1 Surplus Division Output (surplus) to be divided among several agents. • Issues: How to divide? How to produce? How to organize? Plus: adverse selection, moral hazard, ... • Extremes easy: MAX s.t. Reservation Utility + In-centive Constraints • Examples: Joint ownership, Unions, No ownership, Bargaining, Privatization, ..., essentially all the mod-els we had considered without MAX objective; arbi-tration, laws, ... • Objectives? Efficiency, Welfare, Fairness (!). • Mechanisms to achieve: Bargaining, Arbitration, Auc-tions, Rationing, Lotteries, Markets, ... 2 Fair Distribution (Moulin’03) 2.1 Four Principles of Distributive Justice 1. Compensation 2. Reward 3. Exogenous rights 4. Fitness Pluto’s Flute: 4 children 1. (poo r) no toys 2. (worked hard) cleaned and fixed it 3. (owner) father’s flute 4. (efficient user) can play• Compensation: Ex post equality Goal: To equalize distribution of a higher-order charac-teristic Justifies: Disproportional use of resources Examples: Different shares of food for infants, pregnant women, adult males; More medical attention to ill; More attention to Handicapped; affirmative action for socioe-conomically disadvantaged. Justification for macroeconomic redistributive policies (other j..?): tax breaks, welfare support, medical aid. For each i, vi = ui(yi). Choose {yi} to equate vi’s. Egalitarian objective. Mechanisms: handicaps, unequal shares, subsidies to cer-tain groups,... • Reward Unequal treatment is (morally) relevant as a reward (pun-ishment) for actions (behavior). Past sacrifices by soldiers lead to preferential treatment today; prizes for achievements; higher insurance premium for reckless drivers; no organ transplants for criminals. Difficult issue is when the outcome is a product of efforts of multiple agents. How to compensate? (public goods, tragedy of commons) Bargaining outcomes, Shapley value, Exogenous (in ex-pectation) budget breakers,... Another issue: Outside opportunities.• Exogenousrights(equalityexante?) Property rights (and liabilities) Fairness: freedom of speech, religion, access to education (unrelated to IQ), voting rights (unrelated to character-istics, both external (rich, male, educated) and internal (smart, caring, voting)), equal duties, political represen-tation, one share—one vote. Unfairness: order of Priority based on: S eniority, Social standing (cast structure), size of representation,... • Fitness (Efficiency) Who is the best, who values it the most, who in the most need; Child to the true mother, ... a drink to a drunk, ... Distinguish: sum-fitness (MAX SUM) vs efficiency-fitness (Pareto Optimality) Utilitarian Objective. Flute example: ? • Examples Lifeboat (sinking ship): Women and children, Old people, Crew, strong men, “generals”: who first? Food rationing in besieged town; Limited medical resources: Immigration, college admissions, tickets to shows... • Queuing and auctioning How these perform on different criteria? (think over-booked plane) • Political rights (voting) Plato: philosophers should rein• Joint venture (Excess) Teresa (piano) earns $50K alone; David (violin) earns $100K alone; Together: $210K. Split? 1. Proportional solution Stand-alone salaries are proxies for ind ividual contribution yi = xi P xi T 2. Status quo ante solution yi = xi + 1 n ³ T − X xj ´ 3. Equal division (egalitarian) modified: uniform gains solution yi =max{λ, xi}X max{λ, xi} = T. • Joint venture (Deficit) E.g. Bankruptcy Equal division Proportional Division Uniform Losses ...Lotteries3 The Shapley Value The problem of the Commons (a joint production process) Sharing of joint costs or benefits. What is a fair assessment of individual responsibilities or contributions. Extreme: “Without me you are nothing” • Joint venture (revisited) T and D share an office, need good connection. T (D) needs a link that costs cT <cD. (stand-alone costs). There is a single cable outlet. Additional cost δ> 0 to connect both, C = cT +cD+δ. Which solution to use? Comparative statics (suppose the company drives cT to 0). Proportional division: PT =0, PD = cD + δ (full exter-nality). Surely, T has to pay some. Uniform gains: PT = δ, PD = c2 if δ< c2 (equal otherwise). Equal surplus: (sensible) Pi = ci + δ/2. (Status-quo plus Nash B ?)3.1 The Shapley V alue: Definition Cost interpretation: each agent wants one unit of service (equal ex ante ownership) N = {1, 2,...,n}, coalition is a subset S ⊆ N. For each S, there is C(S)–stand-alone cost of serving S. (characteristic function in general) Solution is Expected marginal cost. Let Ai be the set of coalitions NOT containing i; Ai(m) is the set of coalitions from Ai of size m. Shapley Value is xi = nX m=0 X S⊆Ai(m) m!(n − m − 1)! n! [C(S ∪ {i}) − C(S)] . The coefficient comes from an arbitrary order of players in S (those who joined S before i), and of players not in S ∪ {i} (those who join N later). It is presumed that the grand coalition will form. (al-ternative) definition of Shapley value is the average over all possible orders of players of the marginal impact of a given player.• Example: Runway construction Airline A needs short runway only, B medium, Z long. C(A) = 1000,C(B)= C(AB) = 3000, C(Z)= C(ABZ)= C(AZ)= C(BZ) = 6000. How to divide? 6 possible random orders: A : Only when first has marginal cost: xA = 1 6 (1000ABZ + 1000AZB)= 1000 3 ; B : Only when first or second (after A) has added cost: xB = 1 6(2 × 3000B· + 2000ABZ )= 1000 3 + 2000 2 ; C : xZ = 1 6 Ã 2 × 6000Z· + 5 000AZB +3000BZA +2 × 3000··Z ! = 1000 3 + 2000 2 + 3000. 3.2 Stand-alone property of Shapley Value Subadditivity: C(S t T ) ≤ C(S)+ C(T ) Then, C(N) ≤ P C(i); Superadditivity: C(S t T ) ≥ C(S)+ C(T ) Then, C(N) ≥ P C(i). Stand-alone test: C subadditive ⇒ yi ≤ C(i); C super-additive ⇒ yi ≥ C(i). • ShapleyValue meetsS-A test. Lots of other properties (see axioms next)3.3 Shapley V alue: Axiomatic Approach Variety of ways, Original axioms are Equal treatment of equals, Dummy, and Additivity. Generic solution, {γi}n i=1, P γi = C(N). Equal treatment of equals: if i, j are equal (exch) w.r. (C, N), then γi = γj. Dummy: (only axiom that contains reward principle): suppose i is such that for all S ⊆ Ai, C(S ∪ {i}) − C(S)=0,then γi =0. Additivity: (structural invariance)