Bargaining and Coalition Formation Dr James Tremewan (james.tremewan@univie.ac.at) Cooperative Game Theory 1 1 These slides are based largely on Chapter 18, Appendix A of ”Microeconomic Theory” by Mas-Colell, Whinston, and Green.
Cooperative Game Theory Cooperative Games • A cooperative game is a game in which the players have complete freedom of preplay communication and can make binding agreements. • In contrast, in a non-cooperative game no communication is admitted outside the formal structure of the game, and commitments must be self-enforcing , i.e. part of a subgame perfect equilibrium. • There are many solution concepts in cooperative game theory, e.g: • The Nash bargaining solution was an example of a solution concept for a class of cooperative games. • The core includes outcomes that may result from coalitional competition (descriptive). • The Shapley value is motivated as a ”fair” division of surplus (normative). 2/29
Cooperative Game Theory Games in Characteristic Form • Cooperative games are described in characteristic form . The characteristic form summarizes the payoffs available to different coalitions . • The set of players is denoted I = 1 , ..., I . Nonempty subsets S , T ⊂ I are coalitions. • An outcome is a list of utilities u = ( u 1 , ..., u I ) ∈ R I . The relevant coordinates for a coalition S are u S = ( u i ) i ∈ S . • A utility possibility set is a nonempty, closed set U S ⊂ R S where u S ∈ U S and u ′ S ≤ u S implies u ′ S ∈ U S i.e. utility freely disposable. • A game in characteristic form ( I , V ) is a set of players I and a rule V ( · ) that associates to every coalition S ⊂ I a utility possibility set V ( S ) ⊂ R S . 3/29
Cooperative Game Theory Characteristic Form: Examples • Nash bargaining game (using notation from earlier in course): I = { 1 , 2 } ; V ( { 1 } ) = d 1 , V ( { 2 } ) = d 2 , and V ( { 1 , 2 } ) = S . • Three-player example ( I = { 1 , 2 , 3 } ): 4/29
Cooperative Game Theory Superadditive Games • A game is superadditive if two (non-overlapping) coalitions can do at least as well together as alone. • Formally: A game in characteristic form ( I , V ) is superadditive if for any coalitions S , T ⊂ I such that S ∩ T = ∅ we have: if u S ∈ V ( S ) and u T ∈ V ( T ), then ( u S , u T ) ∈ V ( S ∪ T ). • We will look only at superadditive games: as in the bilateral section, we are primarily interested in situations where there are gains from cooperation. 5/29
Cooperative Game Theory Transferable Utility (TU) • Much of the literature focuses on TU games (as will we), i.e. where utility can be transfered costlessly between coalition members. Sufficient conditions for TU are: • Players can make side-payments. • Players utility is linear in money. • Excludes, for example, situations where bribes are illegal. • In TU games, V ( S ) = { u S ∈ R S : � u S i ≤ v ( S ) } for some v ( s ). i ∈ S • i.e. coalition S chooses a joint action to maximise their total utility, denoted v ( S ) which can be allocated amongst S in any way. • v ( S ) is called the worth of coalition S . 6/29
Cooperative Game Theory TU Games: Example Boundaries in TU games are hyperplanes in R S : 7/29
Cooperative Game Theory TU Games: Simplex Representation Normalising utilities such that V ( { i } ) = 0 ∀ i allows us to represent an n-player TU game on an (n-1) dimension simplex: 8/29
Cooperative Game Theory Two Games in Characteristic Form • A three-player game is defined by: • v ( { 1 , 2 , 3 } ) = 1, • v ( { 1 , 3 } ) = v ( { 2 , 3 } ) = 1, v ( { 1 , 2 } ) = 0, • v ( { 1 } ) = v ( { 2 } ) = v ( { 3 } ) = 0. • What do you think will/should happen in this game? • A three-player game is defined by: • v ( { 1 , 2 , 3 } ) = 10, • v ( { 1 , 2 } ) = 10, v ( { 1 , 3 } ) = 3, v ( { 2 , 3 } ) = 2, • v ( { 1 } ) = v ( { 2 } ) = v ( { 3 } ) = 0. • What do you think will/should happen in this game? 9/29
The Core The Core • The set of feasible utility outcomes with the property that no coalition could improve the payoffs of all its members. • In TU games the core is the set of utility vectors u = ( u 1 , ..., u I ) such that: • � u i ≥ v ( S ) ∀ S ⊂ I , and i ∈ S • � u i ≤ v ( I ) i ∈ I • The core may be: • Empty (strategic instability, no useful prediction, e.g. divide-the-dollar majority rules). • Large (makes no useful prediction, e.g. divide-the-dollar unanimity rules). • Non-empty and small (makes sharp prediction). • Note that the core depends only on ordinal utility. 10/29
The Core A TU Game with Non-Empty Core 11/29
The Core A TU Game with Empty Core 12/29
The Shapley Value The Shapley Value • Describes a ”reasonable” or ”fair” division taking as given the strategic realities captured by the characteristic form. • Idea of fairness here is egalitarianism: gains from cooperation should be divided equally. • Two player example ( I , v ) = ( { 1 , 2 } , v ): • Gains from cooperation = v ( I ) − v ( { 1 } ) − v ( { 2 } ). • Sh i ( I , v ) = v ( { i } ) + 1 2 ( v ( I ) − v ( { 1 } ) − v ( { 2 } )) • Can re-write as: • Sh 1 ( I , v ) − Sh 1 ( { 1 } , v ) = Sh 2 ( I , v ) − Sh 2 ( { 2 } , v ), and • Sh 1 ( I , v ) + Sh 2 ( I , v ) = v ( I ), where Sh i ( { i } , v ) = v ( { i } ) • The benefit to player one from the presence of player two is the same as the benefit to player two from the presence of player one. • Now generalize this idea to more players... 13/29
The Shapley Value The Shapley Value • The Shapley value of a game ( I , v ) is the outcome consistent with: • Sh i ( S , v ) − Sh i ( S \{ h } , v ) = Sh h ( S , v ) − Sh h ( S \{ i } , v ), and • � Sh i ( S , v ) = v ( S ), i ∈ S • for every subgame ( S , v ) and all players i , h ∈ S . • In words: the benefit a member of a coalition (player i ) gets from another (player h ) joining is equal to the benefit player h would get if already a member and player i was joining. • This outcome is unique. 14/29
The Shapley Value The Shapley Value: Axioms • The Shapley value can also be derived as the unique value satisfying three axioms (here loosely defined): • Efficiency : � Sh i ( I , v ) = v ( I ), i.e. no utility is wasted. i ∈ S • Symmetry : If ( I , v ) and ( I , v ′ ) are identical except the roles of players i and h are swapped, then Sh i ( I , v ) = Sh j ( S , v ′ ), i.e. labeling doesn’t matter. • Additivity : If a game is in a particular sense the sum of two other games 2 , then Sh ( I , v + w ) = Sh ( I , v ) + Sh ( I , w ). 2 See Shapley (1953) for a precise definition. 15/29
The Shapley Value The Shapley Value • An ”easy” way to calculate the Shapley value: s !( n − s − 1)! • Sh i ( I , v ) = � ( v ( S ∪ { i } ) − v ( S )), n ! S ∈ U • where U is the set of all subsets of players not including i , s = | S | and n = | I | . • Intuition: imagine a coalition of all the players is formed by including one player at a time in a random order, and each player receives all of the added benefit to the coalition at the time they are included. The Shapley value is the expected value of this process if all orders are equally likely. • v ( S ∪ { i } ) − v ( S ) is the value the new player adds. • s ! is number of ways the existing members of S could have arrived. • ( n − s − 1)! is number of ways the remaining players can arrive. • n ! is the number of possible ways of the players arriving overall. 16/29
Examples Glove Market: The Shapley Value • A three-player game is defined by: • v ( { 1 , 2 , 3 } ) = 1, • v ( { 1 , 3 } ) = v ( { 2 , 3 } ) = 1, v ( { 1 , 2 } ) = 0, • v ( { 1 } ) = v ( { 2 } ) = v ( { 3 } ) = 0. • Player 1: • The value P1 adds to the coalition { 3 } is 1, and this happens only if P3 is added first, then P1 second. • P1 adds 0 to any other coalition. • There are six possible orderings so Sh i ( I , v ) = 1 6 . • Player 2: as for P1. 17/29
Examples Glove Market: The Shapley Value • Player 3: • adds 1 to the coalition { 1 } is, and this happens only if P1 is added first, then P3 second. • adds 1 to the coalition { 2 } is, and this happens only if P2 is added first, then P3 second. • adds 1 to the coalition { 1 , 2 } is, and this happens if P1 is added first, P2 second, then P3 third, or if P2 is added first, P1 second, then P3 third. • Sh 3 ( I , v ) = 1 6 · 1 + 1 6 · 1 + 2 6 · 1 = 2 3 • The Shapley value suggests the allocation { 1 6 , 1 6 , 2 3 } • Check! The sum of Sh i ( I , v ) is v ( I )! 18/29
Examples Glove Market: the Core • What is the core of this game?: • Consider an allocation { u 1 , u 2 , u 3 } . • Suppose u 1 > 0. P2 and P3 are better off forming the coalition 2 = u 2 + 1 3 = u 3 + 1 { 2 , 3 } and sharing u ′ 2 u 1 and u ′ 2 u 1 . • Therefore in any allocation in the core u 1 = 0 (similarly u 2 = 0). • No coalition can improve on { 0 , 0 , 1 } for all members. • The core is { 0 , 0 , 1 } . • The strategic environment means P1 and P2 undercut each other leaving zero profits. • The Shapley value ( { 1 6 , 1 6 , 2 3 } ) is more equitable than the core. • However the Shapley value is less equitable than an even split ( { 1 3 , 1 3 , 1 3 } ) because it recognises to some extent the individual contributions (bargaining power?) of the different players. 19/29
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