Axiomatic bargaining. theory

Transcription

1 Axiomatic bargaining theory

2 Objective: To formulate and analyse reasonable criteria for dividing the gains or losses from a cooperative endeavour among several agents. We begin with a non-empty set of alternatives X and a collection I of agents. We will assume that agents tastes are given to us in the form of utility functions u i : X R. Principle of welfare economics: Policy-making should not be paternalistic. 1

3 Implication: Alternatives that cannot be distinguished from the standpoint of agents tastes should not be distinguished by the policy-maker either. So, only agents utility values for the different alternatives should matter. Thus, the relevant constraint for the policymaker is the utility possibility set. 2

4 Definition: The utility possibility set (UPS) is the set U = {(u 1,.., u I ) R I : u 1 u 1 (x),...u I u I (x) for some x X} R L The Pareto frontier of U is formed by the utility vectors u = (u 1,..., u I ) U for which there is no other u = (u 1,..., u I ) U with u i u i for every i and u i > u i for some i. 3

5 The description of a bargaining problem among I agents is composed of 2 elements: a utility possibility set U R I and a threat-point or status-quo point u U. Set U represents the allocations of utility that can be settled on if there is cooperation among the different agents. The point u is the outcome that occurs if there is a breakdown in cooperation. Cooperation requires unanimous participation of all agents. 4

6 Setup completely general with 2 agents We assume that U R I is convex and closed, and that it satisfies the free disposal property U R I + U (i.e., if u u and u U, then u U. We also assume u is interior to U and that {u U : u u } is bounded. A bargaining solution is a rule that assigns a solution vector f(u, u ) U to every bargaining problem (U, u ). 5

7 (Desirable) Properties of a bargaining solution Definition: The bargaining solution is independent of utility origins (IUO) or invariant to independent changes of origins if for any α = (α 1,..., α I ) R I, we have f i (U, u + α) = f i (U, u ) + α i for every i, whenever U = {(u 1 +α 1,..., u I +α I ) : u U} 6

8 The IUO property says the bargaining solution does not depend on absolute scales of utility. We will assume this property holds. We therefore have f i (U, u ) = f(u {u }, 0)+ u. We can normalise our problems to u = 0. Thus f(u) f(u, 0) 7

9 Definition: The bargaining solution is independent of utility units (IUU) (invariant to independent changes of units) if for any β = (β 1,..., β I ) R I with β i > 0 i, we have f i (U ) = β i f i (U) for every i, whenever U = {(β i u 1,...β 1 u I ) : u U} Together, above 2 properties imply that although the bargaining solution uses cardinal information on preferences, it does not involve interpersonal comparisons of utilities. 8

10 Definition: The bargaining solution satisfies the Pareto property (P) or is Paretian if for every U, f(u) is a (weak) Pareto optimum. i.e., there is no u U such that u i > f i (U) for every i. Definition: The bargaining solution satisfies the property of symmetry (S) if whenever U R I is a symmetric set (i.e., U remains unaltered under permutations of the axes), all the entries of f(u) are equal. 9

11 Definition: The bargaining solution satisfies the property of individual rationality (IR) if f(u) 0 (i.e., cooperative solution does not give any agent less than threat point). Definition: The bargaining solution satisfies the property of independence of irrelevant alternatives (IIA) if whenever U U and f(u) U, it follows that f(u ) = f(u). 10

12 The IIA says that if f(u) is the reasonable outcome in U and we consider a U that is smaller than U but retains the feasibility of f(u) (i.e., we only eliminate from U irrelevant alternatives ) then f(u) remains the reasonable outcome. We examine 4 examples of bargaining solutions - all 4 satisfy the Paretian, symmetry, IR and IUO properties. 11

13 1. Egalitarian solution: At the egalitarian solution f e (.), the gains from cooperation are split equally among the agents. i.e., for every bargaining problem U R I, f e (U) is the vector in the frontier of U with all coordinates equal. Every f e (U) maximises the social welfare function Min{u 1,..., u I } on U. It satisfies IIA but violates IUU. 12

14 2. Utilitarian solution: For every U, we now let f u (U) be a maximiser of Σ i u i on U R I +. If U is stricty convex, this point is uniquely defined. On the domain of strictly convex bargaining problems, the IIA property is satisfied. Solution again violates IUU. 13

15 3. Nash solution: Here, f n (U) is the point in U R I + that maximises the product of utilities u 1... u I Σ i lnu i. In the I = 2 case, the Nash solution has the following geometry: f n (U) is the boundary point of U through which we can draw a tangent line with the property that its midpoint in the positive orthant is precisey the given boundary point f n (U). 14

16 Given a bargaining problem U, the Nash solution is the only utility outcome that, for some rescaling of units of utility, coincides simultaneously with the utilitarian and egalitarian solutions. Proposition: The Nash solution is the only bargaining solution that satisfies IUO, IUU, IIA, is Paretian and symmetric. 15

17 4. Kalai-Smordinsky (KS) solution: Given a bargaining problem U R I, denote by u i (U) R the maximum utility value that agent i could attain by means of some vector in U R I +. Suppose agent i has all the bargaining power; then the outcome would give u i (U) to agent i and nothing to the remaining. 16

18 We could therefore argue that if cooperation takes place, then the solution should be the Pareto optimal allocation where are utilities of the different agents are proportional to (u 1 (U),...u I (U)). This is the KS solution f k (U). Geometrically, to find the K-S solution, we solve the problem: max{λ : λ 0, λ(u 1 (U), u 2 (U)) R} 17

19 Applying the Nash bargaining solution to the household bargaining problem we studied a few classes ago... The decision of the household will solve: max x1,x 2 [u h (x 1, x 2 ) u h ][u w (x 1, x 2 ) u w ] subject to p 1 x 1 + p 2 x 2 = Y h + Y w The optimality condition for the household will then be: ( u h / x 1 ).[u w (x 1,x 2 ) u w ]+( u w / x 1 ).[u h (x 1,x 2 ) u h ] ( u h / x 2 ).[u w (x 1,x 2 ) u w ]+( u w / x 2 ).[u h (x 1,x 2 ) u h ] = p 1 p 2 18

20 Three important things to note about this. 1. Neither person has a MRS equal to the relative prices of the two goods, at a bargained household decision, neither person gets his or her individual optimum. 2. In general, one person has a MRS above it, and the other has it below. i.e., one person would rather the household consume more of good one, while the other would prefer that the household consumes more of good two. 19

21 3. Finally, the households MRS between the goods is like a weighted average of the two individuals MRSs, with the weights depending on the relative sizes of the individuals threat points ; if a person has a better outside option, the household puts more weight on his or her preferences. All of these implications seem reasonable, which is why the Nash-bargaining model is an appealing way to model household behavior. 20

22 Example: The two goods are x 1 = his clothing and x 2 = her clothing. The price of each is 1. Mr. Smith cares about only the first, while Mrs. Smith cares about only the second. Their utility functions are simply: u h (x 1, x 2 ) = x 1, u w (x 1, x 2 ) = x 2 Their outside options depend on their private incomes; let s say that u h = Y h and u w = Y w If the private incomes are Y h = 16 and Y w = 9 (respectively), then the problem is to find: 21

23 max x1,x 2 (x 1 16).(x 2 9) subject to x 1 + x 2 = The solution requires that 1.(x 2 3)+0.(x 1 4) 0.(x 2 3)+1.(x 1 4) = 1 x 1 = x From budget constraint, we have x 1 = 25 x 2. Together, this gives us: x 1 = 13; x 2 = 12 22

24 Because Mr. Smith s income is higher, the household spends more on his clothing. If we were to swap the incomes so that Y h = 9 and Y w = 16, then the household would spend more on Mrs. Smith s clothing. Note that the overall income of the household remains unchanged, but who earns it affects how it is spent! 23

25 Shapley s impossibility result All bargaining models analysed thus far presume that bargainers preferences are represented by von Neumann-Morgenstern utility, referred to as cardinal utility. Under such an assumption, two utility functions can be viewed as the same if one can be derived from the other by an affine positive transformation. 24

26 Thus a bargaining solution based on any of the above models should be invariant under any affne positive transformations. However, traditional economic theory considers bargaining problems in which players pref- erences are represented in ordinal. Therefore, ideally, a bargaining solution should be invariant under any order-preserving transformations on utilities - a property referred to as ordinal invariance in the game-theoretic literature 25

27 Ordinal bargaining solutions are more desirable than cardinal solutions because ordinal information about players preferences is easier to elicit than cardinal preferences and the corresponding solutions can be more robust However, Shapley showed that for the twoperson bargaining problem (bilateral bargaining) there is no non-trivial (i.e., strongly individual rational) ordinal solution. 26

28 Example: Two players, A and B, bargain over the partition of a cake. Let x i be the share of the cake in percentage to player i (i = A,B). The set of possible agreements is represented by Ω = {(x A, x B ) : 0 x A 100 and x B = 100 x A }. For each x i [0, 100], u i (x i ) is player i s utility from obtaining a share x i of the cake. Let u A (x A ) = x A and u B (x B ) = x 2 B. Failure to agree is rated 0 by both A and B. 27

29 Nash s bargaining solution to the problem gives the outcome (33.3, 66.7) and Kalai- Smorodinsky s solution gives (38.2,61.8). Both solutions are in favour of player B. This is because player B is less risk-averse. For both Nash s solution and Kalai-Smorodinsky s solution, risk-loving players has advantage in bargaining comparing to risk-neutral and risk-averse players 28

30 Consider an order-preserving transformation τ(x) = sqrtx on player B s utility. The transformed utility of player B becomes linear. Under the new utility scales, both Nash s solution and Kalai-Smorodinsky s solution give (50,50) as the outcome. This means that none of the solutions is ordinally invariant. The non-linearity of utility functions, which expresses the risk posture of a player, determines the outcomes of bargaining but collapses under ordinal transformations. 29