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Investment and market structure in industries with congestion Ramesh Johari November 7, 2005 (Joint work with Gabriel Weintraub and Ben Van Roy)
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Big picture Consider industries where: customer experience degrades with congestion providers invest to mitigate congestion effects Basic question: What should we expect?
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The current situation Current answer: don’t know! Trauma in the backbone industry Unbundling, then bundling of DSL Municipal provision of WiFi access How do engineering facets impact industry structure?
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Outline Background and model Returns to investment The timing of pricing and investment Key results Future work and conclusions
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Basic model Consumers Destination
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Basic model Consumers Destination Total mass = X ; assumed “infinitely divisible”
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Basic model Consumers Destination Providers
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Model 1: “selfish routing” Only considers congestion cost Consumers Destination l1(x1)l1(x1) l2(x2)l2(x2) l3(x3)l3(x3) Congestion cost seen by a consumer
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Model 1: “selfish routing” Consumers split so l 1 ( x 1 ) = l 2 ( x 2 ) = l 3 ( x 3 ) ) Wardrop equilibrium Consumers Destination l1(x1)l1(x1) l2(x2)l2(x2) l3(x3)l3(x3)
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Model 2: Selfish routing + pricing Providers charge price per unit flow Consumers Destination p 1 + l 1 (x 1 ) p 2 + l 2 (x 2 ) p 3 + l 3 (x 3 ) Prices
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Model 2: Selfish routing + pricing Assumes the networks are given Timing: First: Providers choose prices Next: Consumers split so: p 1 + l 1 ( x 1 ) = p 2 + l 2 ( x 2 ) = p 3 + l 3 ( x 3 ) [Recent work on equilibria, efficiency, etc., by Ozdaglar and Acemoglu, Tardos et al., etc.]
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Model 3: Our work Providers invest and price Consumers Destination p 1 + l(x 1, I 1 ) p 2 + l(x 2, I 2 ) p 3 + l(x 3, I 3 )
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Model 3: Our work Providers invest and price Consumers Destination p 1 + l(x 1, I 1 ) p 2 + l(x 2, I 2 ) p 3 + l(x 3, I 3 ) Investment levels
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Model details Cost of investment: C ( I ) Congestion cost: l ( x, I ) Given “total traffic” x and investment I Increasing in x, decreasing in I Given prices p i and investments I i customers split so that: p i + l ( x i, I i ) = p j + l ( x j, I j ) for all i, j Profit of firm i : p i x i - C ( I i )
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Costs Two sources of “cost”: disutility to consumers: congestion cost provisioning cost of providers: investment cost
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Model details: Efficiency Efficiency = minimize total cost: i [ x i l ( x i, I i ) + C ( I i ) ] Total congestion cost in provider i’s network Provider i’s investment cost
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Model details: Efficiency Efficiency = minimize total cost: i [ x i l ( x i, I i ) + C ( I i ) ] Central question: When do we need regulation to achieve efficiency?
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Returns to investment A key role is played by: K ( x, I ) = x l ( x, C -1 ( I ) ) Idea: measure investment in $$$. Fix > 1. K ( x, I ) < K ( x, I ): increasing returns to investment K ( x, I ) > K ( x, I ): decreasing returns to investment
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Returns to investment Increasing returns to investment occur if: one large link has lower congestion than many small links (e.g. statistical multiplexing) marginal cost of investment is decreasing Example: Fiber optic backbone (?)
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Returns to investment Decreasing returns to investment occur if: splitting up investments is beneficial (e.g. many “small” base stations vs. one “large” base station (?) ) marginal cost of investment is increasing
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Increasing returns and monopoly Important (basic) insight: increasing returns to investment ) natural monopoly is efficient ) some regulation needed For the rest of the talk: Assume decreasing returns to investment.
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Timing: pricing and investment When do providers price and invest? Long term investment, then short term pricing? Or, short term investment, and short term pricing?
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Timing: pricing and investment Long term investment + short term pricing: Can be arbitrarily inefficient. (Under-investment first, then price gouging later.)
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Timing: pricing and investment What about simultaneous pricing and investment? i.e., investment decisions are short term and relatively reversible Remarkable fact: Competition is efficient! (in a wide variety of cases…)
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Summary of results In a wide range of models, if a (Nash) equilibrium exists, it is unique, symmetric, and efficient. Sufficient competition is needed to ensure equilibrium exists. With fixed entry cost: competition is asymptotically efficient.
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Efficiency of equilibrium If C ( I ) is convex and: l ( x, I ) = l ( x )/ I, and l ( x, I ) is convex; OR l ( x, I ) = l ( x / I ), and l ( ¢ ) is convex; OR l ( x, I ) = x q / I, for q ¸ 1 Then: At most one Nash equilibrium exists, and it is symmetric and efficient.
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Efficiency of equilibrium Included: l ( x, I ) = x / I : x = total # of bits to transfer I = capacity (in bits/sec) l ( x, I ) = time to completion Not included: M/M/1 delay: l ( x, I ) = 1/( I - x )
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Existence of equilibrium If l ( x, I ) = x q / I and C ( I ) = I, then Nash equilibrium exists iff N ¸ q + 1 ( N = # of providers)
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Entry Suppose: To enter the market, providers pay a fixed startup cost. Then: As the customer base grows, the number of entrants becomes efficient.
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Application: Wi-Fi In Wi-Fi broadband access provision, we see: constant marginal cost of capacity expansion low prices for upstream bandwidth short term investment decisions Would competition be efficient?
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Application: source routing Common argument: Source routing would give providers the right investment incentives Our answer: depends on cost structure depends on timing of pricing and investment
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Back to Clean Slate What is the value of this research? Technology informs investment cost structure Performance objectives inform congestion cost structure Both impact market efficiency
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Open issues Future directions: Ignored contracting between providers Peering relationships Transit relationships Ignored heterogeneity of consumers
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