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 Single source single sink For a large class of buyer demand functions, equilibrium always exists and allocations can often 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 3 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
D EFINE STRUCTURE 1 5 1 C OMPUTER MARKET 2 A D _ MARKETS 3 4 2 4 MONOPOLY 3 M AIN QUESTIONS 5 4 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
D EFINE STRUCTURE 1 5 1 • consider a market with multiple sellers that can be represented by a directed graph G as follows: • Every seller owns an item, which 4 2 is a link in the network. • Every infinitesimal buyer seeks to purchase a path in the network 3 (set of items) connecting some pair of nodes. 5 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
C OMPUTER MARKET 2 5 1 • 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 4 2 complete computer system. 3 6 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
A D _ MARKETS 3 5 1 • In ad-markets, the buyers (advertisers) may want to purchase ads from a satisfactory combination of websites to reach a target audience. 4 2 3 7 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
4 MONOPOLY 5 1 • Although monopolies can cause large inefficiencies in general, our main results for single-source single-sink networks indicate that for several natural demand 4 2 functions the efficiency only drops linearly with M. 3 8 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
M AIN QUESTIONS 5 5 1 1. What conditions on the market structure guarantee equilibrium existence? 2. How efficient are the 4 2 equilibrium allocations and how do they depend on buyer demand and network topology? 3 9 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Model and Equilibrium Concept
Singe-source Single-sink Seller buyers are interested in a single type of good. buyer wants to Each seller e controls a single purchase a path between the good or link in a network G same source node s and sink any quantity x of this good node t. incurring a production cost of C e (x) model Profit Buyer Seller ’ s profit: p e x e − C e (x e ) Every buyer i wants to buyer ’ s utility : v i minus the purchase an infinitesimal total price paid amount of some path connecting a source A sink node for which she Nash Equilibrium receives a value v i 11 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
The Inverse Demand Function
Full information in the large 01 λ( x) = v 02 implies that exactly x amount of buyers value the path at v or larger Example : λ( x) = 1 − x 03 λ( 0.25) = 0.75, one-fourth of the buyers have a value of 0.75 or more for the s-t paths 13 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Single-Source Single-Sink Networked Markets
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. 15 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
UNIFORM: THE NASH EQUILIBRIUM M := {e | (s, t) are disconnected in (V, E − {e})} MAXIMIZES WELFARE Uniform ⊂ polynomial ⊂ concave ⊂ log-concave = MHR POLYNOMIAL: EFFICIENCY DROPS DEAMAND LOGARITHMICALLY AS M INCREASES 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 CONCAVE: THE EFFICIENCY LOSS IS 1 + M/2 16 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Definitions and Preliminaries
Every buyer wants to An instance of our two-stage purchase edges on some s-t game is specified by a path directed graph G = (V, E) 1 8 2 An inverse demand A source and a sink (s, t) function λ (x) 7 3 A cost function C e (x) x amount of buyers hold a value of λ (x) or on each edge more for these paths 6 4 We define M to be the There is a population T of 5 number of monopolies in the infinitesimal buyers market 18 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Solution
𝒚 𝝁 𝒖 𝒆𝒖 − σ 𝒇 C 𝒇 (𝒚 𝒇 ) The social welfare: 𝒖=𝟏 𝒚 𝝁 𝒖 𝒆𝒖 − σ 𝒇∈𝑭 𝒒 𝒇 𝒚 𝒇 THE AGGREGATE UTILITY OF THE BUYERS IS 𝒖=𝟏 THE TOTAL FLOW OR MARKET DEMAND : σ 𝒒∈𝑸 𝒚 𝑸 P is the set of s-t paths UTILITY OF THE SELLERS IS VECTOR OF PRICES ON EACH ITEM P σ 𝒇∈𝑭 (𝑸 𝒇 𝒚 𝒇 − 𝑫 𝒇 (𝒚 𝒇 )) The amount of each edge purchased by the buyers ( x e ) e ∈ E AN ALLOCATION OR FLOW X OF THE AMOUNT OF EACH S-T PATH 20 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
The standard definition of Nash equilibrium for two-stage games
An allocation x is best- A solution (p, x) is a response by the buyers to Nash equilibrium prices p For every feasible If buyers only For any cheapest best-response flow buy the cheapest path P , 𝜇 𝑦 x is a best-response ( 𝑦 𝑓 ′ ) for the new prices, seller's profit paths = σ 𝑓∈𝐹 𝑞 𝑓 cannot increase Result: for every best-response by the buyers, the seller ’ s profit should not increase Price Competition in Networked Markets Romina Jafarian Narges 22 Rezaie
Classes of inverse demand functions
01 - UNIFORM DEMAND: 𝝁 𝒚 = 𝝁 𝟏 𝒈𝒑𝒔 𝒚 ≤ 𝑼 Classes of inverse demand functions 02 - POLYNOMIAL DEMAND: 𝝁 𝒚 = 𝝁 𝟏 𝐛 − 𝒚 𝜷 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 24 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Min-Cost Flows and the Social Optimum
R(x) is the cost σ 𝒇 C 𝒇 (𝒚 𝒇 ) 01 of the min-cost flow of magnitude x ≥ 0 R(x) is continuous, non- 02 decreasing, differentiable, and convex 𝒔 𝒚 = 𝒆 03 𝒆𝒚 𝑺(𝒚) 𝐬 𝐲 = 𝐝 𝒇 (𝒚 𝒇 ) 04 𝒇∈𝐐 Maximizing social welfare is a 05 min-cost flow of magnitude x ∗ satisfying λ(x ∗ ) ≥ r(x ∗ ) 26 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Monotone Price Elasticity (MPE)
An inverse demand function λ(x) is said to have a monotone price • elasticity if its price Non-Trivial Pricing 𝒚|𝝁 ′ 𝒚 | (Non-zero flow) elasticity 𝝁(𝒚) is a non- • Recovery of Production decreasing function of x Costs (Individual which approaches zero as Rationality) x → 0 • Pareto-Optimality • Local Dominance (Robustness to small perturbations) There exists a Nash Equilibrium (𝒒) 𝒇∈𝑭 ,( 𝒚) 𝒇∈𝑭 satisfying some properties 28 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Effect of Monopolies on the Efficiency of Equilibrium
For any demand λ satisfying the MPE condition, ∃ a Nash equilibrium with a 𝑌 𝑓 ) of size x̃ ≤ x ∗ such that Either λ(x̃) − r(x̃) 𝑌 |λ ′ ( ෨ min-cost flow ( ෪ = ෨ 𝑌 ) or 𝑁 ෨ 𝑌 = 𝑌 ∗ , the optimum solution 30 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
While for general functions λ(x) obeying the MPE condition, the efficiency can be exponentially bad, we 01 show that for many natural classes of functions it is much better, even in the presence of monopolies In any network with no monopolies there 02 exists a focal Nash Equilibrium maximizing social welfare The social welfare of the Nash equilibrium is 03 always within a factor of 1 + M of the optimum for MHR λ, and this bound is tight 31 Price Competition in Networked Markets Romina Jafarian Narges Rezaie
Generalizations: Multiple-Source Networks
Recommend
More recommend
