Network Upgrade Game. Qiuhui Li, Qiao Zhao

Transcription

1 Network Upgrade Game Qiuhui Li, Qiao Zhao

2 Outline Two ISPs N-ISPs Declining Upgrade Cost Model

3 Motivation

4 Network ater one upgrade

5 Network ater both upgrade

6 Contribution Formulate this upgrade decision process as a repeated game. Obtain several sub-game perect equilibria(spe) under dierent conditions. Analyze the eect o declining upgrading cost in a continuous model.

7 Two ISPs model In every time period, an ISP makes a decision according to its strategy. δ Discount actor is. Game model or single period: Upgrade Not Upgrade Upgrade R,R a, Not Upgrade,a 0,0

8 SPE and Conditions Upgrade Immediately: Both ISP choose to upgrade in the current period. This is SPE i U Delayed Upgrade: I another ISP upgrade beore period n, it upgrade in the next period. Else it waits until the period n to upgrade. This is SPE i Never Upgrade: Using this strategy, the ISPs will never upgrade. It is SPE i: R 1 δ n a ( δ δ ) R R U n n 1 δ + (1 δ )(1 δ ) U a R > and U >

9 SPE and Conditions cont. No First Upgrade: Both do not upgrade until the other upgrades in the previous period. It s SPE i: R R a + δ U Asymmetric Free-ride: One upgrades immediately and the other never upgrade. This is SPE i R a U

10 SPE and Conditions cont. Mixed Free-ride: I no one has upgraded, both upgrade with probability α. I one has upgraded, the other remains not-upgraded. It s SPE i: R a U Mixed Upgrade: I no one has upgraded, both upgrade with probability α. I one has upgraded, the other also upgrades. It s SPE i: R a δ U R +

11 Proo Upgrade Immediately: Both ISP choose to upgrade in the current period. This is SPE i U R 1 Proo: Let ISP B upgrade immediately. δ Action o A Upgrade at 0 Never upgrade Upgrade at n Payo o A R U + 1 δ In between

12 Proo Delayed Upgrade: I another ISP upgrade beore period n, it upgrade in the next period. Else it waits until the period n to upgrade. This is SPE i n a ( δ δ ) R R U n n 1 δ + (1 δ )(1 δ ) Proo: Let ISP B use delayed upgrade: Action o A Upgrade at 0 Never upgrade Upgrade at n Beore or ater n Payo o A δ R a U + 1 δ δ n n R δ U Less or in between

13 Proo Mixed Free-ride: I no one has upgraded, both upgrade with probability. I one has upgraded, the other remains not-upgraded. It s SPE i: Proo: Let ISP B use Mixed Free-ride. When no one has upgraded, the expected payo o A or upgrading is J R a U R a J0 ( α) = α (1 α) U + The expected payo o A or not upgrading is J J1( α) = α (1 α) δj1( α) + a R (0) J (0) = U 0 J0(1) J1( 1) = U There exist 0< α < 1such that J0( α) = J1( α) and A is indierent between two choices. α

14 Outline Two ISPs N-ISPs Declining Upgrade Cost Model

15 N-ISPs Deine ( j) : ree-rider beneit i j ISPs upgraded (0) = 0 a( j: ) early-adopter beneit i j ISPs upgraded an ( ) = R Unique SPE: every ISP upgrading in the irst period, i: R ( N 1) U 1 δ R i stick to upgrade: U i never upgrade: 1 δ ( N 1) 1 δ >

16 Upgrade Immediately (Strict Dominance) Claim: upgrade is a better decision in any situation, i prove by induction Initial: suppose all but 2 ISPs have upgraded, say A & B. Upgrade Incentive: Dominant: U < min j= 0,..., N 1 a( j+ 1) ( j) A B upgrade reeride upgrade an ( ) an ( 1) U U reeride ( N 1) ( N 2) an ( ) ( 1) ( 1) ( 2) U N, an U N an ( ) ( N 1) an ( 1) ( N 2) U < min(, )

17 Upgrade Immediately (Strict Dominance) Assume all but (i) ISPs upgrade, and the claim is correct. Now, suppose all but (i+1) ISPs upgrade, i k {0,..., i} upgrade in current period, R an ( i+ k) + δ U, A upgrades now R R ( N i + k 1) + δ δu, A waits δ is a upbound Upgrade Incentive: U < So, induction argument shows Every One Upgrade is a SPE i: U < min j= 0,..., N 1 aj ( + 1) ( j) an ( i+ k) ( N i+ k 1), k 0,..., i upgrade is better than ree-rider!

18 Outline Two ISPs N-ISPs Declining Upgrade Cost Model

19 Declining Upgrade Cost Model Continuous Time Model, 2 ISPs here. A B upgrade ree ride Revenues discount actor: t e δ upgrade R, R a, ree ride, a 0, 0 i both/one upgrade at time 0, the total revenue, is: + t R + t a + δ δ t e R, e a, e δ = 0 δ = 0 δ = 0 δ Cost decline at actor: e γ t Includes: declining upgrade costs and discounting actor, The upgrade cost at time t is: e γ t ' U γ > δ

20 Two Critical Time t : the time upgrade is a better decision than ree-ride, t a δt ( R ) γt 1 γu t = arg max e Ue + = log t δ γ δ R : the time the irst to upgrade is a better decision t U = δ = γ δ a δt a γt 1 γ a arg max e Ue log + t Claim: both ISPs have upgraded ater time t, proo as ollows.

21 Theorem Theorem 3: t t a Suppose t t, the only SPE is two ISPs both upgrade at Proo: a R a Step 1: A upgrade at time beore t, B will upgrade at time t t δt R δt δt γt δt R γt arg max e ( e e ) Ue argmax e Ue t t t' + δ δ = = t t' δ Step 2: A upgrade ater time t, B ollows immediately t R t t t t R t arg max e δ ( e δ e δ ) Ue γ arg max e δ Ue γ t' t t' + δ δ = t t' δ = t

22 Theorem 3 (cont.) Step 3: i A consider upgrade beore time, derive his payo as he can induce B s behavior: δt R δt δt a γt δta γt arg max e ( e e ) Ue arg max e Ue min ( t, ) a t t tt + = tt δ δ δ = = t i A consider upgrade ater time, maximize his payo as he can induce B s behavior: t arg max e tt R t 1 U Ue min t, log( ) t δ = γ δ R = γ δ t γ So, upgrade at time is always a best decision when t t t a

23 Theorem Theorem 4: Suppose t < a one ISPs upgrade at, the only SPE is: the other one upgrade at Similar proo with Theorem 3. t t a t Intuition: compete or the time t a, i ail wait to upgrade until time t

24 Time to Upgrade t t Time to upgrade t technique improvemen t speed up the upgrade process technique improvement delay the upgrade time

25 Summary Discrete Time Model: Obtain several sub-game perect equilibrium(spe) under dierent conditions. Discuss the ree riding eect to the upgrading decisions Continuous Time Model: Cost declining model, discuss the eect o technique improvement to the upgrade decision. Technique improvement may delay the upgrading time

26 Thank you! Questions?