Mechanism Design for Network Decongestion: Rebates and Time-of-Day Pricing

Transcription

1 Mechanism Design for Network Decongestion: Rebates and Time-of-Day Pricing Galina Schwartz1 (with Saurabh Amin2, Patrick Loiseau3 and John Musacchio3 ) 1 University of California, Berkeley 2 MIT 3 University of California, Santa Cruz Autumn 2011 TRUST Conference November 2-3, 2011, Hotel Palomar, Washington DC, USA

2 Motivation 2 / 16 Congestion: the Phenomenon wired Internet access wireless Internet access Congestion: the Causes variations in individual user demands (randomness) disparity of public and private incentives (brings tragedy of commons) Congestion: the Management peak-time (congested) off-peak time (not congested) the need to design mechanisms to shape user demand But a problem: users prefer flat (monthly) prices

3 Mechanisms of Congestion Management: Time-of-Day Pricing versus Rebates 3 / 16 Time-of-day (ToD) pricing [more standard] fixed reward: (individual shifted demand) (unit reward) Rebates (random) to users [with a fixed total rebate amount] individual shifted demand probabilistic reward: (total reward) total shifted demand Connections to literature probabilistic reward lottery (i.e., can be interpreted as a lottery) lottery scheduling [Waldspurger & Weihl (1994)] lotteries for roads decongestion [Merugu, Prabhakar & Rama (2009)] lotteries for public good provision [Morgan (2000)]

4 Model of network (with congestion) 2 periods: peak (congested) and off-peak (not congested) Non-atomic users User type θ Θ (delay (& time) preference) x θ fraction shifted to off-peak µ distribution of user types Aggregate user demand: D = Θ dµ(θ) peak demand D p off-peak demand D o D p = D and D o = f o D f p f o off-peak demand D o D p User demand shifted to off-peak period: G = x θ dµ(θ) [public good] Q: Why traffic shifted to off-peak is a public good? Θ 4 / 16

5 Network Load and Network Delay 5 / 16 Classical processor-sharing model (holds for 3G & 4G networks): Expected delay is proportional to 1 1 load h( ) users valuation for decongestion (user dis-utility of delay)(derived from latency) ( ) Dp G h(g) = L (ρ p ) = L CT p

6 User Utilities 6 / 16 Each user chooses x θ [0, x max ] to maximize utility: U θ (x θ, x θ ) U θ = P θ (1 x θ ) + O θ (f o + x θ ) p (1 x θ ) L (ρ p ) (f o + x θ ) L (ρ o ) ignore c θ (x θ ) = Ū θ [P θ (1 x θ ) + O θ (f o + x θ )] P θ peak period utility [no congestion] O θ off-peak period utility [no congestion] Ū θ user utility [no congestion] c θ could interpret as cost of shifting p subscription price (monthly)

7 User Utilities 7 / 16 Choice of x θ [0, x max ] to maximize utility: U θ (x θ, x θ ) = Ū θ + (1 x θ )h(g) c θ (x θ ) p h(g) user valuation of network quality (derived from latency) θ = 0.2 θ = 0.8 h(g) 0.5 c θ (x) G x

8 Social Optimum vs Nash equilibrium (no mechanism) 8 / 16 Aggregate welfare [Total user utility] W = U θ dµ(θ) = (D G)h(G) c θ (x θ )dµ(θ) pd. Θ Θ Social optimum [SO] x = arg max W (x) = x G = xθ dµ(θ) x (eq) θ Nash equilibrium [NE] = arg max U θ (x θ, x (eq) θ ) x θ G (eq) = x (eq) θ dµ(θ)

9 Mechanism design: how to improve user incentives? U θ (x θ, x θ ) = Ū θ + (1 x θ )h(g) c θ (x θ ) p + EM i (x θ, G) Both mechanisms reward users for shifting demand (both are budget-balanced; financed via increased monthly price)

10 Mechanism design: how to improve user incentives? U θ (x θ, x θ ) = Ū θ + (1 x θ )h(g) c θ (x θ ) p + EM i (x θ, G) Both mechanisms reward users for shifting demand

11 Nash Equilibrium and Social Optimum (a.e. = almost everywhere)

12 Nash Equilibrium and Social Optimum (a.e. = almost everywhere)

13 Nash Equilibrium and Social Optimum (a.e. = almost everywhere)

14 Nash Equilibrium and Social Optimum (a.e. = almost everywhere)

15 Nash Equilibrium and Social Optimum U θ (x θ, x θ ) = Ū θ + (1 x θ )h(g) c θ (x θ ) p + ES i (x θ, G) NE: fixed point of G x (resp) θ (G) G (resp) (G) = Θ x (resp) θ (G)dµ(θ) G (resp) G (resp) (Raffle) G (resp) (T.O.D. pricing) G unit reward: r [ToD pricing] R G [Rebate] h (G)(D G) [SO]

16 Nash Equilibrium and Social Optimum U θ (x θ, x θ ) = Ū θ + (1 x θ )h(g) c θ (x θ ) p + ES i (x θ, G) NE: fixed point of G x (resp) θ (G) G (resp) (G) = Θ x (resp) θ (G)dµ(θ) G (resp) G (resp) (Raffle) G (resp) (T.O.D. pricing) G (resp) (SO) unit reward: r [ToD pricing] R G [Rebate] G

17 Imperfect Info: User utilities perturbed Baseline: R = G h (G )(D G ), r = h (G )(D G ) NE and SO coincide But... R, r could be wrong (ex. ISP misspecified user utilities)? perturbation: c θ ( ) c θ ( ) = γ c θ ( ) 12 / 16

18 Imperfect Info: User utilities perturbed Baseline: R = G h (G )(D G ), r = h (G )(D G ) NE and SO coincide

19 Imperfect Info: User utilities perturbed Baseline: R = G h (G )(D G ), r = h (G )(D G ) NE and SO coincide

20 Imperfect Info: User utilities perturbed Baseline: R = G h (G )(D G ), r = h (G )(D G ) NE and SO coincide

21 Imperfect Info: User utilities perturbed Baseline: R = G h (G )(D G ), r = h (G )(D G ) NE and SO coincide

22 Imperfect Info: User utilities perturbed Baseline: R = G h (G )(D G ), r = h (G )(D G ) NE and SO coincide

23 Robustness results ism with unit reward derivative at G closer to SO is

24 Robustness results: c θ ( ) c θ ( ) = γ c θ ( ) Rebate is more robust ToD pricing is more robust G (eq) L G (eq) T (γ) (Raffle) (γ) (T.O.D. pricing) Target value: G (γ) G (eq) L G (eq) T (γ) (Raffle) (γ) (T.O.D. pricing) Target value: G (γ) W (γ) W (eq) L (γ) (Raffle) 0.25 W (γ) W (eq) T (γ) (T.O.D. pricing) Target value: SO renormalized γ W (γ) W (eq) L (γ) (Raffle) 0.03 W (γ) W (eq) T (γ) (T.O.D. pricing) Target value: SO renormalized γ 14 / 16

25 Summary of Results 15 / 16 Mechanisms for peak-time decongestion ToD mechanism Rebate mechanism Baseline NE achieves SO NE achieves SO Imperfect Info total reward unpredictable less robust in many situations of interest total reward predictable more robust in many situations of interest

26 Plans for Future 16 / 16 Extensions Providers [ISP] are included as players. ISP choices access prices rebates financed by reduced cost of network maintenance rebates financed by reduction of per user network cost Possible Applications electricity networks transportation networks