic Algorithm

Basic algorithm is to solve this problem by raising theprices of goods in A − S until arriving the goal.

The execution sequence:

↑, ↑, ↑, E1, ↑, ↑, E2, ↑, E1, ↑, E2, ↑, ↑, ↑, ↑, E2, ↑, · · ·

We can divide it into phases.

↑, ↑, ↑, E1︸︷︷︸

Phase 1

, ↑, ↑, E2, ↑, E1︸︷︷︸

Phase 2

, ↑, E2, ↑, ↑, ↑, ↑, E2, ↑, · · ·︸︷︷︸

···

Each phase is partitioned into iterations which concludewith a new edge entering the equality subgraph.

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Let M be the total money possessed by the

buyers and let f be the max-flow computed innetwork N(p) at current prices p. Thus M − f

is the surplus money with the buyers.

Let U = maxi ,j{ui ,j} and let ∆ = nUn.

Lemma

At the termination of a phase, the prices of goods inthe newly tight set must be rational numbers with

denominator ≤ ∆.

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Lemma

At the termination of a phase, the prices of goods in the newlytight set must be rational numbers with denominator ≤ ∆.

Figure: Consider the newly tight set (S , Γ(S)) as a connected bipartitegraph. We can derive mf (S) = pjc/d ⇒ pj = mf (Γ(S))d/c .

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LemmaEach phase consists of at most n iterations.

Lemma

Consider two phases P and P ′, not necessarily

consecutive, such that good j lies in the newly tightsets at the end of P as well as P ′. Then the increase

in the price of j , going from P to P ′, is ≥ 1/∆2.

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lemma

tight set can be found using n max-flow computations.

Lemma

Basic Algorithm terminates with market clearing prices in atmost M∆2 phases, and executes O(Mn2∆2) max-flowcomputations.

No one discovered a technique to bound the running timein P of this approach.

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Recall: Maximum-Flow problem

Figure: FORD-FULKERSON Method is not in P. Edmonds-Karpalgorithm runs in O(|V ||E |2) time by finding better augmenting path.

How do we ensure that the surplus money of buyersreduces fast enough that the algorithm terminates

in polynomial time?

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Establishing polynomial running time

Like Maximal-Flow problem, the method of finding improvementsolution is critical. The authors proposed a method namedbalanced flow.

For a given flow f in the network N(p), define the surplus of buyeri , γ(p, f ), to be the residual capacity of the edge (i , t) withrespect to f , which is equal to mi minus the flow sent through theedge (i , t).

Define the surplus vector

γ(p, f ) := (γ1(p, f ), γ2(p, f ), · · · , γn(p, f ))

Let |v | denote the l2 norm of vector v .

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Balanced Flow

DefinitionFor any given p, a maximum flow that minimizes

|γ(p, f )| over all choices of f is called a balancedflow. If |γ(p, f )| < |γ(p, f ′)|, then we say f is more

balanced than f ′.

Lemma

All balanced flows in N(p) have the same surplusvector.

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Balanced Flow

Figure: When x = 2, there are some feasible surplus vector, e.g.,v = (20, 20, 20, 0), v ′ = (0, 60, 0, 0). |v | ∼ 35 < |v ′| = 60.

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Residual Network

Figure: For a given p and a flow f in N(p), let R(p, f ) be the residualnetwork of N(p) with respect to the flow f .

Theorem

A maximum-flow f in N(p) is balanced iff it satisfies the followingproperty. If γj(N , f ) < γi (N , f ) then there is no path from node j tonode i in R(f ) − {s, t}.

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Balanced Flow

Figure: Circulation is impossible.

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The Improved algorithm

Figure: Here we skip the detial to find balanced flow. (p.64) 49 / 67



Time Complexity Analysis

Lemma

The number of iterations executed in a phase is at

most n. Moreover, in every phase, there is an

iteration in which surplus of at least one of the

vertices is reduced by at least δn.

Lemma

If p0 and p∗ are surplus vectors before and after a

phase, |p∗|2 ≤ |p0|2(1 − 1n2 ).

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Time Complexity Analysis

Theorem

The algorithm of finding equilibrium prices and allocations for

linear utility functions in Fisher’s model needs to run

O(n4(log n + n log U + log M))

times maximal-flow algorithm.

Since (1 − 1n2 )

n2∼ 1

e∼ 1

2, we can observe that every O(n2)

phases, |γ(p)|2 is reduced by a factor of half. Hence, thenumber of phases is at most

O(n2 lg(∆4M2)) = O(n2(4 lg n + 4n lg U + 2 lg M))

, since M2/2i ≤ 1/∆4.

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Arrow-Debreu model: An Auction-Based Algorithm

Jain proved that Arrow-Debreu’s Model with Linear UtilityFunctions is in P at 2007[4], however it is based on ellipsoidalgorithm. And ellipsoid algorithm is slow in practice.

Garg et al. [3] proposed an auction-based PTAS for the linearcase of the Arrow-Debreu model.

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An Auction-Based Algorithm[3]

1 Initializes the price of each good to be unit, computes theworth of the initial endowment of each agent, and gives thismoney to each agent. All goods are initially fully unsold.

2 LOOP UNTIL (the surplus of the traders becomes sufficientlysmall, i.e., εamin, or all the goods are assigned.)1

2.1 LOOP UNTIL (no agent has surplus money)

2.1.1 Agent i outbid to buy her optimal good j as many as possible,i.e., at price (1 + ε)pj .

2.1.2 If agent i has all good j , update the worth of agents withprice (1 + ε)pj . And Jump to [2].

1amin means the minimum amount of initial endowments which a agent has.53 / 67



Analysis of The Auction-Based Algorithm

Theorem

The performance ratio of above algorithm is (1 − ε)2.

This algorithm is a natural way to achieves equilibrium, i.e., italways sells an agent her optimal goods relative to current prices p.Consider the case that agent i doesn’t run out her money. Let M

denote the total worth of i ’s initial endowment at terminatingprices. Assume that she spent M1 of this. Since the total surplusmoney left at termination is at most εamin,M1 ≥ (1 − ε)M.

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On the other hand, agent i runs out her monet.Some part of good j may have been sold at price (1 + ε)pj toagent i , even though the equilibrium price announced is pj . Henceshe didn’t not get the total utilities M, i.e., the total worth of i ’sinitial endowment at terminating prices. We assume she spentmoney M1. Then we have

Her real utilites =M1

1 + ε≥

(1 − ε)M

1 + ε≥ (1 − ε)2M

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Theorem

The algorithm given above finds an approximate equilibrium for the

linear case of the Arrow-Debreu model in time

O

(mn

ε2log

nvmax

εaminvminlog

vmax

vmin

)

Denote by pmax the maximum price assigned to a good by thealgorithm.

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Each iteration raises the price of a good by a factor of (1 + ε).Hence

O(n log1+εpmax)

How many rounds in an iteration? The total surplus at thebeginning of the iteration is npmax , and the algorithm terminatesas soon as the total surplus is at most εamin. Hence

O(log1+ε

npmax

εamin

)

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O(log1+ε

npmax

εamin) is an upper bound of the number of rounds in an

iteration, although this case only occurs at the case that thisalgorithm terminates at the first iteration.

Note that the ratio of maximum to minimum price of a good isbounded by umax

umin, where ui ,j means utility per unit money of trader

i for good j . Therefore, Pmax is also bounded by umax

umin.

Since each round costs O(m), we have the upper bound of thenumber of rounds

O

(

(n log1+ε

umax

umin

) × (log1+ε

n umax

umin

εamin

) × m

)

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Linear utility functions is too restrictive to beuseful. Concave utility functions are considered

especially useful in economics.

Obtaining a polynomial time algorithm forConcave utility functions is a premier open

problem today.Other interesting market problem, e.g., GoogleAdWord Market.

Advertisers provide bids for keywords they areinterested in. Maximize total revenue of Google.

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References I

Devanur, N. R., Papadimitriou, C. H., Saberi, A., and Vazirani, V. V.

Market equilibrium via a primal-dual-type algorithm.In FOCS ’02: Proceedings of the 43rd Symposium on Foundations of Computer Science (Washington, DC,USA, 2002), IEEE Computer Society, pp. 389–395.

Devanur, N. R., Papadimitriou, C. H., Saberi, A., and Vazirani, V. V.

Market equilibrium via a primal–dual algorithm for a convex program.J. ACM 55, 5 (2008), 1–18.

Garg, R., and Kapoor, S.

Auction algorithms for market equilibrium.In STOC ’04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (New York,NY, USA, 2004), ACM, pp. 511–518.

Jain, K.

A polynomial time algorithm for computing an arrow–debreu market equilibrium for linear utilities.SIAM Journal on Computing 37, 1 (2007), 303–318.

Nisan, N., Roughgarden, T., Tardos, E., and Vazirani, V. V.

Algorithmic Game Theory.Cambridge University Press, New York, NY, USA, 2007.

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How to Find the Tight Set

Let p denote the current price vector (i.e. at x = 1).We first present a lemma that describes how the

min-cut changes in N(xp) as x increases. Define

x∗ = minS 6=∅,S⊆A

mf (Γ(S))

mf (S)

the value of x at which a nonempty set goes tight.Let S∗ denote the tight set at prices x∗p.

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Find the Tight Set

Recall Lemma 1

For given prices p network N(p) satisfies theInvariant 1 iff.

∀S ⊆ A, mf (S) ≤ mf (Γ(S))

if x ≤ x∗, then (s, A ∪ B ∪ t) is a min-cut.

if x > x∗, then (s, A ∪ B ∪ t) is not a min-cut.If (s ∪ A1 ∪ B1, A2 ∪ B2 ∪ t) is a min-cut in

N(xp) then S∗ ⊂ A1.

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Find the Tight Set

Figure: Consider x = 2.1 > x∗ = 2. The cut is impossible containing theedges from goods 3, 4 to buyers 1, 2, 3, since their weights are infinity.Note that 140 < 147, 126 < 180. Since only 10x + 60x exceeds themoney flow of its neighbors, S∗ ⊂ A1.

If we assume x = mf (B)/mf (A) at first, then we can solve itrecursively by above this lemma.

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Balanced Flow

Theorem

Computing the balanced flow needs to run at most n times

maximal-flow algorithm.

Sketch of Proof : Let mavg := (∑

i m(i) −∑

j pj)/|B | . Computethe maximum flow in the equality subgraph after subtracting mavg

from the capacity of each edge adjacent t. Let (S ,T ) be themaximal s-t min-cut in that network. s ∈ S , t ∈ T . If T = {t}then the current maximum flow is balanced. Otherwise, let N1 andN2 be the networks induced by T ∪ {s} and S ∪ {t} respectively.

Repeated by divided and conquer approach.

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Sketch of Proof of Theorem of Balanced Flow

Figure: Let x = 2. We assume that this graph already substract mavg .And we get the min-cut. Then we have m′

avg = (180 − 120)/3 = 20.Hence we have a new graph N ′ with money (80, 40, 0). Compute themin-cut for N ′, we have T ′ = {t} for N ′, hence it is a balanced flow ofN ′. 65 / 67




Claim that the union of balanced flows in N1 and N2

is a balanced flow in N(p).

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Clearly, the number of goods in the biggest piecedrops by at least 1 in each iteration. Therefore, thedepth of recursion is at most n.

Observe that the sub-instance of size 1 doesn’t needto compute max-flow.

Hence, the total computational overhead is nmax-flow computations.

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Introduction to Algorithmic aspect of Market Equlibra

Software

open problems introduction

market of fishers model

market equilibrium

open problems market

walrasian model

exchange model

equilibrium price

fishers model problem