A bargaining approach to the ordinal Shapley rule

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A bargaining approach to the ordinal Shapley rule

• Juan Vidal-Puga

The latest version of this paper can be found at
http//webs.uvigo.es/vidalpuga
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The bargaining problem

• A bargaining problem for a set of players N is a
  pair (d, S) where
• d status quo point in RN
• S is a subset of RN
• comprehensive
• upper compact
• ?S nonlevel

• A rule is a function ? that assigns to each
  bargaining problem (S, d) a point ?(S, d) in RN

if x?? S and y ? x, then y ? S
x ? S x ? y is compact, for all y
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Ordinal transformations

• A is the set of alternatives
• q?A is the status quo
• ?i is a preference order on A for player i
• ui A ? R is a utility representation of ?i for
  each i
• a ?i b iff ui(a) ? ui(a)
• Then, S u(a) a?A and d u(q)
• We assume that S and d comprise all the relevant
  information
• If fi R ? R is an order preserving
  transformation of R
• then
• fi(ui) represents the same preferences for i

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Ordinal solutions

• A rule should not be affected by order-preserved
  changes in the utility functions
• formally
• Let f (fi)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))

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Ordinal solutions

• For N 2, there does not exists any relevant
  ordinal rule (Shapley, 1969)

contradiction!

• For N 3, there exists relevant ordinal rule
  (Shapley, 1969)

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The ordinal Shapley rule
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2
1
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The ordinal Shapley rule
3
2
Sh(S,d) limT?? pT,ij
1
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The ordinal Shapley rule

• Related literature
• Pérez-Castrillo and Wettstein (2002, 2005) study
  (A,q,?i) to define an ordinal rule.
• K?br?s (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
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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.

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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 1FP, 2FR, 3P.

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The non-cooperative game

• 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

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veto
i
j
j
Vetoer ? P The other ? chooses between FP or FR
i
y
veto
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First result

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

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Assumptions

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

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Main result

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