Perspectives on Resource Allocation Kameswari Chebrolu Bhaskaran Raman

Perspectives on Resource Allocation Kameswari Chebrolu, Bhaskaran Raman, Ramesh R. Rao November 11, 2004 ISMA 2004 Workshop on Internet Signal Processing (WISP) 1

Resource Allocation • Resource allocation is a critical issue in the design of communication and computing systems: – Bandwidth (e. g. wireless channel) – CPU (e. g. shared multi-user systems) – Memory, file cache • Demand from multiple sources may exceed the supply – Uncertainty on the supply side (varying channel conditions) – Uncertainty on the demand side (fluctuating traffic) • Mechanisms of control – Admission control, retransmission, differential pricing… • Yet demand may exceed supply – How to divide the available resources? • • Proportional share is often taken to be the definition of fairness in this domain Is Proportional Share really fair? Are there other allocation mechanisms that deserve our attention? ISMA 2004 Workshop on Internet Signal Processing (WISP) 2

Resource Allocation in other Domains • Bankruptcy: – the liquidation value of a bankrupt firm is to be divided among its N creditors • Taxation – the cost of a project is to be divided among N taxpayers • Claims Problem Definition – An amount E has to be divided among a set of N agents with claims (c) adding up to more than E – A division rule R assigns an award (x) to each claimant such that 0 ≤ x ≤ c with awards adding up to E E c 4 c 5 c 1 c 2 c 3 x 6 x 1 x 2 x 3 x 4 x 5 ISMA 2004 Workshop on Internet Signal Processing (WISP) c 6 3

Proportional Rule (PROP) Rule: Explanation: • Make awards proportional to claims Example: E = 150 c = [20, 80, 100] P = [15, 60, 75] ISMA 2004 Workshop on Internet Signal Processing (WISP) 4

Constrained Equal Awards (CEA) Rule: Explanation: • Equality underlies many theories of economic justice • Based on equality but respects upper bounds on awards • Assign equal amounts to all claimants subject to no one receiving more than his claim Example: 150/3 = 50 [20, 50]; 30/2 [20, 65] E = 150 c = [20, 80, 100] P = [20, 65] ISMA 2004 Workshop on Internet Signal Processing (WISP) 5

Constrained Equal Losses (CEL) Rule: Explanation: • Similar in spirit to CEA but focuses on losses claimants incur • Assign equal amount of losses to all claimants subject to no one receiving a negative amount Example: E = 150 c = [20, 80, 100] P = [3. 33, 63. 33, 83. 33] 50/3 = 16 2/3 [3. 33, 63. 33, 83. 33] ISMA 2004 Workshop on Internet Signal Processing (WISP) 6

Talmud Rule: Explanation: • Two regimes are defined based on the half-sum of the claims • If half sum of claims is less than amount to divide, CEA is applied • If more, every one receives half their claim and CEL is applied to the remainder • Note that half-claims are used in the formula instead of claims Example: E = 150 c = [20, 80, 100] P = [10, 60, 80] ISMA 2004 Workshop on Internet Signal Processing (WISP) 7

Talmud Continued • Claim [20, 80, 100] and estate size is 150 • First run CEA on half the claims against the full estate – Run [10, 40, 50] against 150 – This produces [10, 40, 50] with 50 leftover • Then run CEL on remaining half claim against residual estate – Run [10, 40, 50] against a residual estate of size 50 – This produces loss of 16. 66 each in Round 1 or [0, 23. 33, 33. 33] with a residual loss of 6. 66 – Divide residual loss by residual claimants 6. 66/2 = 3. 33 – In round 2 claims reduce further to [0, 20, 30] • Total award = [10, 40, 50] + [0, 20, 30] = [10, 60, 80] ISMA 2004 Workshop on Internet Signal Processing (WISP) 8

Comparison of Rules a) Proportional c) Constrained Equal Losses ISMA 2004 Workshop on Internet Signal Processing (WISP) b) Constrained Equal Awards d) Talmud 9

Properties of Rules • Invariance under Claims Truncation: – If ci > E, replacing ci with E should not effect the chosen awards vector – Satisfied by CEA and Talmud, not by CEL, PROP • Composition Down (Up): – When E is found to be less (more) than initially thought, the award vector should be the same for both of the following – Cancel initial division and apply the rule to the revised problem; or – Consider initial awards as claims on the revised E – Consider residual claim as claims on the increment in E – Satisfied by Proportional, CEA and CEL, not by Talmud • No advantageous Transfer: – No group of agents should receive more by transferring claims among themselves – Only Proportional rule passes this test ISMA 2004 Workshop on Internet Signal Processing (WISP) 10

Properties of Rules Continued • CEA is the only rule such that for each problem – the gap between the smallest amount any claimant receives and the largest such amount is the smallest – the variance of the awards is the least for each problem • CEA is the only rule that guarantees – Equal treatment of equals and – Invariance under claims truncation and – Composition up ISMA 2004 Workshop on Internet Signal Processing (WISP) 11

Strategic Models • Game 1: – Agents propose rules and the various rules are applied to the problem at hand – The claim of each agent is replaced by the maximal amount awarded to him by any one of the rules – The rules are applied to the revised problem and so on – This game has a unique Nash equilibrium which is the awards vector selected by Constrained Equal Awards (CEA) – no player benefits by changing their strategy if the other players keep theirs unchanged ISMA 2004 Workshop on Internet Signal Processing (WISP) 12

Strategic Models • Game 2: – Player A proposes an amount to other players in order – If any player B accepts his offer, B leaves – If a player B rejects the offer made to him, the next stage starts with B making an offer to the next player C; player A is moved to the end of the line – The process continues until only one player is left – No player should be offered an amount – greater than his claim or – the amount that remains to be distributed – As the discount factor of future utilities goes to one, the limit of payoff vectors in this game converges to Constrained Equal Awards (CEA) ISMA 2004 Workshop on Internet Signal Processing (WISP) 13

Experimental Study on the Allocation Mechanisms • Normative judgments: – what people decide to be “fair” in a dispassionate setting • Actual negotiation: – when people negotiate among themselves to arrive at a mutually agreeable allocation • Experiments have shown that people’s decision to allocate is: – Close to PROP in normative judgments – Close to CEA in actual negotiation • Conjecture: – CEA is better suited to resource allocation – Leads to “better” user satisfaction ISMA 2004 Workshop on Internet Signal Processing (WISP) 14

Comments and Conclusions • Consider a communication link with rate 100 kbps – Suppose claims are 40 kbps and 160 kbps – PROP allocates 20 kbps and 80 kbps – CEA allocates 40 kbps and 60 kbps – In CEA, the 40 kbps demand is as important as the first 40 kbps demand of the larger 160 kbps claim • How to Satisfy users? – Min-Claim-First satisfies maximum number of users, but starves larger claimants – PROP rewards inflated claims – CEA may make better sense • Conclusions – It is debatable if proportional share is a good definition of fairness – A different set of rules advocated in bankruptcy claims problem – We advocate further study of CEA in networking given its interesting properties • References – “Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey”, William Thomson. Mathematical Social Sciences, 45, (2003) 249 -257 – “Dividing Justly in Bargaining Problems with Claims”, working paper Gachter, Riedl 2004 ISMA 2004 Workshop on Internet Signal Processing (WISP) 15