Lecture: Topics in Cooperative Game Theory

Lecture: Topics in Cooperative Game Theory Martin Kohl University Leipzig November 5 2010 1 / 25

Overview PotSh Selfdu MCP Rec ExiSPS Potential of the Shapley value Selfduality Marginal Contributions Recursion formulas Implementation Backward induction An Implementation of the Shapley value 2 / 25

The potential to the Shapley value PotSh Selfdu MCP Rec ExiSPS De nition A potential P for TU games is an operator that assigns to any TU game (N, v ) a number P (N, v ) 2 R such that (i) P (, v ) = 0, i (ii) i2n hp (N, v ) P Nn fig, v j N nfig = v (N). Theorem There is a unique potential fortu games, which satis es P (N, v ) P Nn fig, v j N nfig = Sh i (N, v ), i 2 N. 3 / 25

The potential to the Shapley value #2 Proof. Uniqueness: Let P, Q be two potentials. We show P = Q. PotSh Selfdu MCP Rec ExiSPS Induction basis: jnj = 1. by (i+ii), P (fig, v ) = Q (fig, v ) = v (fig). Induction hypothesis (H): P = Q for jnj k Induction step: let jnj = k + 1. this implies P (N, v ) (ii) = v (N) + P Nn fig, v j N nfig i2n H = Q (N, v ) jnj 4 / 25

The potential to the Shapley value #3 PotSh Selfdu MCP Rec ExiSPS existence: consider the operator P given by this gives P (, v ) = 0 and P (N, v ) := λ T (v ) jt j T N,T 6= P (N, v ) P (Nn fig, v j N ) i2n λ = T (v ) jt j = i2n T N,T 6= i2n T N,T 6=,i2T λ T (v ) jt j = Sh i (N, v ) = v (N) i2n i2n T N,T 6=,i /2T λ T (v ) jt j 5 / 25

Selfduality De nition PotSh Selfdu MCP Rec ExiSPS The dual game (N, v ) of a TU game (N, v ) is de ned by v (S) = v (N) v (NnS) for S N. (1) Interpretation: Lose of the great coalition, if the players of S leave. De nition A value φ for a TU game (N, v ) is called selfdual, if φ(n, v ) = φ(n, v ). (2) 6 / 25

Selfduality PotSh Selfdu MCP Rec ExiSPS Theorem The Shapley value is selfdual. Proof:For i 2 N and K Nn fig, we have Further, MC v i (K ) = v (K [ fig) v (K ) = v (N) v (Nn (K [ fig)) (v (N) v (NnK )) Sh i (N, v ) = K N nfig = v (NnK ) v ((NnK ) n fig) = MC v i ((NnK ) n fig). = K N nfig jk j! (jnj jk j 1)! MC v i (K ) jnj! (jnj jk j 1)! jk j! MCi v ((NnK ) n fig) jnj! = Sh i (N, v ). 7 / 25

Marginal Contributions Property 1 De nition PotSh Selfdu MCP Rec ExiSPS Let S N and v 2 V (N). The TU game (N n S, v S ) de ned by v S (T ) = v (S [ T ) v (S) for all T N n S (3) is called the S - marginal game of (N, v ). Interpretation: The rst players in a rank order of N are the players of coalition S. If the coalition T joins S, v S (T ) describes the contribution of T to S. Problem Show: For any S N and any monotonic game v 2 V (N), v S 2 V (NnS) is nonnegative. 8 / 25

Marginal Contributions Property 2 PotSh Selfdu MCP Rec ExiSPS De nition A value φ su ces the marginal contributions property, if for any TU game (N, v ) and any i, j 2 N, i 6= j Theorem φ i (N, v ) φ i (N n fjg, v j ) = φ j (N, v ) φ j (N n fig, v i ). The Shapley value su ces the marginal contributions property. 9 / 25

Recursion formulas for the Shapley value 1 Theorem PotSh Selfdu MCP Rec ExiSPS For all v 2 V(N) and i 2 N, Sh i (N, v ) = 1 1 (v (N) v (Nn fig) + jnj jnj Sh i Nn fjg, v j2n nfig N nfjg. Interpretation: The Shapley value of i is the sum of the marginal contribution to Nn fig (i is the last player in the rank order) and the sum of all Shapley values, such that another player is the last player. 10 / 25

Recursion formulas for the Shapley value 2 PotSh Selfdu MCP Rec ExiSPS Proof: For jnj = 1 the claim is immediate. Let now jnj > 1.The Shapley value satis es BC and E. By BC we get Sh i (N, v ) Sh j (N, v ) = Sh i Nn fjg, v N nfjg Sh j Nn fig, v N nfig. By summing up over j 2 Nnfig we get Sh i Nn fjg, v N nfjg j2n nfig (jnj 1) Sh i (N, v ) Sh N nfig (N, v ) Sh N nfig Nn fig, v N nfig = 11 / 25

Recursion Formulas for the Shapley value 3 Adding a zero term leads to PotSh Selfdu MCP Rec ExiSPS jnj Sh i (N, v ) Sh N (N, v ) = Sh i Nn fjg, v N nfjg j2n nfig Sh N nfig Nn fig, v Using e ciency jnj Sh i (N, v ) v (N) = Sh i Nn fjg, v N nfjg j2n nfig Hence Sh i (N, v ) = 1 1 (v (N) v (Nn fig) + jnj jnj and we are done. Sh i Nn fjg, v j2n nfig N nfig. v (Nn fig) N nfjg. 12 / 25

Recursion Formulas for the Shapley value 4 Theorem PotSh Selfdu MCP Rec ExiSPS For all v 2 V(N) and i 2 N, Sh i (N, v ) = 1 jnj v (fig) + 1 jnj Sh i Nn fjg, v j, j2n nfig Proof For j 2 N and S 2 Nnfjg : v j (S) = v j (Nnj) v j (Nn fjg ns) = v (N) v (fjg) v (NnS) + v (fjg) = v (N) v (NnS) = v (S). 13 / 25

Recursion Formulas for the Shapley value 5 By using this equation and the self duality of the Shapley value we get PotSh Selfdu MCP Rec ExiSPS and we are done. Sh(N, v ) = Sh(N, v ) 1 = jnj (v (N) v (Nn fig)) + = = 1 jnj Sh i Nn fjg, v N nfjg j2n nfig 1 (v (N) v ( ) v (N) + v (fig)) + jnj 1 jnj Sh i Nn fjg, v j j2n nfig 1 jnj v (fig) + 1 jnj Sh i Nn fjg, v j j2n nfig 14 / 25

Implementation of cooperative solutions PotSh Selfdu MCP Rec ExiSPS Cooperative game theory: allocation rules, but no description of the bargaining Noncooperative game theory: equilibrium concepts (Nash, Pareto) and description of bargaining, but uniqueness is not guaranteed. De nition Given a cooperative game. A cooperative value is implemented, if there is a noncooperative game, derived by the cooperative game, such that the payo of the cooperative value is an equilibrium outcome. De nition A cooperative value is fully implemented if the payo of the cooperative game is the unique equilibrium payo of the noncooperative game. special case of mechanism design (compare Bolton/Dewatripont: Contract Theory, Mas Colell/Whinston/Green: Microeconomic Theory) 15 / 25

An example for an implementation PotSh Selfdu MCP Rec ExiSPS Theorem If there is perfect recall and no moves of nature, every strategy combination, which is derived by backward induction, is a subgame perfect equilibrium. Kongo/Funaki/Tijs: New Axiomatizations and an implementation of the Shapley value Full Implementation of the Shapley value multistage game! subgame perfect equilibria (derived by backward induction) proofs use the recursion formulas 16 / 25

Recap Backward induction Given a game in tree form, for example chicken game: PotSh Selfdu MCP Rec ExiSPS 1 geradeaus fahren 2 2 geradeaus fahren geradeaus fahren ausweichen ausweichen ausweichen ( 0,0) ( 4,2) ( 2,4) ( 3,3) 17 / 25

Recap Backward induction PotSh Selfdu MCP Rec ExiSPS Starting with the last decisions/actions the other players know, that the last deciding player choses his/her best action. 1 geradeaus fahren 2 2 geradeaus fahren geradeaus fahren ausweichen ausweichen ausweichen ( 0,0) ( 4,2) ( 2,4) ( 3,3) 18 / 25

Model 1 PotSh Selfdu MCP Rec ExiSPS Given a TU - Game (N, v ) 2 V(N), a noncooperative game Γ(N, v ) is de ned by: (a) If jnj = 1, the player obtains v (i) and the game is nished. (b) If there is more than one player, the game has 3 stages: (b1) Each player i 2 N o ers bids bj i 2 R to every other player j 2 Nnfig Let α 2 arg max i j6=i bj i j6=i b j i be chosen randomly. α pays bj α to every player j 6= α. 19 / 25

Model 2 PotSh Selfdu MCP Rec ExiSPS (b2) Player α o ers xj α 2 R to player j 2 Nnfαg. (b3) Let there be σ = (σ 1,..., σ n 1 ) a random order of Nnfαg. First the player σ n 1 responds to the o er made by player α. Responses are either accepting or rejecting. In case player σ n 1 accepts, player σ n 2 responds and so on. If every player accepts the o er, an agreement is reached. If there is some rejection, no agreement is reached. If an agreement is reached, proposer α pays his o ers to the players in Nnfαg and in return α obtains v (N). The game stops. If an agreement is not reached, the proposer obtains v (α) and has to leave the game. The remaining players continue the bargaining in the game Γ(Nnα, v α ). 20 / 25

Existence of subgame perfect strategies 1 PotSh Selfdu MCP Rec ExiSPS Theorem Γ(N, v ) produces the Shapley payo for (N, v ) in any subgame perfect equilibrium. This will be shown by few minor theorems Theorem There is a strategy combination which produces the Shapley payo for (N, v ) Proof: Consider the following strategy: (b1) Each player i 2 N announces b i j = Sh j (N, v ) Sh j (Nni, v i ) for j 6= i. (b2) Proposer α o ers xj α = Sh j (Nnα, v α ) for j 2 Nnα (b3) Responder j accepts o er if and only if xj α Sh j (Nnα, v α ) By using the e ciency axiom we can easily see, that the generated payo is the Shapley value. 21 / 25

Existence of subgame perfect strategies 2 PotSh Selfdu MCP Rec ExiSPS Theorem The de ned strategy combination is a subgame perfect equilibrium Proof: By induction with respect to the number of players. If jnj = 1, the theorem is correct. Assume the theorem holds for n players. (b3) If no other player after player j rejects the o er, one optimal reaction is accepting o ers x α j Sh j (Nnα, v α ), because the player compares x α j and the payo in the reduced game, which is Sh j (Nnα, v α ) by induction. So the strategies are Nash equilibria in stage (b3). (b2) By considering the chosen strategies the proposer α obtains v (N) j6=α Sh j (Nnα, v α ) = v (N) v α (Nnα) = v (α).for improving this payo, the proposer has to lower at least one o er. But the chosen strategies imply, that there will be no acceptance of the lowered o er and the proposer s payo is v (α) which is not better o. Hence, the strategies are Nash equilibria in stages (b2) and (b3). 22 / 25

Existence of subgame perfect strategies 3 PotSh Selfdu MCP Rec ExiSPS (b1) First we have to analyze which player will become the proposer. The netbids are B i = Sh j (N, v ) Sh j (Nnfig, v i ) j6=i j6=i = 0 (MCP) Sh i (N, v ) Sh i (Nnfjg, v j ) Every player is chosen as the proposer with probability of n 1, but we have shown the outcome does not depend on which player is chosen as the proposer. If player i changes the o ers bj i to b j i = bj i + a j and j6=i a j < 0, the player will not be appointed as proposer.if j6=i a j 0, the player will be appointed as proposer (with a positive probability), but the player s payo will not increase, such that the strategy is optimal and all strategies yield to a Nash equilibrium in the whole game, which means the strategy combination is subgame perfect. 23 / 25

Uniqueness of subgame perfect payo s Theorem PotSh Selfdu MCP Rec ExiSPS Γ(N, v ) produces the Shapley value payo for (N, v ) in any subgame perfect equilibrium. This means the noncooperative game fully implements the Shapley value payo. 24 / 25

Show the following claims: PotSh Selfdu MCP Rec ExiSPS Theorem In any subgame starting in b2) the proposer obtains v (α) and each of the other players obtains his/her Shapley value of the game (Nnα, v α ) in any subgame perfect equilibrium. Show now the theorem of uniqueness of the payo s in subgame perfect equilibria: Theorem The payo of any subgame perfect equilibrium coincides with the Shapley value payo. Hint: In any subgame perfect equilibrium we have j6=i b i j = j6=i b j i. Besides the payo s are the same regardless of who is chosen as a proposer. 25 / 25