Perspectives on Resource Allocation

1. Perspectives on Resource Allocation Kameswari Chebrolu, Bhaskaran Raman, Ramesh R. Rao November 11, 2004

2. 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?

3. Resource Allocation in other Domains E c3 c4 c5 c6 c1 c2 x3 x4 x5 x6 x1 x2 • 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

4. Proportional Rule (PROP) Rule: Explanation: • Make awards proportional to claims Example: E = 150 c = [20,80,100] P = [15,60,75]

5. 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: E = 150 c = [20,80,100] P = [20,65,65] 150/3 = 50 [20,50,50]; 30/2 [20,65,65]

6. 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]

7. 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] E = 150 c = [20,80,100] P = [10,60,80]

8. 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]

9. Comparison of Rules a) Proportional b) Constrained Equal Awards c) Constrained Equal Losses d) Talmud

10. 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

11. 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

12. 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

13. 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)

14. 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

15. 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