PR PRIC ICE E CO COMP MPETITIO ETITION N IN IN NE NETWORKED - - PowerPoint PPT Presentation

PR PRIC ICE E CO COMP MPETITIO ETITION N IN IN NE NETWORKED WORKED MA MARKET RKETS: S: HOW DO MO HOW DO MONO NOPO POLIES LIES IM IMPACT CT SO SOCI CIAL AL WE WELF LFARE? ARE?

Romina Jafarian Narges Rezaie

Introduction

Introduction

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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Single source single sink For a large class of buyer demand functions, equilibrium always exists and allocations can

• ften be close to optimal.

Evil monopoly monopolies may not be as ‘evil’ as they are made out to be Multiple source

In the absence of monopolies, mild assumptions on the network topology guarantee an equilibrium that maximizes social welfare

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DEFINE STRUCTURE

1

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COMPUTER MARKET

2

AD_MARKETS

3

MONOPOLY

4

MAIN QUESTIONS

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Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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DEFINE STRUCTURE

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• consider a market with multiple

sellers that can be represented by a directed graph G as follows:

• Every seller owns an item, which

is a link in the network.

• Every infinitesimal buyer seeks to

purchase a path in the network (set of items) connecting some pair of nodes.

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COMPUTER MARKET

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• in a computer market each link

could represent some component (e.g., a processor or video card) and buyers require a set of parts to assemble a complete computer system.

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AD_MARKETS

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• In ad-markets, the buyers

(advertisers) may want to purchase ads from a satisfactory combination of websites to reach a target audience.

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MONOPOLY

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• Although monopolies can cause

large inefficiencies in general, our main results for single-source single-sink networks indicate that for several natural demand functions the efficiency only drops linearly with M.

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MAIN QUESTIONS

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• 1. What conditions on the market

structure guarantee equilibrium existence?

• 2. How efficient are the

equilibrium allocations and how do they depend on buyer demand and network topology?

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Model and Equilibrium Concept

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model Singe-source Single-sink

buyers are interested in a single type of good. buyer wants to purchase a path between the same source node s and sink node t.

Profit

Seller’s profit: pexe − Ce(xe) buyer’s utility : vi minus the total price paid

Nash Equilibrium Seller

Each seller e controls a single good or link in a network G any quantity x of this good incurring a production cost of

Ce(x)

Buyer

Every buyer i wants to purchase an infinitesimal amount of some path connecting a source A sink node for which she receives a value vi

The Inverse Demand Function

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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Full information in the large

λ(x) = v

implies that exactly x amount of buyers value the path at v or larger

Example : λ(x) = 1 − x

λ(0.25) = 0.75, one-fourth of the buyers have a value of 0.75 or more for the s-t paths

02 03 01

Single-Source Single-Sink Networked Markets

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• There exists a Nash Equilibrium pricing

in every market under a very mild assumption on the demand function. we call such a solution a focal equilibrium.

• We further prove the uniqueness of focal

equilibria.

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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M := {e | (s, t) are disconnected in (V, E−{e})} Uniform ⊂ polynomial ⊂ concave ⊂ log-concave = MHR If the inverse demand function has a monotone hazard rate (MHR), the loss in efficiency at equilibrium is bounded by a factor of 1+m

DEAMAND UNIFORM: THE NASH EQUILIBRIUM MAXIMIZES WELFARE POLYNOMIAL: EFFICIENCY DROPS LOGARITHMICALLY AS M INCREASES CONCAVE: THE EFFICIENCY LOSS IS 1 + M/2

Definitions and Preliminaries

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There is a population T of infinitesimal buyers Every buyer wants to purchase edges on some s-t path An inverse demand function λ(x) x amount of buyers hold a value of λ(x) or more for these paths We define M to be the number of monopolies in the market

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An instance of our two-stage game is specified by a directed graph G = (V, E) A source and a sink (s, t) A cost function Ce(x)

• n each edge

Solution

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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VECTOR OF PRICES ON EACH ITEM P THE TOTAL FLOW OR MARKET DEMAND : σ𝒒∈𝑸 𝒚𝑸

P is the set of s-t paths

THE AGGREGATE UTILITY OF THE BUYERS IS ׬

𝒖=𝟏 𝒚

𝝁 𝒖 𝒆𝒖 − σ𝒇∈𝑭 𝒒𝒇𝒚𝒇 AN ALLOCATION OR FLOW X OF THE AMOUNT OF EACH S-T PATH UTILITY OF THE SELLERS IS σ𝒇∈𝑭(𝑸𝒇𝒚𝒇 − 𝑫𝒇(𝒚𝒇))

The amount of each edge purchased by the buyers (xe) e∈E

The social welfare:׬

𝒖=𝟏 𝒚

𝝁 𝒖 𝒆𝒖 − σ𝒇 C𝒇(𝒚𝒇)

The standard definition of Nash equilibrium for two-stage games

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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Result: for every best-response by the buyers, the seller’s profit should not increase

An allocation x is best- response by the buyers to prices p

If buyers only buy the cheapest paths For any cheapest path P ,𝜇 𝑦 = σ𝑓∈𝐹 𝑞𝑓

A solution (p, x) is a Nash equilibrium

x is a best-response For every feasible best-response flow (𝑦𝑓

′) for the new

prices, seller's profit cannot increase

Classes of inverse demand functions

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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Classes of inverse demand functions

01 - UNIFORM DEMAND: 𝝁 𝒚 = 𝝁𝟏 𝒈𝒑𝒔 𝒚 ≤ 𝑼

02 - POLYNOMIAL DEMAND: 𝝁 𝒚 = 𝝁𝟏 𝐛 − 𝒚𝜷 𝒈𝒑𝒔 𝜷 ≥ 𝟐

03 - CONCAVE DEMAND: 𝝁′ 𝒚 Is a non-increasing function of x

04 - MONOTONE HAZARD RATE (MHR) DEMAND:

𝝁′(𝒚) 𝝁(𝒚) is non-increasing

Example function: λ(x) = e −x

Min-Cost Flows and the Social Optimum

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02

R(x) is continuous, non- decreasing, differentiable, and convex

05

Maximizing social welfare is a min-cost flow of magnitude x∗ satisfying λ(x∗) ≥ r(x∗)

01

R(x) is the cost σ𝒇 C𝒇(𝒚𝒇)

• f the min-cost flow of

magnitude x ≥ 0

04

𝐬 𝐲 = ෍

𝒇∈𝐐

𝐝𝒇(𝒚𝒇)

03

𝒔 𝒚 = 𝒆 𝒆𝒚 𝑺(𝒚)

Monotone Price Elasticity (MPE)

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There exists a Nash Equilibrium (𝒒)𝒇∈𝑭,(෥ 𝒚)𝒇∈𝑭 satisfying some properties

• Non-Trivial Pricing

(Non-zero flow)

• Recovery of Production

Costs (Individual Rationality)

• Pareto-Optimality
• Local Dominance

(Robustness to small perturbations) An inverse demand function λ(x) is said to have a monotone price elasticity if its price elasticity

𝒚|𝝁′ 𝒚 | 𝝁(𝒚) is a non-

decreasing function of x which approaches zero as x → 0

Effect of Monopolies on the Efficiency of Equilibrium

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For any demand λ satisfying the MPE condition, ∃ a Nash equilibrium with a min-cost flow (෪ 𝑌𝑓) of size x̃ ≤ x ∗ such that Either λ(x̃) − r(x̃)

𝑁

= ෨ 𝑌|λ′ ( ෨ 𝑌) or ෨ 𝑌=𝑌∗, the optimum solution

Price Competition in Networked Markets Romina Jafarian Narges Rezaie

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02

In any network with no monopolies there exists a focal Nash Equilibrium maximizing social welfare

01

While for general functions λ(x) obeying the MPE condition, the efficiency can be exponentially bad, we show that for many natural classes of functions it is much better, even in the presence of monopolies

The social welfare of the Nash equilibrium is always within a factor of 1 + M of the optimum for MHR λ, and this bound is tight

03

Generalizations: Multiple-Source Networks

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We now move on to more general networks where different buyers have different Si-ti paths that they wish to connect and the demand function can be different for different sources A multiple-source single-sink series-parallel network with no monopolies admits a welfare. maximizing Nash Equilibrium for any given demand

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We show additional conditions on both the network topology and demand that lead to efficient equilibria, even in the presence of monopolies There exists a fully efficient equilibrium in multiple-source multiple - sink networks with Uniform demand buyers at each source if one of the following ,is true: (i) Buyers have a large demand and production costs are strictly convex. (ii) Every source node is a leaf in the network

In telecommunication networks

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Q&A SESSION

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THANK YOU!