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Contents A public good model for p2p An Asymptotically Optimal Scheme Simple contribution policies with exclusions for P2P File Sharing An application to file sharing heterogeneous file popularity stability Panayotis

  1. Contents • A public good model for p2p An Asymptotically Optimal Scheme • Simple contribution policies with exclusions for P2P File Sharing • An application to file sharing – heterogeneous file popularity – stability Panayotis Antoniadis Athens University of Economics and Business – group formation Costas Courcoubetis Athens University of Economics and Business – adaptation Richard Weber Centre for Mathematical Sciences, Cambridge, UK • Conclusions A model for p2p file sharing I mplications • Implication: “free market” solution is inefficient • Resource allocation in p2p file sharing is a public good – each peer maximizes own net benefit problem – actions affect others – all peers benefit from the contribution of any single peer – hence private optimum differs from social optimum – downloading a file by one peer does not prevent another peer • Need regulation: use prices or rules to influence behaviour from downloading the same file (no congestion effects) – incentives for each peer reflect the effect it has on others – but contribution is costly – example of a rule: downloads = uploads – positive externality creates an incentive to free-ride on efforts of others – a peer’s incentive is to offer a few files in the common pool and • Problem: optimal design requires information on user types requests lots of downloads from others – under full info: personalized price/rule for each peer θ θ – “first-best” policy u ( Q ), is iid with distributi on H benefit = i i peer i : f cost = = payment in “kind” • Existing approaches based on heuristics i n peers – reciprocity based punishments/rewards

  2. What to do? Large systems are simpler! • How can the system/planner/network manager get the • Size helps! required information to design optimal contribution rules? – simplifies mechanism, limits per capita efficiency loss – if lucky, can gather personalized data about users • Theorem: A very simple mechanism – otherwise, users must be given incentives to reveal relevant information to planner “contribute f if join, 0 otherwise” is nearly optimal when the network is large • Mechanism Design: set prices/rules to encourage users to • Why? act truthfully, maximize social welfare – in a large network it is hard to get people pay more than – Well-developed economic theory; but solutions typically a minimum • very complex, dependent on fine details • Other major benefits: • require large amounts of info to be passed to centre • “second-best” policy – Low informational requirements, easy to apply in a large • Approximations? class of examples Some formulas for SW… Theorem − θ 1 H ( ) * , θ No contributions, system of size Q * Q Let maximize ∑ θ ⎛ ⎞ 1 1 θ E [ ] ∫ = − ⎜ ⎟ i P max n yh ( y ) dy u ( Q ) c ( Q ) ⎛ ⎞ ⎝ ⎠ ∫ 1 θ ∈ ≥ θ = − [ 0 , 1 ], Q 0 ⎜ ⎟ SW n yh ( y ) dy u ( Q ) c ( Q ) ⎝ ⎠ 0 − θ θ − ≥ n [ 1 H ( )] u ( Q ) c ( Q ) 0 subject to Fixed contributions covering cost, system of size Q Then, the policy: − θ 1 H ( ) ⎛ ⎞ ∫ 1 = − ⎜ ⎟ SW n yh ( y ) dy u ( Q ) c ( Q ) h = θ * * ⎝ ⎠ each participating peer contributes f u ( Q ) θ 1 0 θ ≤ ≤ + − θ θ − ≥ P P P o ( n ) achieves n [ 1 H ( )] u ( Q ) c ( Q ) 0 SB expected number fee P = efficiency of second-best policy SB of participants

  3. Example File sharing • Q : expected number of distinct files = = θ 1 / 2 u ( Q ) 0 . 6 Q , c ( Q ) Q , uniform in [ 0 , 1 ] θ i u ( Q ) f • peer i : utility = , cost = = number of i i ⎛ ∫ ⎞ 1 − files provided to the system ⎜ ⎟ 2 max n ydy 0 . 6 Q Q ⎝ ⎠ θ θ ∈ ≥ [ 0 , 1 ], Q 0 f • randomly chosen from N files i − θ θ − ≥ 2 s . t . n [ 1 ] 0 . 6 Q Q 0 ∑ − ≈ − = F / N Q ( F ) N ( 1 e ), where F f i θ = = = * * 2 2 Solution: 1 / 4 , Q 0 . 0126 n , SW 0 . 006328 n ⎛ ∫ ⎞ 1 − ⎜ ⎟ max n yh ( y )) dy u ( Q ( F )) F • satisfaction of cost coverage constraint: • Solve ⎝ ⎠ θ θ ∈ ≥ [ 0 , 1 ], F 0 reduction of SW by 43% − θ θ − ≥ n [ 1 H ( )] u ( Q ( F )) F 0 subject to The function F(Q) Heterogeneous file popularity F ( Q ) F : popular content • General case: u ( F , F ), c ( F , F ) 1 1 2 1 2 F N =10,000 : less popular content * * 2 f 1 , f – specify contributions 2 + = + • Interesting case: u ( aF F ), c ( F , F ) bF F 1 2 1 2 1 2 a = b • Then, provide both types only if * f • Optimum contribution is a scalar + = * f , f s . t . af f f – a peer can provide any combination 1 2 1 2 , – measuring rate of uploads is a good proxy Q

  4. Stability Group formation (1/ 5) * f • Assume contribution fixed • Assume peers of different sub-types θ • Participation decision: based on prior expectation A ~ [ 0 , 0 . 5 ] • Type A: (e.g. ISDN users) i regarding total content availability F θ B ~ [ 0 . 5 , 1 ] • Type B: (e.g. DSL users) i * F • Will be reached? • Do they gain more by – forming 2 distinct groups vs X X content + k k 1 − = y b ( y ) 0 A B forming a larger group? + k k 1 rounds – being distinguished by the system = * X b ( X , f ) in the larger group? A+B + k 1 k F F = * 1 X ≥ F F stability if 2 0 1 Group formation (2/ 5) Group formation (3/ 5) θ A ~ [ 0 , 0 . 5 ] • Group A: (e.g. ISDN users) i As a function of the percentage of each group in the mix θ B • Group B: (e.g. DSL users) ~ [ 0 . 5 , 1 ] i Assume that the percentage of each group in the mix is 50% (n=1000) Welfare Group A Group B Total Distinct groups 3296 35156 38452 Indistinguishable 6976 44792 51768 (+ 111%) (+ 27%) Distinguishable 31249 31250 62500 (+ 848%) (-11%)

  5. Group formation (4/ 5) Group formation (5/ 5) = • How to provide better incentives for both types to a c ( F ) m F , m = # of participants Adding a “congestion” cost: combine and reveal their types? Group A – reduce cost of heavy users by limiting upload rates Group B – reduce fees of heavy users 2.2 • Offer sets of tariffs (versioning) 1.15 2 – allow self-selection 1.8 1.1 1.6 • Model difference in cost for uploading 1.4 1.05 1.2 – higher-cost peers benefit in a larger group when types can be distinguished 200 400 600 800 200 400 600 800 0.8 N N Adaptation Conclusions • Fixed contribution schemes can be efficient ( ⋅ H ) • What if not known? • Result to tractable optimization problems • In general incentive to shade declarations • Provide useful insight to many interesting questions • Information regarding user types may be strategic • Repeated game formulation: in each round, peer i θ θ • Open issues: samples from H , declares i i – more complex cost structures – truth-telling equilibrium – adaptation – multiple round games – practical application • Check also … – Market Management of P2P Systems (MMAPPS) • http://www.mmapps.org – AUEB Network Economics and Services Group • http://nes.aueb.gr/p2p.html

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