A bargaining approach to the ordinal Shapley rule Juan Vidal-Puga The latest version of this paper can be found at

A bargaining problem for a set of players N is a pair (d, S) where  d: status quo point in R N  S is a subset of R N comprehensive upper compact  S nonlevel The bargaining problem A rule is a function  that assigns to each bargaining problem (S, d) a point  (S, d) in R N if x  S and y  x, then y  S {x  S : x  y } is compact, for all y These functions are always well-defined

Ordinal transformations A is the set of alternatives q  A is the status quo  i is a preference order on A for player i u i : A  R is a utility representation of  i for each i  a  i b iff u i (a)  u i (a) Then, S = {u(a) : a  A} and d = u(q) We assume that S and d comprise all the relevant information If f i : R  R is an order preserving transformation of R then f i (u i ) represents the same preferences for i

Ordinal solutions A rule should not be affected by order-preserved changes in the utility functions formally: Let f = (f i ) i  N a vector of order preserving transformation of R Define f(S) = {f(x) : x  S} A rule  is ordinal if f(  (S,d)) =  (f(S),f(d)) S d  f (S) f (d) f()f()

Ordinal solutions For |N| = 2, there does not exists any relevant ordinal rule (Shapley, 1969) For |N| = 3, there exists relevant ordinal rule (Shapley, 1969) contradiction!

The ordinal Shapley rule 1 2 3

1 2 3 Sh(S,d) = lim T  p T,ij

The ordinal Shapley rule Related literature:  Pérez-Castrillo and Wettstein (2002, 2005) study (A,q,  i ) to define an ordinal rule.  Kιbrιs (2001, 2002) characterizes axiomatically the ordinal Shapley rule.  Safra and Samet (2004a, 2004b) extend the ordinal Shapley value for more than 3 players.  Calvo and Peters (2005) study mixed situations with “ordinal players” and “cardinal players”. Our objective is to find a non-cooperative game such that the Shapley rule arises in equilibrium

The non-cooperative game Idea:  Two players negotiate in two steps: First they bargain to reach a pre-agreement If they fail to reach a final agreement, the pre-agreement in implemented.  At the middle of this process (just after the pre-agreement has been reached), the third player makes a counter-offer to one of the other players (she chooses whom)  If the counter-offer is rejected, the negotiation between the first two players goes on.  Once two players reach an agreement, the other player can veto this agreement.

The non-cooperative game The game is played in T rounds. If agreement is not reached after round T, the players receive the status quo payoff d. In each round, the players follows certain roles:  First proposer (FP),  First responder (FR),  Pivot (P). The roles change in each round. In the first round, assume wlog 1=FP, 2=FR, 3=P.

The non-cooperative game 1 2 y,i i j j i y z 33 zx 3 x xy zx veto In case of veto, a round passes by (in the last round, d is implemented) We can always recognize: The vetoer The player who made the vetoed offer “The other” Vetoer  P “The other”  chooses between FP or FR

First result Theorem: There exists a subgame perfect equilibrium (SPE) whose payoff allocation is p T,13 Corollary: As T increases, there exists a collection of SPE whose payoff allocations approach the ordinal Shapley rule.

Assumptions 1.If one player is indifferent between to veto or not to veto, she strictly prefers no to veto. 2.If the pivot (say k ) is indifferent when choosing i, and x k is strictly less than the maximum x k in SPE, then she strictly prefers to choose the first responder.

Main result Theorem: Under Assumptions 1 and 2, there exists a unique SPE whose payoff allocation is p T,13 Corollary: As T increases, these SPE payoff allocations approach the ordinal Shapley rule.